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[ "A Comment on \"Cycles and Instability in a Rock-Paper-Scissors Population Game: A Continuous Time Experiment\"", "A Comment on \"Cycles and Instability in a Rock-Paper-Scissors Population Game: A Continuous Time Experiment\"" ]	[ "Zhijian Wang [email protected] \nExperimental Social Science Laboratory\nZhejiang University\n\n", "Siqian Zhu \nExperimental Social Science Laboratory\nZhejiang University\n\n\nChu Kochen Honors College\nZhejiang University\n\n", "Bin Xu \nExperimental Social Science Laboratory\nZhejiang University\n\n" ]	[ "Experimental Social Science Laboratory\nZhejiang University\n", "Experimental Social Science Laboratory\nZhejiang University\n", "Chu Kochen Honors College\nZhejiang University\n", "Experimental Social Science Laboratory\nZhejiang University\n" ]	[]	The authors (Cason, Friedman and Hopkins, Review of Economic Studies, 2014) claimed a conclusion that the treatments (using simultaneous matching in discrete time) replicate previous results that exhibit weak or no cycles. After correct two mathematical mistakes in their cycles tripwire algorithm, we research the cycles by scanning the tripwire in the full strategy space of the games and we find significant cycles missed by the authors. So we suggest that, all of the treatments exhibit significant cycles.	null	[ "https://arxiv.org/pdf/1311.2506v3.pdf" ]	55,340,980	1311.2506	afc251747a300e1566aa7cb812029bc05beb3060	A Comment on "Cycles and Instability in a Rock-Paper-Scissors Population Game: A Continuous Time Experiment" 19 Nov 2013 Zhijian Wang [email protected] Experimental Social Science Laboratory Zhejiang University Siqian Zhu Experimental Social Science Laboratory Zhejiang University Chu Kochen Honors College Zhejiang University Bin Xu Experimental Social Science Laboratory Zhejiang University A Comment on "Cycles and Instability in a Rock-Paper-Scissors Population Game: A Continuous Time Experiment" 19 Nov 20131. 3 Public Administration College, Zhejiang Gongshang University. 4 SKLTP, ITP, Chinese Academy of Sciences. correspondence:JEL numbers: C72C73C92D83; Keywords: experimentslearningcyclesmixed equilibriumdiscrete time The authors (Cason, Friedman and Hopkins, Review of Economic Studies, 2014) claimed a conclusion that the treatments (using simultaneous matching in discrete time) replicate previous results that exhibit weak or no cycles. After correct two mathematical mistakes in their cycles tripwire algorithm, we research the cycles by scanning the tripwire in the full strategy space of the games and we find significant cycles missed by the authors. So we suggest that, all of the treatments exhibit significant cycles. The existence of cycles in mixed equilibrium games is a cutting edge question in the field crossing game theory [1] and evolutionary game theory [2] for decades. The finding of cycles's existences by quantitative measurement in controlled experiments reported by Cason, Friedman and Hopkins [3] is a milestone-like contribution in the field. But one of the conclusions in their article attracts our attention. In their article [3], they claimed that control treatments (using simultaneous matching in discrete time) replicate previous results that exhibit "weak or no cycles". This depended on the results --The stable discrete time treatments (SD) do not exhibit clear cyclical behavior indicated by CRI's not significantly different from 0 (see their Table 3). Two mathematical mistakes in the measurement algorithm were found when we checked their results above. 1 We corrected the mistakes (see Appendix). Using the refined measurement, if setting the start point (α, β) of the tripwire for counting cycles (see Fig. ??) at ( 1 4 , 1 4 ) --Nash equilibrim (NE) of the games as [3], CRI of SD will still not be significantly different from 0 (see up panel in Table 1 comparing with their Table 3). So, for SD, we agree that, there are only weak or no cycles around NE. However, if we set (α, β) at (0.23, 0.26) for SD-Mixed and at (0.22, 0.40) for SD-Pure, using the CRI measurement as criterion still, 2 we can find, both CRI of SD are significantly different from 0 which indicates cycles' existence (see low panel in Table 1). Meanwhile, if using the accumulated counting number of cycles C as index (see the Eq.(2) in [4] for detail explanation) instead of CRI, we also find that allC (of the experimental blocks) for the treatments are significantly different from 0 (see Table 2). This, again, indicates the cycles' existence. So, for SD, we suspect that their conclusion of "weak or no cycle". We suggest that the treatments (using simultaneous matching in discrete time) exhibit significant cycles instead of "weak or no cycles". 3 Appendix CRI was defined as [3] CRI= CCT −CT CCT +CT , and CCT and CT can be interpreted as follows. Supposing the Poincare section ("tripwire") is the segment between P c :≡(α, β) as shown in Fig. A1 and P e =(α, 0), and a transit is a directional segment from state (x 1 , y 1 ) observed at t to state (x 2 , y 2 ). These two segments could cross at X as Number of Number of Cycle Game Condition (α, β) Counter-Clock Clockwise Rotation p-value -wise Transits Transits Index (CRI) S Continuous-Instant ( 1 4 , 1 4 ) 25.2 5.5 0.65 0.028 S Continuous-Slow ( 1 4 , 1 4 ) 9.4 0.8 0.87* 0.027 S Discrete-Mixed (SD-Mixed) ( 1 4 , 1 4 ) 2.3 1.2 0.38 0.116 S Discrete-Pure (SD-Pure) ( 1 4 , 1 4 ) 1.0 1.0 0.13 0.753 Ua Continuous-Instant ( 1 4 , 1 4 ) 31.9 0.9 0.94* 0.027 Ua Continuous-Slow ( 1 4 , 1 4 ) 8.2 0.0 1.00* 0.014 Ua Discrete-Mixed ( 1 4 , 1 4 ) 2.3 0.2 0.79* 0.027 Ua Discrete-Pure ( 1 4 , 1 4 ) 1.9 0.3 0.79* 0.027 U b Continuous-Slow ( 1 4 , 1 4 ) 0.3 8.5 -0.93* 0.028 S Discrete-Mixed (SD-Mixed) (X :≡ (X x , X y ) = α, y 1 + (y 2 − y 1 )(α − x 1 ) x 2 − x 1 .(1) Accordingly, CCT = Ct>0 C t and CT = Ct<0 \|C t \| in which the C t value of the transit (at time t) should be 4 Condition 1 C t = 0 if X ∈ (P c , P e ] ∪ x 2 = x 1 Condition 2 C t = 1 if X ∈ (P c , P e ] ∩ x 2 > α > x 1 C t = −1 if X ∈ (P c , P e ] ∩ x 2 < α < x 1 Condition 3 C t = 1 2 if X ∈ (P c , P e ] ∩ x 2 > x 1 ∩ (x 1 = α ∪ x 2 = α) C t = − 1 2 if X ∈ (P c , P e ] ∩ x 2 < x 1 ∩ (x 1 = α ∪ x 2 = α) At the same time, the accumulated counting number of cycles is C:= C t (exactly the same as Eq. (2) in [4]). According to [4], when C serves as an index, the criterion for cycles existence is that: if C is Figure A1: Illustration of the tripwire for cycles counting (Poincare section) [3]. A tripwire for cycles counting is a line segment constructed between the reference point (α, β) and the simplex edge, as the vertical dashed line extending below the point (α, β). The right tripwire is set as the reference point (α, β)=(0.82, 0.10). The authors [3] set (α, β) = ( 1 4 , 1 4 ) (the Nash equilibrium), used the tripwire (middle) and observed weak and no cycles for stable discrete time treatments (SD). The arrow indicates a transit from (x1, y1) to (x2, y2) crossing the left tripwire at C. significantly different from 0, cycles exist; alternatively, no cycle. There are two mathematical mistakes in the measurement algorithm (see Result 3.do in their paper's supplement). Mistake-1: For Condition 2, a necessary condition for C t = ±1 were set as y1+y2 2 < β without Eq.(1). Mistake-2: For Condition 3, C t were set as 0. These mistakes need to be corrected. Table A1 exhibits both results, without algorithm refined and with algorithm refined. Obviously, without refined algorithm we cannot find cycles even if we set (α, β) at (0.22, 0.40). On contrary, with the refined algorithm, the existence of cycles can be verify by both CRI and C. This comparison implies that the algorithm correction is necessary. Programs for the replication of the results in this paper are provided as supplements. Mean Transits and CRI (Update theTable 3in[3] with Refined Measurement). Denotes CRI Index significantly (p-value < 5%) different from 0 according to 2-tailed Wilcoxon test. The accumulated counting number of cycles (C) in each block. † B indicates block, number 1 ∼ 6 indicates the block number in the related treatment. ‡C indicates the mean accumulated counting number of cycles of an experimental block. Denotes C index significantly (p-value < 5%) different from 0 according to 2-tailed Wilcoxon test.0.23, 0.26) 2.3 1.1 0.41* 0.035 S Discrete-Pure (SD-Pure) (0.22, 0.40) 4.1 3.3 0.14* 0.028 Table 1 Game Condition (α, β) B1 † B2 B3 B4 B5 B6C ‡ p-value S Continuous-Instant ( 1 4 , 1 4 ) 128 53 120 74 58 157 98.3* 0.028 S Continuous-Slow ( 1 4 , 1 4 ) 49 47 45 55 24 38 43.0* 0.028 S Discrete-Mixed (SD-Mixed) (0.23, 0.26) 3 0 3 2 9 18 5.8* 0.035 S Discrete-Pure (SD-Pure) (0.22, 0.40) 6 4 2 4 8 1 4.2* 0.027 Ua Continuous-Instant ( 1 4 , 1 4 ) 158 114 218 195 175 70 155.0* 0.028 Ua Continuous-Slow ( 1 4 , 1 4 ) 45 53 48 39 28 34 41.2* 0.028 Ua Discrete-Mixed ( 1 4 , 1 4 ) 9 16 10 6 6 17 10.7* 0.027 Ua Discrete-Pure ( 1 4 , 1 4 ) 5 11.5 1.5 13 5.5 11 7.9* 0.028 Ua Discrete-Pure ( 1 4 , 1 4 ) -52 -29 -43 -49 -40 -33 -41.0* 0.028 Table 2: Table A1: Explanation of the Necessary of the Correction for the Measurement Algorithm. Denotes CRI Index significantly (p-value < 5%) different from 0 according to 2-tailed Wilcoxon test.Game Algorithm Index Mean Condition (α, β) refined chosed Index p-value (before/after) CRI/C value S Discrete-Pure (SD-Pure) (0.22, 0.40) before CRI 0.07 0.173 (0.22, 0.40) after CRI 0.14 0.028 (0.22, 0.40) before C 2.17 0.113 (0.22, 0.40) after C 4.2* 0.027 We thank the authors'[3] confirmation on this point during ESA-NA (2014) Conference.2 Our pilot results suggest that using angular motion of the transits (e.g. the θ in Eq.(10) in[4]) are efficient observation for cycles too. We wish to return to these future. For a careful description for this measurement, see the Eq.(2) for the accumulated counting number C for cycles in[4]. We thank the authors'[3] confirmation on this point. (No.Y3KF261CJ1) and Philosophy & Social Sciences Planning Project of Zhejiang Province (13NDJC095YB). 3We would like to point out follows.(1)The α, β for the two SD is close to the actual mean observation of the aggregated social strategy which seems to tell us that, cycles are actually rounding actual mean observation instead of NE. (2) Their treatments of RPS games are economical because only 6 groups of 8 subjects and 15 minutes are needed. If cycle could be obtained, such treatments could be exemplified classroom experiments for teaching evolutionary game theory.AcknowledgmentsWe thank John Ledyard, Charles Plott, Hai-Jun Zhou, Daniel Friedman and Qiqi Cheng for helpful discussion. This work was supported by Grants from 985 at Zhejiang University, SKLTP of ITP CAS The work of John Nash in game theory. Harold W Kuhn, C John, Reinhard Harsanyi, Selten, W Jorgen, Eric Weibull, Van Damme, Economic Sciences. Kuhn, Harold W, John C Harsanyi, Reinhard Selten, Jorgen W Weibull, and Eric van Damme. 1997. "The work of John Nash in game theory." Economic Sciences, 1991-1995 :160. Evolution and the Theory of Games. J M Smith, Cambridge university pressSmith, J.M. 1982. Evolution and the Theory of Games. Cambridge university press. Cycles and Instability in a Rock-Paper-Scissors Population Game: a Continuous Time Experiment. T N Cason, D Friedman, E Hopkins, 10.1093/restud/rdt023Review of Economic Studies. 81Cason, T. N., D. Friedman, and E. Hopkins. 2014. "Cycles and Instability in a Rock-Paper- Scissors Population Game: a Continuous Time Experiment." Review of Economic Studies 81, doi:10.1093/restud/rdt023. Cycle frequency in standard Rock-Paper-Scissors games: Evidence from experimental economics. B Xu, H.-J Zhou, Z Wang, 10.1016/j.physa.2013.06.039Physica A: Statistical Mechanics and its Applications. 39220Xu, B., H.-J. Zhou, and Z. Wang. 2013. "Cycle frequency in standard Rock-Paper-Scissors games: Evidence from experimental economics." Physica A: Statistical Mechanics and its Applications 392 (20):4997 -5005, doi:10.1016/j.physa.2013.06.039.	[]
[ "Sequential sampling models in computational psychiatry: Bayesian parameter estimation, model selection and classification", "Sequential sampling models in computational psychiatry: Bayesian parameter estimation, model selection and classification" ]	[ "Thomas V Wiecki " ]	[]	[]	Current psychiatric research is in crisis. In this review I will describe the causes of this crisis and highlight recent efforts to overcome current challenges. One particularly promising approach is the emerging field of computational psychiatry. By using methods and insights from computational cognitive neuroscience, computational psychiatry might enable us to move from	null	[ "https://arxiv.org/pdf/1303.5616v1.pdf" ]	16,953,238	1303.5616	8a06002b7ec0cb1ffc8f83e10d2a9284908e0b68	Sequential sampling models in computational psychiatry: Bayesian parameter estimation, model selection and classification February 7, 2014 22 Mar 2013 Thomas V Wiecki Sequential sampling models in computational psychiatry: Bayesian parameter estimation, model selection and classification February 7, 2014 22 Mar 20131 Current psychiatric research is in crisis. In this review I will describe the causes of this crisis and highlight recent efforts to overcome current challenges. One particularly promising approach is the emerging field of computational psychiatry. By using methods and insights from computational cognitive neuroscience, computational psychiatry might enable us to move from a symptom-based description of mental illness to descriptors based on objective computational multidimensional functional variables. To exemplify this I will survey recent efforts towards this goal. I will then describe a set of methods that together form a toolbox of cognitive models to aid this research program. At the core of this toolbox are sequential sampling models which have been used to explain diverse cognitive neuroscience phenomena but have so far seen little adoption in psychiatric research. I will then describe how these models can be fitted to subject data and highlight how hierarchical Bayesian estimation provides a rich framework with many desirable properties and benefits compared to traditional optimization-based approaches. Finally, non-parametric Bayesian methods provide general solutions to the problem of classifying mental illness within this framework. Part I: Motivation Imagine going to a doctor because of chest-pain that has been bothering you for a couple of weeks. The doctor would sit down with you, listen carefully to your description of symptoms and prescribe medication to lower blood pressure in case you have a heart condition. After a couple of weeks your pain has not subsided. The doctor now prescribes medication against reflux which finally seems to help. In this scenario not a single medical analysis (e.g. EKG, blood work or a gastroscopy) was performed and medication with potentially severe side-effects prescribed on a trial-and-error basis. While highly unlikely to occur if you walked into a primary care unit with these symptoms today, this scenario is actually a realistic description if you had mental problems and had seen a psychiatrist instead. There are several reasons for this discrepancy in sophistication between psychiatry and other fields of medicine. The main cause is that by definition, mental illness affects the brain -the most complex biological system yet encountered. Compared to the level of scientific understanding achieved on other organs of the human body such as the heart, our understanding of the normally functioning brain is still, arguably, in its infancy. Despite this complexity concerted efforts in the brain sciences have lead to an explosion of knowledge and understanding about the healthy and diseased brain in the last decades. The discovery of highly effective psychoactive drugs in the 50s and 60s raised expectations that psychiatry would progress in a similar fashion. Unfortunately, in retrospect it appears that these discoveries were serendipitous in nature as little progress has been made since then (e.g Hyman, 2012). This lack of progress also caused many major pharmaceuticals companies like AstraZeneca and GlaxoSmithKline to withdraw from psychiatric drug development and close large research centers (Nutt and Goodwin, 2011;Cressey, 2011). In sum, psychiatry is a field in crisis (Poland et al., 1994;Hyman, 2012;Sahakian et al., 2010). As outlined in more detail below, the main reason for this crisis is a lack of measurable quantitative descriptors of mental illness. This lack results from an explanatory gap of how basic neurobiological aberrations result in complex disorders of the mind (Montague et al., 2011;Hyman, 2012). In part II I will review current challenges in psychiatry and recent efforts to overcome them. Several examples from the domain of decision making show the promise of moving away from symptom-based description of mental illness and instead formulate objective, quantifiable computational biomarkers as a basis for further psychiatric research. Part III introduces a computational cognitive toolbox that is suited to construct these computational biomarkers. Sequential sampling models serve as a case study for how computational models, when fit to behavior, have successfully been used to identify and quantify latent neurocognitive processes in healthy humans. Bayesian methods provide a resourceful framework to fit these models to behavior and establish individualized descriptors of neurocognitive function. After establishing the validity of these models to provide neurocognitive descriptors of individuals, I will review how clustering techniques can be used to construct a map of individual differences based on these neurocognitive descriptors. In sum, the objective of this review is to outline a research program to map the domain of neuropsychiatric disease. In order to maintain a clear focus on this objective there are certain relevant issues not addressed here-within. While of critical importance to psychiatric patients, I will not discuss the clinical, pharmaceutical, environmental, social or developmental aspects of mental illness or rehabilitation programs. Moreover, I will treat mental illness as a disease of the brain with a focus on dysfunctional neurocircuitry (Insel, 2010). Current challenges in psychiatry While the current crisis in psychiatry has complex causes that are deeply rooted in existing classification systems, one of the core problems is what has been identified by Montague et al. (2011) as the explanatory gap. This gap refers to our lack of understanding of the causal processes linking genes, molecular, cellular, neurocircuitry and cognition to psychiatric symptoms. This explanatory gap, coupled with the "(almost) unreasonable effectiveness of psychotropic medication" (Montague et al., 2011) gave many a false premise to expect progress without understanding. This explanatory gap suggests a new approach to research of mental disorders which aims to link cognitive and pure neuroscience to clinical symptoms without the restrictions of prior classification schemes (Poland and Von Eckardt, 2013;Cuthbert and Insel, 2010;Robbins et al., 2012). I will next specify problems current classification systems introduce followed by recent efforts to address them. Diagnostic and Statistical Manual of Mental Disorders For decades the Diagnostic and Statistical Manual of Mental Disorders (DSM) has been the basis of clinical diagnosis, treatment and research of mental illness. At its core, the DSM defines distinct disease categories like schizophrenia (SZ) and depression. These categories are mainly derived from translating subjective experience to objective symptomatology (Nordgaard et al., 2012) assuming unspecified biological, psychological, or behavioral dysfunctions (Poland et al., 1994). While of certain value to clinicians the DSM is specifically designed to also serve as a classification system for scientific research with the goal of more easily translating results directly into clinical practice. While this translational research goal is commendable, decisions regarding systematic classification are more often based on perceptions of clinical utility rather than scientific merit (Poland and Von Eckardt, 2013). As a consequence, DSM-based research programs failed to deliver consistent, replicable and specific results (Kendell and Jablensky, 2003;Andreasen, 2007;Regier and Narrow, 2009;Kendler et al., 2009;Cuthbert and Insel, 2010;Hyman, 2010). Heterogeneity and Comorbidity One major problem of contemporary psychiatry classification is the heterogeneus symptomatology of patients receiving identical diagnoses. One striking example of this is SZ where one must show at least 2 out of 5 symptoms to receive a diagnosis (Heinrichs, 2001). It is thus possible to have patients with completely different symptomatology being diagnosed as schizophrenic. Comorbidity is defined as the co-occurrence of multiple diseases in one individual. Importantly, we must differentiate between two relevant types of comorbidity: (i) True comorbidity is a result of independent disorders co-occurring; (ii) artificial comorbidity on the other hand is a result of separately classifying disorders that have a similar pathogenic cascade. It is now widely documented (Markon, 2010;Krueger and Markon, 2006) that "comorbidity between mental disorders is the rule rather than the exception, invading nearly all canonical diagnostic boundaries." (Buckholtz and Meyer-Lindenberg, 2012). The authors further note that "It is important to understand that comorbidity in psychiatry does not imply the presence of multiple diseases or dysfunctions but rather reflects our current inability [to formulate] a single diagnosis to account for all symptoms." 1 Together, these issues belie the assumption that DSM-disorders represent distinct and independent categories with a unique pathological cascade (Krueger and Markon, 2006;Hyman, 2010;Cuthbert and Insel, 2010;Andreasen, 2007). Part II: Potential Solutions As outlined above, the short-comings of the current DSM manual are well documented and a need for improvement has been recognized. In the following I will outline current efforts to address these challenges. Research Domain Criteria Project The Research Domain Criteria Project (RDoC) is an initiative by the National Institute for Mental Health (NIMH) Neurocognitive phenotyping In a recent review article, Robbins et al. (2012) suggest the use of neurocognitive endophenotypes to study psychiatric disease: "Neurocognitive endophenotypes would furnish more quantitative measures of deficits by avoiding the exclusive use of clinical rating scales, and thereby provide more accurate descriptions of phenotypes for psychiatric genetics or for assessing the efficacy of novel treatments. The use of such measures would likely also facilitate and improve the use of informative animal models in psychiatry by focusing on cognitive and neural processes that can often be investigated in parallel across species. Defining such endophenotypes might cut across traditional psychiatric classification, and hence begin to explain the puzzle of apparent comorbidities." Of particular interest are three studies that use such neurocognitive endophenotypes by constructing dimensional functional profiles (MFPs) from summary statistics of a battery of various neuropsychological tasks to identify subtypes of ADHD (Durston et al., 2008;Sonuga-Barke, 2005;Fair et al., 2012). Durston et al. (2008) argues that there are distinct pathogenic cascades along which abnormalities in at least three different brain circuits can lead to similar symptomatology. Specifically, abnormalities in dorsal frontostriatal, orbito-frontostriatal, or fronto-cerebellar circuits can lead to impairments of cognitive control, reward processing and timing, respectively. Core deficits in one or multiple of these brain networks can thus result in a clinical diagnosis of ADHD (see figure 1) and provides a compelling explanation for the heterogeneity of the ADHD patient population. Preliminary evidence for and ADHD patients. The clustering is achieved by the application of graph theory. Interestingly, the authors find that HC and ADHD is not the predominant dimension along which clusters form. Instead, the authors uncover different functional profiles that apply to both, HC and ADHD patients. Critically, a classifier trained to predict HC and ADHD subjects inside of individual profiles achieved better performance than a classifier trained on the aggregated data. In addition, the abnormalities in cognitive functions of ADHD patients were different across different clusters (e.g. one cluster might show differences in response inhibition while another one shows differences in RT variability; see figure 2b). In other words, this implies that the overall population clusters into different cognitive profiles. ADHD affects individuals differently based on which cognitive profile they exhibit. Importantly, this study suggests that the source of heterogeneity may not only be distinct pathogenic cascades being labeled as the same disorder but may actually be a result of the inherent heterogeneity present in the overall population -healthy and diseased. The above mentioned studies all exemplify the danger of lumping subjects at the level of disease and treating them as one homogeneous category with a single, identifiable pathological cascade. Instead, these studies use MFPs to find an alternative characterization of subjects independent of their DSM While a clear improvement on previous research efforts that use the DSM diagnosis as the sole descriptor this approach still has problems. First, although there is less reliance on DSM categories, these studies still use the diagnostic label for recruiting subjects. It could be imagined, for example, that patients with similar impulse control disorders like OCD or Tourette's have abnormalities in similar brain circuits; thus, if only OCD patients are recruited a critical part of the picture might be missed. Second, the cognitive task battery only covers certain aspects of cognitive function. Other tasks that for example measure working memory or reinforcement learning would be a useful addition. Finally, performance on each individual task is assessed by an aggregate performance score. Recent behavioral and neuropsychological findings, however, suggest that executive control in a single task may instead be more accurately characterized as a collection of related but separable abilities (Baddeley, 1966;Collette et al., 2005), a pattern referred to as the unity and diversity of executive functions (Duncan et al., 1997;Miyake et al., 2000). Most cognitive tasks rely on a concerted and often intricate interaction of various neural networks and cognitive processes (see e.g. Collins and . This task impurity problem (Burgess, 1997;Phillips, 1997) complicates identification of separate brain circuits based solely on MFPs. In sum, while cognitive phenotypes provide a useful framework for measuring brain function there is still ambiguity when using behavioral scores that present an aggregate measure of various brain networks. This issue is also discussed by Buckholtz and Meyer-Lindenberg (2012) in relation to comorbidity: "The fact that a brain circuit can be involved in multiple cognitive domains helps explain why diverse psychiatric disorders can exhibit common deficits and symptoms (comorbidity)." This "common symptom, common circuit" model of psychopathology is illustrated by figure 3. Disentangling these transdiagnostic patterns of psychiatric symptoms thus requires identification and measurement of underlying brain circuits. While the authors propose the use of functional imaging studies and genetic analysis I will discuss how computational modeling can contribute to disambiguate the multiple pathways leading to behavioral features. Computational psychiatry How have other brain sciences dealt with one-to-many mapping problem trying to dissociate behavior on a cognitive task with brain circuits? Computational models at different levels of abstraction have had tremendous impact on the field of cognitive neuroscience. The aim is to construct a model based on integrated evidence from neuroscience and psychology to explain neural activity as well as cognitive behavior. While more detailed biologically inspired models such as biophysical and neural network models are generally more constrained by neurobiology they often have many parameters which make it very difficult to fit them directly to human behavior. More abstract, process based models on the other hand often have fewer parameters that allow them to be fit directly to data at the cost of being less detailed about the neurobiology. Critically, all of these models allow for increased specificity in the identification of different neuronal and psychological processes that are often lumped together when analyzing task behavior based on summary statistics. Using computational models to infer dysfunctional latent processes in the brain is the premise of the newly emerging field of computational psychiatry. In their groundbreaking review, Montague et al. (2011) define the goal for computational psychiatry of "extracting computational principles around Figure 3: Pathonegic cascade from Brain Circuit (BC) over multiple related cognitive processes (C1-C3) to symptoms (Sa-Si). "Some of these symptoms will constitute diagnostic criteria for categorical disorder A but not disorder B (yellow shading), and some symptoms will be relatively selective for disorder B but not disorder A (red shading). However, the plurality of symptoms will overlap the two diagnostic categories (transdiagnostic symptoms, orange shading). This highlights the idea that connectivity circuits convey cognitive and symptom domain-specific, but disorder-general, genetic risk for mental illness." Reproduced from Buckholtz and Meyer-Lindenberg (2012). which human cognition and its supporting biological apparatus is organized. Achieving this goal will require new types of phenotyping approaches, in which computational parameters are estimated (neurally and behaviorally) from human subjects and used to inform the models. This type of large-scale computational phenotyping of human behavior does not yet exist." Also, see figure 4 for a visual depiction. Based on this premise, Maia and Frank (2011) identify computational models as a "valuable tool in taming [the complex pathological cascades of mental illness] as they foster a mechanistic understanding that can span multiple levels of analysis and can explain how changes to one component of the system (for example, increases in striatal D2 receptor density) can produce systems-level changes that translate to changes in behavior". Moreover, the authors define three concrete strategies for how computational models can be used to study brain dysfunction (see also figure 5) given a model of normal function: • Deductive approach: Established neuronal models can be tested for how pathophysiologically plausible alterations in connectivity or neurotransmitter levels (e.g. dopamine is known to be reduced in Parkinson's disease) affect system level activations and behavior. This is essentially a bottom-up approach as it involves the study of how known or hypothesized neuronal changes affect higher-level functioning. • Abductive approach: Computational models can be used to infer neurobiological causes from known behavioral differences. In essence, this is a top-down approach which tries to link behavioral consequences back to underlying latent causes. • Quantitative abductive approach: Parameters of a computational model are fit to a subjects' behavior on a suitable task or task battery. Different parameter values point to differences in underlying neurocircuitry of the associated subject or subject group. These parameters can either be used comparatively to study group differences (e.g. healthy and diseased) or as a regressor with e.g. symptom severity. This approach is more common with abstract models than with neural network models as the former have often fewer parameters and thus can be more easily fit to data. Case studies in the domain of decision making One key area in which computational models have had tremendous success in elucidating how the different cognitive and neurobiological gears work together is the domain of decision making. In addition, many mental illnesses can be characterized by aberrant decision making of one sort or another (Maia and Frank, 2011;Wiecki and Frank, 2010;Montague et al., 2011). In the following I will review recent cases where computational models of decision making have been used to better understand brain disorders. Computational models of reinforcement learning Parkinson's Disease Our first case study concerns Parkinson's disease (PD). Its most visible symptoms affect the motor system as manifest as hypokinesia, bradykinesia, akinesia, rigidity, tremor and progressive motor degeneration. However, recently, cognitive symptoms have received increased attention (e.g., Cools, 2005;Frank, 2005;Moustafa et al., 2008;Cunha et al., 2009). PD is an intriguing neuropsychiatric disorder because its pathogenic cascade is well identified to be the cell death of Figure 5: Different approaches computational models can inform mental health research. Given a computational model of normal function (a), research can provide a mechanistic bridge from neural abnormalities to explain or compare behavioral differences in a deductive approach (b). Contrary, in an abductive approach (c) behavioral differences are used to infer underlying neuronal abnormalities. Similarly, computational models can be quantitatively fit to behavior to infer underlying causes (i.e. quantitative abductive approach; d). See text for more information. Reproduced from Maia and Frank (2011). midbrain dopaminergic neurons in the substantia nigra pars compacta (SNc), part of the basal ganglia (BG) (Kish et al., 1988). Neural network models of the BG (Frank, 2005 interpret this brain network as an adaptive action selection device that conditionally gates internal or external actions based on their previous reward history. DA is critically involved in learning from rewards and punishments which actions to facilitate and which actions to suppress in the future (Ljungberg et al., 1992;Schultz, 1998;Waelti et al., 2001;Pan et al., 2005;Bayer et al., 2007;Roesch et al., 2007;Sutton and Barto, 1990;Barto, 1995;Schultz et al., 1997). Behavioral reinforcement learning tasks show that the chronic low levels of DA in PD patients result in a bias towards learning from punishment at the cost of learning from rewards (Frank et al., 2004;Cohen and Frank, 2009). In extension, we have argued that PD is not a motor disorder per se but rather an action selection disorder in which the progressive decline of motor and cognitive function can be interpreted in terms of aberrant learning not to select actions (Wiecki and Frank, 2010;Wiecki et al., 2009). In this case study, an existing biological model of normative brain function was paired with a known and well localized neuronal dysfunction to extend our understanding of the symptomatology of a brain disorder. Note, however, that the model was not fit to data quantitatively. In the terminology established by Maia and Frank (2011), this is an example of the deductive approach in which the model provides a mechanistic bridge that explains how abnormal behavior can result from neurocircuit dysfunctions. Depression Our second case study involves how depression may be understood as an action planning disorder. Planning a series of future actions is an exponentially complex problem as each individual action can have different outcomes which themselves enable new actions essentially forming a decision tree. One way to deal with this complexity is to prune the decision tree and not consider certain actions (see figure 6). Recently, Huys et al. (2012) proposed a model of how humans perform this approximation. Briefly, the authors suggest that actions that would lead to comparatively bad outcomes will not be further considered and pruned from the decision tree (note that this is provably not optimal as early bad outcomes could be more than accounted for later on). To test this theory the authors test healthy subjects on a novel behavioral task that requires execution of a sequence of actions each associated with winning or losing a certain number of points. Critically, in one task condition, Figure 6: Depiction of decision trees with 2 possible actions (pressing of 'U' key (green) or 'I' key (yellow). Each action (starting at the top node) leads to winning or losing points and subsequent action choice. After 3 subsequent actions points are aggregated. Critically, subjects experience rewards and have to learn the reward structure (which remains stable across a number of trials) to maximize rewards. Complete enumeration of a decision tree (left) has exponential complexity and is thus computationally infeasible. Thus, the decision tree must be pruned at a certain point. One possible strategy is to prune once a large negative reward is encountered (as depicted). Reproduced from Huys et al. (2012). subjects had two action paths available to them: (i) one path in which a very large loss occurred early that, however, was more than accounted for later on; and (ii) another less favorable one action path without an early loss but lower cumulative rewards overall. Participants overwhelmingly chose the non-optimal action sequence that did not result in an early large loss suggesting that humans prune the decision tree once actions are encountered leading to unfavorable outcomes, even in scenarios where this strategy does not result in optimal performance. The authors formulate, fit and compare various computational algorithmic models with different pruning strategies. The model that provided the best fit (measured with an integrated BIC score that penalizes model complexity; see below) had three parameters: specific pruning, general pruning and reward sensitivity. Specific pruning represents the probability of a participant to stop evaluating sub-trees that are associated with large losses while general pruning represents the probability to stop considering sub-trees irrespective of rewards. Reward sensitivity captured the tendency to evaluate a loss of e.g. -140 to be of larger absolute magnitude than a win of 140 points although they cancel each other (i.e. loss aversion). Participants were also given questionaires to assess their (sub-clinical) levels of depression (BDI). Intriguingly, this depression rating correlated with the specific pruning parameter of the model (see figure 7). This effect was specific to this parameter as there was no such correlation with the general pruning parameter. In other words, subjects with higher depressive ratings were quicker to discard plans that lead to bad outcomes early on -sometimes missing large rewards available to them later on. Finally, the authors speculate that the specific pruning depends on the serotonergic transmitter system. Schizophrenia Despite SZ being the focus of intense research over the last decades, no single theory of its underlying neural causes has been able to explain the diverse set of symptoms that can lead to a SZ diagnosis. The symptomatology is structured in terms of positive symptoms like psychosis, negative symptoms like anhedonia which refers to the inability to experience pleasure from activities usually found enjoyable such as social interaction. Recent progress has been made by the application of RL models to understand individual symptoms or a single symptom category (e.g. negative symptoms) rather than SZ as a whole (Waltz et al., 2011;Gold et al., 2008Gold et al., , 2012Strauss et al., 2011). In a recent behavioral study using a RL task, Waltz et al. (2007) found that SZ patients show reduced performance in selecting previously rewarded stimuli compared with HCs. Moreover, this performance deficit is most pronounced in patients with severe negative symptoms. Notably, SZ and HC did not differ in their ability to avoid actions leading to negative outcomes. However, this behavioral analysis did not allow to differentiate whether SZ patients were impaired at learning from positive outcomes or from a failure in representation of the prospective reward values during decision making. This dichotomy in learning vs representation is also present in two types of RL models -actor-critic and Q-learning models (Sutton and Barto, 1998). An actor-critic model consists of two modules: an actor and a critic. The critic learns the expected rewards of states and trains the actor to perform actions that lead to better-than expected outcomes. Q-learning models on the other hand have an explicit representation of the outcomes that are associated with each action. Thus, while a Q-learning model chooses actions based on their absolute reward values, an actor-critic chooses actions based on whether they lead to better-than-expected outcomes. In a follow-up study, Gold et al. (2012) administered a new task that paired a neutral stimulus in one context with a positive and in another context with a negative stimulus. While the neutral stimulus has the same value of zero in both contexts, it is known that DA signals reward prediction errors (RPE) that drive learning in the BG are coding outcomes relative to the expected reward (Sutton and Barto, 1990;Barto, 1995;Schultz et al., 1997). Thus, in the negative context, receiving nothing is better than expected and will result in a positive RPE, driving learning in the BG to select this action in the future (Frank, 2005). In a test period in which no rewards were presented, participants had to choose between an action that had been rewarding and one that had simply avoided a loss. Both actions should have been associated with better-than-expected outcomes. An actor-critic model should thus show a tendency to select the neutral stimulus while a Q-learning model with explicit representation of the reward contingencies should mainly select the one with a higher reward. Intriguingly, when both of these models were fit to participant data, the actor-critic model produced a better fit for SZ patients with high degree of negative symptoms while HC and SZ with low negative symptoms were better fit by a Q-learning model. In other words, the high negative symptoms group largely based decisions on learned stimulus-response associations instead of expected reward values. Notably, HC and the low negative symptom group did not differ significantly in their RL behavior. Moreover, this also rules out possibly confounding effects of antipsychotic drugs, as both patient groups were similarly medicated. In a related line of work, Strauss et al. (2011) tested HC and SZ patients on a reinforcement learning task that allowed subjects to either adopt a safe strategy and exploit the rewards of actions with previously experienced rewards, or, to explore new actions with perhaps even higher payoffs. Moreover, linking the results to prior neural network modeling efforts (Frank and Claus, 2006), as described above, also points to the OFC as a promising target for further investigation as a neural source of negative symptoms in SZ patients. Computational models of response inhibition Besides RL, response inhibition is another widely studied phenomenon in cognitive neuroscience of relevance to mental illness. Response inhibition is required when actions in the planning or execution stage are no longer appropriate and must be suppressed. The antisaccade task is one such task that is often used in a psychiatric setting (e.g Aichert et al., 2012;Fukumoto-Motoshita et al., 2009). It requires subjects to inhibit a prepotent response to a salient stimulus and instead saccade to the opposite side (Hallett, 1979). A wealth of literature has demonstrated reduced performance of psychiatric patients with disorders including attentiondeficit/hyperactivity disorder (ADHD) (Nigg, 2001;Oosterlaan et al., 1998;Schachar and Logan, 1990), obsessive compulsive disorder (OCD) (Chamberlain et al., 2006;Menzies et al., 2007;Penadés et al., 2007;Morein-Zamir et al., 2009), schizophrenia (SZ) (Huddy et al., 2009;Bellgrove et al., 2006;Badcock et al., 2002), Parkinson's disease (PD) (van Koningsbruggen et al., 2009) and substance abuse disorders (Monterosso et al., 2005;Nigg et al., 2006). However, as demonstrated by Wiecki and Frank (2013), even a supposedly simple behavioral task such as the antisaccade task requires a finely orchestrated interplay between various brain regions including frontal cortex and basal ganglia. It thus can not be said that decreased accuracy in this task is evidence of response inhibition deficits per se as the source of this performance impairment can be manifold. In follow-up work, we have formulated a psychological process model which summarizes the higherlevel computations of the neural network and has fewer parameters. In an attempt to bridge these two levels of abstraction, we fit this process model to the outputs of this neural network model for which the biological modulations can be tightly controlled (Wiecki and Frank, 2010). Interestingly, by modulating different biological parameters in the neural network model and recovering which parameter of the process model was affected by this modulation, we were able to associate high-level computational processes with their neural correlates. The hope is that these associations, once validated, allow us to infer specific neural aberrations from behavioral performance. In sum, computational models like the DDM allow mapping of behavior to psychological processes and could thus be categorized as the computational abductive approach. However, ambiguity of how psychological processes relate to the underlying neurocircuitry still have to be disambiguated. By combining different levels of modeling these ambiguities can be better identified and studied. Ultimately, this might allow development of tasks that use specific conditions (e.g. speed-accuracy trade-off, reward modulations and conflict) to disambiguate the mapping of psychological processes to their neurocircuitry. Using biological process models to test different hypotheses about the behavioral and cognitive effects of neurocircuit modulations would correspond to the deductive approach. In other words, by combining the research approaches outlined by Maia and Frank (2011) we can use our understanding of the different levels of processing to inform and validate how these levels interact in the healthy and dysfunctional brain. In sum, there are a few example studies which applied established computational models to identify model parameters (which aim to describe a single cognitive function) and relate it to the severity of a specific clinical symptom. In the following, I will review sequential sampling models and Bayesian methods and show how they are well suited to solve many of the issues encountered in computational psychiatry. Part III: Quantitative Methods Part I described the current issues in psychiatry. Several examples presented in part II highlighted computational psychiatry as an area with a lot of potential to solve these problems. In part III I will review several quantitative methods that can solve the problems associated with quantifying cognitive function. Specifically, sequential sampling models present a versatile tool to model cognitive function. Fitting these models to data -especially with the small number of trials often found in clinical experiments -is a challenge well addressed by hierarchical Bayesian models that share statistical power by assuming similarity between subjects. However, there are two limitations with this basic approach. (i) Traditional inference methods require a likelihood function which is often intractable for more nuanced formulations of sequential sampling models. Likelihood free methods solve this problem as they only require a generative process from which the likelihood is estimated. (ii) While the similarity assumption made by hierarchical Bayesian modeling is reasonable we can not know the exact form of this similarity ahead of time. Bayesian mixture models address this problem by inferring clusters from the data. While traditional methods like Gaussian Mixture Models require specification of the number of clusters to be found in the data ahead of time, Bayesian non-parametrics relax this restriction and infer the number of clusters from the data. In the following I will focus on how each quantitative method helps to solve the above mentioned issues. Mathematical details can be found in the appendix. Finally, while these methods are described with the motivation of estimating sequential sampling models, the Bayesian methods are applicable more broadly (e.g. to the estimation of RL models). Sequential Sampling models Cognition spans many mental processes that include attention, social cognition, memory, emotion, and reasoning, to name a few. As outlined above, RL models have already proven to be a valuable tool in explaining neuropsychological disorders and their symptoms. A computational psychiatric framework that aims to explain the multi-faceted domain of mental illness must thus include computational cognitive neuroscience models that cover a broad range of cognitive processes. As such a review is outside of our scope (but see O'Reilly and Munakata (e.g. 2000)) I will focus on sequential sampling models as an illustrative example for how these models work, how they have been applied to study normal and aberrant neurocognitive phenomena, how they can be fit to data using Bayesian estimation and how subgroups of subjects can be inferred using mixture models. Sequential sampling models (SSMs) (e.g. Townsend and Ashby, 1983) have established themselves as the de-facto standard for modeling data from simple decision making tasks (e.g. Smith and Ratcliff, 2004). Each decision is modeled as a sequential extraction and accumulation of information from the environment and/or internal representations. Once the accumulated evidence crosses a threshold, a corresponding response is executed. This simple assumption about the underlying psychological process has the intriguing property of reproducing reaction time distributions and choice probability in simple two-choice decision making tasks. SSMs generally fall into one of two classes: (i) diffusion models which assume that relative evidence is accumulated over time and (ii) race models which assume independent evidence accumulation and response commitment once the first accumulator crossed a boundary (e.g. LaBerge, 1962;Vickers, 1970). While there are many variants of these models they are often closely related on a computational level and sometimes mathematically equivalent under certain assumptions (Bogacz et al., 2006). As such, I will restrict discussion to two exemplar models from each class widely used in the literature: the drift diffusion model (DDM) (Ratcliff and Rouder, 1998;Ratcliff and McKoon, 2008) belonging to the class of diffusion models and the linear ballistic accumulator (LBA) (Brown and Heathcote, 2008) belonging to the class of race models. Drift Diffusion Model The DDM models decision making in two-choice tasks. Each choice is represented as an upper and lower boundary. A drift-process accumulates evidence over time until it crosses one of the two boundaries and initiates the corresponding response (Ratcliff and Rouder, 1998;Smith and Ratcliff, 2004). The speed with which the accumulation process approaches one of the two boundaries is called the drift rate and represents the relative evidence for or against a particular response. Because there is noise in the drift process, the time of the boundary crossing and the selected response will vary between trials. The distance between the two boundaries (i.e. threshold) influences how much evidence must be accumulated until a response is executed. A lower threshold makes responding faster in general but increases the influence of noise on decision making while a higher threshold leads to more cautious responding. Reaction time, however, is not solely comprised of the decision making process -perception, movement initiation and execution all take time and are summarized into one variable called non-decision time. The starting point of the drift process relative to the two boundaries can influence if one response has a prepotent bias. This pattern gives rise to the reaction time distributions of both choices (see figure 8; mathematical details can be found in the appendix). Later on, the DDM was extended to include inter-trial variability in the drift-rate, the non-decision time and the starting point in order to account for two phenomena observed in decision making tasks -early and late errors. Models that take this into account are referred to as the full DDM (Ratcliff and Rouder, 1998). Linear Ballistic Accumulator The Linear Ballistic Accumulator (LBA) model belongs to the class of race models (Brown and Heathcote, 2008). Instead of one drift process and two boundaries, the LBA contains one drift process for each possible response with a single boundary each. Thus, the LBA can model decision making when more than two responses are possible. Moreover, unlike the DDM, the LBA drift process has no intra-trial variance. RT variability is obtained by including inter-trial variability in the drift-rate and the starting point distribution (see figure 9). Note that the simplifying assumption of a noiseless drift-process simplifies the math significantly leading to a computationally faster likelihood function Figure 8: Trajectories of multiple drift-processs (blue and red lines, middle panel). Evidence is accumulated over time (x-axis) with drift-rate v until one of two boundaries (separated by threshold a) is crossed and a response is initiated. Upper (blue) and lower (red) panels contain histograms over boundary-crossing-times for two possible responses. The histogram shapes match closely to that observed in reaction time measurements of research participants. Figure 9: Two linear ballistic accumulators (left and right) with different noiseless drifts (arrows) sampled from a normal distribution initiated at different starting points sampled from uniform distribution. In this case, accumulator for response alternative 1 reaches criterion first and gets executed. Because of this race between two accumulators towards a common threshold these model are called race-models. Reproduced from Donkin et al. (2011). for this model. In a simulation study it was shown that the LBA and DDM lead to similar results as to which parameters are affected by certain manipulations (Donkin et al., 2011). Relationship to cognitive neuroscience SSMs were originally developed from a pure information processing point of view and primarily used in psychology as a high-level approximation of the decision process. More recent efforts in cognitive neuroscience have simultaneously (i) validated core assumptions of the model by showing that neurons indeed integrate evidence probabilistically during decision making (Smith and Ratcliff, 2004;Gold and Shadlen, 2007) and ( Multiple routes to decision threshold modulation have been identified. Decision threshold in the speed-accuracy trade-off is modulated by changes in the functional connectivity between pre-SMA and striatum (Forstmann et al., 2010a). Neural network modeling Ratcliff and Frank, 2012) validated by studies of PD patients with a deep-brain-stimulator (DBS) in their subthalamic nucleus (STN) (Frank et al., 2007a) suggest that this node is implicated in raising the decision threshold when there is conflict between two options associated with similar rewards. This result was further corroborated by Cavanagh et al. (2011) who found that frontal theta power (as measured by electroencelophagraphy and thought to correspond to conflict (Cavanagh et al., 2012)) is correlated with decision threshold increase on a trial-by-trial basis. As predicted, this relationship was broken in PD patients with DBS turned on (but, critically, not when DBS was turned off thus showing the effect is not a result of the disease). In other words, by interfering with STN function through stimulation we were able to show that this brain area is causally involved in decision threshold modulation despite intact experience of conflict (as measured by theta power). Interestingly, these results provide a computational cognitive explanation for the clinical symptom of impulsivity observed in PD patients receiving DBS (Frank et al., 2007a). Application to computational psychiatry Despite its long history, the DDM has only recently been applied to the study of psychopathology. For example, threat/no-threat categorization tasks (e.g. "Is this word threatening or not?") are used in anxiety research to explore biases to threat responses. Interestingly, participants with high anxiety are more likely to classify a word as threatening than low anxiety participants. One hypothesis assumes that this behavior results from an increased response bias towards threatening words in anxious people (Becker and Rinck, 2004;Manguno-Mire et al., 2005;Windmann and Krüger, 1998). Using DDM analysis, White (2009) showed that instead of a response bias (or a shifted starting-point in DDM terminology), anxious people actually showed a perceptual bias towards classifying threatening words as indicated by an increased DDM drift-rate. In a recent review article, White et al. (2010) use this case-study to highlight the potential of the DDM to elucidate research into mental disease. Note that in this study the authors did not attempt to hypothesize about the underlying neural cause of this threat-bias. While there is some evidence that bias in decision making is correlated with activity in the parietial network (Forstmann et al., 2010b) this was not tested in respect to threatening words. Ultimately, I suggest that this research strategy should be applied to infer neural correlates of psychological DDM decision making parameters using functional methods like fMRI to the study neuropsychopathology (as outlined above). The DDM has also been successfully used to show that ADHD subjects were less able to raise their decision threshold when accuracy demands were high (Mulder et al., 2010). Interestingly, the amount by which ADHD subjects failed to modulate their decision threshold correlated strongly with patients' impulsivity/hyperactivity rating. Moreover, this correlation was specific to impulsivity as no correlation was found between decision threshold modulation and inattentiveness. In sum, SSMs show great promise as a tool for computational psychiatry. However, their applicability depend on the ability to accurately estimate them to construct individual CMDFs. In the following I will review different parameter estimation techniques. Special focus will be given to Bayesian methods. Finally, once SSMs can be fit accurately the question arises how to construct a map of mental illness based on these CMDF. Towards this goal I will review clustering methods that can be expressed in the Bayesian framework. Parameter estimation To identify computational biomarkers in a variable clinical population with the DDM it is critical to have robust and sensitive estimation methods. In the following I will describe traditional parameter estimation methods and their pitfalls. I will then argue how Bayesian estimation provides a complete framework that avoids these pitfalls. Maximum Likelihood and χ 2 Traditionally, fitting of computational models is treated as an optimization problem in which an objective function is minimized. Different objective functions for the DDM have been proposed. Most common is the quantile method which calculates the quantiles of the DDM likelihood and uses the χ 2 statistic as the objective to compare it to the quantiles observed in the data (Ratcliff and Tuerlinckx, 2002 Psychological experiments often test multiple subjects on the same behavioral task. Models are then either fit to individual subjects or to the aggregated group data. Both approaches are not ideal. When models are fit to individual subjects we neglect any similarity the parameters are likely to have. While we do not necessarily have to make use of this property to make useful inference if we have lots of data, the ability to infer subject parameters based on the estimation of other subjects generally leads to more accurate parameter recovery (Sofer et al.) in cases where little data is available as is often the case in clinical and neurocoognitive experiments. One alternative is to aggregate all subject data into one meta-subject and estimate one set of parameters for the whole group. While useful in some settings, this approach is unsuited for the setting of computational psychiatry as individual differences play a huge role. Hierarchical Bayesian models Statistics and machine learning have developed efficient and versatile Bayesian methods to solve various inference problems (Poirier, 2006). More recently, they have seen wider adoption in applied fields such as genetics (Stephens and Balding, 2009) Bayesian methods require specification of a generative process in form of a likelihood function that produced the observed data x given some parameters θ. By specifying our prior belief we can use Bayes formula to invert the generative model and make inference on the probability of parameters θ: P (θ\|x) = P (x\|θ) * P (θ) P (x)(1) Where P (x\|θ) is the likelihood and P (θ) is the prior probability. Computation of the marginal likelihood P (x) requires integration (or summation in the discrete case) over the complete parameter space Θ: P (x) = Θ P (x\|θ) dθ(2) Note that in most scenarios this integral is analytically intractable. Sampling methods like Markov-Chain Monte Carlo (MCMC) (Gamerman and Lopes, 2006) circumvent this problem by providing a way to produce samples from the posterior distribution. These methods have been used with great success in many different scenarios (Gelman et al., 2003) and will be discussed in more detail below. Another nice property of the Bayesian method is that it lends itself naturally to a hierarchical design. In such a design, parameters for one distribution can themselves come from a different distribution which allows chaining together of distributions of arbitrary complexity and map the structure of the data onto the model. This hierarchical property has a particular benefit to cognitive modeling where data is often scarce. We can construct a hierarchical model to more adequately capture the likely similarity structure of our data. As above, observed data points of each subject x i,j (where i = 1, . . . , S j data points per subject and j = 1, . . . , N for N subjects) are distributed according to some likelihood function f \|θ. We now assume that individual subject parameters θ j are normal distributed around a group mean with a specific group variance (λ = (µ, σ) with hyperprior G 0 ) resulting in the following generative description: µ, σ ∼ G 0 () (3) θ j ∼ N (µ, σ 2 ) (4) x i,j ∼ f (θ j )(5) See figure 10 for the corresponding graphical model description. Another way to look at this hierarchical model is to consider that our fixed prior on θ from formula (1) is actually a random variable (in our case a normal distribution) parameterized by λ which leads to the following posterior formulation: Figure 10: Graphical notation of a hierarchical model. Circles represent continuous random variables. Arrows connecting circles specify conditional dependence between random variables. Shaded circles represent observed data. Finally, plates around graphical nodes mean that multiple identical, independent distributed random variables exist. P (θ, λ\|x) = P (x\|θ) * P (θ\|λ) * P (λ) P (x) (6) i = 1, . . . , S j j = 1, . . . , N θ j x i,j λ Note that we can factorize P (x\|θ) and P (θ\|λ) due to their conditional independence. This formulation also makes apparent that the posterior contains estimation of the individual subject parameters θ j and group parameters λ. Several inference methods to estimate the posterior distribution have been developed. For details on commonly used sampling algorithms, see the appendix. In sum, hierarchical Bayesian estimation leverages similarity between individual subject to share statistical power and increase sensitivity in our parameter estimation. However, note that in our computational psychiatry application the homogeneity assumption that all subjects come from the same normal distribution is almost certainly violated (see above). To deal with the heterogeneous data often encountered in psychiatry I will discuss mixture models further down below. Next, I will describe algorithms to estimate this posterior distribution. Likelihood-free methods Most models in cognitive neuroscience and mathematical psychology like the DDM are described by a latent generative process (see e.g. (7)). However, this generative description is usually ill suited for inference as it does not directly provide us with a closed-form likelihood function p(x\|θ) of how observed data (e.g. the wiener first passage time; see above) arise from this generative process. While the DDM is used partly because it has a tractable likelihood functions, many interesting considerations require models for which a generative process but no tractable likelihood can be specified. Recent examples of these efforts include changes of mind as new evidence is processed (Resulaj et al., 2009), the influence of attention (Krajbich and Rangel, 2011), and reward-based decision making given conflict in values of alternative actions (Cavanagh et al., 2011;Ratcliff and Frank, 2012). In these cases, a likelihood function must instead be approximated by simulation from the generative process. This type of inference is commonly called likelihood free. It can be expected that the full spectrum of cognitive function will be relevant in computational psychiatry. Being able to formulate (and estimate) drift-diffusion models very flexibly potentially allows its application to new domains. For a detailed explanation of these methods, see the appendix. Model selection Computational models often allow formulation of several plausible accounts of cognitive behavior. We anticipate that this problem will also occur in computational psychiatry where multiple theories of cognitive dysfunction must be tested. One way to differentiate between these various plausible hypothesis as expressed by alternative models is model comparison. In the following I will review various methods and metrics to compare hierarchical models. The most critical property for model comparison is that model complexity gets penalized because more complex models have greater explanatory power by design. Several model comparison measures have been devised. I refer the reader to the appendix section for mathematical details. Mixture Models In this section I will review different mixture models that allow estimation of clusters in data in a Bayesian framework. These are relevant to our objective as they (i) deal with the heterogeneity encountered in computational psychiatry and (ii) have the potential to bootstrap a new psychiatric classification system based on measurable, quantitative, computational endophenotypes. Because we are describing a toolbox using hierarchical Bayesian estimation techniques I will focus this section on mixture models as they are easily integrated into this framework. Where possible, I will highlight connections to more traditional clustering methods like k-means. Gaussian Mixture Models GMMs assume parameters to be distributed according to one of several Gaussian distributions (i.e. clusters). Specifically, given the number of clusters k, each cluster mean and variance gets estimated from the data. This type of model is capable of solving our above identified problem of assuming heterogeneus subjects to be normally distributed: by allowing individual subject parameters to be assigned to different clusters we allow estimation of different sub-groups in our patient and healthy population. Note, however, that the number k of how many clusters should be estimated must be specified a-priori in a GMM and remain fixed for the course of the estimation. This is problematic as we do not necessarily know how many sub-groups to expect in advance. As we will see below, Bayesian non-parametrics solve this issue by inferring the number of clusters from data. Dirichlet Process Gaussian Mixture Models Dirichlet processes Gaussian mixture models (DPGMMs) belong to the class of Bayesian non-parametrics (Antoniak, 1974). They can be viewed as a variant of GMMs with the critical difference that they assume an infinite number of potential mixture components (see Gershman and Blei (2012) for a review). Mixture models can infer sub-groups when the data is heterogeneous as is generally the case in patient populations. While the mindset describing these methods was their application towards the SSM their applicability is much more general than that. For example, the case-studies described above which used, among others, RL models to identify differences between HC and psychiatric patients could easily be embedded into this hierarchical Bayesian mixture model framework I outlined here. There are multiple benefits to such an approach. First, computational models fitted via hierarchical Bayesian estimation provide a tool to accurately describe the neurocognitive functional profile of individuals. Second, the mixture model approach is ideally suited to deal with the heterogeneity in patients but also healthy controls (Fair et al., 2012). Third, by testing psychiatric patients with a range of diagnoses (as opposed to most previous research studies that only compare patients with a single diagnosis, e.g. SZ, to controls) we might be able to identify shared pathogenic cascades as suggested by Buckholtz and Meyer-Lindenberg (2012). Conclusions As outlined above, computational psychiatry is an emerging field that shows great promise to understand aberrant biological processes in mental disease and address current challenges encountered in mental health research. By fitting computational models to behavioral data we can construct computational multi-dimensional features to replace symptom-based classification as implemented by the DSM. Decision making appears to provide a good framework for studying psychiatric disease as many disorders show abnormalities in core decision making processes. Sequential sampling models have a good track record in describing individual differences and can be linked to neuronal processes. Hierarchical Bayesian estimation provides a compelling toolbox to fit these models directly to data as it (i) provides an uncertainty measure; (ii) allows estimation of individual and group-level parameters simultaneously; (iii) allows for direct model comparison; (iv) can be used in scenarios where a likelihood can not be easily formulated; and (v) enable deconstruction of symptoms by identifying latent clusters. For example, impulsivity is a core symptom of impulse control disorders like ADHD, OCD, Tourette syndrome, substance abuse and eating disorder (Robbins et al., 2012). Computational cognitive models have already started to deconstruct this broadly defined behavioral symptom and identified separate pathways that can all lead to alterations in impulse control (Dalley et al., 2011) including reduced motor inhibition (Chamberlain et al., 2006(Chamberlain et al., , 2008 early temporal discounting of future rewards, insensitivity towards negative relative to positive outcomes (Frank et al., 2007b;Cockburn and Holroyd, 2010), or an inability to adjust the decision threshold appropriately (Mulder et al., 2010;Cavanagh et al., 2011;Frank et al., 2007a). Ultimately, the hope is to find a way to describe and diagnose psychiatric disease based on objective computational neurocognitive markers rather than the current subjective symptom-based approach. I believe that this combination of computational tools described here-within is powerful enough to lead the charge towards a new level of understanding of mental illness based on identifiable and reproducible neurocognitive computational multi-dimensional features. Appendix The following serves as a reference for the mathematical details of the methods motivated above. Drift-Diffusion Model Mathematically, the DDM is defined by a stochastic differential equation called the Wiener process with drift: dW ∼ N (v, σ 2 )(7) where v represents the drift-rate and σ the variance. As we often only observe the response times of subjects we are interested in the wiener first passage time (wfpt) -the time it takes W to cross one of two boundaries. Assuming two absorbing boundaries of this process and through some fairly sophisticated math (see e.g. Smith, 2000) it is possible to analytically derive the time this process will first pass one of the two boundaries (i.e. the wiener first passage time; wfpt). This probability distribution 2 then serves as the likelihood function for the DDM. Bayesian Inference Empirical Bayesian Approximation Empirical Bayes can be regarded as an approximation of equation (6). To derive this approximation consider P (θ\|x) which we can calculate by integrating over P (λ): P (θ\|x) = P (x\|θ) P (x) P (θ\|λ)P (λ) dλ(8) Now, if the true distribution P (θ\|λ) is sharply peaked, the integral can be replaced with the point estimate of its peak λ : 2 the wfpt will not be a distribution rather than a single value because of the stochasticity of the wiener process P (θ\|x) P (x\|θ)P (θ\|λ ) P (x\|λ ) Note, however, that λ depends itself on P (θ\|x). One algorithm to solve this interdependence is Expectation Maximization (EM) (Dempster et al., 1977). EM is an iterative algorithm that alternates between computing the expectation of P (θ\|x) (this can be easily done by Laplace Approximation (Azevedo-filho and Shachter, 1994)) and then maximizing the prior point estimate λ based on the current values obtained by the expectation step. This updated point estimate is then used in turn to recompute the expectation. The algorithm is run until convergence or some other criterion in reached. This approach is used for example by Huys et al. (2012) to fit their reinforcement learning models. Markov-Chain Monte-Carlo As mentioned above, the posterior is often intractable to compute analytically. While Empirical Bayes provides a useful approximation, an alternative approach is to estimate the full posterior by drawing samples from it. One way to achieve this is to construct a Markov-Chain that has the same equilibrium distribution as the posterior (Gamerman and Lopes, 2006). Algorithms of this class are called Markov-Chain Monte Carlo (MCMC) samplers. One common and widely applicable algorithm is Metropolis-Hastings (Chib and Greenberg, 1995;Andrieu et al., 2003). Assume we wanted to generate samples θ from the posterior p(θ\|x). In general, we can not sample from p(θ\|x) directly. Metropolis-Hastings instead generates samples θ t from a proposal distribution q(θ t \|θ t−1 ) where the next position θ t only depends on the previous position at θ t−1 (i.e. the Markov-property). For simplicity we will assume that this proposal distribution is symmetrical; i.e. q(θ t \|θ t−1 ) = q(θ t−1 \|θ t ). A common choice for the proposal distribution is the Normal distribution, formally: θ t ∼ N (θ t−1 , σ 2 )(10) The proposed jump to θ t is then accepted with probability α: α = min(1, p(θ t \|x) p(θ t−1 \|x) )(11) In other words, the probability of accepting a jump depends on the probability ratio of the proposed jump position θ t to the previous position θ t−1 . Critically, in this probability ratio, the intractable integral in the denominator (i.e. p(x) = p(x\|θ) dθ) cancels out. This can be seen by applying Bayes formula (1): p(θ t \|x) p(θ t−1 \|x) = p(x\|θ t )p(θ t ) p(x) p(x\|θ t−1 )p(θ t−1 ) p(x) = p(x\|θ t )p(θ t ) p(x\|θ t−1 )p(θ t−1 )(12) Thus, to calculate the probability of accepting a jump we only have to evaluate the likelihood and prior, not the intractable posterior. Note that θ 0 has to be initialized at some position and can not directly be sampled from the posterior. From this initial position, the Markov chain will explore other parts of the parameter space and only gradually approach the posterior region. The first samples generated are thus not from the true posterior and are often discarded as "burn-in". Note moreover that once the algorithm reaches a region of high probability it will continue to explore lower probability regions in the posterior, albeit with lower frequency. This random-walk behavior is due to the probability ratio α which allows Metropolis-Hastings to also sometimes accept jumps from a high probability position to a low probability position. Another common algorithm is Gibbs sampling that iteratively updates each individual random variable conditional on the other random variables set to their last sampled value (e.g Frey and Jojic, 2005). Starting at some configuration θ 0 , the algorithm makes T iterations over each random variable θ i . At each iteration t each random variable is sampled conditional on the current (t − 1) value of all other random variables that it depends on: θ t i ∼ p(θ (t) i \|θ (t−1) i =j )(13) Critically, θ Note that while Gibbs sampling never rejects a sample (which often leads to faster convergence and better mixing), in contrast to Metropolis-Hastings, it does require sampling from the conditional distribution which is not always tractable. Likelihood free methods Several likelihood-free methods have emerged in the past (for a review, see Turner and Van Zandt (2012)). Instead of an analytical solution of the likelihood function, these methods require a sampling process that can simulate a set of data points from a generative model for each θ. We will call the simulated data y and the observed data x. Approximate Bayesian Computation (ABC) relies on a distance measure ρ(x, y) that compares how similar the simulated data y is to the observed data x (commonly, this distance measure relies on summary statistics). We can then use the Metropolis-Hastings algorithm introduced before and change the acceptance ration α (11) to use ρ(x, y) instead of a likelihood function. α =      min(1, p(θ t ) p(θ t−1 ) ) if ρ(x, y) ≤ 0 0 if ρ(x, y) ≥ 0(14) where 0 is an acceptance threshold. Large 0 will result in higher proposal acceptance probability but a worse estimation of the posterior while small 0 will lead to better posterior estimation but slower convergence. An alternative approach to ABC is to construct a synthetic likelihood function based on summary statistics (Wood, 2010). Specifically, we sample N r multiple data sets y 1,...,Nr from the generative process. We then compute summary statistics s 1,...,Nr for each simulated data set 3 . Based on these summary statistics we then construct the synthetic likelihood function to evaluate θ (see figure 11 for an illustration): p(x\|θ) N (S(x); µ θ , Σ θ )(15) This synthetic likelihood function based on summary statistics can then be used as a drop-in replacement for e.g. the Metropolis-Hastings algorithm outlined above. Figure 11: Construction of a synthetic likelihood. To evaluate parameter vector θ, N r data sets y 1,...,Nr are sampled from the generative model. On each sampled data set summary statistics s 1,...,Nr are computed. Based on these summary statistics a multivariate normal distribution is approximated with mean µ θ and covariance matrix Σ θ . The likelihood is approximated by evaluating summary statistics of the actual data on the log normal distribution with the estimated µ θ and Σ θ . Reproduced from (Wood, 2010). Model Comparison Deviance Information Criterion The Deviance Information Criterion (DIC) is a measure which trades off model complexity and model fit (Spiegelhalter et al., 2002). Several similar measures exist such as Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). However, both these measures use the number of parameters as a proxy for model complexity. While a reasonable approximation to the complexity of non-hierarchical models, the relationship between model parameters (some of which are latent) and complexity in hierarchical models is more intricate. The DIC measure instead infers the number of parameters from the posterior. The DIC is computed as follows: DIC =D + pD (16) where pD =D −D(17) D is the posterior mean of the deviance (i.e. −2 * log(likelihood)) andD is a point estimate of the deviance obtained by substituting in the posterior means. Loosely,D represents how well the model fits the data on average whileD captures the deviance at the best fitting parameter combination. pD then acts as a measure related to the posterior variability and used as a proxy for the effective number of parameters. Complex models with many parameters will tend to have higher posterior variability and thus result in increased pD penalization. Note that the only parameters that affectD directly in our hierarchical model (equation 6) are the subject parameters θ i . Thus, DIC estimates model fit based on how well individual subjects explain the observed data. BIC The Bayesian Information Criterion (BIC) is defined as follows: BIC = −2 * logp(x\|θ M L ) + k * log(n)(18) where k is the number of free parameters, n is the number of data points, x is the observed data and logp(x\|k) is the likelihood of the parameters given the data (Schwarz, 1978). While BIC can not directly be applied to hierarchical models (as outlined above), it is possible to integrate out individual subject parameters (e.g. Huys et al., 2012): logp(x\|θ M L ) = i log p(x i \|h)p(h\|θ M L ) dh(19) where x i is the data belonging to the ith subject. The resulting score is called integrated BIC. Since the subject parameters are integrated out, integrated BIC estimates how well the group parameters are able to explain the observed data. Bayes Factor Another measure to compare two models is the Bayes Factor (BF) (Kass and Raftery, 1995). It is defined as the ratio between the marginal model probabilities of the two models: BF = p(x\|M 1 ) p(x\|M 2 ) = p(θ 1 \|M 1 )p(x\|θ 1 , M 1 ) dθ 1 p(θ 2 \|M 2 )p(x\|θ 2 , M 2 ) dθ 2(20) The magnitude of this ratio informs the degree one should belief in one model compared to the other. As BF integrates out subject and group parameters this model comparison measure should be used when different classes of models are to be compared in their capacity to explain observed data. Mixture Models Gaussian Mixture Models Mixture models infer k number of clusters in a data set. The assumption of normal distributed clusters leads to a Gaussian Mixture Model (GMM) with a probability density function as follows: p(x\|π, µ 1,...,K , σ 1,...,K ) = K k=1 π k N (x i \|µ k , σ 2 k ) Each observed data point x i can be created by drawing a sample from the normal distribution selected by the unobserved indicator variable z i which itself is distributed according to a multinomial distribution π: µ k , σ k ∼ G 0 () (22) z i ∼ π (23) x i ∼ N (µ zi , σ 2 zi )(24) where the base measure G 0 defines the prior for µ k and σ k . To simplify the inference it is often advisable to use a conjugate prior for these paramters. For example, the normal distribution is the conjugate prior for a normal distribution with known variance: µ k ∼ N (µ 0 , σ 0 )(25) In a similar fashion, we can assign the mixture weights a symmetric Dirichlet prior: π ∼ Dir( α K , . . . , α K )(26) Note that the GMM assumes a mixture distribution on the level of the observed data x i . However, in our relevant case of a multi-level hierarchical model we need to place the mixture at the level of the latent subject parameters instead of the observed data. As before, we use the subject index j = 1, . . . , N . µ k , σ k ∼ G 0 ()(27) π ∼ Dir(α) (28) z j ∼ Categorical(π)(29) θ j ∼ N (µ zj , σ 2 zj ) (30) x i,j ∼ f(θ j )(31) Where f denotes the likelihood function. Interestingly, the famous K-Means clustering algorithm is identical to a Gaussian Mixture Model (GMM) in the limit σ 2 → 0 (Kulis et al., 2012). K-Means is an expectation maximization (EM) algorithm that alternates between an expectation step during which data points are assigned to their nearest cluster centroids and a maximization step during which new cluster centroids are estimated. This algorithm is repeated until convergence is reached (i.e. no points are reassigned to new clusters). Dirichlet Process Gaussian Mixture Models p(x\|π, µ 1,...,∞ , σ 1,...,∞ ) = ∞ k=1 π k N (x i \|µ k , σ 2 k ) As above, we specify our generative mixture model: µ k , σ k ∼ G 0 ()(33)z i ∼ Categorical(π)(34) x i ∼ N (µ zi , σ 2 zi ) with the critical difference of replacing the hyperprior π with the stick breaking process (Sethuraman, Figure 12: Left: Stick-breaking process. At each iteration (starting from the top) a π is broken off with relative length ∼ Beta(1, α). Right: Histogram over different realizations of the stick-breaking process. As can be seen, higher values of hyperprior α lead to a more spread out distribution. Taken from Eric Sudderth's PhD thesis. 1991): π ∼ StickBreaking(α) The stick-breaking process is a realization of a Dirichlet process (DP). Specifically, π = {π k } ∞ k=1 is an infinite sequence of mixture weights derived from the following process: β k ∼ Beta(1, α) (37) π k ∼ β k * k−1 l=1 (1 − β l )(38) with α > 0. See figure 12 for a visual explanation. The Chinese Restaurant Process (CRP) -named after the apparent infinite seating capacity in Chinese restaurants -allows for a more succinct model formulation. Consider that customers z i are coming into the restaurant and are seated at table k with probability: p(z i = k\|z 1,...,n−1 , α, K) = n k + α/K n − 1 + α where k = 1 . . . K is the table and n k is the number of customers already sitting at table k (see figure 13 for an illustration). It can be seen that in the limit as K → ∞ this expression becomes: p(z i = k\|z 1,...,n−1 , α) = n k n − 1 + α Thus, as customers are social, the probability of seating customer z i to table k is proportional the number of customers already sitting at that table. This desirable clustering property is also known as the "rich get richer". Note that for an individual empty table k at which no customer has been seated (i.e. n k = 0) the probability of seating a new customer to that table goes to 0 in the limit as K → ∞. However, at the same time the number of empty tables approaches infinity. Consider that we have so far seated L customers to tables and the set Q contains all empty tables such that there are \|Q\| = K − L empty tables in the restaurant. The probability of seating a customer z i at an empty table becomes: p(z i ∈ Q\|z 1,...,n−1 , α) = α n − 1 + α As can be seen, the probability of starting a new table is proportional to the concentration parameter α. Intuitively, large values of the dispersion parameter α lead to more clusters being used. Thus, while the Stick-Breaking process sampled mixture weights from which we had to infer cluster assignments, the CRP allows for direct sampling of cluster assignments. The resulting model can then be written as: µ k , σ k ∼ G 0 ()(39) z 1,...,N ∼ CRP(α) (40) x i ∼ N (µ zi , σ 2 zi ) (41) Figure 13: Illustration of the Chinese Restaurant Process. Customers are seated at tables with parameters θ. The more customers are already seated at a table, the higher the probability that future customers are seated at the same table (i.e. clustering property). Taken from Gershman and Blei (2012). Finally, in a hierarchical group model we would need to place the infinite mixture on the subject level rather than the observed data level: µ k , σ k ∼ G 0 () (42) z j ∼ CRP(α)(43) θ j ∼ N (µ zj , σ 2 zj ) (44) x i,j ∼ F(θ j )(45) See figure 14 for a graphical model description. Note that while the potential number of clusters is infinite, any realization of this process will always lead to a finite number of clusters as we always have finite amounts of data. However, this method allows the addition (or subtraction) of new clusters as new data becomes available. Figure 1 : 1this hypothesis is provided by Sonuga-Barke (2005) who used principal component analysis (PCA) on multi-dimensional functional profiles (based on a neuropsychological task battery) of ADHD patients and identified 3 distinct sub-types co-varying on timing, cognitive control, and reward. A similar approach of identifying clusters in the ADHD population using MFPs was taken by Fair et al. (2012). While similar in spirit to Sonuga-Barke (2005) and Durston et al. (2008), Fair et al. (2012) do not only look at differences in the patient population but in both, healthy controls (HC) Abnormalities in distinct brain areas (bottom level) can lead to different cognitive impairments (2nd level) and result in an ADHD diagnosis (top level). Figure reproduced from Durston et al. (2008). Figure 2 : 2Profile differences between ADHD and healthy controls (TDC) on based on functional descriptors. Lower left corner describes which each point along the x-axis represents. Upper left corner represents overall profile-differences. Right side shows differences between ADHD and HC functional profiles inside various clusters (i.e. profiles). As can be seen, group-differences vary over different profiles. Reproduced fromFair et al. (2012).classification that is (i) quantitatively measurable and (ii) a closer approximation to the underlying neurocircuitry(Robbins et al., 2012). Figure 4 : 4Overview of computational psychiatry. Different computational tools like computational modeling contribute to insights in mental health research. Reproduced from Montague et al. (2011). Figure 7 : 7Correlation between specific pruning parameter λ S and residual depression rating (BDI). Each circle represents one subject. Diameter of the circle corresponds to estimation uncertainty. See text for further model details. Reproduced fromHuys et al. (2012). In sum, Huys et al. (2012) proposed an algorithmic model of normative cognitive computation. By quantitatively fitting the model to behavioral data on a novel task and regressing an independent clinical variable (rating on the depression scale) with the fitted parameter values the authors are able pinpoint the cognitive computation underlying a clinically significant symptom. In terms of the Maia and Frank (2011) terminology, Huys et al. (2012) used a quantitative abductive approach. develop a computational model that can be fit to subjects' behavior and recover how individual subjects balance this exploration-exploitation trade-off. Intriguingly, applying this model to SZ patients,Strauss et al. (2011) found that patients with high anhedonia ratings where less willing to explore their environment and uncover potentially better actions. This result suggests a reinterpretation of the computational cognitive process underlying anhedonia. For example, one might assume that the lack of engagement of social activities of anhedonistic patients results from an inability to experience pleasure and as a consequence, a failure to learn the positive value of social interaction.Instead, this study suggests that anhedonia is a result of an inability to consider the prospective benefit of doing something that might lead to better outcomes.In sum,Gold et al. (2012) andStrauss et al. (2011) used a quantitative abductive approach to infer aberrant computational cognitive processes in RL in a subgroup of SZ patients. By grouping subjects according to symptom severity instead of diagnosis the authors addressed the problem of heterogeneity. ii) applied this model to understand and describe neural correlates of cognitive processes (e.g.Forstmann et al., 2010a;Cavanagh et al., 2011). and psychology (e.g.Clemens et al., 2011). One reason for this Bayesian revolution is the ability to quantify the certainty one has in a particular estimation. Moreover, hierarchical Bayesian models provide an elegant solution to the problem of estimating parameters of individual subjects outlined above. Under the assumption that participants within each group are similar to each other, but not identical, a hierarchical model can be constructed where individual parameter estimates are constrained by group-level distributions(Nilsson et al., 2011;Shiffrin et al., 2008). are treated as constant. The sampled value of θ (t) i will then be treated as fixed while sampling the other random variables. Figure 14 : 14Graphical model representation of the hierarchical Dirichlet process mixture model. Group parameters λ k = (µ k , σ k ). See text for details. NC Andreasen. DSM and the death of phenomenology in America: an example of unintended conse- arXiv:1303.5616v1 [q-bio.NC] 22 Mar 2013 Current challenges in psychiatry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Diagnostic and Statistical Manual of Mental Disorders . . . . . . . . . . . . . . . 6 Research Domain Criteria Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Neurocognitive phenotyping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Computational psychiatry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 Case studies in the domain of decision making . . . . . . . . . . . . . . . . . . . 14 Sequential Sampling models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.1.1 Drift Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.2 Linear Ballistic Accumulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1.3 Relationship to cognitive neuroscience . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1.4 Application to computational psychiatry . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2.1 Maximum Likelihood and χ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2.2 Hierarchical Bayesian models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.3 Likelihood-free methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Mixture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4.1 Gaussian Mixture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.2 Dirichlet Process Gaussian Mixture Models . . . . . . . . . . . . . . . . . . . . . 33 Drift-Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 Bayesian Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2.1 Empirical Bayesian Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2.2 Markov-Chain Monte-Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.3 Likelihood free methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.4 Model Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.4.1 Deviance Information Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.4.2 BIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.4.3 Bayes Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.5 Mixture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.5.1 Gaussian Mixture Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.5.2 Dirichlet Process Gaussian Mixture Models . . . . . . . . . . . . . . . . . . . . . 431 Contents 1 Part I: Motivation 4 1.1 2 Part II: Potential Solutions 7 2.1 3 Part III: Quantitative Methods 22 3.1 4 Conclusions 34 5 Appendix 35 5.1 ). RDoC improves on previous research efforts based on the DSM in the following ways. First, as the name implies it is conceptualized as a research framework only and is thus clearly separated from clinical practice. Second, RDoC is completely agnostic about DSM categories. Instead of a top-down approach which aims at identifying neural correlates of psychiatric disease, RDoC suggests a bottom-up approach that builds on the current understanding of neurobiological underpinnings of different cognitive processes and link those to clinical phenomena. Third, the RDoC research program integrates different levels of analysis like imaging, behavior and self-reports.At its core, RDoC is structured into a matrix with columns representing different "units of analysis" and rows for research domains. The units of analysis include genes, molecules, cells, circuits, physiology, behavior, and self-reports. Research domains are clustered into negative and positive valence systems, cognitive systems, systems for social processes and arousal/regulatory systems. Each of these domains is further subdivided into distinct processes; for example, cognitive systems include attention, perception, working memory, declarative memory, language behavior and executive control.Despite clear improvements over previous DSM-based research programs, the RDoC initiative currently lacks consideration of computational descriptors(Poland and Von Eckardt, 2013). As I will 1 SeeBorsboom et al. (2011) for an alternative explanation based on causal relationships between symptoms. outline below, computational methods show great promise to help link different levels of analysis, elucidate clinical symptoms and identify sub-groups of healthy and patient populations. ). 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[ "Chaos and Complexity of quantum motion", "Chaos and Complexity of quantum motion" ]	[ "Tomaž Prosen \nPhysics Department\nFaculty of mathematics and physics\nUniversity of Ljubljana\nJadranska 19SI-1000LjubljanaSlovenia\n" ]	[ "Physics Department\nFaculty of mathematics and physics\nUniversity of Ljubljana\nJadranska 19SI-1000LjubljanaSlovenia" ]	[]	The problem of characterizing complexity of quantum dynamics -in particular of locally interacting chains of quantum particles -will be reviewed and discussed from several different perspectives: (i) stability of motion against external perturbations and decoherence, (ii) efficiency of quantum simulation in terms of classical computation and entanglement production in operator spaces, (iii) quantum transport, relaxation to equilibrium and quantum mixing, and (iv) computation of quantum dynamical entropies. Discussions of all these criteria will be confronted with the established criteria of integrability or quantum chaos, and sometimes quite surprising conclusions are found. Some conjectures and interesting open problems in ergodic theory of the quantum many problem are suggested.	10.1088/1751-8113/40/28/s02	[ "https://arxiv.org/pdf/0704.2247v2.pdf" ]	17,608,120	0704.2247	e08e799b2fa02004d35aa2b239ea3756c2b994ab	Chaos and Complexity of quantum motion 7 May 2007 Tomaž Prosen Physics Department Faculty of mathematics and physics University of Ljubljana Jadranska 19SI-1000LjubljanaSlovenia Chaos and Complexity of quantum motion 7 May 2007Invited review article for special issue of Journal of Physics A on Quantum Informationnumbers: 0545Pq0545Jn0367Mn0560Gg The problem of characterizing complexity of quantum dynamics -in particular of locally interacting chains of quantum particles -will be reviewed and discussed from several different perspectives: (i) stability of motion against external perturbations and decoherence, (ii) efficiency of quantum simulation in terms of classical computation and entanglement production in operator spaces, (iii) quantum transport, relaxation to equilibrium and quantum mixing, and (iv) computation of quantum dynamical entropies. Discussions of all these criteria will be confronted with the established criteria of integrability or quantum chaos, and sometimes quite surprising conclusions are found. Some conjectures and interesting open problems in ergodic theory of the quantum many problem are suggested. Introduction This article will be about discussing the possibilities to characterize randomness and dynamical complexity in quantum mechanics and relating this issue to the questions of non-equilibrium statistical mechanics. We shall try to illustrate, mainly by presenting various numerical examples, a possible 'cyclist approach' [1] towards the quantum manybody problem which is inspired by experiences gained in studies of quantum and classical chaos of one or few particles (see e.g. [2,1]). Solving the many body problem in quantum mechanics presents a major challenge from its very beginnings. And along this way, many ingenious important analytical and efficient numerical methods of solution have been developed, for example Bethe ansatz [3], later generalized and interpreted as the quantum inverse scattering problem [4], real space renormalization group methods [5], quantum Monte Carlo [6], density matrix renormalization group (DMRG) [7], and more recently various quantum informationtheoretic based time-dependent DMRG (tDMRG) [8,9,10]. The ultimate aim of any of these methods is to efficiently find analytical or numerical approximation to the solution for some of the physical observables in the quantum many body problem, however many methods work only under some specific conditions which are not always well understood. For example, Bethe ansatz and quantum inverse scattering work only for a small subset of problems which are completely integrable, and which may often have very non-generic non-equilibrium properties, such as e.g. ballistic transport at high temperatures [11]. Quantum Monte Carlo techniques represent a very successful set of numerical methods which can yield thermodynamic equilibrium averages for generic (non-integrable) systems, however they are practically useless for computing nonequilibrium, or time dependent quantities, such as transport coefficients. Traditional DMRG method [7] is provably very successful for computing accurate ground state expectation values of almost any physical observables of one-dimensional interacting systems. And tDMRG [8,10] promised to extend the success of DMRG to time dependent physics. Indeed, the first numerical experiments looked very promising, but after a closer look one may realize that they have all been applied to a rather special subset of interacting systems and to a rather special subset of initial states. For generic (non-integrable) interacting systems or for sufficiently complex initial states, tDMRG should fail to provide an efficient computation as shall be discussed below. In our view this is to be expected, and represents an intrinsic characteristic of quantum complexity of such system and should correspond to many body extensions of the phenomena of quantum chaos [2] where spectra and eigenvector coefficients can be described by statistical ensembles of random matrices. In the past two, almost three decades there has been a lot of activity in the so called field of quantum chaos, or quantum chaology [12], where people were trying to understand the essential and significant features of quantum systems which behave chaotically in the classical limit. Classical chaos can be defined in terms of positive (algorithmic, Kolmogorov-Chaitin) complexity of systems' orbits. Still, the question whether such definition of complexity can be in an intuitively sensible way translated to quantum mechanics failed to be answered in spite of many efforts. It seems that quantum systems of finite (one or few body) chaotic systems are generically more robust against imperfections [13] than their classical counterparts, which is consistent with a simple illustration based on wave-stability of unitary quantum time evolution [14]. Nevertheless, it has been suggested that exponential sensitivity to initial conditions, the essential characteristics of classically chaotic systems, has many fingerprints in quantum mechanics. For example, in one of the pioneering works on quantum Loschmidt echoes (or fidelity decay), Jalabert and Pastawski [15] have shown that in the semiclassical regime, decay of system's sensitivity to external perturbation, as defined by fidelity, is exponential with the rate which can be related to classical Lyapunov exponents. However, this is only true in sufficiently semi-classical regime, where effective Planck constant is smaller than effective strength of perturbation, and where fidelity can be essentially explained classically [16]. On the other hand, in purely quantum regimes, quantum fidelity decays in completely different manner than the classical fidelity, and quite surprisingly displays slower decay for systems with stronger decay of temporal correlations [13]. Throughout this paper we shall discuss various ergodic properties of a simple toy model of non-integrable interacting quantum many body system, namely kicked Ising (KI) chain [17], and its time-independent version, which (both) undergo a transition from integrable to quantum chaotic regimes when the direction of the (kicking) magnetic field is changed. By simulating the dynamics of the model in terms of tDMRG we find that it can be performed efficiently only in the integrable regimes. Being interested in statistical mechanics of the model we shall describe numerical experiments addressing the questions on the relationship between the onset of quantum chaos in KI model and its quantum mixing property, namely the nature of relaxation to equilibrium, and the properties of non-equilibrium steady state. On one hand we argue that the regime of quantum chaos essentially corresponds to the regime of quantum ergodicity and quantum mixing where diffusive transport laws are valid. On the other hand, we conjecture that the transition to non-ergodic regime may occur before the system parameters reach the integrable point (even in thermodynamic limit), and that non-ergodic to ergodic transition can be characterized with order parameters which change discontinuously at a critical value of system parameter. We argue that the process of relaxation to equilibrium in quantum chaotic (mixing) case can be described in terms of quantum analogues of Ruelle resonances [18]. Furthermore, we conjecture that the eigenvectors of this process possess a certain scale invariance which can be described by simple power laws. We also discuss a possibility of numerical calculation of quantum dynamical entropies [19,20,21] in a non-trivial setting of KI model, and find, quite remarkably, that positivity of such dynamical entropies does not correspond to any other measures of quantum chaos, namely quantum dynamical entropies appear to be positive even in the integrable regimes. About two thirds of the material presented in this paper constitutes a review of a selection of recent results related to quantum chaos in many-body systems, with a flavor of quantum information. However, about one third of the material, constituting a major part of section 7, is new and original and has not yet been published before. The paper is organized as follows. In short section 2 we review basic definitions of complete integrability and quantum chaos, in particular in the context of many-body systems. In section 3 we introduce KI model which will serve us as a very convenient and efficient toy model to numerically demonstrate all the phenomena discussed in this paper. In section 4 we discuss the robustness of quantum systems to external perturbations and decoherence in open quantum systems, mainly as characterized by quantum Loschmidt echoes and entanglement between the system and the environment. In section 5 we discuss the time efficiency of best (known) classical simulation, namely using tDMRG, of locally interacting 1D quantum systems and its possible dependence on integrability of the model. In section 6 we relate standard criteria of quantum chaos to normal (diffusive) versus anomalous (ballistic) transport and discuss a simulation of a quantum heat current in non-equilibrium steady state. In section 7 we discuss a problem of quantum relaxation to equilibrium, i.e. the quantum mixing property, and quantitative characterization of quantum dynamical complexity. This section contains a large portion of original intriguing numerical results which support few interesting and perhaps surprising conjectures. In section 8 we conclude and discuss some open problems. Integrability versus chaos The central issue of this paper is to verify and demonstrate to what extend the complete integrability affects non-equilibrium properties of quantum many-body systems and their dynamical complexities, and conversely, whether (strong) non-integrability generically coincides with the established criteria of quantum chaos. Let us start by giving some established definitions (see e.g. [1,2,22,4]) of the basic terms needed to understand the issue. A classical Hamiltonian many-body system with L degrees of freedom is completely integrable, if there exist L functionally independent global smooth phase space functions (integrals of motion) which are mutually in involution, i.e. all Poisson brackets among them vanish. In such a case global canonical transformation to canonical action-angle variables can be constructed, and dynamics can be explicitly solved -at least in principle. For locally interacting infinite systems (L → ∞) analytic methods for an explicit construction of integrals of motion and canonical action-angle variables are known which usually go under a common name of inverse scattering method. Explicit solution by a mapping to an iso-spectral Schrodinger problem in terms of inverse scattering technique is usually understood as a definition of complete integrability in such context. Definition of quantum complete integrability is less unique. Nevertheless, algebraic, non-commutative versions of the inverse scattering technique exist and can be applied to some quantum interacting lattices in one-dimension, generalizing the famous Bethe ansatz solution of the Heisenberg spin 1/2 chain, and this is used as the most general known definition of quantum complete integrability. Other versions of integrability for finite L quantum systems have been proposed but they do not correspond to the integrability of the underlying classical limit, if the latter exists, so these ideas will not be considered in this paper. It has not been generally accepted yet, though demonstrated in many occasions, that completely integrable quantum systems constitute only a small subset of physical models and posses many exceptional (non-generic) non-equilibrium dynamical properties, like for example anomalous transport at finite temperatures (see e.g. [11].) On the other extreme of ergodic hierarchy we have chaotic systems. In classical Hamiltonian dynamics of few particles, chaos is best defined in terms of positive algorithmic (Kolmogorov) complexity of systems' trajectories, or equivalently, by exponential sensitivity to initial conditions. However, bounded quantum systems of finite number of particles cannot be dynamically complex as their excitation spectrum is discrete, and hence the evolution is necessarily quasi-periodic (or almost periodic). Still, quite surprisingly, even for such systems certain dynamical properties are random and universal, if the underlying classical limit is sufficiently strongly chaotic [12,2]. But genuine dynamical complexity may emerge in thermodynamic limit. However, there is still no completely satisfactory definition of dynamical chaos for infinite quantum systems. The commonly used working definition is the reference with the random matrix theory [2], namely the many-body quantum system is said to be quantum chaotic if its excitation spectrum or some other dynamical properties can be (on certain energy, or time scale) well described by ensembles of random Hermitian matrices with appropriate symmetry properties. Toy model Throughout this paper we shall use, either for illustration of theoretical results, or for numerical experimental studies, the following 1D locally interacting quantum lattice system, namely a chain of L qubits, or spin 1/2 particles, coupled with nearest neighbour Ising interaction and periodically kicked with spatially homogeneous, but arbitrarily oriented magnetic field. In a suitable dimensionless units our model can be written in terms of a three-parameter periodic time-dependent Hamiltonian H(J, h x , h z ; t) = L−1 j=0 Jσ x j σ x j+1 + δ p (t)(h x σ x j + h z σ z j ) .(1)δ p = m∈Z δ(t − m) is a unit-periodic Dirac delta and σ x,y,z j are the usual Pauli spin variables on a finite lattice j ∈ Z L = {0, 1, . . . L − 1}, satisfying the commutation (Lie) algebra [σ α j , σ β k ] = γ 2iε αβγ δ jk σ γ j . Sometimes it will turn fruitful if we extend the set of Pauli matrices by identity matrix and assign them a numerical superscript σ α j , α ∈ Z 4 , namely σ 0 j ≡ ½, σ 1 j ≡ σ x j , σ 2 j ≡ σ y j , σ 3 j ≡ σ z j . The finite chain will often be treated with periodic boundary conditions, σ α L ≡ σ α 0 , and sometimes the thermodynamic limit (TL) L → ∞ will be considered, in particular when we shall study the statistical mechanics of (1) in section 7. Although kicked Hamiltonian one-particle models have been very popular in the field of nonlinear dynamics and quantum chaos for decades, for example the Chirikov's kicked rotator model [23], the use of kicked systems in quantum many body physics has been so far very limited. If one is not only interested in zero temperature (ground state) or low temperature physics, then as we shall try to demonstrate in this review, kicked many-body models like (1) provide simpler and clearer phenomenological picture of global dynamics than time-independent models. The main reason is that since energy is not a conserved quantity, the entire Hilbert space of many-body configurations is accessible for non-integrable dynamics, and the notions of quantum ergodicity and mixing (see e.g. Ref. [24] for definitions and further references) can be defined more clearly than in the time-independent, autonomous setting. Perhaps the first kicked interacting quantum lattice has been introduced in Ref. [25]. Even if in traditional solid state physics such kicked dynamics would represent un-physically strong excitations, one has to realize that kicked quantum chains could be attractive options as benchmark models of quantum state manipulation and quantum computation in optical lattices. The Kicked Ising (KI) model (1) has been defined for the first time in Ref. [17], generalizing the integrable KI model with transverse field introduced in Ref. [26]. Clearly, as shown there [26] for the case of transverse field, h x = 0, the time-dependent model (1) can be considered completely integrable since it can be solved explicitly, for example by Wigner-Jordan-Bogoliubov transformation to non-interacting spinless fermions, and a large class of its time correlation functions can be calculated explicitly. In addition, an infinite sequence of local conserved quantities (integrals of motion) can be constructed in such a case. There is another, trivial completely integrable regime of KI model, namely the case of longitudinal field, h z = 0. Yet another, more non-trivial completely integrable regime of KI model is found when the magnitude of dimensionless field is a multiple of π/2, namely h = h 2 x + h 2 z = nπ/2, n ∈ Z, since then the magnetic kick can be considered as a multiple of π rotation, and generated by a slightly obscure set of non-interacting spinless fermions. However, in a general case of titled magnetic field when both components h x and h z are non-vanishing, and 2h/π is non-integer, the model is non-integrable, and is conjectured to be not amenable to exact analytical methods. As discussed in the following sections, non-integrable KI model can display a variety of regimes according to the criteria of quantum chaos, quantum ergodicity and quantum mixing. In fact, recently the spectral statistics of KI model in strongly non-integrable regime has been studied in detail and random matrix behaviour has been clearly confirmed at short energy ranges [27]. Due to kicked nature of interaction, the evolution propagator of KI model for one unit period of time (the so-called Floquet operator), starting just before the kick, can be constructed explicitly in terms of a time-ordered product U(J, h x , h z ) = T exp −i 1 0 dtH(J, h x , h z ; t − 0) (2) = exp(−iJ j σ x j σ x j+1 ) exp(−i j (h x σ x j + h z σ z j )) (3) = j U ′′ j,j+1 (J) j U ′ j (h x , h z ).(4) The last line suggests a simple efficient quantum protocol to simulate KI model in terms of simple 1-qubit U ′ (h x , h z ) = exp(−i(h x σ x + h z σ z )) and 2-qubit U ′′ (J) = exp(−iJσ x ⊗ σ x ) quantum gates. If we write a compact Kicked Ising 2-qubit gate W (J, h x , h z ) = U ′′ (J) · (U ′ (h x , h z ) ⊗ ½)(5) and introduce a left-to-right ordered operator product, namely + j A j ≡ · · · A 1 A 2 A 3 · · ·, then we can write KI protocol as a simple string of W −gates U(J, h x , h z ) = + j W j,j+1 (J, h x , h z ).(6) The KI model has a better known autonomous limit, namely time-independent Ising chain in a tilted magnetic field H ′ (J, h x , h z ) = lim τ →0 1 τ H(τ J, τ h x , τ h z ; t) = j (Jσ x j σ x j+1 + h x σ x j + h z σ z j ),(7) which is again non-integrable unless the field is transverse h x = 0, or longitudinal h z = 0. Decoherence and fidelity Loschmidt echoes One of the central questions about the dynamics of complex quantum systems is their robustness against small imperfections in the Hamiltonian. While it is clear that due to unitarity the quantum evolution is stable against variation of initial states [14], it is not so clear how stable it is against variation of the Hamiltonian, either being static or time-dependent -perhaps even noisy. Let us write the Hamiltonian as H δ = H 0 + δV , where H 0 is the unperturbed Hamiltonian, V is a Hermitian perturbation operator and δ is a small perturbation parameter. Peres [28] proposed the following measure of stability of quantum evolution: Let us start from some fixed initial state \|ψ , and write the time evolutions of this state generated with the unperturbed and perturbed Hamiltonian, as \|ψ 0 (t) = U 0 (t)\|ψ and \|ψ δ (t) = U δ (t)\|ψ , respectively. Then the stability is characterized by fidelity, i.e. the squared overlap between these two states F (t) = \| ψ 0 (t)\|ψ δ (t) \| 2 = \| ψ\|U † 0 (t)U δ (t)\|ψ \| 2 .(8) There are two alternative interpretations of quantum fidelity: (i) One can interpret (8) as a quantum Loschmidt echo, namely the probability that the initial state \|ψ , which is propagated forward with perturbed evolution U δ (t), and after that propagated backwards in time with unperturbed evolution U † 0 (t) = U 0 (−t), is again measured in the same (initial) state. Alternatively (ii) is just the square modulus of expectation value of the unitary echo operator U 0 (−t)U δ (t) which is the quantum propagator in the interaction picture. There have been three main theoretical approaches to understanding of fidelity decay in quantum dynamical systems: 4.1.1. Semi-classical approach. In a seminal work Jalabert and Pastawski derived quantum Loschmidt echo for systems which posses well defined classical limit. They have shown that under certain conditions, namely that the perturbation strength is large enough -typically larger than appropriately scaled Planck constant, and that initial states have certain classical interpretations -like coherent states, position states, etc, the quantum Loschmidt echo decays exponentially F (t) ∼ exp(−λt)(9) with the rate λ which is perturbation independent and typically equals the Lyapunov exponent of the underlying classical dynamics. More recently a completely classical interpretation of this so-called Lyapunov decay has been given [16] in terms of the classical Loschmidt echo and a theory for exponents λ has been developed in terms of the full spectrum of Lyapunov exponents for classical dynamical systems with few [16] or many [29] degrees of freedom. Time dependent perturbation theory and linear response approximation. In the opposite regime, where the scaled Planck constant is bigger than the perturbation strength δ, one can use time-dependent quantum perturbation theory to second order to derive a simple linear response formula for fidelity decay [17,13] F (t) = 1 − δ 2 2 t 0 dt ′ t 0 dt ′′ C(t ′ , t ′′ ) + O((δ/ ) 4 )(10) in terms of 2-point time correlation function of the perturbation C(t ′ , t ′′ ) = Ṽ (t ′ )Ṽ (t ′′ ) − Ṽ (t ′ ) Ṽ (t ′′ )(11) whereṼ (t) = U 0 (−t)V U 0 (t) is the perturbation operator in the interaction picture and . = ψ\|.\|ψ is an expectation value in the initial state. From this formula -which can be viewed as a kind of Kubo-like linear response theory for dissipation of quantum information, an interesting conclusion can be drawn: Fidelity decays faster for systems with slower decay of temporal correlations, or alternatively phrased, quantum system is more robust against external perturbations if it relaxes to equilibrium faster. One can actually go beyond the second order perturbation theory, and sum up the Born series for fidelity to all orders in many specific situations [13]. Since we are here mainly interested in qubit (spin 1/2) chains, we shall only review specific results for Kicked Ising chain [17]. We shall discuss three different specific values of system parameters, in all three we take J = 1.0, h z = 1.4: (a) Integrable regime of transverse field h x = 0, (b) weakly non-integrable regime with h x = 0.4, and (c) strongly nonintegrable regime with h x = 1.4. All the results, correlation functions and fidelity, are for random initial states, which can be interpreted as pure states of maximum quantum information, and averaging over random states is equivalent to a tracial state . = 2 −L tr(.). We consider the evolution operator (3) and perturb it by changing the magnetic field such that the perturbation is generated by the transverse component of the magnetization M = j σ z j , namely U δ (t) = [U(J, h x , h z ) exp(−iδM)] t . In the integrable case, or in general, in non-ergodic case, where the correlation function C(t ′ , t ′′ ) = M(t ′ )M(t ′′ ) (note that M(t) ≡ 0) approaches a non-vanishing plateau value as \|t ′ − t ′′ \| → ∞, namely time averaged fluctuation of magnetization D M = lim t→∞ (1/t 2 ) t 0 dt ′ t 0 dt ′′ C(t ′ , t ′′ ) is non-vanishing D M = 0, one can sum up Born series to all orders, yielding a Gaussian decay of fidelity F (t) = exp(−(t/τ ne ) 2 ), τ ne = D −1/2 δ −1 .(12) The only assumptions are that t is long enough for the time average in the definition of D M to converge, and short enough that fidelity is still above the finite size plateau value F * ∼ 1/2 L . Note that D M can be considered as an analog of Drude weight or charge stiffness in the linear response solid state transport theory (see e.g. [11]). In figs.(1,2) (a,b) one can observe the correlation plateaus of correlation functions for integrable and weakly non-integrable cases and respective good agreement with a Gaussian decay of fidelity (12). Note that in the integrable case the correlation function and the plateau D M have been calculated also analytically [26]. Note that for comparing different lattice sizes L a size-scaled value of the perturbation strength δ ′ = δ L/L 0 , with L 0 = 24, has been fixed rather than δ itself. One the other hand, for sufficiently strong integrability breaking, say in case (c) of KI model, the correlation function C(t ′ − t ′′ ) = C(t ′ , t ′′ ) decays to zero in TL, which can be interpreted as quantum mixing behviour. It has been found that quantum mixing behaviour typically corresponds to random matrix (quasi)energy level statistics, see e.g. Ref. [24] and the subsequent sections of the present paper. We shall call such behaviour the regime of quantum chaos. Here again, Born series for fidelity can be summed up to all orders, yielding an exponential decay [17] where σ = t 0 C(t ′ )dt ′ is a transport coefficient. The assumptions for validity of (13) are that t is larger than a characteristic mixing time scale on which C(t) decays and short enough so that finite size effects in quantum correlation function C(t) does not yet affect the transport coefficient σ. This regime of fidelity decay is sometimes referred to as the Fermi Golden Rule regime. Again, as demonstrated in figs. (1,2)c the agreement of numerical data for KI model with the theory is excellent. F (t) = exp(−t/τ m ), τ m = 1/(2σδ 2 )(13) Note that decay time of fidelity scales as ∝ δ −2 for ergodic and mixing dynamics, and as ∝ δ −1 for non-ergodic dynamics, making the latter much more sensitive to small perturbations. Random matrix theory (RMT). Complex quantum systems can often be well described by statistical ensembles of Hamiltonians [30]. Assuming that H 0 and V both belong to canonical ensembles of Gaussian random matrices, one can evaluate ensemble averages of the fidelity and relate C(t) to the spectral form factor of the underlying random matrix ensemble. The perturbative (linear response) theory has been developed in Ref. [31], see also Ref. [32,33]. Furthermore a non-perturbative (super-symmetric) averaging has been successfully applied to obtain exact expressions for average fidelity amplitude in the most interesting cases [34]. RMT theory of fidelity has been successfully applied to chaotic quantum systems and even to several experimental situations [35]. Following more applied philosophy, groups from Toulouse and Como have performed a series of numerical experiments [36] analyzing the robustness of several reasonable models of quantum computer hardware under small imperfections, being either due to (static) unwanted inter-qubit coupling or due to stochastic (noisy) unwanted coupling to the environment, when performing quantum algorithms simulating various toy models of classical and quantum single-particle chaos [33], like for example quantum kicked rotator [37]. The numerical results are in line with a general linear response theory, stating that static perturbations are in general more dangerous than noisy ones. Decoherence and entanglement between weakly coupled systems Similar thinking as in the previous subsection can be applied to perhaps even more fundamental problem of quantum physics, to the problem of decoherence [38]. Here we shall limit ourselves to an abstract unitary model of decoherence, where we treat a complete unitary evolution over two subsystems, a central system C, and an environment E, and then address a relevant information about the central system (i.e. the part of the system which is of physical interest) by partially tracing over the environment. Such a discussion can indeed be followed with a close link to the problem of Loschmidt echoes, by writing the total Hamiltonian H δ = H 0 + δV as an ideal (unperturbed) separable evolution H 0 = H C ⊗ ½ E + ½ C ⊗ H E perturbed by a small coupling V between the system and the environment. We shall also assume that we start from initial pure state which is a product state \|ψ = \|ψ C ⊗\|ψ E . We are interested in the properties of a generally mixed state of the central system obtained by partial tracing over the environment ρ C (t) = tr E [U δ (−t)\|ψ ψ\|U δ (t)](14) Then, under an ideal evolution, the state of the system remains a product state at all times, so the state of the central system remains pure and there is no decoherence. Decoherence is usually characterized in terms of decaying off-diagonal matrix elements of ρ C in a suitable pointer state basis, for example in the eigenbasis of the central system Hamiltonian H E . In fact for certain special forms of the perturbation operator V , the magnitudes of off-diagonal matrix elements of ρ C can be literally written as fidelities, or quantum Loschmidt echoes, in the evolution of the environment perturbed by systemenvironment coupling [39]. In such framework, the Lyapunov regime of fidelity decay corresponds exactly to the Lyapunov growth of decoherence discussed by Zurek and Paz [40]. However, another interesting indicator of decoherence is the growth of bi-partite entanglement between the system and the environment. Perhaps this notion is even more general since it does not depend on a particular choice of the pointer state basis. Since the state of the universe (central system + environment) is always pure, the characterization of entanglement is easy, say in terms of Von Neuman entropy of ρ C , S(t) = −tr C ρ C log ρ C , or linear entropy S 2 (t) = − log tr C ρ 2 C (t) which is a negative logarithm of purity P (t) = trρ 2 C (t). Usually, the quantities S(t) and S 2 (t) essentially give equivalent results -namely they can both be interpreted as the logarithm of an effective rank of the state ρ C , but the linear entropy, or purity, is more amenable to analytical calculations. Again considering chaotic models for the central system and the environment, Miller and Sarkar [41] (see also [42]) have been able to observe the 'Lyapunov regime' of entanglement growth, namely S(t) is for sufficiently strong perturbations δ found to grow linearly S(t) = ht with the rate h which is perturbation independent and given by (the sum of positive) classical Lyapunov exponents of the unperturbed, separated (sub)systems. On the other hand, in the purely quantum regime of small perturbation δ, again typically smaller than effective Planck constant, one can use time-dependent perturbation theory and derive perturbation dependent entanglement entropy [43,44], namely the purity can be explicitly expressed as P (t) ∼ 1 − δ 2 C(t) where C(t) is a particular integrated correlation function of the perturbation. In this regime we again find that quantum systems, and quantum environments, which display faster relaxation to equilibrium, are more robust against decoherence due to typical couplings. For studies of bi-partite entanglement, in particular in KI model system, see Refs. [45,46]. There exists even closer connection to fidelity theory, namely one can prove a general inequality [43,47] stating that purity is always bounded by the square of fidelity (F (t)) 2 ≤ P (t).(15) In other words: log(1/F (t) 2 ) gives an upper bound for the growth of the linear entanglement entropy, S 2 (t) ≤ −2 log F (t). In a slightly different context, bit-wise entanglement between a pair of qubits of a quantum register representing a time dependent quantum state, where the rest of the register is considered as an environment, has been demonstrated to be an indicator of quantum chaos [48] and even a signature of classical chaos [49]. Efficiency of classical simulations of quantum systems In the theory of classical dynamical systems there is a fundamental difference between integrable and chaotic systems as outlined in section 2. Chaotic systems, having positive algorithmic complexity, unlike the integrable ones, cannot be simulated for arbitrary times with a finite amount of information about their initial states. Computational complexity of individual chaotic trajectories is linear in time, however, if one wants to describe statistical states (phase space distributions) or observables of chaotic classical systems, up to time t, exponential amount of computational resources is needed, typically ∝ exp(ht) where h is the Kolmogorov's dynamical entropy related to exponential sensitivity to initial conditions. But on the other hand, how difficult is it to simulate isolated and bounded quantum systems of many interacting particles using classical resources? In analogy with the classical (chaotic) case, we might expect that the best classical simulation of typical quantum systems (in TL) is exponentially hard, i.e. the amount of computing resources is expected to grow exponentially with time. Even though there is no exponential sensitivity to initial conditions in quantum mechanics, there is a tensor-product structure of the many-body quantum state space which makes its dimension to scale exponentially with the number of particles, as opposed to linear scaling in the classical case, and due to presence of entanglement generic quantum time evolution cannot be reduced to (efficient) classical computation in terms of non-entangled (classical like) states. Here we propose the idea to use the computation complexity of best possible classical simulation of quantum dynamics, as a measure of quantum algorithmic complexity. This section essentially reviews the article [50]. We note that our proposal is different from existing proposals of quantum algorithmic complexity [51,52,53,54], namely we consider merely the classical complexity of (best) classical simulations of quantum states. In the sense of Mora and Briegel [54], quantum algorithmic complexity per unit time of initially simple timeevolving quantum states propagated by locally interacting Hamiltonian H is clearly vanishing, since an approximate quantum circuit which reproduces the state after time t is a simple repetition of Trotter-Suzuki decomposition of exp(−iHδt). Time dependent density matrix renormalization group: How far can it go? Recently, a family of numerical methods for the simulation of interacting many-body systems has been developed [8] which is usually referred to as time-dependent densitymatrix-renormalization group (tDMRG), and which has been shown to provide an efficient classical simulation of some interacting quantum systems. Of course, it cannot be proven that tDMRG provides the best classical simulation of quantum systems, but it seems that it is by now the best method available. Simulations of locally interacting one-dimensional quantum lattices were actually shown rigorously to be efficient in the number L of particles [55] (i.e., computation time and memory resources scale as polynomial functions of L at fixed t, whereas here we are interested in the scaling of computation time and memory with physical time t (in TL, L = ∞), referred to as time efficiency. In this section we address the question of time efficiency implementing tDMRG for a family of Ising Hamiltonian (7) which undergoes a transition from integrable (transverse Ising) to non-integrable quantum chaotic regime as the magnetic field is varied. We mainly consider time evolution in operator spaces [10], say of density matrices of quantum states, or (quasi) local operator algebras. Note that time evolution of pure states is often ill defined in TL [56]. As a quantitative measure of time efficiency we define and compute the minimal dimension D ǫ (t) of matrix product operator (MPO) representation of quantum states/observables which describes time evolution up to time t within fidelity 1 − O(ǫ). Our main question concerns possible scaling of D ǫ (t) for different types of dynamics, and indeed we shall demonstrate a correspondence between, respectively, quantum chaos or integrability, and exponential or polynomial growth of D ǫ (t). The key idea of operator-valued tDMRG [10] is to represent any operator in a matrix product form, O MPO = s j tr(A s 0 0 A s 1 1 · · · A s L−1 L−1 )σ s 0 0 σ s 1 1 · · · σ s L−1 L−1 ,(16) in terms of 4L matrices A s j j of fixed dimension D. The number of parameters in the MPO representation is 4LD 2 and for sufficiently large D it can describe any operator. In fact, the minimal D required equals to the maximal rank of the reduced super-density-matrix over all bi-partitions of the chain. The advantage of MPO representation lies in the fact that doing an elementary local one or two qubit unitary transformation O ′ = U † OU can be done locally, affecting only a pair of neighboring matrices A s j j . We will illustrate evolution of Ising chain (7), with open boundary conditions, for two different magnetic field values: (i) an integrable (regular) case H R = H ′ (1, 0, 2) with transverse magnetic field, and (ii) non-integrable (quantum chaotic) case H C = H ′ (1, 1, 1) with45 • tilted magnetic field. To confirm that H C , and H R , indeed represent generic quantum chaotic, and regular, system, respectively, we calculated level spacing distribution (LSD) of their spectra (shown in fig. 3). LSD is a standard indicator of quantum chaos [2]. It displays characteristic level repulsion for strongly non-integrable quantum systems, whereas for integrable systems there is no repulsion due to existence of conservation laws and quantum numbers. Evolution by tDMRG proceeds as described [2]. Eigenenergies ∈ [−9, 9] were used and statistics for even and odd parity states were combined. in [8,10,50] using an approximate Trotter-Suziki factorization, for some time step δt, of the evolution operator U(t) = exp(−iHt) in terms of 2-qubit gates. And for each two qubit gates, the matrices A s j j can be updates using a singular value decomposition with some truncation error η. Interestingly, it has been found [50] that the sum of all truncation errors up to time t, denoted by η(t) (provided δt is small enough so that the Trotter-Suzuki factorization error is negligible) is simply proportional to fidelity error, namely 1 − F (t) ≈ cη(t)/δt,(17) where F (t) = \|tr{O MPO (t)O exact (t)}\| 2 \|tr{O 2 MPO (t)}\|\|tr{O 2 exact (t)}\| ,(18) and numerical constant c ∼ 1 does not dependent on either δt, D or L. The central quantity we are going to study is D ǫ (t) which is the minimal dimension D of matrices A s i i in order for the total truncation error η(t) to be less than some error tolerance ǫ, for evolution to time t. In numerical experiments shown we take ǫ = 10 −4 except for simulating thermal states of quenched Hamiltonians where ǫ = 10 −6 . Simulating pure states For a reference we start by investigating the evolution of pure states following the basic tDMRG algorithm [8]. We studied time efficiency of simulation of pure states in Schrödinger picture, for which many examples of efficient applications exist, however all for initial states of rather particular structure, typically corresponding to low energy sectors of few quasi-particle excitations or to low dimensional invariant subspaces. Treating other, typical states, e.g. eigenstates of unrelated Hamiltonians, linear combinations of highly excited states, or states chosen randomly in the many-particle Hilbert space, we found that, irrespectively of integrability of dynamics, tDMRG is not time-efficient, i.e. D ǫ (t) typically grows exponentially in time even in the integrable case of transverse field (consistently with a linear growth of entanglement entropy [57]). Numerical results are summarized in fig. 4. Simulating local observables We continue by discussing the time efficiency of operator-valued tDMRG method using MPOs (16). Let us first study the case where the initial operator is a local operator in the center of the lattice O(0) = σ s L/2 , s ∈ {x, y, z}. In the integrable case time evolution O(t) can be computed exactly in terms of Jordan-Wigner transformation and Toeplitz determinants [58], however for initial operators with infinite index ‡ like e.g for σ x,y L/2 , L → ∞, the evolution is rather complex and the effective number of terms (Pauli group elements) needed to span O(t) grows exponentially in t. In spite of that, our numerical simulations shown in fig. 5 strongly suggest the linear growth D ǫ (t) ∼ t for initial operators with infinite index. Quite interestingly, for initial operators with finite ‡ Index of a product operator O [Sect.2, 1st of Refs. [58]] is half the number of fermi operators in Jordan-Wigner transformation of O and is a conserved quantity for H R . . 0 + \|01 . . . 0 + · · · \| 00 . . . 1 )/ √ L and \|GHZ = (\|00 . . . 0 + \|11 . . . 1 )/ √ 2) . Note that the full squares corresponds to the same example as studied in Ref. [9]. index, D ǫ (t) saturates to a finite value, for example D ǫ (∞) = 4 for σ z L/2 , or D ǫ (∞) = 16 for σ z L/2−1 σ z L/2 . In non-integrable cases the rank has been found to grow exponentially, D ǫ (t) ∼ exp(h q t) with exponent h q which does not depend on ǫ, properties of O(0) or L, for large L. For H = H C we find h q = 1.10. Simulating extensive observables In physics it is often useful to consider extensive observables, for instance translational sums of local operators, e.g. the Hamiltonian H or the total magnetization M s = L−1 j=0 σ s j . As opposed to local operators, extensive initial operators, interpreted as Wlike states in operator space, contain some long-range 'entanglement' so one may expect that tDMRG should be somewhat less efficient than for local operators. Indeed, in the integrable case we find for extensive operators with finite index that D ǫ (t) does no longer saturate but now grows linearly, D ǫ (t) ∼ t, whereas for extensive operators with infinite index the growth may be even somewhat faster, most likely quadratic D ǫ (t) ∼ t 2 but clearly slower than exponential. In the non-integrable case, we again find exponential growth D ǫ (t) ∼ exp(h q t) with the same exponent h q = 1.10 as for local initial observables. The results are summarized in fig. 6. imaginary time tDMRG from initial identity operator using the same MPO rank D as it is later used for real time dynamics. We find, consistently with previous results, that at high temperature (β ≪ 1) the rank D ǫ (t) grows very slowly, perhaps slower than linear, in the integrable case, and exponentially D ǫ (t) ∼ exp(h q t), in the non-integrable case. Interestingly, at lower temperatures we find exponential growth in both cases, even in the integrable one. This is not unreasonable as the initial (thermal) state can be expanded in a power series in β and the higher orders H p 0 become less local with longer entanglement range as we increase the power p. These results are summarized in fig. 7. Simulating thermal states Quantum chaos and far from equilibrium quantum transport In this section we would like to demonstrate the connection between quantum chaos, (non)integrability and transport in non-equilibrium steady states of interacting quantum chains [59]. Within the linear response theory, the property of quantum mixing (which is typically implied by quantum chaos, i.e. by the validity of random matrix level statistics [24], see also correlation functions in sect.4), is typically synonymous for normal quantum transport since it implies finite Kubo transport coefficients [60] (provided temporal correlation functions decay fast enough). However, here we would like to address the connection between quantum chaos and transport in far-from-equilibrium steady state, which may be beyond applicability of linear response. In particular, we are interested in the validity [61] of the Fourier law J = −κ∇T in quantum chains, relating the macroscopic heat flux J to the temperature gradient ∇T . To investigate this problem one has to deal with a finite open system connected to heat baths. Here we consider an interacting quantum spin-1/2 chain (7) which exhibits the transition from integrability to quantum chaos as a parameter, e.g. the magnetic field, is varied. The standard treatment of this problem is based on the master equation, thus limiting numerical investigations to relatively small system sizes. By using this method, in an interesting paper [62], the decay of current correlation function in a model of non-integrable chain of quantum spins is computed. However, these results were not fully conclusive and the conclusions rely on linear response theory. Also, in [63] Lindblad formalism was used to study the validity of Fourier's law for different type of spin-spin interaction. Here we describe a different approach (see Ref. [59] for details) namely we follow the evolution of the system described by a pure state, which is stochastically coupled to an idealized model of heat baths. Stochastic coupling is realized in terms of a local measurement at the boundary of the system and stochastic but unitary exchange of energy between the system and the bath. By this method we have been able to perform very effective numerical simulations which allow to observe a clear energy/temperature profile and to measure the heat current J. Again we consider Ising spin chain in the magnetic field (7) of size L, where the first and the last spin are coupled to thermal baths at temperatures T l and T r , respectively. In the non-integrable regime where the spectral statistics is described by RMT -the regime of quantum chaos -we found very accurate Fourier law scaling J/∆T ∝ 1/L, where L is the size of the chain. In the integrable and near-integrable (non-ergodic) regimes instead, we found that the heat transport is ballistic J ∝ L 0 . Let us describe the numerical simulations. Again we consider the autonomous model (7) [59]. Let us now turn to study the energy transport in this model system. To this end we need to couple both ends of the chain of spins to thermal baths at temperatures T r , T l . We have devised a simple way to simulate this coupling, namely the state of the spin in contact with the bath is statistically determined by a Boltzmann distribution with parameter T . Our model for the baths is analogous to the stochastic thermal baths used in classical simulations [64] and we thus call it a quantum stochastic thermal bath. In the representation basis of σ z n the wave function at time t can be written as \|ψ(t) = where s n = 0, 1 represents the up, down state of the n-th spin, respectively. The wave function at time t is obtained from the unitary evolution operator U(t) = exp(−iHt). The interaction with the bath is not included in the unitary evolution. Instead, we assume that the spin chain and the bath interact only at discrete times with period τ at which the states of the leftmost (s 0 ) and the rightmost (s L−1 ) spins are stochastically reset. Thus, the evolution of the wave function from time t to time t + τ can be represented as \|ψ(t + τ ) = Ξ(β l , β r )U(τ )\|ψ(t) ,(20) where Ξ(β l , β r ) represents the stochastic action of the interaction with the left and right baths at temperatures β −1 l and β −1 r respectively. The action of Ξ(β l , β r ) takes place in several steps: • (i)µ(β j ) = e β j h e −β j h + e β j h ; j ∈ {l, r} .(22) • (iv) Finally, the wave function is rotated back to the σ z n basis, \|ψ → e iα(σ y 0 +σ y L−1 )/2 \|ψ . This completes the description of the interaction with the quantum stochastic bath. This interaction thus (periodically) resets the value of the local energy hσ l,r of the spins in contact with the baths. Therefore, the value of τ controls the strength of the coupling to the bath. We have found that, in our units, τ = 1 provides an optimal choice. We have nevertheless performed simulations for other values of τ with qualitatively similar results. In particular, for weak couplings (τ 1) the heat conductivity does not depend on the coupling strength . Note that our method does not correspond to a standard stochastic unraveling of a master equation for the density operator (e.g., in the Lindblad form) [65]. However, we have tested that, using our prescription, averages over the ensemble of quantum trajectories or time averages of one given quantum trajectory are sufficient to reconstruct a density matrix operator ρ = \|ψ(t) ψ(t)\| that correctly describes the internal thermal state of the system in and out of equilibrium. For each run the initial wave-function \|ψ(0) of the system is chosen at random. The system is then evolved for some relaxation time τ rel after which it is assumed to fluctuate around a unique steady state. Measurements are then performed as time averages of the expectation values of suitable observables. We further average these quantities over different random realizations of "quantum trajectories". In order to compute the energy profile we write the Hamiltonian (7) in terms of local energy density operators H n : H n = Jσ x n σ x n+1 + h 2 · ( σ n + σ n+1 ) .(23) such that the total Hamiltonian (7) can be written as H = n H n apart from the boundary corrections. First we have performed equilibrium simulations in order to show that time averaged expectation values of the local energy density can be used to determine canonical local temperature. To this end we set the left and right baths to the same temperature T . For low T , the energy per site E = (1/L) H saturates to a constant which, together with the entire energy profile H n , is determined by the ground state. However, for larger T > 1, the energy profile is constant within numerical accuracy, and numerical simulations give E ∼ −1/T , all results being almost independent of L for L ≥ 6. The numerical data for E(T ) can be well approximated with a simple calculation of energy density for a two-spin chain (L = 2) in a canonical state at temperature T , namely E can (T ) = trH 0 e −H 0 /T /tre −H 0 /T . Therefore, if the temperatures of both baths are in high T regime, then we can define the local temperature via the relation T ∝ −1/E. We stress that equilibrium numerical data shown are insensitive to the nature of dynamics (consistent with results of Ref. [66]), whether being chaotic, regular or intermediate. In fig. 8 we show the energy profile H n for an out of equilibrium simulation of the chaotic chain. In all non-equilibrium simulations, the temperatures of the baths were set to T l = 5 and T r = 50. After an appropriate scaling the profiles for different sizes L collapse to the same curve. More interesting, in the bulk of the chain the energy profile is in very good approximation linear. In contrast, we show that in the case of the integrable (inset I) and intermediate (inset II) chains, no energy gradient is created which is a characteristic of ballistic transport. We now define the local current operators through the equation of continuity: ∂ t H n = i[H, H n ] = −(J n+1 − J n ), requiring that J n = −i[H n , H n−1 ]. Using eqs. (23) and (7) the local heat current operators are explicitly given by J n = h z J σ x n−1 − σ x n+1 σ y n . In fig. 9 we plot J/∆E as a function of the size L of the system for sizes up to L = 20. The mean current J is calculated as an average of J n over time and space n. The energy difference was obtained from the energy profile as ∆E = H L−3 − H 2 . Two spins near each bath have been discarded in order to be in the bulk regime. Since ∆L = L − 5 is an effective size of the truncated system, the observed 1/∆L dependence confirms that the transport is normal (diffusive). Moreover, also the quantity J/∆T , where ∆T = −1/ H L−3 + 1/ H 2 , shows the correct scaling with the size L. On the other hand, in integrable and intermediate chains we have observed that the average heat current does not depend on the size J ∝ L 0 , clearly indicating the ballistic transport. Quantum relaxation and complexity in a toy model: Kicked Ising Chain Let us now come back to the kicked Ising model (1) and try to consider some very elementary but fundamental questions considering its dynamics and non-equilibrium statistical mechanics. Observing data of fig. 1 in section (4) one can conclude that the model perhaps displays an interesting order to chaos, or non-ergodicity to ergodicity transition when the integrability breaking parameter is increased. Now we would like to inspect this transition more closely, and in particular understand the rate of relaxation to equilibrium in the ergodic and mixing case. We should stress right from the start that we are unable to prove any non-trivial statements about the model, but we can provide many suggestive numerical experiments which can be performed in a rather efficient way. We have learned in section 5 that it may be more fruitful to consider time evolution in the operator algebra spaces instead of in the spaces of pure states. Let us go now a bit deeper into this subject. For some related results on high-temperature relaxation in isolated conservative many-body quantum systems see e.g. Refs. [67,68,69,70,71]. Time automorphism Time automorphism on unital quasi-local C * algebra (see e.g. [72] for introduction into the subject) A Z , T : A Z → A Z of an infinite KI lattice, for one period of the kick, can be explicitly constructed by the following observations. Formally, for any A ∈ A Z , TA := U † AU, where U is given by either (3,4,6). Let A [m,n] , with m ≤ n, denote a finite, local 4 n−m+1 dimensional algebra on a sub-lattice [m, n] ⊂ Z, which is spanned by operators σ sm m σ s m+1 m+1 · · · σ sn n . It is straightforward to prove that dynamics is strictly local T : A [m,n] → A [m−1,n+1] .(24) In other words, the homomorphism T (24) is a simple nontrivial example of a quantum cellular automaton as defined by Schumacher and Werner [56]. A Z can also be treated as a Hilbert space with respect to the following inner product (A\|B) = ω(A † B), where ω(A) is a tracial state, ω(A) = 2 −(n−m+1) trA for A ∈ A [m,n] . This Hilbert space can be in fact considered as a 1D lattice of 4-level quantum systems (qudits with d = 4) with the orthonormal basis \| . . . , s −1 , s 0 , s 1 , . . .) ≡ · · · σ s −1 −1 σ s 0 0 σ s 1 1 · · · labeled by an infinite sequence of 4-digits s j ∈ Z 4 . Restricting for a moment to a dimer lattice A [j,j+1] we can write the adjoint action of a 2-qubit gate W (5) in terms of a 16 × 16 unitary matrix W j,j+1 \|s j , s j+1 ) = W † σ s j j σ s j+1 j+1 W = r j ,r j+1 ∈Z 4 \|r j , r j+1 )W (r j ,r j+1 ),(s j ,s j+1 ) (25) where, very explicitly W (r 1 ,r 2 ),(s 1 ,s 2 ) = 1 4 tr (σ r 1 ⊗ σ r 2 )W † (σ s 1 ⊗ σ s 2 )W .(26) We should note that the map is unital W\|0, 0) = \|0, 0), and that due to anti-unitary symmetry of KI model, the matrix W is real. We extend the map W j,j+1 to entire algebra j+1] . Now, following the protocol (6) we finish the construction of the time automorphism as a string of right-to-left ordered 2-qudit (with d = 4) gates A Z by W j,j+1 (A ⊗ B) = W j,j+1 (A) ⊗ B, for any A ∈ A [j,j+1] , B ∈ A Z−[j,T = − j∈Z W j,j+1 .(27) Explicitly, for any local observable A = sm,s m+1 ,...,sn a (sm,s m+1 ,...,sn) \|s m , s m+1 , . . . , s n ) ∈ A [m,n] , we have the following Algorithm 1: (i) Set an initial vector: a The "infinite-temperature" time correlation function of (traceless) local quantum observables A, B ∈ A [m,n] can be written as C BA (t) = (B\|T t A), where t ∈ Z. An interesting question is, up to what time t the C BA (t) can be computed numerically exactly with a finite computer register [m − l, n + l] of r = n − m + 1 + 2l qudits of size 4 r ? Due to locality (24) of time homomorphism one can easily prove that (B\|T t A) = (B\|T t [m−l,n+l] A), for t ≤ 2l(29) hence the correlation functions are computable exactly up to time 2l, and as we shall see later, truncated correlation function C BA (t) = (B\|T t [m−r,n+r] A) often well approximates C BA (t) even at later times, or even its asymptotic decay. Let us continue our discussion by considering time evolution for translationally invariant extensive (TIE) observables. Given some quasi-local observable A ∈ A Z we shall construct the corresponding TIE observable by a formal mapping, A → F(A) = x∈Z S x (A) in terms of lattice translation automorphisms S x : A Z → A Z , S x (σ s j ) = σ s j+x . The image of the entire quasi-local algebra under this mapping Z = F(A Z ), is not a C * algebra, but it is a linear space which can be again turned into a Hilbert space with the following inner product ((X\|Y )) = lim n→∞ 1 2n + 1 (P [−n,n] (X)\|P [−n,n] (Y ))(30) where the domain of projector P [−n,n] is extended to Z by continuity. Orthonormal basis of Z is given by TIE observables Z (c 0 ,c 1 ,...,c r−1 ) = F(σ c 0 0 σ c 1 1 · · · σ c r−1 r−1 ), for orders r = 1, 2, . . ., and for uniqueness of notation, requiring c 0 , c r−1 = 0. We shall interchangeably represent finite sequences of 4-digits with integers, (c 0 , c 1 , . . . c r−1 ) ≡ c = r−1 j=0 c j 4 j . Let Z r be 3 × 4 r−1 dimensional subspace spanned by TIE observables Z c with order ≤ r, i.e. for c having at most r base-4 digits, so we have an inclusion sequence Z 1 ⊂ Z 2 . . . ⊂ Z. Since time and space automorphisms commute TS x = S x T, one can immediately extend the time map onto the space of TIE observables,T : Z → Z by continuity. Formally, we haveTF = FT. Furthermore, locality (24) implieŝ T : Z r → Z r+2 ,(31) so we can write a simple adaptation of Algorithm 1 for explicit construction of a time map of an arbitrary finite-order TIE observable Y = c =0 (mod 4) 0<c<4 r y c Z c ∈ Z r : Algorithm 2: (i) Take the following pre-image of the TIE observable A = c =0 (mod 4) 0<c<4 r y c \|4c) ∈ A [1,r] , namely F(A) = Y . (ii) Compute a c of T(A) = 0≤c<4 r+2 a c \|c) ∈ A [0,r+1] according to Algorithm 1. (iii) Transforming back to Z the result readŝ T(Y ) = F(T(A)) = c =0 (mod 4) 0<c<4 r+2 y ′ c Z c , y ′ c =      a c + a 4c + a 16c , c < 4 r ; a c + a 4c , 4 r ≤ c < 4 r+1 ; a c , c ≥ 4 r+1 .(32) Let us further define the natural truncations of TIE space to order r,P r : Z → Z as orthogonal projectionsP r (X) = c =0 (mod 4) 0<c<4 r Z c ((Z c \|X)), and truncated time evolution operatorsT r =P rT : Z r → Z r , which are naturally implemented on a computer by simply truncating overflowing coefficients y ′ c . Physically interesting question now concerns computation of time correlation functions between a pair of finite order (say q) TIE observables X, Y ∈ Z q , namely C Y X (t) = ((Y \|T t X)), for example in fig. 1 we have shown the case of X = Y = Z 3 . As a consequence of locality (24), and translational invariance, we find that the truncated evolution on Z r , reproduces correlation functions exactly ((B\|T t A)) = ((B\|T t r A)), for t ≤ r − q. Relaxation and quantum Ruelle resonances In classical mechanics of chaotic systems one typically observes that states (phasespace densities) develop small details in the course of time evolution at an exponential average rate. Consequently, introducing a small stochastic noise of strength ǫ to Perron-Frobenius operator (PFO, i.e. Liouvillian propagator for discrete time dynamics), makes it non-unitary and shifts its spectrum inside the unit circle. Typically, the effect of noise is equivalent to an ultraviolet cutoff -truncation of PFO -at the Fourier scale k ∼ 1/ǫ, and often the leading eigenvalues of truncated PFO -the so-called Ruelle resonancesremain frozen inside the unit circle in the limit ǫ → 0 (or k → ∞) [73]. For a general introduction to relaxation phenomena in classical Hamiltonian dynamics see e.g. [74]. Let us now draw some some analogies with our quantum setting. We have seen that that the evolutionT somehow most closely resembles Liouvillian evolution of classical Hamiltonian dynamics. In Hilbert space topology, operatorT is unitary and its spectrum lies on a unit circle, just like in the case of classical PFO. However, truncated (3 × 4 r−1 ) × (3 × 4 r−1 ) matricesT r represent natural "ultraviolet" cutoff truncations for increasing orders r. Let us check numerically if some eigenvalues of these matrices remain frozen when r → ∞. Indeed, as we demonstrate in fig. 10, we find several eigenvalues which converge as r increases in the case of strongly non-integrable (quantum chaotic) case, with a gap between an eigenvalue of maximal modulus and the unit circle, whereas in the integrable case a set of r eigenvalues touch the unit circle (actually eigenvalue 1 is rfold degenerate). Numerical results suggest the following speculative conclusions. Let e −qn be the converged (frozen) eigenvalues ofT r , and {Θ R n }, {Θ L n } the corresponding right and left eigenvectors, respectively. Then for arbitrary pair X, Y ∈ Z, the time correlation function can be expressed in terms of spectral decomposition (see e.g. [74]) C Y X (t) ∼ n w n e −qnt , w n = ((Y \|Θ R n ))((Θ L n \|X)) ((Θ L n \|Θ R n )) . The above relation is the contribution of the point spectrum and is exact if the spectrum is pure-point. However, in classical cases one may quite typically have various singular components and branch cuts [74]. Note that the denominator ((Θ L n \|Θ R n )) is finite although both vectors should have infinite l 2 norm ((Θ L n \|Θ L n )) = ∞, ((Θ R n \|Θ R n )) = ∞, for any eigenvalue away from the unit circle, Re q n = 0. There is a simple relation between the spectrum of PFO and the ergodic properties of dynamics: (i) If there is a spectral gap, i.e. there exists λ > 0 such that for all n, \|e −qn \| ≤ exp(−λ) < 1, then dynamics is exponentially mixing, C Y X (t) ≤ C exp(−λt). (ii) If some eigenvalues are on the unit circle, meaning that the corresponding eigenvector coefficients should be in l 2 , then the system is non-mixing since there are correlation functions which do not decay. (iii) If some eigenvalues are at 1 then the system is non-ergodic since the correlation functions may have non-vanishing time-averages. If Q n is a complete set of orthonormalized eigenvectors corresponding to eigenvalue 1, ((Q n \|Q m )) = δ n,m (and note that since we are on the unit circle: Q R n = Q L n ) then D X := C XX (t) = n \|((X\|Q n ))\| 2 .(35) The latter (iii) happens in generic completely integrable quantum lattices, where Q n correspond to an infinite sequence of conservation laws [75]. Furthermore, we have a strong numerical evidence that also in certain non-integrable quantum lattices [76], and also in KI model [17], one has a regime where few normalizable ('pseudo-local') but not local (like in integrable models) conservation laws exist. This situation we call the regime of intermediate dynamics and is characterized by a non-vanishing stiffness D X = 0 signalling ballistic transport. In figure 11 exact time evolution C L (t) = 1 L MU −t L MU t L on a finite lattice of length L with periodic boundary conditions, (2) iteration of truncated TA matrix on infinite lattice C r (t) = ((M\|T t r M)), and (3) asymptotics based on (few) leading eigenvalue resonance(s) (using formula (34) in terms of q n and w n .). We note that the leading eigenvalue and eigenvector of truncated TAT r can most efficiently be computed using our Algorithm 2 as a key step of an iterative powermethod. In this way we were able to perform calculations of the leading Ruelle resonances up to r = 15 in contrast to full diagonalization of truncated matricesT r which were feasible only up to r = 7. Let us observe the structure of the eigenvector coefficients v L,R c = ((Z c \|Θ L,R n )) corresponding to the leading eigenvalue. Numerical results (see fig. 12, see also subsect.7.5 later) strongly suggest self-similar behaviour upon multiplying the code c by 4 which is a consequence of the fractal structure of the transfer matrix (see illustration in Ref. [77,18]). The stiffness D X and the spectral gap ∆ = \|1 − e −q 1 \| may be considered as order parameters, characterizing a particular kind of dynamical phase transition, namely the transition from non-ergodic dynamics -ordered phase, where D X = 0 for a typical X §, or ∆ = 0, to an ergodic and mixing dynamics -disordered phase, where D X = 0, for all traceless X ∈ Z, or ∆ > 0. We know that KI model is non-ergodic in integrable regimes. Let us consider a fixed transverse field case h x = 0, and start to switch on a small amount of longitudinal field h x . The interesting question is whether the transition happens for infinitesimal integrability breaking parameter h x in TL, or at a finitecritical field. In fig. 13, we fix J = 0.7 and plot a two dimensional phase diagram of the spectral gap ∆(h x , h z ). It is clear that we have different behaviours in different regions of parameter space, for example we identify two: (i) Type I transition: if the transverse field is roughly on the interval h z ∈ [0.7, 1.2], then the spectral gap opens in the fastest possible manner which is allowed by a h x → −h x symmetry and the analyticity of the problem, namely ∆ ∝ h 2 x . See fig. 14. (ii) Type II transition: if the initial transverse field h z < 0.7, or h z > 1.2, then the gap opens up in a much more abrupt -perhaps a discontinuous way. We give an example by scanning the diagonal transition, i.e. putting h x = h z and increasing h x from zero. Numerical results, shown in fig.15 cannot be made fully conclusive, but they are not inconsistent with a conclusion that an abrupt transition to ergodic behaviour takes place at h x = h z ≈ 0.3. Translationally invariant conservation laws as matrix product operators There is another possibly interesting way of characterizing the transition, i.e. in terms of pseudo-local translationally invariant conservation laws [76]. Such conservation laws are the square normalizable elements Q ∈ Z, which are mapped onto themselves under the dynamicsT(Q) = Q. We shall first make a non-trivial variational MPO ansatz for elements of Z, namely let us take an auxiliary vector space C D = C D 1 ⊕ C D 2 ⊕ C D 3 , where D = D 1 + D 2 + D 3 . Then any operator Q, which is formally written in terms of MPO on the infinite spin chain Q = ...s −1 s 0 s 1 ...∈Z 4 ( a L · · · A s −1 A s 0 A s 1 · · · a R ) · · · σ s −1 −1 σ s 0 0 σ s 1 1 · · ·(36) where A s ∈ C D×D , a L , a R ∈ C D have the block matrix form (for k = 1, 2, 3): A 0 =   1 0 0 0 1 0 0 0 E 0   , A k =   0 * * 0 0 0 0 * E k   , a L =   * 0 0   , a R =   0 * 0   ,(37) represents a translationally invariant pseudo-local operator, i.e. an element of Z, provided that \|\|E 0 \|\| < 1 and \|\|E 1 \|\| 2 + \|\|E 2 \|\| 2 + \|\|E 3 \|\| 3 < 1, where \|\|.\|\| is a spectral matrix norm and * 's stand for arbitrary matrices/vectors. Of course, converse cannot be generally true, not any element of Z can be written as MPO (36) with finite D, but still there are elements of the form (36,37) which are not in Z r for any finite r. There exist a straightforward algorithm which performs time evolution on MPO data (37), namelyT(Q) is also of the form (36,37) with dimension D ′ ≤ 2D. We shall now make the following simple numerical experiment. Let us fix D, setting D 1 = D 2 = 1 representing the simplestT-invariant subclass of (36,37), and optimize (maximize) the fidelity-like quantity This is exactly what we observe in KI model following a line of type II transition (see fig. 16). F (Q) = \|((Q\|TQ))\| ((Q\|Q)) ,(38) Operator-space entanglement measures and complexity of time evolution Numerical results of section 5 suggested that operator space entanglement measures can be used to characterize the complexity of time evolution, namely the minimal required rank D ǫ of MPO ansatz is simply related to entanglement entropy of a time-evolving local observable, which is interpreted as a Hilbert space vector. In order to make things as simple and precise as possible we go back to the time evolution automorphism T over the quasi-local spin algebra A Z . Let us take some truncation order r = 2n + 1 and consider the truncated map T (t) = − log P (m,n) (t) where P (m,n) (t) = tr[R (m,n) (t)] 2 is a purity of reduced super-density matrix. In fig. 17 we plot the entanglement purity P (0,n) (t) -for close to symmetric bipartition m = 0 where the entanglement is expected to be maximal -for three different characteristic cases of KI model, which will be in the following referred to as: quantum chaotic (QC), J = 0.7, h x = 0.9, h z = 0.9, integrable (IN), J = 0.7, h x = 0, h z = 0.9, and non-ergodic (NE) non-integrable case, J = 0.7, h x = 0.2, h z = 0.2. We find that in QC case purity decreases exponentially P (0,n) (t) = exp(−h q t), meaning S 2 (t) = h q t, where the exponent h q is independent of the initial observable A(0) and asymptotically independent of r. On the other hand, in IN case P (0,n) (t) does not decay at all so the resulting dynamical entropy h q = 0, whereas in NE case P (0,n) (t) decays slowly, likely slower than exponentially. Scaling invariance and the problem on semi-infinite lattice The eigenvectors ofT corresponding non-unimodular eigenvalues seem to exhibit a certain scaling invariance ( fig. 12). Here we would like to explore this property in a little bit more detail. For that purpose we again explore the map T [−n,n] on the quasi-local algebra A [−n,n] since the representation of dynamics is conceptually simpler (Algorithm 1) than dynamicsT 2n+1 on Z 2n+1 . In particular, it is worth to mention that if one traces out an additional qudit after each time step, then the dynamics is exact on a closed set of 4 n−m+1 × 4 n−m+1 super-density matrices and has a simple explicit form in terms of a completely positive matrix map R (m+1,n+1) (t + 1) = tr 0 T n−m R (m,n) (t) ⊗ E 00 T † n−m ,(40)T r = − 0≤k≤r ½ ⊗k 4 ⊗ W ⊗ ½ ⊗(r−k) 4 ,(41) namely no truncation is needed since at time t + 1 we are describing an exact observable on A [−n−1,n+1] . We write (tr 0 R) j,k := 3 s=0 R s+4j,s+4k for tracing out the least significant qudit, and E 00 = \|0)(0\| is an elementary 4×4 projector. Note that T n−m is just a matrix of T [m,n] in the canonical basis \|s). It is rather trivial to exactly solve this dynamics for a small finite n − m, however this does not yield physically very useful information about the KI dynamics. One would wish to study the correct TL by first taking n → ∞, and only after that m, t → ∞, however this task seems almost computationally intractable. Still, we were able to make some modest numerical experiments exploring this question, suggesting that for sufficiently strong integrability breaking (say QC case) the asymptotic matrix R (m,∞) (∞) has a remarkable scale invariance if we coarse-grain it by tracing over its 4 × 4 blocks: R (m+1,∞) (∞) = tr 0 R (m,∞) (∞) = ζR (m,∞) (∞),(42) where ζ is some scaling factor. This seems to be true for both orders of the limits t → ∞, n → ∞, although better numerical results have been obtained for the 'incorrect' limit, namely letting the number of iterations t → ∞ for a finite register size n, and then checking the convergence of results with increasing n. NE cases we find non-vanishing plateaus in the correlation function, i.e. non-vanishing stiffness D + := C + = 0 which signals non-ergodicity and existence of local (for integrable cases) and pseudo-local (for non-ergodic and non-integrable cases, e.g. NE) conservation laws of T + . We note that the asymptotic correlation decay in non-integrable cases, say QC and NE, seems quite insensitive to increasing truncation order n -indicating that the leading eigenvalues of T + [0,n] remain frozen when increasing n. Note that the asymptotic exponents of correlation decay for a semi-infinite chain are not the same as for an infinite one, i.e. the point spectra of T + [0,n] andT n are in general different, however we have some indications to believe that their phase diagrams should agree, namely ergodic regimes in the KI model on semi-infinite chain model are in one-to-one correspondence with ergodic regimes of the model on infinite chain. A ⊗ B) j+j ′ d,k+k ′ d = A j,k B j ′ ,k ′ . However, the most remarkable feature of dynamics T + is the following. The dynamical equation for R (m+1,n+1) (t + 1) in terms of R (m,n) (t) is exactly the same as for doubly-infinite lattice, namely eqs. (40,41). Hence also the conjecture (42) on scaling invariance of R (m,∞) (∞) should be the same for the two lattice topologies. However, numerical computations are much easier and thus the results are more suggestive for the semi-infinite case. Figure 19. Scaling of the partial norms u(p) = R (m0+p,n) 0,0 (t) in log scale (units are arbitrary) versus the partial tracing index p, for cases QC (a), NE (b) and IN (c). Diamonds, stars, and squares represent data for the semi-infinite KI lattice with truncation size n = 15, (diamonds, stars) and n = 11 (squares), all for m 0 = 4. Finite number of time steps t = t * = 17 (diamonds), just before the absorbing boundary affects any of the data shown, is compared to steady state observable t = ∞ (stars). In the non-integrable cases (a,b), data are compared also with steady state t = ∞ simulation of the same local initial observable A(0) = σ x 0 on the two-sided (doublyinfinite) KI lattice with truncation size n = 7 (r = 15), and m 0 = 2 (triangles). Let us now discuss some numerical results. We have always started from local initial operator A(0) = σ x 0 . We took n as large as allowed by existing computing resources, namely n = 15 for the semi-infinite chain and n = 7 for the infinite chain, and compared the data for asymptotic matrices R (m,n) (∞) (in numerics t has been chosen such that the results converged, typically t ≈ 100) with finite time data R (m,n) (t * ) where time t * was set as large as allowed so that the data were still exact and no truncation was needed, typically t * ≈ n. In all cases, numerical results were quite insensitive to small changes in truncation order n. First we have computed the scaling of the principal matrix element, or the partial norms R (42), should asymptotically scale as ∝ ζ m (see fig.19). If asymptotic dynamics t → ∞ is determined by normalizable eigenvectors of T + , which necessarily correspond to uni-modular eigenvalues, then we should have ζ = 1. In QC case a clear scaling was observed for both topologies (Z and Z + ) with the same exponent ζ, however the exponent was slightly different for R (m,n) (∞) and R (m,n) (t * ). In the cases of nonergodic dynamics (NE and IN) the results for two topologies were quite different. For semi-infinite topology we find very clearly that ζ = 1 indicating that A(t * ) and even A(∞) can be written as l 2 convergent sums of local operators. As a more quantitative test of conjecture (42) we compare the upper-left ('most important') 16 × 16 block of the super density matrix scaled to a unit principal element R p j,k (t) ≡ R (m 0 +p,n) j,k (t)/R (m 0 +p,n) 0,0 (t). In fig.20 we plot the diagonal elements R p j,j (t) for different p, while in fig.21 we plot 2D charts of the entire scaled density matrices R p j,k (t). Indeed we find for QC case that the matrix R p j,k (t) is practically insensitive to increasing truncation (p) and to topology of the lattice (semi-infinite versus infinite), both for finite Figure 20. Scaling of the diagonal elements of scaled reduced super-density matrices R p j,j (t) for initial tracing index m 0 = 6, and p = 1 (diamonds), p = 2 (stars), p = 3 (squares), p = 4 (triangles), for non-integrable cases of KI chain on semi-infinite lattice truncated at n = 15, namely for case QC (a,b), and case NE (d,e). Finite time data at t = t * = 17 kicks are shown in (a,d), while steady state observables t = ∞ are analyzed in (b,e), all starting from initial observable σ x 0 . For comparison, asymptotic steady state t = ∞ data for two-sided, doubly infinite KI chain, truncated at r = 13, and with initial tracing index m 0 = 2, are shown in (c,f), namely for QC case (c) and NE case (f). Note that data in plot (b) are practically exactly overlapping. time t = t * and 'steady-state' t = ∞. On the other hand, in non-ergodic cases, the scaling (42) is typically broken. However, it seems to be observed in the steady state (t = t * ) of NE case, which is (in our setting) probably a non-physical but still quite robust effect due to truncation (a kind of absorbing boundary condition). Summarizing this subsection, we conjectured that in the regime of quantum chaos reduced super-density matrices of time-evolved observables typically obey the scaling law (42) with the exponent ζ which only depends on global dynamics on quasi-local algebra of observables and not on a particular choice of initial observable. We expect that accurate numerical calculations of exponent ζ would be possible within a certain quantum dynamic renormalization group scheme, however its precise formulation at present remains an open problem. Figure 21. Scaling of scaled reduced super density matrices R p j,k (t) -the grayness level is proportional to \|R p j,k (t)\| -for initial tracing index m 0 = 4, and p = 1, . . . , 6 (columns plots), for six different cases, QC, NE, and IN of truncated semi-infinite KI lattice (n = 15), at finite time t = t * = 17, and asymptotic steady state t = ∞ (row plots), always starting form initial observable σ x 0 . 16 × 16 'most important' matrix elements are plotted with the matrix site j = k = 0 at lower-left corner. Dynamical entropies In section 5, further elaborated in subsection 7.4, we have proposed an entanglement in operator space of quasi-local algebra as a possible new measure of quantum algorithmic complexity. Here we would like to compare this briefly to some established proposals of quantum dynamical entropy, such as for example CNT entropy [19] or AF entropy [20], both being ideally suited for a quantum dynamical system formulated in terms of time automorphism T, tracial invariant state ω, and quasi-local C * algebra A Z . We shall here briefly review AF entropy which is conceptually simpler. Let us start by taking a set of say k elements A α , α = 1, . . . k of A Z which form a partition of unity, namely α A * α A α = ½. Following the dynamics up to integer time t, A α (t) = T t A α , we construct a set of k t elements depending on a multi-index α = (α 0 , . . . , α t−1 ), namely A (t) α = A α 0 (0)A α 1 (1) · · · A α t−1 (t − 1) = A α 0 TA α 1 T · · · TA α t−1 .(45) From the homomorphism property and unitality of dynamics it follows that A ρ (t) α,β = ω [A (t) β ] * A (t) α .(46) ρ (t) can clearly be interpreted as a density matrix pertaining to dynamically generated partition, and its Von Neumann entropy generated per unit time defines the AF entropy S AF = sup {Aα} lim sup t→∞ − 1 t tr ρ (t) log ρ (t) ,(47) where, strictly speaking, supremum over all possible generating partitions has to be taken. It is not surprising that practical evaluation of S AF is impossible except for rather trivial cases, such as dynamics generated by shift automorphism S 1 [20]. However, one can easily show that a related linear AF entropy S AF 2 = sup {Aα} lim sup t→∞ − 1 t log tr[ρ (t) ] 2 ,(48) is tractable much more easily, while the behaviour of S AF and S AF 2 is likely to be similar in practice. The key observation is to write the purity P AF (t) = tr[ρ (t) ] 2 = α,β ρ (t) If ½ is a unit element in A Z then ½×½ is a unit element inÃ Z , and purity can be written using a transfer-matrix-like approach as P AF (t) =ω [TK] t (½ × ½) .(50) The idea can be worked out in detail for a complete generating partition of a local subalgebra A [−q+1,q] of size k = 4 2q . There it turns out that, due to locality of dynamics (24), the resulting purity is independent of q ≥ 1, i.e. the supremum (48) is already achieved by a rather modest partition of 4 2 = 16 elements, and the mapTK can be factored into a direct product of two maps acting separately on two independent copies of a semi-infinite lattice. Leaving out some technical details of derivation, the final result reads as follows. Let us consider a time dependent 4 t × 4 t matrix Q(t), representing a state onÃ Z + = A Z + × A Z + , with initial value Q(0) = Q 0,0 (0) = 1, and dynamics given by the following completely positive matrix map Q(t + 1) = tr [0,1] T t+1 [E 00 ⊗ ½ 4 ⊗ Q(t) ⊗ E 00 ] T † t+1 ,(51) where (tr [0,1] R) j,k = 15 s=0 R s+16j,s+16k traces out two least important qudits, and unitary time evolution matrix T t is given in (41). Then the purity, and linear dynamical entropy (LDE), are simply given as P AF (t) = [Q 0,0 (t)] 2 , S 2 (t) = −2 log Q 0,0 (t), respectively. The asymptotic increase of S 2 (t) per unit time yields the linear AF entropy (48). Again, for practical calculations it is convenient to consider truncation of matrices Q(t) after each iteration (51), say at dimension 4 r . In fig. 22 we plot LDE for different cases of KI dynamics, and we observe that LDE is always clearly growing linearly ∝ t, so the AF entropy is always strictly positive, even in non-ergodic (NE) and integrable (IN) cases, and that the results are robust and stable against changing the truncation order r. Perhaps this finding appears surprising, but one has to bear in mind that AF and CNT entropies can be positive even for simpler dynamics, such as quasi-free flows on C * algebras. For ergodic dynamical systems on C * algebras one can use Shannon-Mcmillan-Breiman (SMB) theorem (for classical SMB theorem see e.g. [78], and [79] for a possible quantum extension), which states that for typical sequences α, multi-time correlation function (MTCF) should decay exponentially C(t) = ω(A (t) α ) ∼ exp(−ht)(52) where the exponent h, which should be essentially independent of α, is equal to a dynamical entropy of the map T with respect to an invariant state ω. For an interesting application of SMB theorem in the context of quantum dynamical chaos see Ref. [80]. In our numerical experiments we considered truncated dynamics T [−n,n] , writing truncation order as r = 2n + 1, and computed two kinds of MTCF: (i) For a uniform sequence α = (1, 1, . . .), where A 1 = σ x 0 (or A 1 = σ x 0 σ x 1 in which case the truncated lattice was placed as [−n, n + 1], hence r = 2n + 2), we estimated MTCF (52) as (A 1 \|T [−n,n] A 1 T [−n,n] · · · T [−n,n] A 1 ). (ii) For a random sequence α, we took a complete generating partition {\|s −q , . . . s q )} on A [−q,q] with k = 4 l elements, for l ≡ 2q + 1 = 1 or l = 3, and computed an average square modulus of MTCF \|C(t)\| 2 = \|(A α 0 \|T [−n,n] A α 1 T [−n,n] · · · T [−n,n] A α t−1 )\| 2 over 20 randomly sampled sequences α. In both cases (i,ii) we have found a rather good agreement between log \|C(t)\| 2 and LDE S 2 (t) for the case of quantum chaotic dynamics ( fig. 22). For average random MTCF we IN (c)), compared to − ln \|C(t)\| 2 where C(t) are MTCF with obervables X = σ x 1 and XX = σ x 1 σ x 2 at different cutoff orders r (symbols), and to average − ln \|C(t)\| 2 over random MTCF sampled over 20 sequences of random Pauli observables of lengths l = 1 and l = 3 (curve-symbols). found rather good agreement even for non-ergodic cases (NE and IN), however MTCF for the uniform sequence exhibited big oscillating fluctuations there, indicating simply that SMB theorem does hold for non-ergodic dynamics of KI model. Summarizing, it seems that AF entropy, even though it very cleanly generalizes the concept of Kolmogorov-Sinai classical dynamical entropy to quantum dynamical systems, cannot be used as an indicator of quantum chaos or an indicator of computational complexity of quantum dynamical systems. Nevertheless, it has to be stressed that quantum dynamical entropies can be related to the notion of quantum algorithmic complexity, mentioned in section 5, by a quantum version of Brudno theorem established in Ref. [81]. Conclusions and open questions In the present article we have reviewed several possible approaches to describe dynamic instabilities, relaxation phenomena, and computation complexity in the simple model of one-dimensional non-integrable locally interacting quantum many body dynamics. We have argued that the essential non-equilibrium statistical properties of quantum dynamical systems -in the absence of idealized external baths -may be crucially related to the integrability of the system, or in the complementary case, to the existence of the regime of quantum chaos. The general flavor which remains after such studies is that the non-integrable quantum many body problem at high temperature will preclude any exact and complete solution by its very nature. Still it is hoped that a more complete ergodic theory of such systems could be developed, allowing for example for exact calculation of relaxation rates, scaling exponents of resonance eigenvectors, etc. It has been shown in many cases that integrable quantum systems have anomalous non-equilibrium statistical mechanics at high temperature, for example they exhibit ballistic transport. This can easily be understood as being a consequence of existence of (an infinite sequence of) exact local conservation laws which prevents quantum ergodicity, similarly as existence of canonical action variables in classical integrable systems prevents classical ergodicity. An interesting open question is the following: How strong integrability breaking perturbation is needed, in generic cases, to break all the exact conservation laws and yield normal (diffusive) transport? In other words: A quantum KAM theory is needed! Numerical experiments shown in this paper suggest an interesting possibility, namely that in some cases a finite, critical perturbation strength is required. the grants P1-0044, and J1-7347 of Slovenian Research Agency. Figure 1 . 1Correlation decay for three cases of finite KI model: (a) integrable h x = 0, (b) intermediate h x = 0.4, and (c) ergodic h x = 1.4, whereas J = 1.0, h z = 1.4, and for different sizes L = 20, 16, 12 [solid-dotted connected curves, almost indistinguishable in (a,b)]. Circles in (a) show exact result for L = ∞. Figure 2 . 2Average fidelity amplitude (absolute value of it) for three cases of finite KI: (a) integrable h x = 0, (b) intermediate h x = 0.4, and (c) ergodic h x = 1.4, whereas J = 1.0, h z = 1.4, and for different sizes L = 20, 16, 12, and different scaled perturbations δ ′ . Chain curves give theoretical predictions[17,13]. Figure 3 . 3Nearest neighbor LSD for H C (left) and H R (right) for L = 12. Dashed curves are p(s) = sπ/2 exp (−π 2 s 2 /4) (left) and p(s) = exp (−s) (right), typical for chaotic and regular systems, respectively Figure 4 . 4MPS rank D ǫ (t) for simulating pure states with integrable transverse Ising model H R , except full squares which are for Heisenberg XX chain, starting from the initial states indicated in the legend (explanation: \|W = (\|10 . . Figure 5 . 5In the last set of numerical experiments we consider time efficiency of the evolution of a thermal initial state O(t) = Z −1 exp(−βH 0 ) under a sudden change of the Hamiltonian at t = 0, namelyH(t < 0) = H 0 = H ′ (1, 0, 1), H(t > 0) = H 1 . Again, D ǫ (t) for local initial operators. We consider three cases O(0) = σ x,y,z L/2(empty circles, squares and triangles), for non-integrable evolution H C , and four cases, O(0) = σ x,y L/2 (full squares, diamonds), σ z L/2−1 σ y L/2 (full triangles), all with infinite index, and O(0) = σ z L/2−1 σ z L/2 (full circles) with index 2, for integrable evolution H R . In the inset we plot the data for the non-integrable case H C in semi-logarithmic scale, and the full line in the inset illustrates exponential growth ∝ e 1.1t . Full squares and diamonds are for L = 40, otherwise L = 20. Figure 6 .Figure 7 . 67D ǫ (t) for extensive initial operators. For both Hamiltonians H C , H R we take O(0) = j σ x j (empty, full squares) with infinite index, and O(0) = H ′ (1, 0, 1) (empty, full circles) with index 1. For H R we also show the case O(0) = j σ z j σ z j+1 + σ y j σ y j+1 (full diamonds) with index 1 and 2. In the semi-log inset we illustrate exponential increase ∝ e 1.1t (full straight line) for H C and polynomial ∼ t 2 (full curve) for H R . For full circles L = 64, otherwise L = 32. two situations: in the first case we consider change after which the Hamiltonian remains integrable, H 1 = H ′ (1, 0, 2) = H R , while in the other case the change breaks integrability of the Hamiltonian, H 1 = H ′ (1, 1, 1) = H C . The initial state is prepared by means of D ǫ (t) for thermal states of H 0 with β = 0.01 (β = 0.05 in inset), for evolution with H C (open symbols) and H R (full symbols) at L = 40. Solid curves again indicate exponential increase ∝ e 1.1t . at three particular cases: (i) quantum chaotic case H = H ′ (−2, 2, 3.375) at which LSD agrees with RMT and thus corresponds to the regime of quantum chaos, (ii) integrable case H = H ′ (−2, 0, 3.375), at which LSD is close to the Poisson distribution, and (iii) intermediate case H = H ′ (−2, 2, 7.875) at which the distribution LSD shows and intermediate character of weak level repulsion and exponential tail s 0 ,s 1 ,...,s L−1 C s 0 ,s 1 ,...,s L−1 (t)\|s 0 , s 1 . . . s L−1 , The wave function is first rotated by the angle α = arctan(h x /h z ) to the eigenbasis of the components σ l = h · σ 0 /h, σ r = h · σ L−1 /h of the edge spins along the field h = (h x , 0, h z ), that is \|ψ → e −iα(σ y 0 +σ y L−1 )/2 \|ψ . Here, h = \| h\| stands for the magnitude of the magnetic field.• (ii) A local measurement of the observables σ l , σ r is performed. Then the state of the spins at the borders collapses to a state (after choosing s * 0 , s * L−1 , we put all coefficients C s 0 ,s 1 ,...,s L−1 with (s 0 , s L−1 ) = (s * 0 , s * L−1 ) to zero. • (iii) The new state of the edge spins (s 0 , s L−1 ) is stochastically chosen. After this action simulating the thermal interaction with the baths each of the edge spins is set to down, (up) state with probability µ,(1 − µ). If the new state of s 0 , or s L−1 , is different than the one after step (ii), then a simple unitary spin flip is performed to the wave-function. The probability µ(β) depends on the canonical temperature of each of the thermal baths: Figure 8 . 8Out of equilibrium energy profile H n for the chaotic chain. The temperatures of the baths T l = 5 and T r = 50, are both in the high temperature regime. Results for chains of size L = 8 (crosses), L = 14 (open circles) and L = 20 (solid circles) are shown. The dashed line was obtained from a linear fit of the data for L = 20 for the L − 4 central spins. Insets (I) and (II) show the energy profile for the integrable and intermediate cases respectively, for L = 15. Figure 9 . 9Size dependence of the energy current in the chaotic chain with T l = 5 and T r = 50. We show J/∆E (open circles) and J/∆T (open squares). The dashed lines corresponds to 1/∆L scaling. In the inset, the size dependence of the energy current is shown for the integrable (solid circles) and the intermediate (solid squares) cases. m−1 ,sm,...,sn,s n+1 ) = δ s m−1 ,0 a (sm,...,sn) δ s n+1 ,0 . (ii) For k = 0, 1, . . . , n − m + m−1 ,sm,...,sn,s n+1 ) = r,r ′ ∈Z 4 W (s m−1+k ,s m+k ),(r,r ′ ) a (k) (s m−1 ,...,s m+k−2 ,r,r ′ ,s m+k+1 ,...,s n+1 ) (28) (iii) The result is TA = sm,s m+1 ,...,sn a (n−m+2) (s m−1 ,sm,...,s n+1 ) \|s m−1 , s m , . . . s n+1 ). The algorithm produces exact result in (n − m + 2)4 n−m+4 multiplications and about the same number of additions. Let P [m,n] : A Z → A [m,n] denote a linear orthogonal projector, satisfying P [m,n] (A ⊗ E) = (½\|E)A if A ∈ A [m,n] ,E ∈ A Z−[m,n] . Let us define a truncated time evolution operator T [m,n] = P [m,n] T : A [m,n] → A [m,n] which we can actually implement on a computer with a finite memory register. Few remarks are in order: (i) Truncated translationally invariant time evolutionT r is perhaps more natural object to study than truncated local time evolution T [m,n] , for a simple reason that the truncationP r commutes with a shift S x , while P [m,n] does not. (ii) A space Z can be identified with translationally invariant linear functionals over A Z , namely X(A) = ((X\|FA)), X ∈ Z, A ∈ A Z . We have (T † X)(A) = X(TA) and (TX)(A) = X(T † A) where Hermitian adjoint mapsT † and T † simply correspond to time reversed dynamics. (iii) Convex subspace of positive translationally invariant functionals W ⊂ Z is an interesting invariant subspace of physical states,TW ⊆ W. Figure 10 . 10The spectra of truncated transfer operatorsT r , for r = 5, 6, 7 in strongly non-integrable case J = 0.7, h x = 0.5, , h z = 1.1 (left) and integrable case J = 0.7, h x = 0.0, h z = 1.1 (right), lying inside complex unit circle (thin arcs). The points in upper/lower unit semi-disks correspond to positive/negative paritŷ RZ (c0,c1,...,c l−1 ) = Z (c l−1 ,...,c1,c0) eigenvectors. Arrows point at converged positions of the leading eigenvalue e −q1 . Figure 11 . 11we compare the time autocorrelation function of the transverse magnetization M = j∈Z σ z j = Z 3 , computed in three different ways: Correlation function of the transverse magnetization C(t) = ((M \|M (t))), in the mixing case J = 0.7, h x = 0.5, h z = 1.1, computed from finite system dynamics for different sizes L (symbols), and from truncated adjoint propagatorsT r of infinite systems (curves) for different truncation orders r. The chain line indicates the asymptotics based on the leading quantum Ruelle resonance. Figure 12 .Figure 13 . 1213Eigenvector Θ R 1 = c v c Z c , of the leading eigenvalue (closest to unit circle) has statistically selfsimilar structure, when expanded in Z c . We plot the modulus of coefficients v c (in log scale) of the right eigenvector versus the log (in base 4) of the integer code c. Dashed line indicates power law scaling c −ν with slope ν = 0.32. In the inset we plot partial scalar products u m = c =0 (mod 4)4 m−1 ≤c<4 m −1 ((Θ L 1 \|Z c ))((Z c \|Θ R 1 )) with the corresponding left eigenvector within fixed orders m. Two dimensional numerical phase diagram for kicked Ising lattice at cutoff order r = 7. Gray level indicates the spectral gap log ∆ ofT r as a function of h x and h z at fixed J = 0.7. White regions correspond to ∆ < 10 −6 . Figure 14 .Figure 15 . 1415Type I transition: Spectral gap ofT r as a function of h x at J = 0.7, h z = 1.1 at different truncation orders r. In the inset we indicate a line of transition in 2D phase diagram. Type II transition: Type I transition: Spectral gap ofT r as a function of h x = h z at J = 0.7 at different truncation orders r. In the inset we indicate a line of transition in 2D phase diagram. Figure 16 . 16within this class of operators. Let us write an operator which maximizes F (Q) for a given D as Q D . Note that 1 − F (Q D ) gives a strong-topology measure of conservation of Optimized fidelity (38) -in log scale -for approximate conservation laws within MPO spaces of different fixed dimensions D (indicated in the legend) for an infinite KI lattice along the diagonal type II transition with h x = h z and J = 0.7. A simple stochastic search has been used to maximize fidelity F (Q). Note the similarity with the gap curve shown infig.15.observable Q D in one step of time evolution, so F (Q D ) = 1 only for exact conservation laws. Increasing D may improve fidelity, if pseudo local conservation laws exist to which Q D may converge, however in ergodic and mixing situation where no exact pseudo-local conservation laws exist, increasing D should have no significant effect to fidelity F (Q D ). Figure 17 . 17[−n,n] . Starting with local operators on a single site A [0,0] this truncated map is exact up to time t = n. Writing time evolved observable at any instant of time as A(t) = T t [−n,n] A(0) = s a (t) (s −n ,...sn) \|s −n , . . . , s n ) in terms of a 'wave-function' a (t) (s −n ,...,sn) , and partitioning a sub-lattice at m, −n < m ≤ n, as [−n, n] = [−n, m − 1] ∪ [m, n], we can define 4 n−m+1 × 4 n−m+1 reduced super-density Purity P (0,n) (t) of reduced operator space density matrix for KI lattice, with different cutoff sizes r = 2n + 1 = 15, 13, 11, and for two initial centered local operators X = σ x 0 and Z = σ z 0 . Note exponential decay of purity for QC case and saturation or slow decay of purity for non-ergodic cases (IN, NE). matrix as R (m,n) (sm,...sn),(s ′ m ,...,s ′ n ) (t) = s −n ,...,s −n ,...,s m−1 ,sm,...sn) a (t) * (s −n ,...,s m−1 ,s ′ m ,...s ′ n ) . (39) Since A(t) is interpreted as a 'pure state', namely a vector from A [−n,m−1] ⊗ A [m,n] , the operator space entanglement is most simply characterized either by Von Neuman entropy S (m,n) (t) = −trR (m,n) log R (m,n) or linear entropy S (m,n) 2 Before discussing numerical results we note another useful observation. Let us define and briefly study KI chain on a semi-infinite lattice Z + = [0, ∞], with the Hamiltonian (1) for L = ∞ and open boundary condition on the left edge. Now we consider a quasilocal algebra A Z + . The time automorphism is again strictly local T + : A [0,n] → A [0,n+1] , and can be written as a semi-infinite product T + = − j∈Z + W j,j+1 . In the definition of the truncated time map T + [0,n] = P [0,n] T + : A [0,n] → A [0,n] truncation is needed only on the right edge. Note that due to this property, simulation of local observables (localized near the edge of the lattice) is twice as efficient than on doubly-infinite lattice, meaning that with the same size of computer register one can exactly simulate for twice longer times. For example, computation of correlation functions C + BA (t) := (B\|[T + ] t A) is exact (B\|[T + ] t A) = (B\|[T + [0,n] ] t A), for t ≤ 2(n − q), if A, B ∈ A [0,q] . (43) Numerically inspecting the correlation functions of a simple local spin A = B = σ x 0 infig. 18we find clear asymptotic exponential decay for QC case, whereas for IN and When labeling tensor product matrix elements we shall always follow a convention that left factors are labelled with less significant digits, namely ( Figure 18 . 18Decay of correlations C + (t) = σ x 0 (t)σ x 0 for semi-infinite KI lattice, with different cutoff sizes n = 15, 13, 11, and for different cases (QC, NE, IN), all indicated in the figure. Note that the two curves for NE case are practically overlapping, and that all curves for different r's are exactly overlapping until t = 2r smaller . Splitting the truncated semi-lattice as [0, n] = [0, m − 1] ∪ [m, n] and following the time evolution of an observable A(t) = [T + [0,n] ] t A(0) = s a (t) (s 0 ,...sn) \|s 0 , . . . , s n ) in terms of a 'super-wavefunction' a (t) (s 0 ,...,sn) , one can again define the reduced super-density matrix as R (m,n) (sm,...sn),(s ′ m ,...,s ′ n ) (t) = s 0 ,...,s 0 ,...,s m−1 ,sm,...sn) a (t) * (s 0 ,...,s m−1 ,s ′ m ,...s ′ n ) . ...s m−2 ,s m−1 ∈Z 4 \|a (t) (...,s m−2 ,s m−1 ,0,0...) \| 2 which, assuming ½. Hence using an invariant state ω one can form a positive, Hermitian, trace-one, k t × k t matrix of dynamics over a product algebraÃ Z = A Z × A Z , with time automorphism T(A×B) = T(A)×T(B) and an invariant stateω(A×B) = ω(A)ω(B). To this product structure we add a linear mapK :Ã Z →Ã Z depending on a generating partition {A α } K(B) = k α,β=1(A β × A α )B(A * α × A * β ). 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[ "Dynamics of Field Induced Polarization Reversal in Strained Perovskite Ferroelectric Films with c-oriented Polarization", "Dynamics of Field Induced Polarization Reversal in Strained Perovskite Ferroelectric Films with c-oriented Polarization" ]	[ "Laurent Baudry \nInstitute of Electronics, Microelectronics and Nanotechnology (IEMN)-DHS Départment\nUMR CNRS 8520\nUniversité des Sciences et Technologies de Lille\n59652Villeneuve d'Ascq CedexFrance\n", "Igor A Luk'yanchuk \nLaboratory of Condensed Matter Physics\nUniversity of Picardie Jules Verne\n80039AmiensFrance\n\nL. D. Landau Institute for Theoretical Physics\nMoscowRussia\n", "Anna Razumnaya \nPhysics Department\nSouthern Federal University\n344090Rostov on DonRussia\n" ]	[ "Institute of Electronics, Microelectronics and Nanotechnology (IEMN)-DHS Départment\nUMR CNRS 8520\nUniversité des Sciences et Technologies de Lille\n59652Villeneuve d'Ascq CedexFrance", "Laboratory of Condensed Matter Physics\nUniversity of Picardie Jules Verne\n80039AmiensFrance", "L. D. Landau Institute for Theoretical Physics\nMoscowRussia", "Physics Department\nSouthern Federal University\n344090Rostov on DonRussia" ]	[]	The field-induced polarization reversal in c-oriented ferroelectric phase of strained perovskite film has been studied. We show that in additional to the conventional longitudinal switching mechanism, when c-oriented polarization vector changes its modulus, the longitudinal-transversal and transversal mechanisms when the perpendicular component of polarization is dynamically admixed are possible. The later process can occurs either via the straight-abrupt or initially-continues polarization turnover scenario. We specified the obtained results for the case of PbTiO3 and BaTiO3 ferroelectrics and propose the experimental methods for their investigation.	10.1103/physrevb.91.144110	[ "https://arxiv.org/pdf/1403.4191v2.pdf" ]	119,249,658	1403.4191	eeebce777052a0f09dfa437b1c23c22d8884d580	Dynamics of Field Induced Polarization Reversal in Strained Perovskite Ferroelectric Films with c-oriented Polarization 24 Mar 2014 Laurent Baudry Institute of Electronics, Microelectronics and Nanotechnology (IEMN)-DHS Départment UMR CNRS 8520 Université des Sciences et Technologies de Lille 59652Villeneuve d'Ascq CedexFrance Igor A Luk'yanchuk Laboratory of Condensed Matter Physics University of Picardie Jules Verne 80039AmiensFrance L. D. Landau Institute for Theoretical Physics MoscowRussia Anna Razumnaya Physics Department Southern Federal University 344090Rostov on DonRussia Dynamics of Field Induced Polarization Reversal in Strained Perovskite Ferroelectric Films with c-oriented Polarization 24 Mar 2014(Dated: March 25, 2014) The field-induced polarization reversal in c-oriented ferroelectric phase of strained perovskite film has been studied. We show that in additional to the conventional longitudinal switching mechanism, when c-oriented polarization vector changes its modulus, the longitudinal-transversal and transversal mechanisms when the perpendicular component of polarization is dynamically admixed are possible. The later process can occurs either via the straight-abrupt or initially-continues polarization turnover scenario. We specified the obtained results for the case of PbTiO3 and BaTiO3 ferroelectrics and propose the experimental methods for their investigation. Dynamical switching properties of ferroelectrics are essential for their application in the memory-storage devices [1]. The underlying mechanism of polarization reversal is of special interest for the mostly used pseudocubic perovskite crystals that, depending on the orientation of polarization P = (P 1 , P 2 , P 3 ) can exhibit tetragonal, orthorhombic or rhombohedral structural phases in the ferroelectric state of the bulk material [2]. The situation is more diverse in case of substrate-deposited perovskite ferroelectric films in which the substrate-provided deformation makes the lattice constant c in z-direction (perpendicular to the film surface) different from the in-plane lattice constants a = b already in the hightemperature paraelectric phase with P = 0. In particularly, Pertsev, Zembilgotov and Tagantsev [3,4] studied the effect of substrate clamping on PbTiO 3 and BaTiO 3 films and proposed that at least four structural phases can exist in strain-temperature, u m -T phase diagram (Fig. 1). The so-called c-phase with P = (0, 0, P 3 ) occurs at high compressive strains whereas the aa-phase with P = (P 1 , P 1 , 0) is realized at high tensile strains. Either ac-phase with P = (P 1 , 0, P 3 ) or r-phase with P = (P 1 , P 1 , P 3 ) can occur at low strains. These phases are thermodynamically stable and separated by continuous (thin) or discontinuous (bold) transition lines in Fig. 1. In the present letter we study the uniform polarization switching in PbTiO 3 and BaTiO 3 oxides induced by the applied electric field and demonstrate that the situation is even more rich. Additional phases can dynamically appear during the polarization reversal. We restrict ourselves to the c-phase region of u m -T phase diagram and consider the switching process when the initially uporiented polarization P = (0, 0, P 3 ) decreases and then suddenly drops down under the oppositely applied field E = (0, 0, E) with E < 0. To describe the PbTiO 3 and BaTiO 3 materials we use the renormalized Landau-Devonshire functional given in Regions of the longitudinal (l ), longitudinaltransversal (lt) and transversal (t) switching regimes and corresponding separating lines L0 and L1 on the phase diagrams of strained films of PbTiO3 (a) and BaTiO3 (b), adopted from Ref. [3]. Stability line L2 determines the type of transversal switching. Close location of L2 and L1 for BaTiO3 implies that it occurs according the initially-continuous turnover of polarization (tc), whereas the absence of this line for PbTiO3 means that transversal switching is straight-abrupt (ta) [3], for which the account of the six-order terms is known to be important [5][6][7]: G (P, E, T, u m ) = a * 1 P 2 1 + P 2 2 + a * 3 P 2 3 + a * 11 P 4 1 + P 4 2 +a * 33 P 4 3 + a * 13 P 2 1 + P 2 2 P 2 3 + a * 12 P 2 1 P 2 2 + a 123 P 2 1 P 2 2 P 2 3 +a 112 P 4 1 P 2 2 + P 2 3 + P 4 3 P 2 1 + P 2 2 + P 4 2 P 2 1 + P 2 3 +a 111 P 6 1 + P 6 2 + P 6 3 + u 2 m s 11 + s 12 − EP 3 .(1) The last term in Eq. (1) presents the field-driving interaction with electric polarization. The renormalized coefficients a * 1 , a * 3 , a * 11 , a * 33 , a * 13 and a * 12 depend on the misfit strain u m and temperature T whereas other coefficients s 11 , s 12 , a 111 , a 112 and a 123 correspond to its bulk homologous, as was explicitly specified in Ref. [3]. Note that several alternative approaches were proposed to establish the u m -T phase diagram of BaTiO 3 [8][9][10]. Their results are competitive with [3,4] mostly in relative location of r-and ac-phases. This minor difference is not essential for our consideration and can be easily taken into account for each particular case. In what follows, we consider the competition between the switching-induced ac and r phases. By substitution of the corresponding order parameters P = (P 1 , 0, P 3 ) and P = (P 1 , P 1 , P 3 ) in (1) we obtain the following effective functional: G = b 1 2 P 2 1 + b 3 2 P 2 3 + b 11 4 P 4 1 + b 33 4 P 4 3 + b 13 2 P 2 1 P 2 3 (2) + b 113 2 P 4 1 P 2 3 + b 133 2 P 2 1 P 4 3 + b 111 6 P 6 1 + b 333 6 P 6 3 − EP 3 , where b 1 = 2a * 1 , b 3 = 2a * 3 , b 11 = 4a * 11 , b 13 = 2a * 13 , b 33 = 4a * 33 , b 111 = 6a 111 , b 113 = 2a 112 , b 133 = 2a 112 , b 333 = 6a 111 for ac-phase case and b 1 = 4a * 1 , b 3 = 2a * 3 , b 11 = 8a * 11 + 2a * 12 , b 13 = 4a * 13 , b 33 = 4a * 33 , b 111 = 12a 111 + 12a 112 , b 113 = 2a 123 + 4a 112 , b 133 = 4a 112 , b 333 = 6a 111 for r-phase case. Our approach is inspired by that given by Iwata and Ishibashi [11] for the case of cubic (unstrained) lattice in paraelectric phase. It was demonstrated that depending on the strength of the polarization-lattice coupling, two reversal mechanisms are possible. For strong cubic anisotropy the switching occurs like in uniaxial one-component ferroelectrics by dynamical change of the modulus of the longitudinal polarization component P 3 . For weak anisotropy the transversal component P 1 virtually admixes to P 3 during the process. Such polarization-rotation scenario can, for instance, occurs in PbZr x Ti 1−x O 3 compounds when the anisotropic coupling is soften just as the composition parameter x approaches the morphotropic point x ≃ 0.44 from above. The distinguishing feature of the substrate-deposited films from the bulk cubic case is the strain-induced uniaxial anisotropy that is reflected both by the splitting of the critical temperatures in the second order P 2 1 and P 2 3 terms and by accounting for the 6th-order cross-coupling terms. To understand the dynamical mechanism of polarization reversal we should catch the critical field at which the switching instability occurs. Application of an opposite electric field leads to the decrease of c-oriented polarization which stays yet positive until the critical field is reached. At this stage the field-driven polarization evolution, P 3 (E) is given by the one-component variational equation: ∂G ∂P 3 P1,2=0 = b 3 P 3 + b 33 P 3 3 + b 333 P 5 3 − E = 0. (3) The value of the critical field at which polarization switching starts can be obtained from the loss of the positive definiteness of the Hessian matrix H ij = ∂ 2G ∂Pi∂Pj , presented in the extremal point of initial equilibrium P 1 = 0, P 3 = P 3 (E) as: H 33 = b 3 + 3b 33 P 2 3 + 5b 333 P 4 3 ,(4)H 11 = b 1 + b 13 P 2 3 + b 133 P 4 3 ,(5)H 13 = H 31 = 0.(6) where the dependence P 3 (E) is given by Eq. (3). Upon field increase the competition occurs between the longitudinal and transversal critical fields E (l) and E (t) , determined by the conditions H 33 P 3 E (l) = 0 and H 11 P 3 E (t) = 0. Importantly, the switching occurs at the instability field E (l) or E (t) which is attained first and the further scenario of polarization vector evolution is determined by the occurring type of instability. (i) For E (l) < E (t) the longitudinal (l ) switching instability is realized first and the polarization vector reverses its direction by change of the amplitude of P 3 from positive to negative, passing through P 3 = 0. (ii) For E (t) < E (l) the transversal (t ) switching instability is realized first and the component P 1 is admixed to P 3 after the beginning of the reversal process, just above E (t) . Polarization switching has therefore the rotational constituent, like in the Iwata and Ishibashi model. (iii) There can exist also the mixed longitudinaltransversal (lt) regime when the polarization reversal starts according to longitudinal scenario at E = E (l) but the transversal component P 1 virtually appears at the later stage of the process. The polarization evolution in l, lt and t switching regimes is sketched in Fig. 2. We presume that they are separated by crossover lines L 0 and L 1 in u m -T phase diagram and find the condition of their existence. The t-type switching can have either initially-continous (t c ) or straight-abrupt (t a ) character as will be specified later. According to the given above consideration the transversal component P 1 can dynamically admix to the component P 3 during polarization reversal if the polarization-dependent Hessian matrix element H 11 becomes negative in course of the switching. This occur e.g., when coefficient b 1 is negative. Then, when the dropping-down polarization goes through the state with vanishing P 3 , the element H 11 , according Eq. (5) acquires the negative sign in the vicinity of P 3 = 0. The polarization vector will experience the instability towards the transversal deviation and the lt regime will be realized. Therefore the crossover line L 0 between l and lt regimes is given by the condition: L 0 : b 1 (u m , T ) = 0.(7) Noteworthy that the line L 0 can be found in u m -T phase diagram as the prolongation of the paraelectric aa phase transition line located in u m > 0 region into the u m < 0 region. The condition of crossover between lt and t switching regimes can be found by equating the critical fields E (l) and E (t) or, what is equivalent and easier, by equating the corresponding longitudinal and transversal critical polarizations P (l) 3 = P 3 E (l) , P (t) 3 = P 3 E (t) calculated at these fields. The latter can be found from Eqs. P (l) 2 3 = 9b 2 33 − 20b 3 b 333 1/2 − 3b 33 10b 333 ,(8)P (t) 2 3 = b 2 13 − 4b 1 b 133 1/2 − b 13 2b 133 .(9) Condition P (l) 3 = P (t) 3 determines the crossover line L 1 between lt and t regimes: L 1 : b 3 b 13 − 3b 1 b 33 5b 1 b 333 − b 3 b 133 = 5b 1 b 333 − b 3 b 133 3b 33 b 133 − 5b 13 b 333 .(10) To be more specific we delimit the location of l, lt and t switching regimes and corresponding crossover lines L 0 and L 1 on phase diagram of strained PbTiO 3 and BaTiO 3 Fig. 1 (a). The cross and circle markers indicate the beginning of the longitudinal and transversal polarization reversal process correspondingly. films using the taken from [3] strain and temperature dependencies of coefficients of functional (2) and examining separately the cases of transitions through the ac and r phases. In the case of PbTiO 3 all these regimes are clearly visible and are located inside the region of thermodynamically stable c-phase as shown in Fig. 1 (a). To study the transient polarization dynamics we select the representative points for each transition region (points A, B and C in Fig. 1 (a)) and numerically solve the Landau-Khalatnikov kinetic equations. L i dP i dt = − δG δP i ,(11) for each polarization component P i = P i (t). Here L i are the corresponding damping coefficients. The results are presented in Fig. 3 in form of experimentally measurable longitudinal and transversal polarization currents I l = dP3 dt and I t = dP1 dt . Point A is selected for the l-switching region at u m = −0.0025 and T = 500 • C. As it follows from Fig. 3 (a) the polarization current has only the longitudinal component that is characteristic for the longitudinal switching regime. Point B corresponds to the lt-switching region and is taken at u m = −0.0025, T = 250 • C. As shown in Fig. 3 (b) both components of polarization current are observed but the transversal one is excited after the longitudinal one and vanishes earlier than the longitudinal one. Point C is taken in the t-switching region at u m = −0.0025 and T = 0 • C. As shown in Fig. 3 (c) the longitudinal and transversal polarization currents are excited simultaneously. In the case of BaTiO 3 [ Fig. 1 (b)] one can observe only the l and t-switching regimes. The lt-switching regime is difficult to detect because of the very close location of the lines L 0 and L 1 . An important issue for the t-type switching is the dynamical behavior of polarization just upon reaching the transversal instability field. Under certain conditions the intermediately-stable ac-or r-phase can be induced just above E (t) . Then, the continuous (as function of the field) turnover of polarization through this phase will precede the abrupt rotational drop-down. To distinguish between the shown in Fig. 2 initially-continous (t c ) and straight-abrupt (t a ) transversal switching process we study the global stability of functional (2) with respect to small deviations ∆P 1 , ∆P 3 about the equilibrium point P (t) 1 = 0 and P (t) 3 = P 3 E (t) exactly at E = E (t) . This is a peculiar problem since at E = E (t) the coefficient H 11 before (∆P 1 ) 2 is equal to zero and the higher-order terms should be taken into account. Following the catastrophe theory we keep only the most relevant terms and present the expansion of (2) as: G ≈ 1 2 ρ (∆P 3 ) 2 + µ (∆P 3 ) (∆P 1 ) 2 + 1 2 λ (∆P 1 ) 4 ,(12) where ρ = H 33 (see (5) 3 3 and λ = 1 12 ∂ 4G ∂P 4 1 = 1 2 b 33 . Transformation x = (∆P 1 ) 2 and z = (∆P 3 ) maps the problem onto the study of quadratic functional 1 2 λx 2 + µxy + 1 2 ρy 2 . The last one is globally unstable at λρ > µ 2 that provides the straight-abrupt switching at E E (t) . At λρ < µ 2 this functional is locally stable and at the initial stage of reversal process the P 1 component develops continuously as a function of the field. Using the given above definition of λ, ρ, µ and excluding P according Eq. (9) we, after some algebra, present the line L 2 , separating these two regimes on the u m -T phase diagram by equation: ), µ = 1 2 ∂ 3G ∂P3∂P 2 1 = b 13 P (t) 3 + 2b 133 P (t)L 2 : P R = Q 2 .(13) with P = b 3 b 33 b 13 − 3b 1 b 2 33 + 2b 1 b 2 13 − 8b 2 1 b 133 ,(14)Q = 5b 1 b 33 b 333 − b 3 b 33 b 133 , R = 3b 2 33 b 133 − 2b 2 13 b 133 + 8b 1 b 2 133 − 5b 13 b 33 b 333 . The t c and t a switching regions being located below and above this line correspondingly. Thoughtful analysis of equation (13) for BaTiO 3 case shows that the line L 2 is located very close to the line L 1 (see Fig. 1 b) which means that "t c -switching" always occurs through the intermediate field-induced ac-phase. In contrast, the line L 2 does not exist in c-phase region of the phase diagram of PbTiO 3 [ Fig. 1 (a)] which implies the "t a -switching" through the intermediate r-phase takes place. In this letter we have demonstrated the existence of different polarization reversal regimes in strained pseudocubic ferroelectric PbTiO 3 and BaTiO 3 films. Depending on the temperature and on the misfit strain one can distinguish the polarization reversal governed by the longitudinal, transversal or mixed longitudinal-transversal switching regimes. All three mechanisms can be observed in PbTiO 3 compounds. In BaTiO 3 compounds only the longitudinal and transversal mechanisms can be detected. The later occurs through the intermediate acphase with initial continuous turnover of the polarization vector as function of the field. The dynamic appearance of the transversal polarization during transition can be observed by the time-resolved piezo-force microscopy or by the in-field Raman spectroscopy sensitive to the polarization vector variation. Note, however that situation can be even more complex if the 180 o ferroelectric domains exist in the initial c-phase or/and 90 o transversal ferroelastic domains emerge during the switching process. The study of such scenarios can be done on the basis of presented above calculations. This work was supported by FP7 ITN-NOTEDEV and IRSES-SIMTECH MC mobility programs. FIG. 1 . 1FIG. 1. Regions of the longitudinal (l ), longitudinaltransversal (lt) and transversal (t) switching regimes and corresponding separating lines L0 and L1 on the phase diagrams of strained films of PbTiO3 (a) and BaTiO3 (b), adopted from Ref. [3]. Stability line L2 determines the type of transversal switching. Close location of L2 and L1 for BaTiO3 implies that it occurs according the initially-continuous turnover of polarization (tc), whereas the absence of this line for PbTiO3 means that transversal switching is straight-abrupt (ta) FIG. 2 . 2Reversal of the polarization vector as function of increasing with time switching electric field during longitudinal (l), longitudinal-transversal (lt), transversal straightabrupt (ta) and transversal initialy-continuous (tc) switching. Solid lines present the thermodynamically stable field-induced states whereas the dashed lines present the dynamicallyvirtual states appearing during the abrupt switching process. FIG. 3 . 3Time dependence of the longitudinal, I l = dP 3 dt and transversal, It = dP 1 dt polarization currents for (a) transversal (t), (b) longitudinal-transversal (lt) and (c) longitudinal (l ) switching regimes for PbTiO3. Panels (a), (b) and (c) correspond to the points A, B and C in . * Laurent, [email protected]* [email protected] Advanced microelectronics. J F Scott, Ferroelectric Memories. SpringerJ. F. Scott, Ferroelectric Memories, Advanced micro- electronics, Springer, 2000. M E Lines, A M Glass, Principles and Applications of Ferroelectrics and Related Materials, Oxford University Press. M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials, Oxford Univer- sity Press, 1977. . N A Pertsev, A G Zembilgotov, A K Tagantsev, Phys. Rev. Lett. 801988N. A. Pertsev, A. G. Zembilgotov, and A. K. Tagantsev, Phys. Rev. Lett. 80, 1988 (1998). . N A Pertsev, A G Zembilgotov, A K Tagantsev, Ferroelectrics. 22378N. A. Pertsev, A. G. Zembilgotov, and A. K. Tagantsev, Ferroelectrics 223, 78 (1999). . A J Bell, L E Cross, Ferroelectrics. 59197A. J. Bell and L. E. Cross, Ferroelectrics 59, 197 (1984). . Y L Li, L E Cross, L Q Chen, J. Appl. Phys. 9864101Y. L. Li, L. E. Cross, and L. Q. Chen, J. Appl. Phys. 98, 064101 (2005). . Y L Wang, J. Appl. Phys. 101104115Y. L. Wang et al., J. Appl. Phys. 101, 104115 (2007). . O Diéguez, Phys. Rev. B. 69212101O. Diéguez et al., Phys. Rev. B 69, 212101 (2004). . B.-K Lai, I A Kornev, L Bellaiche, G J Salamo, Applied Physics Letters. 86B.-K. Lai, I. A. Kornev, L. Bellaiche, and G. J. Salamo, Applied Physics Letters 86, (2005). . V B Shirokov, Yu I Yuzyuk, B Dkhil, V V Lemanov, Phys. Rev. B. 75224116V. B. Shirokov, Yu. I. Yuzyuk, B. Dkhil and V. V. Le- manov Phys. Rev. B 75, 224116 (2007) . M Iwata, Y Ishibashi, Jpn. J. Appl. Phys. 385670M. Iwata and Y. Ishibashi, Jpn. J. Appl. Phys. 38, 5670 (1999).	[]
[ "Barium abundance in red giants of NGC 6752 Non-local thermodynamic equilibrium and three-dimensional effects", "Barium abundance in red giants of NGC 6752 Non-local thermodynamic equilibrium and three-dimensional effects" ]	[ "V Dobrovolskas [email protected] \nVilnius University Astronomical Observatory\nM. K.Čiurlionio 29LT-03100VilniusLithuania\n", "A Kučinskas \nVilnius University Astronomical Observatory\nM. K.Čiurlionio 29LT-03100VilniusLithuania\n\nInstitute of Theoretical Physics and Astronomy\nVilnius University\nGoštauto 12, Vilnius01108LTLithuania\n", "S M Andrievsky \nDepartment of Astronomy and Astronomical Observatory\nOdessa National University and Isaac\nNewton Institute of Chile Odessa branch\n65014Shevchenko Park, OdessaUkraine\n\nGEPI\nObservatoire de Paris\nCNRS\nUniversité Paris Diderot\nPlace Jules Janssen92190MeudonFrance\n", "S A Korotin \nDepartment of Astronomy and Astronomical Observatory\nOdessa National University and Isaac\nNewton Institute of Chile Odessa branch\n65014Shevchenko Park, OdessaUkraine\n", "T V Mishenina \nDepartment of Astronomy and Astronomical Observatory\nOdessa National University and Isaac\nNewton Institute of Chile Odessa branch\n65014Shevchenko Park, OdessaUkraine\n", "P Bonifacio \nGEPI\nObservatoire de Paris\nCNRS\nUniversité Paris Diderot\nPlace Jules Janssen92190MeudonFrance\n", "H.-G Ludwig \nZentrum für Astronomie\nUniversität Heidelberg\nLandessternwarte, Königstuhl 1269117HeidelbergGermany\n", "E Caffau \nGEPI\nObservatoire de Paris\nCNRS\nUniversité Paris Diderot\nPlace Jules Janssen92190MeudonFrance\n\nZentrum für Astronomie\nUniversität Heidelberg\nLandessternwarte, Königstuhl 1269117HeidelbergGermany\n" ]	[ "Vilnius University Astronomical Observatory\nM. K.Čiurlionio 29LT-03100VilniusLithuania", "Vilnius University Astronomical Observatory\nM. K.Čiurlionio 29LT-03100VilniusLithuania", "Institute of Theoretical Physics and Astronomy\nVilnius University\nGoštauto 12, Vilnius01108LTLithuania", "Department of Astronomy and Astronomical Observatory\nOdessa National University and Isaac\nNewton Institute of Chile Odessa branch\n65014Shevchenko Park, OdessaUkraine", "GEPI\nObservatoire de Paris\nCNRS\nUniversité Paris Diderot\nPlace Jules Janssen92190MeudonFrance", "Department of Astronomy and Astronomical Observatory\nOdessa National University and Isaac\nNewton Institute of Chile Odessa branch\n65014Shevchenko Park, OdessaUkraine", "Department of Astronomy and Astronomical Observatory\nOdessa National University and Isaac\nNewton Institute of Chile Odessa branch\n65014Shevchenko Park, OdessaUkraine", "GEPI\nObservatoire de Paris\nCNRS\nUniversité Paris Diderot\nPlace Jules Janssen92190MeudonFrance", "Zentrum für Astronomie\nUniversität Heidelberg\nLandessternwarte, Königstuhl 1269117HeidelbergGermany", "GEPI\nObservatoire de Paris\nCNRS\nUniversité Paris Diderot\nPlace Jules Janssen92190MeudonFrance", "Zentrum für Astronomie\nUniversität Heidelberg\nLandessternwarte, Königstuhl 1269117HeidelbergGermany" ]	[]	Aims. We study the effects related to departures from non-local thermodynamic equilibrium (NLTE) and homogeneity in the atmospheres of red giant stars, to assess their influence on the formation of Ba II lines. We estimate the impact of these effects on the barium abundance determinations for 20 red giants in Galactic globular cluster NGC 6752. Methods. One-dimensional (1D) local thermodynamic equilibrium (LTE) and 1D NLTE barium abundances were derived using classical 1D ATLAS9 stellar model atmospheres. The three-dimensional (3D) LTE abundances were obtained for 8 red giants on the lower RGB, by adjusting their 1D LTE abundances using 3D-1D abundance corrections, i.e., the differences between the abundances obtained from the same spectral line using the 3D hydrodynamical and classical 1D stellar model atmospheres. The 3D-1D abundance corrections were obtained in a strictly differential way using the 3D hydrodynamical and classical 1D codes CO 5 BOLD and LHD. Both codes utilized identical stellar atmospheric parameters, opacities, and equation of state.Results. The mean 1D barium-to-iron abundance ratios derived for 20 giants are [Ba/Fe] 1D LTE = 0.24 ± 0.05(stat.) ± 0.08(sys.) and [Ba/Fe] 1D NLTE = 0.05 ± 0.06(stat.) ± 0.08(sys.). The 3D-1D abundance correction obtained for 8 giants is small (∼ +0.05 dex), thus leads to only minor adjustment when applied to the mean 1D NLTE barium-to-iron abundance ratio for the 20 giants, [Ba/Fe] 3D+NLTE = 0.10 ± 0.06(stat.) ± 0.10(sys.). The intrinsic abundance spread between the individual cluster stars is small and can be explained in terms of uncertainties in the abundance determinations. Conclusions. Deviations from LTE play an important role in the formation of barium lines in the atmospheres of red giants studied here. The role of 3D hydrodynamical effects should not be dismissed either, even if the obtained 3D-1D abundance corrections are small. This result is a consequence of subtle fine-tuning of individual contributions from horizontal temperature fluctuations and differences between the average temperature profiles in the 3D and 1D model atmospheres: owing to the comparable size and opposite sign, their contributions nearly cancel each other. This fine-tuning is characteristic of the particular set of atmospheric parameters and the element investigated, hence should not necessarily be a general property of spectral line formation in the atmospheres of red giant stars.	10.1051/0004-6361/201118398	[ "https://arxiv.org/pdf/1203.3124v2.pdf" ]	73,555,919	1203.3124	fa6a712d28fbfc531c2a190bc153909c4dd7c30f	Barium abundance in red giants of NGC 6752 Non-local thermodynamic equilibrium and three-dimensional effects 27 Mar 2012 May 5, 2014 V Dobrovolskas [email protected] Vilnius University Astronomical Observatory M. K.Čiurlionio 29LT-03100VilniusLithuania A Kučinskas Vilnius University Astronomical Observatory M. K.Čiurlionio 29LT-03100VilniusLithuania Institute of Theoretical Physics and Astronomy Vilnius University Goštauto 12, Vilnius01108LTLithuania S M Andrievsky Department of Astronomy and Astronomical Observatory Odessa National University and Isaac Newton Institute of Chile Odessa branch 65014Shevchenko Park, OdessaUkraine GEPI Observatoire de Paris CNRS Université Paris Diderot Place Jules Janssen92190MeudonFrance S A Korotin Department of Astronomy and Astronomical Observatory Odessa National University and Isaac Newton Institute of Chile Odessa branch 65014Shevchenko Park, OdessaUkraine T V Mishenina Department of Astronomy and Astronomical Observatory Odessa National University and Isaac Newton Institute of Chile Odessa branch 65014Shevchenko Park, OdessaUkraine P Bonifacio GEPI Observatoire de Paris CNRS Université Paris Diderot Place Jules Janssen92190MeudonFrance H.-G Ludwig Zentrum für Astronomie Universität Heidelberg Landessternwarte, Königstuhl 1269117HeidelbergGermany E Caffau GEPI Observatoire de Paris CNRS Université Paris Diderot Place Jules Janssen92190MeudonFrance Zentrum für Astronomie Universität Heidelberg Landessternwarte, Königstuhl 1269117HeidelbergGermany Barium abundance in red giants of NGC 6752 Non-local thermodynamic equilibrium and three-dimensional effects 27 Mar 2012 May 5, 2014Received ; acceptedAstronomy & Astrophysics manuscript no. ngc6752˙Ba˙abn˙v6 c ESO 2014stars: late-type -stars: abundances -stars: atmospheres -globular clusters: individual -techniques: spectroscopic Aims. We study the effects related to departures from non-local thermodynamic equilibrium (NLTE) and homogeneity in the atmospheres of red giant stars, to assess their influence on the formation of Ba II lines. We estimate the impact of these effects on the barium abundance determinations for 20 red giants in Galactic globular cluster NGC 6752. Methods. One-dimensional (1D) local thermodynamic equilibrium (LTE) and 1D NLTE barium abundances were derived using classical 1D ATLAS9 stellar model atmospheres. The three-dimensional (3D) LTE abundances were obtained for 8 red giants on the lower RGB, by adjusting their 1D LTE abundances using 3D-1D abundance corrections, i.e., the differences between the abundances obtained from the same spectral line using the 3D hydrodynamical and classical 1D stellar model atmospheres. The 3D-1D abundance corrections were obtained in a strictly differential way using the 3D hydrodynamical and classical 1D codes CO 5 BOLD and LHD. Both codes utilized identical stellar atmospheric parameters, opacities, and equation of state.Results. The mean 1D barium-to-iron abundance ratios derived for 20 giants are [Ba/Fe] 1D LTE = 0.24 ± 0.05(stat.) ± 0.08(sys.) and [Ba/Fe] 1D NLTE = 0.05 ± 0.06(stat.) ± 0.08(sys.). The 3D-1D abundance correction obtained for 8 giants is small (∼ +0.05 dex), thus leads to only minor adjustment when applied to the mean 1D NLTE barium-to-iron abundance ratio for the 20 giants, [Ba/Fe] 3D+NLTE = 0.10 ± 0.06(stat.) ± 0.10(sys.). The intrinsic abundance spread between the individual cluster stars is small and can be explained in terms of uncertainties in the abundance determinations. Conclusions. Deviations from LTE play an important role in the formation of barium lines in the atmospheres of red giants studied here. The role of 3D hydrodynamical effects should not be dismissed either, even if the obtained 3D-1D abundance corrections are small. This result is a consequence of subtle fine-tuning of individual contributions from horizontal temperature fluctuations and differences between the average temperature profiles in the 3D and 1D model atmospheres: owing to the comparable size and opposite sign, their contributions nearly cancel each other. This fine-tuning is characteristic of the particular set of atmospheric parameters and the element investigated, hence should not necessarily be a general property of spectral line formation in the atmospheres of red giant stars. Introduction Red giants in Galactic globular clusters (GGCs) carry a wealth of important information about the chemical evolution of individual stars and their harboring populations. Owing to their intrinsic brightness, they are relatively easily accessible to highresolution spectroscopy, thus are particularly suitable for tracing the chemical evolution histories of intermediate age and old stellar populations. Unsurprisingly, a large amount of work has been done in this direction in the past few decades (for a review see, e.g., Gratton et al. 2004;Carretta at al. 2010), which has resulted, for example, in the discoveries of abundance anti-correlations for Na-O (Kraft 1994;Gratton et al. 2001;Carretta et al. 2009a), Mg-Al (see, e.g., Carretta et al. 2009b), Li-Na (Pasquini et al. 2005;Bonifacio et al. 2007), and correlation for Li-O (Pasquini et al. 2005;Shen et al. 2010). Although the GGC stars display a scatter in their light element abundances, there is generally no spread in the abundances of iron-peak and heavier elements larger than the typical measurement errors (≈0.1 dex). The only known exceptions are ω Cen (Suntzeff & Kraft 1996;Norris et al. 1996) and M 54 (Carretta et al. 2010b), which do show noticeable star-tostar variations in the iron abundance. However, it is generally accepted that they are not genuine GGCs but instead remnants of dwarf galaxies. The first cluster where significant start-tostar variation in heavy element abundances was detected was M 15 (Sneden et al. 1997(Sneden et al. , 2000Otsuki et al. 2006;Sobeck et al. 2011). Roederer & Sneden (2011) found that the abundances of heavy elements La, Eu, and Ho in 19 red giants of M 92 indicate that there are also significant star-to-star variations. The latter claim, however, was questioned by Cohen (2011), who found no heavy element abundance spread larger than ∼ 0.07 dex in 12 red giants belonging to M 92. The primary formation channels of the 1 s-process elements are the low-and intermediate-mass asymptotic giant branch (AGB) stars, thus the information about the variations in heavy element abundances may shed light on the importance of AGB stars to the chemical evolution of GGCs. Chemical inhomogeneities involving the light elements in GGCs are the result of the products of a previous generation of stars. The nature of the stars producing these elements, or 'polluters' as they are often called, remains unclear. The main contenders are rapidly rotating massive stars, that pollute the cluster through their winds (Decressin et al. 2007) or AGB stars (D'Ercole et al. 2011, and references therein). A second order issue is whether the "polluted" stars are coeval and only their photospheres are polluted or they are true second-generation stars formed from the polluted material. The evidence of multiple main-sequences and sub-giant branches in GGCs (see Piotto 2008Piotto , 2009, for reviews) strongly supports the latter hypothesis, although some contamination of the photospheres may still be possible. Most of the abundance studies in GGCs have made the assumption of local thermodynamic equilibrium (LTE). Nonequilibrium effects may become especially important at low metallicity owing to the lower opacities (e.g., overionization by UV photons; see, e.g., Asplund 2005;Mashonkina et al. 2011, for more details). Deviations from LTE also occur because of the lower electron number density in the lower metallicity stellar atmospheres, which in turn decreases the electron collision rates with atoms and ions. Since most GGCs have metallicities that are significantly lower than solar, it is clearly desirable to derive abundances using the non-LTE (NLTE) approach. Nevertheless, real stars are neither stationary nor onedimensional (1D), as assumed in the classical 1D atmosphere models that are routinely used in stellar abundance work. A step beyond these limitations can be made by using three-dimensional (3D) hydrodynamical atmosphere models that account for the three-dimensionality and non-stationarity of stellar atmospheres from first principles. Recent work has shown that significant differences may be expected between stellar abundances derived using 3D hydrodynamical and classical 1D model atmospheres (Collet et al. (2007(Collet et al. ( , 2009González Hernández et al. (2009);Ramírez et al. (2009);Behara et al. (2010); Dobrovolskas et al. (2010); Ivanauskas et al. (2010); see also Asplund (2005) for a review of earlier work). These differences become larger at lower metallicities and at their extremes may reach 1 dex (!). It is thus timely to re-analyze in a systematical and homogeneous way the abundances of various chemical elements in the GGCs, employing for this purpose state-of-the-art 3D hydrodynamical atmosphere models together with NLTE analysis techniques. A step towards this was made in our previous work, where we derived 1D NLTE abundances of Na, Mg, and Ba in the atmospheres of red giants belonging to GGCs M10 and M71 (Mishenina et al. 2009). We found that in the case of the red giant N30 in M71 the 3D-1D abundance corrections for Na, Mg, and Ba, were minor and did not exceed 0.02 dex. In this study, we extend our previous work and derive 1D NLTE abundances of barium in the atmospheres of 20 red giants that belong to the Galactic globular cluster NGC 6752. The analysis is done using the same techniques as in Mishenina et al. (2009). We also derive the 3D-1D LTE abundance corrections for the barium lines in 8 red giants and apply them to correct the 1D barium abundances for the 3D effects. Finally, we quantify the influence of both NLTE and 3D-related effects on the formation of barium lines. The paper is organized as follows. In Sect. 2, we describe the observational material used in the abundance analysis. The procedure of barium abundance determinations is outlined in Sect. 3, where we also provide the details of the LTE/NLTE analysis and the determination of the 3D-1D abundance corrections. A discussion of our derived results is presented in Sect. 4 and the conclusions are given in Sect. 5. Observational data We used reduced spectra of 20 red giants in NGC 6752 available from the ESO Science Archive 1 . The high resolution (R = 60 000) spectral material was acquired with the UVES spectrograph at the VLT-UT2 (programme 65.L-0165(A), PI: F. Grundahl). Spectra obtained during the three individual exposures were co-added to achieve the average signal-to-noise ratio S /N ≈ 130 at 600.0 nm. Observations were taken in the standard Dic 346+580 nm setting that does not include the Ba II 455.403 nm resonance line. The other three Ba II lines at 585.369, 614.173, and 649.691 nm (see Table 2) are all found in the upper CCD of the red arm covering the range 583-680 nm. More details of the spectra acquisition and reduction procedure are provided by Yong et al. (2005). All the red giants studied in this work are located at or below the red giant branch (RGB) bump. Abundance analysis Atmospheric parameters and iron abundances Continuum normalization of the observed spectra and equivalent width (EW) measurements were made using the DECH20T 2 software package (Galazutdinov 1992), where the EWs were determined using a Gaussian fit to the observed line profiles. Stellar model atmospheres used in the abundance determinations were calculated with the Linux port version (Sbordone et al. 2004;Sbordone 2005) of the ATLAS9 code (Kurucz 1993), using the ODFNEW opacity distribution tables from Castelli & Kurucz (2003). Models were computed using the mixing length parameter α MLT = 1.25 and microturbulence velocity of 1 km s −1 , with the overshooting option switched off. The LTE abundances were derived using the Linux port version (Sbordone et al. 2004;Sbordone 2005) of the Kurucz WIDTH9 3 package (Kurucz 1993;Kurucz 2005;Castelli 2005). The effective temperature, T eff , was determined under the assumption of excitation equilibrium, i.e., by requiring that the derived iron abundance should be independent of the excitation potential, χ (Fig. 1, upper panel). To obtain the value of surface gravity, log g, we required that the iron abundances determined from the Fe I and Fe II lines would be equal. The microturbulence velocity, ξ t , was determined by requiring that Fe I lines of different EWs would provide the same iron abundance (Fig. 1, lower panel). The derived effective temperatures, gravities, and microturbulence velocities of individual stars agreed to within 60 K, 0.2 dex, and 0.16 km s −1 , respectively, with those determined by Yong et al. (2005). The LTE iron abundances for all stars in our sample were derived using 50-60 neutral iron lines (Table A.1; note that the iron abundance derived from the ionized lines was required to match that of neutral iron, i.e., to obtain the estimate of surface gravity, thus it is not an independent iron abundance measurement). To minimize the impact of NLTE effects on the iron abundance determinations, we avoided neutral iron lines with the excitation potential χ < 2.0 eV. Oscillator strengths and damping constants for all iron lines were retrieved from the VALD database (Kupka et al. 2000). The obtained iron abundances are provided in Table 1. The contents of the table are as follows: the star identification and its coordinates are given in Cols. 1-3, effective temperatures and iron abundance derivatives relative to the excitation potential are in columns 4 and 5, respectively, the adopted microturbulence velocity and iron abundance derivative relative to the equivalent width are in columns 6 and 7, respectively, the adopted values of log g are in column 8, iron abundances obtained from Fe I and Fe II lines are in columns 9 and 10, respectively, and the difference between them is in column 11. The mean iron abundance obtained for the 20 stars is [Fe/H] = −1.60 ± 0.05, which is in excellent agreement with [Fe/H] = −1.62 ± 0.02 obtained by Yong et al. (2005). One-dimensional LTE abundances of barium One-dimensional (1D) LTE barium abundances were derived from the three Ba II lines centered at 585.3688 nm, 614.1730 nm, and 649.6910 nm. Damping constants and other atomic parameters of the three barium lines are provided in Table 2. The line equivalent widths were measured with the DECH20T software (Table 3, columns 2-4). Hyperfine splitting of the 649.6910 nm line was not taken into account in the 1D LTE analysis. The derived barium abundances and barium-to-iron abundance ratios are given in Table 3, columns 5 and 7, respectively. We note that the barium line at 614.1730 nm is blended with a neutral iron line located at 614.1713 nm. To estimate how this Wiese & Martin (1980); b natural broadening constant, from Mashonkina & Bikmaev (1996); c Stark broadening constant, from Kupka et al. (2000); d van der Waals broadening constant, from Korotin et al. (2011) affects the accuracy of the abundance determination, we synthesized the barium 614.1730 nm line with and without the blending iron line, for all stars in our sample. The comparison of the equivalent widths of blended and non-blended lines reveals that the contribution of the iron blend never exceeds ∼ 2.4 %, or ≤ 0.05 dex in terms of the barium abundance. The contribution of the iron blend to the EW of the 614.1730 nm line was thus taken into account by reducing the measured equivalent widths of this barium line by 2.4 % for all stars. We would like to point out, however, that in the 1D NLTE analysis the barium abundances were derived by fitting the synthetic spectrum to the observed line profile, thus the influence of the iron blend at 614.1713 nm was properly taken into account. Assessment of the abundance sensitivity on the atmospheric parameters yields the following results: -Change in the effective temperature by ±80 K leads to a change in the barium abundance measured from the three Ba II lines by ∓0.03 dex; -Change in the surface gravity by ±0.1 dex changes the barium abundance by ∓0.02 dex; -Change in the microturbulence velocity, ξ t , by ±0.1 km s −1 changes the barium abundance by ∓0.07 dex. Since barium lines in the target stars are strong and situated in the saturated part of the curve of growth, it is unsurprising that the uncertainty in the microturbulence velocity is the largest contributor to the uncertainty in the derived barium abundance. The total contribution from the individual uncertainties in T eff , log g, and ξ t leads to the systematic uncertainty in the barium abundance determinations of ∼ 0.08 dex. We note, however, that the latter number does not account for the uncertainty in the equivalent width determination and thus only provides the lower limit to the systematic uncertainty (e.g., 5 percent in the equivalent width determination leads to the barium abundance uncertainty of ∼ 0.1 dex). The obtained mean 1D LTE barium abundance for the sample of 20 stars in NGC 6752 is A(Ba) 1D LTE = 0.80±0.09±0.08 and the barium-to-iron ratio is [Ba/Fe] 1D LTE = 0.24 ± 0.05 ± 0.08. In both cases, the first error is a square root of the variance calculated for the ensemble of individual abundance estimates of 20 stars. The second error is the systematic uncertainty in the atmospheric parameter determination. The difference between the individual barium abundances derived in a given star using the three barium lines is always below ∼ 0.1 dex. One-dimensional NLTE abundances of barium The one dimensional (1D) NLTE abundances of barium were determined using the version of the 1D NLTE spectral synthesis code MULTI (Carlsson 1986) modified by Korotin et al. (1999). The model atom of barium used in the NLTE spectral synthesis calculations was taken from Andrievsky et al. (2009). To summarize briefly, it consisted of 31 levels of Ba I, 101 levels of Ba II (n < 50) and the ground level of Ba III. In total, 91 bound-bound transitions were taken into account between the first 28 levels of Ba II (n < 12, l < 5). Fine structure was taken into account for the levels 5d 2 D and 6p 2 P 0 , according to the prescription given in Andrievsky et al. (2009). We also accounted for the hyperfine splitting of the barium 649.6910 nm line. Isotopic splitting of the barium lines was not taken into account. Owing to the low ionization potential of neutral barium (∼ 5.2 eV), Ba II is the dominant ionization stage in the line-forming regions of investigated stars, with n(Ba I)/n(Ba II) 10 −4 . It is therefore safe to assume that none of the Ba I transitions may noticeably change the level populations of Ba II (cf. Mashonkina et al. 1999). Further details about the barium model atom, the assumptions used, and implications involved can be found in Andrievsky et al. (2009) and Korotin et al. (2011). The solar abundances of iron and barium were assumed to be log A(Fe) ⊙ = 7.50 and log A(Ba) ⊙ = 2.17 respectively, on the scale where log A(H) ⊙ = 12. These abundances were determined using the Kurucz Solar Flux Atlas (Kurucz et al. 1984) and the same NLTE approach as applied in this study. A typical fit of the synthetic line profiles to the observed spectrum is shown in Fig. 2, where we plot synthetic and observed profiles of all three barium lines used in the analysis. The elemental abundances and barium-to-iron abundance ratios derived for the individual cluster giants are provided in Table 3 (columns 6 and 8, respectively). The mean derived 1D NLTE barium-to-iron ratio for the 20 cluster red giants is [Ba/Fe] 1D NLTE = 0.05 ± 0.06 ± 0.08. The first error is the square root of the variance in [Ba/Fe] 1D NLTE estimates obtained for the ensemble of 20 stars, thus measures the star-to-star variation in the barium-to-iron ratio. The second error is the systematic uncertainty resulting from the stellar parameter determination (see Section 3.1). The individual line-toline barium abundance scatter was always significantly smaller than 0.1 dex. We find that barium lines generally appear stronger in NLTE than in LTE, which leads to lower NLTE barium abundances. This is in accord with the results obtained by Short & Hauschildt (2006) for the metallicity of NGC 6752, and similar to the trends obtained for cool dwarfs by Mashonkina et al. (1999). The NLTE-LTE abundance corrections for the three individual barium lines are always very similar, with the differences being within a few hundredths of a dex. 3D-1D barium abundance corrections We have used the CO 5 BOLD 3D hydrodynamical (Freytag et al. 2012) and LHD 1D hydrostatic (Caffau & Ludwig 2007) stellar atmosphere models to investigate how strongly the formation of barium lines may be affected by convective motions in the stellar atmosphere. The CO 5 BOLD code solves the 3D equations of radiation hydrodynamics under the assumption of LTE. The model assumes a cartesian coordinate grid. For a detailed description of the CO 5 BOLD code and its applications, we refer to Freytag et al. (2012). Since we did not have CO 5 BOLD models available for the entire atmospheric parameter range covered by the red giants in NGC 6752, we estimated the importance of 3D hydrodynamical effects only for stars on the lower RGB. For this purpose, we used a set of 3D hydrodynamical CO 5 BOLD models with T eff = 5000 K and log g = 2.5, at four different metallicities, [M/H]= 0.0, -1.0, -2.0, and -3.0 4 . Allowing for the error margins of ∼ 100 K in the effective temperature and ∼ 0.25 dex in gravity, we assumed that the effective temperature and gravity of this model set is representative of the atmospheric parameters of the stars NGC 6752-08, and NGC 6752-19 to NGC6752-30 (8 objects, see Table 1). For these stars, the extreme deviations from the parameters of the 3D model are ∆T eff ∼ 110 K and ∆ log g ∼ 0.26. These differences would only have a marginal effect on the uncertainty in the abundance estimates, i.e., the systematic uncertainty for the 3D barium abundance derivations would only increase from ±0.08 dex quoted in Sect. 3.2 to ±0.10 dex. The 3D hydrodynamical models were taken from the CIFIST 3D model atmosphere grid (Ludwig et al. 2009). The model pa- To illustrate the differences between the 3D hydrodynamical and 1D classical stellar model atmospheres, we show their temperature stratifications at the metallicity of [M/H] = −2.0, which is the closest to that of NGC 6752 (Fig. 3, upper panel). In the same figure, we also indicate the typical formation depths of the three barium lines. It is obvious that at these depths, the temperature of the 3D hydrodynamical model fluctuates very strongly, especially in the outer atmosphere, as indicated by the RMS value of horizontal temperature fluctuations (∆T RMS = (T − T 0 ) 2 x,y,t , where T 0 is the temporal and horizontal temperature average obtained on surfaces of equal optical depth). As we see below, differences in the atmospheric structures lead to differences in the line formation properties and henceforth to differences in barium abundances obtained with the 3D hydrodynamical and 1D classical model atmospheres. Twenty 3D snapshots (i.e., 3D model structures at different instants in time) were selected to calculate the Ba II line profiles. The snapshots were chosen in such a way that the statistical properties of the snapshot sample (average effective temperature and its r.m.s value, mean velocity at the optical depth unity, etc.) would match as close as possible those of the entire ensemble of the 3D model run. The 3D line spectral synthesis was performed for each individual snapshot and the resulting line profiles were averaged to yield the final 3D spectral line profile. The influence of convection on the spectral line formation was estimated by means of 3D-1D abundance corrections. The 3D-1D abundance correction is defined as the difference between the abundance A(Y) derived for a given element Y from the same observed spectral line using the 3D hydrodynamical and classical 1D model atmospheres, i.e., (Caffau et al. 2011). This abundance correction can be separated into two constituents: (a) correction owing to the horizontal temperature inhomogeneities in the 3D model, 3D , and (b) correction owing to the differences between the temperature profiles of the average 3D and 1D models, 1D . Abundances corresponding to the subscript 3D were derived using the average 3D models, which were obtained by horizontally averaging 3D model snapshots on surfaces of equal optical depth. Spectral line profiles were calculated for each average 3D structure corresponding to individual 3D model snapshots. These line profiles were averaged to yield the final 3D profile, which was used to derive the ∆ 3D −1D abundance corrections. The full abundance correction was then ∆ 3D−1D ≡ ∆ 3D− 3D + ∆ 3D −1D . Spectral line synthesis for all three models, i.e., 3D, 3D , and 1D, was made using the Linfor3D code 5 . ∆ 3D−1D = A(Y) 3D − A(Y) 1D∆ 3D− 3D = A(Y) 3D −A(Y)∆ 3D −1D = A(Y) 3D − A(Y) The barium lines in the target stars are strong (cf. Table 3) and thus the derived 3D-1D abundance corrections are sensitive to the microturbulence velocity, ξ t , of the comparison 1D model. The 3D-1D abundance corrections were therefore calculated using the equivalent widths and microturbulence velocities of the target stars derived in Sect. 3.1 and 3.2. Furthermore, cubic interpolation was used to interpolate between the 3D-1D abundance corrections derived at four different metallicities to obtain its value at the metallicity of the cluster, [Fe/H] = −1.6. The cubic interpolation between the four values of metallicities was chosen because of the nonlinear dependence of the 3D-1D abundance corrections on metallicity. The results are provided in Table 5, which contains the ∆ 3D−1D and ∆ 3D− 3D abundance corrections for the three individual Ba II lines (columns 2-4), the 3D-1D abundance correction for each star (i.e., averaged over the three barium lines, column 5), the microturbulence velocity used with the 1D comparison model (column 6, from Sect. 1), the 3D LTE barium abundances (column 7), the 3D LTE barium-to-iron ratio (column 8), and finally both the 1D NLTE barium-to-iron ratio before (column 9, from Sect. 3.3) and after correction for the 3D effects (column 10). Abundance corrections are sensitive to the choice of the 1D microturbulence velocity and line strength, therefore stars with very similar atmospheric parameters may have different abundance corrections. This is, for example, the case for NGC 6752-19 and NGC 6752-30. These two stars have largest and smallest microturbulence velocities in the entire sample, respectively, and NGC6752-19 has slightly stronger barium lines than NGC 6752-30 (Table 3). This leads to noticeably different abundance corrections, despite both stars having very similar effective temperatures and gravities (Table 5). Yong et al. (2005) and those obtained here is somewhat concerning, especially since both studies were based on the same set of UVES spectra, while the atmospheric parameters and iron abundances of individual stars employed by us and Yong et al. (2005) agree very well (Sect. 3.1). Moreover, the comparison of the equivalent width measurements obtained by us and Yong et al. (2005) also shows good agreement. One would thus also expect good agreement in the derived barium abundances -which is unfortunately not the case. We therefore felt it was important to look into the possible causes of this discrepancy. To this end, we first obtained the 1D LTE barium abundances using the MULTI code. This independent abundance estimate was made using the same procedure as for the 1D NLTE abundance derivations, i.e., by fitting the observed and synthetic line profiles of the three Ba II lines, with the difference that in this case the line profile calculations performed with MULTI were done under the assumption of LTE. The mean barium-to-iron abundance ratio obtained in this way, [Ba/Fe] = 0.22 ± 0.06 ± 0.08, agrees well with the value derived in Section 3.2 ( [Ba/Fe] = 0.24 ± 0.05 ± 0.08). In their abundance determinations, Yong et al. (2005) used an older version of the ATLAS models (Kurucz 1993). The differences between these ATLAS models and those used in our work is that (a) different opacity tables were used in the two cases (i.e., ODFNEW from Castelli & Kurucz 2003, with our models), and (b) the ATLAS models of Kurucz (1993) were calculated with the overshooting parameter switched on, while in our case the overshooting was switched off. To check the influence of these differences on the abundance derivations, we obtained the 1D LTE barium abundance using the older atmosphere models of Kurucz (1993), with the atmospheric parameters and iron abundances derived in Sect. 3.1. In this case, the mean derived barium-toiron abundance ratio was [Ba/Fe] 1D LTE = 0.23 ± 0.05 ± 0.08, i.e., the effect of differences in the model atmospheres was only ∼ 0.01 dex. The change in the barium abundances owing to differences in the atomic parameters (line broadening constants, oscillator strengths) used in the two studies was more significant, i.e., the abundances derived by us using the atomic parameters of Yong et al. (2005) were ∼ 0.1 dex lower. However, this still leaves a rather large discrepancy, ∼ 0.15 dex, between the barium-to-iron ratios obtained by us and Yong et al. (2005), for which we unfortunately cannot find a plausible explanation. As in the case of the 1D LTE abundances, the extent of the star-to-star variations in the derived 1D NLTE barium-to-iron ratio, [Ba/Fe] 1D NLTE = 0.05 ± 0.06 ± 0.08, is small and can be fully explained by the uncertainties in the abundance determination. The 1D NLTE barium-to-iron ratio derived here is similar to the value [Ba/Fe] 1D NLTE = 0.09 ± 0.20 obtained for two red giants in M10 by Mishenina et al. (2009). The elemental ratios obtained in the two studies are thus very similar, although one should keep in mind that the estimate of Mishenina et al. (2009) is based on only two stars. The metallicities of the two clusters are very similar too, [Fe/H] = −1.56 in the case of M10 (Harris 1996(Harris , 2010 and [Fe/H] = −1.60 for NGC 6752 (Sect. 3.1). Galactic field stars typically show no pronounced dependence of [Ba/Fe] on metallicity, although the scatter at any given metallicity is large (Sneden et al. 2008). One may therefore conclude that, taken into account the high [Ba/Fe] spread in field stars, the [Ba/Fe] ratio derived here is comparable to those seen in Galactic field stars and other globular clusters of similar metallicity. The 3D-corrected 1D NLTE barium abundance in NGC 6752 The 3D-1D barium abundance corrections obtained for the eight stars in NGC 6752 (see Section 3.4 above) provide a hint of the net extent to which the 3D hydrodynamical effects may influence spectral line formation (and thus, the abundance determinations) in their atmospheres (Table 5). In the case of all red giants investigated, the corrections are small, -0.03 to +0.15 dex, and the mean abundance correction for the eight stars is ∆ 3D−1D = 0.05. We note though that the individual contributions to the abundance correction, ∆ 3D− 3D and ∆ 3D −1D , are substantial (∼ ±0.1 dex) but often because of their opposite sign nearly cancel and thus the resulting abundance correction is significantly smaller (Table 5). This clearly indicates that the role of convection-related effects on the spectral line formation in these red giants cannot be neglected, even if the final 3D-1D abundance correction, ∆ 3D−1D , is seemingly very small. The mean 3D LTE barium-to-iron abundance ratio obtained for the eight red giants is [Ba/Fe] 3D LTE = 0.28 ± 0.07 ± 0.10. The 3D LTE barium abundance measurements made for a given star from the three barium lines always agree to within ≈ 0.03 dex. In the case of all twenty giants studied here, the mean 1D NLTE barium-to-iron ratio corrected for the 3D-related effects is [Ba/Fe] 3D+NLTE = 0.10 ± 0.08 ± 0.10 and therefore is only slightly different from the 1D NLTE value obtained in Sect. 3.3. However, the positive sign of the 3D-1D abundance differences indicates that in the spectra of red giants in NGC 6752 the three studied Ba II lines will be weaker in 3D than in 1D, in contrast to what is generally seen in red giants at this metallicity (cf. Collet et al. 2007;Dobrovolskas et al. 2010). For the Ba II lines, the 3D-1D abundance corrections are sensitive to the choice of microturbulence velocities in the 1D models: an increase in the microturbulence velocity by 0.10 km s −1 leads to an increase of 0.07 dex in the 3D-1D abundance correction. At the same time, the 1D abundance itself decreases by roughly the same amount. The result is that although the 3D correction is sensitive to microturbulence, the 3D corrected abundance is not. Conclusions We have derived the 1D LTE and 1D NLTE abundances of barium for 20 red giant stars in the globular cluster NGC 6752. The mean barium-to-iron abundance ratios are [Ba/Fe] 1D LTE = 0.24 ± 0.05 ± 0.08 and [Ba/Fe] 1D NLTE = 0.05 ± 0.06 ± 0.08 (the first error measures the star-to-star variation in the abundance ratio and the second is the systematic uncertainty in the atmospheric parameter determination, see Sect. 3.1). Individual barium-to-iron abundance ratios show little star-to-star variation, which leads us to conclude that there is no intrinsic barium abundance spread in the RGB stars at or slightly below the RGB bump in NGC 6752. This conclusion is in line with the results obtained in other studies, for stars in both this and other GGCs (Norris & Da Costa 1995;James et al. 2004;Yong et al. 2005). The derived 1D NLTE barium-to-iron abundance ratio is comparable to the one observed in Galactic halo stars of the same metallicity (Sneden et al. 2008). It is also similar to the mean barium-to-iron abundance ratio obtained by Mishenina et al. (2009) for 2 red giants in the Galactic globular cluster M10. We therefore conclude that the barium-to-iron abundance ratios obtained here generally agree with those seen in the oldest Galactic populations and are not very different from those observed in halo stars. We have also obtained 3D LTE barium abundances for 8 red giants on the lower RGB in NGC 6752. The mean 3D LTE barium abundance, [Ba/Fe] 3D LTE = 0.28 ± 0.07 ± 0.10, is only 0.05 dex higher than that obtained for these stars in 1D LTE. This small 3D-1D correction leads to very minor adjustment of the mean 1D NLTE barium-to-iron ratio for the 20 investigated giants, [Ba/Fe] 3D+NLTE = 0.10 ± 0.08 ± 0.10. It would be misleading, however, to conclude that the role of the 3D effects in the formation of the barium lines in the atmospheres of red giants in NGC 6752 is minor. As a matter of fact, we have found that the 3D-1D abundance corrections owing to horizontal temperature inhomogeneities in the 3D model (i.e., ∆ 3D− 3D correction) and differences in the temperature profiles between the average 3D and 1D models (∆ 3D −1D correction) are substantial and may reach ∼ ±0.1 dex (Table 5). However, their sign depends on the line strength, and owing to this subtle fine-tuning their sum is significantly smaller, from -0.03 to 0.02 dex, which for this given set of atmospheric and atomic line parameters maintains the size of the 3D-1D abundance corrections at the level of the errors in the abundance determination. Fig. 2 . 2Fit of the synthetic NLTE profiles of the Ba II lines (solid line) to the observed spectrum of the red giant NGC6752-23 (dots). Synthetic line profiles corresponding to the abundances 0.1 dex higher/lower than the best-fit value are shown as dashed lines. Fig. 3 . 3Top panel: Temperature stratification in a single snapshot of the 3D hydrodynamical CO 5 BOLD model at the metallicity [M/H]= -2. Gray shaded area shows the temperature probability density on a logarithmic scale, with darker shades meaning a higher probability of finding a particular temperature value in the 3D model simulation box. The solid red line shows mean temperature stratification of the 3D model and the dashed red line is the 1D LHD model temperature stratification. Horizontal bars show the optical depth intervals where 90% of the line equivalent width is formed: black bars correspond to the 3D model while blue dashed correspond to the 1D. Numbers next to the bars designate the wavelength of the given Ba II line in nm. Bottom panel: RMS value of horizontal temperature fluctuations in the 3D model (black line) and temperature difference between the mean 3D and 1D models (blue dashed line). rameters are summarized inTable 4. The physical size of the 3D model box was chosen so that it would accommodate at least ten convective cells in the horizontal plane. Monochromatic opacities used in the model calculations were grouped into five opacity bins for [M/H] = 0.0 and six opacity bins for [M/H] = −1.0, −2.0, −3.0 models. The 1D LHD models were calculated for the same set of atmospheric parameters using the same equation of state, opacities, and chemical composition as in the case of the 3D hydrodynamical models. Table 1 . 1Target stars, their adopted atmospheric parameters, and their derived iron abundances.Note: Fe II abundances were adjusted to match the abundances determined from Fe I lines, in order to estimate the surface gravities of the target stars. The difference between the corresponding abundance ratios,[Fe/H] I , and [Fe/H] II , is thus only indicative of the goodness of the gravity estimates.Star RA Dec. T eff , K dA/dχ ξ t , km s −1 dA/dEW log g, [Fe/H] I [Fe/H] II A(Fe I) -A(Fe II), (2000) (2000) ×10 −3 , ×10 −3 , [cgs] dex dex/eV dex/pm NGC 6752-1 19:10:47 -60:00:43 4749 -2.0 1.37 -1.9 1.95 -1.53 -1.50 -0.03 NGC 6752-2 19:11:11 -60:00:17 4779 -3.1 1.35 -0.2 1.98 -1.54 -1.52 -0.02 NGC 6752-3 19:11:00 -59:56:40 4796 6.9 1.37 -0.2 2.03 -1.59 -1.60 0.01 NGC 6752-4 19:11:33 -60:00:02 4806 -4.9 1.42 0.8 2.04 -1.59 -1.58 -0.01 NGC 6752-6 19:10:34 -59:59:55 4804 -7.6 1.40 0.3 1.97 -1.58 -1.58 0.00 NGC 6752-7 19:10:57 -60:00:41 4829 -11.1 1.39 0.2 2.10 -1.77 -1.75 -0.02 NGC 6752-8 19:10:45 -59:58:18 4910 0.9 1.31 -0.3 2.25 -1.60 -1.60 0.00 NGC 6752-9 19:10:26 -59:59:05 4824 5.8 1.38 -0.9 2.26 -1.61 -1.59 -0.02 NGC 6752-10 19:11:18 -59:59:42 4836 -1.4 1.34 0.3 2.13 -1.56 -1.55 -0.01 NGC 6752-11 19:10:50 -60:02:25 4870 1.6 1.33 0.9 2.13 -1.56 -1.56 0.00 NGC 6752-12 19:10:20 -60:00:30 4841 3.5 1.36 0.4 2.15 -1.60 -1.63 0.03 NGC 6752-15 19:10:49 -60:01:55 4850 0.6 1.35 -1.3 2.19 -1.57 -1.56 -0.01 NGC 6752-16 19:10:15 -59:59:14 4848 -0.8 1.35 0.5 2.06 -1.63 -1.63 0.00 NGC 6752-19 19:11:23 -59:59:40 4928 -4.0 1.45 0.1 2.23 -1.64 -1.63 -0.01 NGC 6752-20 19:10:36 -59:56:08 4929 2.5 1.25 -0.1 2.33 -1.56 -1.59 0.03 NGC 6752-21 19:11:13 -60:02:30 4904 -10.3 1.34 0.0 2.33 -1.62 -1.61 -0.01 NGC 6752-23 19:11:12 -59:58:29 4956 1.7 1.28 1.9 2.35 -1.57 -1.55 -0.02 NGC 6752-24 19:10:44 -59:59:41 4948 -2.8 1.19 1.3 2.28 -1.63 -1.63 0.00 NGC 6752-29 19:10:17 -60:01:00 4900 1.9 1.31 -2.0 2.24 -1.68 -1.63 -0.05 NGC 6752-30 19:10:39 -59:59:47 4943 -4.5 1.18 -0.3 2.42 -1.64 -1.62 -0.02 mean -1.60 -1.60 σ 0.05 0.05 Table 2 . 2Atomic parameters of the barium lines used in this work.λ a , nm χ a , eV log gf a log γ rad b log γ 4 Ne c log γ 6 N H d 585.3688 0.604 -1.000 8.20 -5.460 -7.190 614.1730 0.704 -0.076 8.20 -5.480 -7.470 649.6910 0.604 -0.377 8.10 -5.480 -7.470 a Table 3 . 3Measured equivalent widths of the barium lines and the derived barium abundances in individual red giants in NGC 6752. Star EW (585.4nm), EW (614.2nm), EW (649.7nm), A(Ba) 1D LTE , A(Ba) 1D NLTE , [Ba/Fe] 1D LTE , [Ba/Fe] 1D NLTE , Fig. 1. [Fe/H] abundance ratios derived from Fe I lines for the star NGC 6752-23, plotted versus the excitation potential (top) and line equivalent width (bottom). Linear fits to the data are shown as solid lines. Note that there is a slight correlation between the best-fit slopes in the two panels: adjusting the temperature by zeroing the slope in the upper panel slightly changes the slope in the lower panel, while changes to the microturbulence velocity influence the slope in the upper panel. This correlation does not allow us to obtain zero-valued slopes in both panels simultaneously. The adopted atmospheric parameters of this star are T eff = 4956 K, log g = 2.35, ξ t = 1.28 km s −1 , and [Fe/H] = −1.57.pm pm pm dex dex dex dex NGC 6752-1 8.21 12.54 12.87 0.92 0.63 0.28 -0.01 NGC 6752-2 8.43 12.45 12.08 0.91 0.65 0.28 0.02 NGC 6752-3 7.21 11.84 11.65 0.75 0.50 0.17 -0.08 NGC 6752-4 8.25 12.07 12.26 0.86 0.68 0.28 0.10 NGC 6752-6 8.23 11.96 12.80 0.87 0.60 0.28 0.01 NGC 6752-7 7.13 10.72 11.32 0.60 0.44 0.20 0.04 NGC 6752-8 7.01 11.29 11.37 0.84 0.72 0.27 0.15 NGC 6752-9 7.50 11.65 12.02 0.85 0.70 0.29 0.14 NGC 6752-10 7.57 11.58 12.16 0.87 0.67 0.26 0.06 NGC 6752-11 7.59 11.62 11.13 0.82 0.61 0.21 0.00 NGC 6752-12 7.01 11.44 11.34 0.76 0.61 0.19 0.04 NGC 6752-15 7.13 11.24 11.35 0.78 0.60 0.18 0.00 NGC 6752-16 7.36 11.51 11.59 0.80 0.58 0.26 0.04 NGC 6752-19 6.84 10.76 11.03 0.69 0.57 0.16 0.04 NGC 6752-20 6.77 11.16 10.81 0.87 0.75 0.26 0.14 NGC 6752-21 7.29 10.74 11.09 0.82 0.64 0.27 0.09 NGC 6752-23 7.17 10.71 11.21 0.88 0.72 0.28 0.12 NGC 6752-24 6.03 9.79 9.57 0.68 0.53 0.14 -0.01 NGC 6752-29 6.35 10.06 10.52 0.67 0.51 0.18 0.02 NGC 6752-30 6.39 9.92 10.57 0.82 0.66 0.29 0.13 mean 0.80 0.62 0.24 0.05 σ 0.09 0.08 0.05 0.06 2 3 4 5 -2.0 -1.8 -1.6 -1.4 -1.2 [Fe/H] , eV d[Fe/H]/d = 1.7×10 -3 dex/eV 0 2 4 6 8 10 12 -2.0 -1.8 -1.6 -1.4 -1.2 [Fe/H] EW, pm d[Fe/H]/dEW = 1.9×10 -3 dex/pm NGC 6752-23 Table 4 . 4Parameters of the 3D hydrodynamical CO 5 BOLD atmosphere models used in this work.T eff , K log g [M/H] Grid dimension, Mm resolution x × y × z x × y × z 4970 2.5 0 573×573×243 160×160×200 4990 2.5 −1 573×573×245 160×160×200 5020 2.5 −2 584×584×245 160×160×200 5020 2.5 −3 573×573×245 160×160×200 Table 5 . 5The 3D LTE barium abundances of red giants in NGC 6752.Star ∆ line 3D−1D , ∆ line 3D− 3D , dex ∆ 3D−1D ξ t A(Ba) [Ba/Fe] [Ba/Fe] [Ba/Fe] BaII5854 BaII6142 BaII6497 dex km s −1 3D LTE 3D LTE 1D NLTE 3D+NLTE NGC 6752-08 0.05 0.00 0.08 -0.01 0.06 -0.03 0.06 1.31 0.90 0.33 0.15 0.21 NGC 6752-19 0.13 0.08 0.18 0.11 0.15 0.07 0.15 1.45 0.84 0.31 0.04 0.19 NGC 6752-20 0.01 -0.04 0.04 -0.05 0.02 -0.07 0.02 1.25 0.89 0.28 0.14 0.16 NGC 6752-21 0.07 0.02 0.10 0.02 0.08 -0.01 0.08 1.34 0.90 0.35 0.09 0.17 NGC 6752-23 0.02 -0.03 0.06 -0.02 0.04 -0.05 0.04 1.28 0.92 0.32 0.12 0.16 NGC 6752-24 -0.02 -0.07 -0.01 -0.08 -0.03 -0.10 -0.02 1.19 0.66 0.12 -0.01 -0.03 NGC 6752-29 0.05 0.00 0.08 0.01 0.06 -0.02 0.06 1.31 0.73 0.24 0.02 0.08 NGC 6752-30 -0.03 -0.08 -0.01 -0.09 -0.02 -0.11 -0.02 1.18 0.80 0.27 0.13 0.11 mean: 0.83 0.28 0.08 0.13 sigma: 0.09 0.07 0.06 0.08 4. Discussion 4.1. One-dimensional LTE and NLTE barium abundances in NGC 6752 Generally, the mean 1D LTE barium-to-iron abundance ratio ob- tained in this work, [Ba/Fe] 1D LTE = 0.24 ± 0.05 (random) ± 0.08 (systematic), agrees well with the 1D LTE abundance ratios derived for this cluster by other authors. For example, James et al. (2004) derived [Ba/Fe] = 0.18±0.11 from 9 main- sequence and 9 subgiant stars in NGC 6752. We note that the mean barium-to-iron ratio of James et al. (2004) based on the measurements of only subgiant stars is [Ba/Fe] = 0.25 ± 0.08, i.e., in this case the agreement with our LTE estimate is even better. The barium-to-iron ratios obtained by Norris & Da Costa (1995) and Yong et al. (2005) are lower, [Ba/Fe] = 0.00 ± 0.13 and [Ba/Fe] = −0.06 ± 0.13, respectively. The disagreement between the results of http://archive.eso.org/eso/eso archive adp.html 2 http://www.gazinur.com/DECH-software.html 3 http://wwwuser.oat.ts.astro.it/castelli/sources/width9.html. 2 Dobrovolskas et al.: Barium abundance in NGC 6752 Dobrovolskas et al.: Barium abundance in NGC 6752 The models at four metallicities were needed to interpolate the 3D-1D abundance corrections at the metallicity of NGC 6752. http://www.aip.de/˜mst/Linfor3D/linfor 3D manual.pdf Acknowledgements. We warmly thank David Yong for his support and valuable discussions during re-analysis of his spectroscopic data of NGC 6752. We also thank the referee Ian C. Short for many useful recommendations, which helped to improve the paper significantly. This work was supported in part by grants from the bilateral Lithuanian -Ukrainian program (M/39-2009), the Lithuanian Science Council (TAP-35/2010, TAP-52/2010, and MIP-101/2011) and from the SCOPES grant No. IZ73Z0-128180/1. 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P., Sneden, C., et al. 2011, AJ, 141, 175 . N B Suntzeff, R P Kraft, AJ. 1111913Suntzeff, N. B., & Kraft, R. P. 1996, AJ, 111, 1913 Wavelengths and transition probabilities for atoms and atomic ions: Part 2. Transition probabilities. W L Wiese, G A Martin, NSRDS-NBS. 68Wiese, W. L., & Martin, G. A. 1980, in Wavelengths and transition probabilities for atoms and atomic ions: Part 2. Transition probabilities, NSRDS-NBS Vol. 68 . D Yong, F Grundahl, P E Nissen, H R Jensen, D L Lambert, A&A. 438875Yong, D., Grundahl, F., Nissen, P. E., Jensen, H. R., & Lambert, D. L. 2005, A&A, 438, 875 Barium abundance in NGC 6752, Online Material p 1 Species λ, nm χ, eV log gf Fe I 585. Dobrovolskas, 22187 4.548 -1.330Dobrovolskas et al.: Barium abundance in NGC 6752, Online Material p 1 Species λ, nm χ, eV log gf Fe I 585.22187 4.548 -1.330 . I Fe, 586.23530 4.549 -0.058Fe I 586.23530 4.549 -0.058 . Fe , I 590.56720 4.652 -0.730Fe I 590.56720 4.652 -0.730 . I Fe, 591.62474 2.453 -2.994Fe I 591.62474 2.453 -2.994 . I Fe, 592.77891 4.652 -1.090Fe I 592.77891 4.652 -1.090 . I Fe, 593.01799 4.652 -0.230Fe I 593.01799 4.652 -0.230 . I Fe, 593.46549 3.928 -1.170Fe I 593.46549 3.928 -1.170 . Fe I 595, 3.984 -1.440Fe I 595.27184 3.984 -1.440 . I Fe, 597.67750 3.943 -1.310Fe I 597.67750 3.943 -1.310 . I Fe, 602.40580 4.548 -0.120Fe I 602.40580 4.548 -0.120 . I Fe, 602.70509 4.076 -1.089Fe I 602.70509 4.076 -1.089 . I Fe, 605.60047 4.733 -0.460Fe I 605.60047 4.733 -0.460 . Fe I 606, 2.608 -1.53054822Fe I 606.54822 2.608 -1.530 . I Fe, 607.84910 4.795 -0.424Fe I 607.84910 4.795 -0.424 . Fe I 608, 2.223 -3.573Fe I 608.27106 2.223 -3.573 . I Fe, 609.66653 3.984 -1.930Fe I 609.66653 3.984 -1.930 . Fe , I 612.79066 4.143 -1.399Fe I 612.79066 4.143 -1.399 . I Fe, 613.66153 2.453 -1.400Fe I 613.66153 2.453 -1.400 . Fe I 613, 69947Fe I 613.69947 2.198 -2.950 . I Fe, 613.76917 2.588 -1.403Fe I 613.76917 2.588 -1.403 . Fe I 615, Fe I 615.16181 2.176 -3.299 . Fe , I 615.77284 4.076 -1.260Fe I 615.77284 4.076 -1.260 . Fe I 617, 223 -2.88033356Fe I 617.33356 2.223 -2.880 . I Fe, 618.02042 2.727 -2.586Fe I 618.02042 2.727 -2.586 . I Fe, 618.79904 3.943 -1.720Fe I 618.79904 3.943 -1.720 . I Fe, 619.15584 2.433 -1.417Fe I 619.15584 2.433 -1.417 . Fe , I 620.03129 2.608 -2.437Fe I 620.03129 2.608 -2.437 . I Fe, 621.34303 2.223 -2.482Fe I 621.34303 2.223 -2.482 . Fe I 621, Fe I 621.92810 2.198 -2.433 . I Fe, 623.07230 2.559 -1.281Fe I 623.07230 2.559 -1.281 . Fe I 623, 3Fe I 623.26412 3.654 -1.223 . I Fe, 624.06462 2.223 -3.233Fe I 624.06462 2.223 -3.233 . I Fe, 624.63188 3.602 -0.733Fe I 624.63188 3.602 -0.733 . Fe I 625, 2.404 -1.687Fe I 625.25554 2.404 -1.687 . I Fe, 626.51340 2.176 -2.550Fe I 626.51340 2.176 -2.550 . I Fe, 627.02250 2.858 -2.464Fe I 627.02250 2.858 -2.464 . Fe , I 630.15012 3.654 -0.718Fe I 630.15012 3.654 -0.718 . Fe I 632, Fe I 632.26855 2.588 -2.426 . Fe I 633, 53308Fe I 633.53308 2.198 -2.177 . Fe I 633, 686 -0.85668243Fe I 633.68243 3.686 -0.856 . I Fe, 634.41491 2.433 -2.923Fe I 634.41491 2.433 -2.923 . I Fe, 635.50290 2.845 -2.350Fe I 635.50290 2.845 -2.350 . Fe , I 638.07433 4.186 -1.376Fe I 638.07433 4.186 -1.376 . Fe I 639, 36013Fe I 639.36013 2.433 -1.432 . Fe , I 640.00012 3.602 -0.290Fe I 640.00012 3.602 -0.290 . I Fe, 641.16493 3.654 -0.595Fe I 641.16493 3.654 -0.595 . I Fe, 641.99496 4.733 -0.240Fe I 641.99496 4.733 -0.240 . Fe I 642, Fe I 642.13508 2.279 -2.027 . I Fe, 643.08464 2.176 -2.006Fe I 643.08464 2.176 -2.006 . Fe I 647, 2.559 -2.94256244Fe I 647.56244 2.559 -2.942 . Fe I 648, Fe I 648.18703 2.279 -2.984 . I Fe, 649.49805 2.404 -1.273Fe I 649.49805 2.404 -1.273 . I Fe, 649.64666 4.795 -0.570Fe I 649.64666 4.795 -0.570 . Fe I 651, 2.831 -2.46083671Fe I 651.83671 2.831 -2.460 . I Fe, 659.38705 2.433 -2.422Fe I 659.38705 2.433 -2.422 . Fe , I 660.91103 2.559 -2.692Fe I 660.91103 2.559 -2.692 . I Fe, 663.37497 4.558 -0.799Fe I 663.37497 4.558 -0.799 . Fe I 667, 2.692 -1.41879870Fe I 667.79870 2.692 -1.418 . Fe I 670, 2.758 -3.16035674Fe I 670.35674 2.758 -3.160 . I Fe, 675.01525 2.424 -2.621Fe I 675.01525 2.424 -2.621 . Fe , I 680.68449 2.727 -3.210Fe I 680.68449 2.727 -3.210 . 599.13760 3.153 -3.540Fe II. Fe II 599.13760 3.153 -3.540 . 608.41110 3.199 -3.780Fe II. Fe II 608.41110 3.199 -3.780 . 614.92580 3.889 -2.720Fe II. Fe II 614.92580 3.889 -2.720 . 624.75570 3.892 -2.310Fe II. Fe II 624.75570 3.892 -2.310 . Fe , II 636.94620 2.891 -4.160Fe II 636.94620 2.891 -4.160 . 641.69190 3.892 -2.650Fe II. Fe II 641.69190 3.892 -2.650 . I I Fe, 643.26800 2.891 -3.520Fe II 643.26800 2.891 -3.520 . 645.63830 3.903 -2.100Fe II. Fe II 645.63830 3.903 -2.100 . I I Fe, 651.60800 2.891 -3.320Fe II 651.60800 2.891 -3.320	[]
[ "Critical magnetic field in AdS/CFT superconductor", "Critical magnetic field in AdS/CFT superconductor" ]	[ "Eiji Nakano [email protected]†[email protected] \nDepartment of Physics and Center for Theoretical Sciences\nNational Taiwan University\n106TaipeiTaiwan\n\nGesellschaft für Schwerionenforschung\nGSI\nD-64291DarmstadtGermany\n", "Wen-Yu Wen \nDepartment of Physics and Center for Theoretical Sciences\nNational Taiwan University\n106TaipeiTaiwan\n" ]	[ "Department of Physics and Center for Theoretical Sciences\nNational Taiwan University\n106TaipeiTaiwan", "Gesellschaft für Schwerionenforschung\nGSI\nD-64291DarmstadtGermany", "Department of Physics and Center for Theoretical Sciences\nNational Taiwan University\n106TaipeiTaiwan" ]	[]	We have studied a holographically dual description of superconductor in (2 + 1)-dimensions in the presence of applied magnetic field, and observed that there exists a critical value of magnetic field, below which a charged condensate can form via a second order phase transition.	10.1103/physrevd.78.046004	[ "https://arxiv.org/pdf/0804.3180v2.pdf" ]	119,112,189	0804.3180	b17560c7d91a9c5e5ad5cf2f73ccd58c2e5ea933	Critical magnetic field in AdS/CFT superconductor 11 May 2008 Eiji Nakano [email protected]†[email protected] Department of Physics and Center for Theoretical Sciences National Taiwan University 106TaipeiTaiwan Gesellschaft für Schwerionenforschung GSI D-64291DarmstadtGermany Wen-Yu Wen Department of Physics and Center for Theoretical Sciences National Taiwan University 106TaipeiTaiwan Critical magnetic field in AdS/CFT superconductor 11 May 2008* Electronic address: We have studied a holographically dual description of superconductor in (2 + 1)-dimensions in the presence of applied magnetic field, and observed that there exists a critical value of magnetic field, below which a charged condensate can form via a second order phase transition. I. INTRODUCTION The holographic correspondence between a gravitational theory and a quantum field theory, first emerged under the AdS/CFT correspondence [1], has been proved useful to study various aspects of nuclear physics such as RHIC and condensed matter phenomena, particularly in those recent studies [2,3,4,5,6]. In the papers [7,8], the author proposed a gravity model in which Abelian symmetry of Higgs is spontaneously broken by the existence of black hole. This mechanism was recently incorporated in the model of superconductivity and critical temperature was observed [9], and later on non-Abelian gauge condensate [10]. In this paper, we would like to extend the work to include the magnetic field and will show the existence of critical magnetic field as expected from physics of superconductor. To implement a magnetic field at finite temperature, we introduce a Reissner-Nordstrom charged black hole and a condensate through a charged scalar field. In the superconducting phase, the scalar field takes different values at the horizon for different condensate expectation value at the boundary, indicating the existence of a scalar hair; while in the normal phase, vanishing scalar field tells the ordinary tale of a black hole with no hair. II. THE MODEL WITH APPLIED MAGNETIC FIELD Several important unconventional superconductors, such as the cuprates and organics, are layered in structure and interesting physics can be captured by studying a (2+1) dimensional system. We are now interested in building up a gravity model (in coupled with other matter fields) in (3+1) dimensions which is holographically dual to the desired planar system which develops superconductivity below critical temperature and critical magnetic field. We start with a model composed of the gravity sector and the matter sector. The gravity sector is given by the following Lagrangian density, e −1 L g = R − 6 L 2 − 1 4 F µν F µν ,(1) together with a solution of magnetically charged black hole in AdS 4 , where [11] ds 2 = −f (r)dt 2 + dr 2 f (r) + r 2 (dx 2 + dy 2 ),(2)f (r) = r 2 L 2 − M r + H 2 r 2 .(3) Through the paper we set radius of curvature L = 1 for numerical computation. By assumption the only nonzero electro-magnetic field is the magnetic component F xy = H r 2 , of which the energy density at any fixed radius coordinate r is always finite and constant, that is, F µν F µν ∝ H 2 . This serves the purpose of constant applied magnetic field at the boundary. The black hole is censored by a horizon provided the condition 27M 4 − 256H 6 ≥ 0 and the temperature, as a function of M and H, is determined via the relation T = f ′ (r + ) 4π ,(4) where r + is the most positive root of f (r) = 0 (outer horizon). We expect that the gravity sector, implied by its given name, can be easily obtained from a pure gravity theory of higher dimensions by appropriate reduction. For the matter sector, we will use the Ginzburg-Landau (GL) action for a Maxwell field and a charged complex scalar, which does not back react on the metric [8,9], e −1 L m = − 1 4 F ab F ab + 2 L 2 \|Ψ\| 2 − \|∂Ψ − iAΨ\| 2 .(5) This action differs from the usual GL theory by two places: the coefficient of \|Ψ\| 2 term appears to be negative in both ordinary and superconducting phase, and a \|Ψ\| 4 term is not included. The AdS bulk geometry, however, plays the role of stabilization and we still expect some kind of Higgs mechanism triggered outside the horizon [8]. Enough for our purpose, we will also assume the planar symmetry ansatz for the scalar potential A t = Φ(r) and the complex scalar Ψ(r), where we have already fixed the phase to be constant. Then we need to solve a pair of coupled second order differential equations Ψ ′′ + ( f ′ f + 2 r )Ψ ′ + Φ 2 f 2 Ψ + 2 L 2 f Ψ = 0, Φ ′′ + 2 r Φ ′ − 2Ψ 2 f Φ = 0 (6) with appropriate boundary conditions at the horizon and at asymptotic infinity. They can be solved numerically regardless of difficulty which appears in finding nontrivial analytic solutions. In particular, for normalizable scalar potential, we require at the horizon [8,9] Ψ ′ Ψ r=r + = −2r + 3r 2 + − H 2 r 2 + , Φ(r + ) = 0.(7) Nevertheless we still have freedom for two parameter family of solutions by assigning Φ ′ and Ψ at the horizon, therefore we have a scalar hair from black hole for non-vanishing Ψ. At the boundary, the solutions behave like Ψ = Ψ (1) r + Ψ (2) r 2 + · · · , Φ = µ − ρ r + · · · ,(8) where µ and ρ are interpreted as chemical potential and charge density in the dual field theory. We are interested in the case where either Ψ (1) or Ψ (2) vanishes for stability concern at asymptotic AdS region, then read off the pairing operator O dual to Ψ from the bulkboundary coupling [9], O i = √ 2Ψ (i) .(9) To gain a better intuition of how a condensate is realized in this gravity setup, we may investigate the effective mass of Ψ field along the radius direction, that is We recall that there exists the Bretenlohner-Freedman (BF) bound [12], i.e. m 2 L 2 > −9/4, which guarantees that the AdS vacuum is stable under perturbations of Ψ. We observe that in the Figure 1 that provided fixed temperature and boundary condition at the horizon, the effective mass dives below the BF bound for wider range of r for smaller magnetic field. In the other words, condensate happens more easily while the magnetic field is smaller. This implies the existence of critical magnetic field below which the condensation can take place. m 2 ef f (r) = − 2 L 2 − Φ 2 f .(10) III. CRITICAL MAGNETIC FIELD In the normal phase, we always have solutions to the equations (6), that is Ψ = 0 and Φ = µ − ρ r ; while in the superconducting phase, we may have nontrivial Ψ(r) and its boundary value serves as an order parameter for condensate. In the absence of applied magnetic field, for any fixed ρ, there exists a critical temperature T c , above which there is no more nontrivial solution [9]. In the presence of applied magnetic field, however, the Meissner effect is expected and there exists both T c and a critical magnetic field H c , above which the nontrivial solution is again not admissible. As argued in the previous section, we expect that the stronger applied magnetic field H is, the lower is critical temperature T c . This statement is supported by our numerical results for O 2 as shown in the Figure 2. The operator O 2 corresponds to a pair of fermions, while O 1 to a pair of bosons [9]. We have also found similar results for O 1 only at a small H region. In the Figure 3 we also plot the phase diagram of critical magnetic field against critical temperature. IV. DISCUSSION In this paper, we have considered a hybrid model for AdS/CFT superconductors in the presence of magnetic field. Several comments are in order: At first, a magnetic field is provided in the gravity sector as a background, independent of the probed sector. We argue that this is perfectly fine as long as we only consider a constant magnetic field at the boundary. Secondly, the matter sector has no back reaction to the gravity sector, therefore the equation of motion for total Lagrangian is not satisfied. Although this may not be crucial to the occurrence of superconducting phase, it is still interesting to investigate a fully back-reacted action which can be derived from some higher-dimensional theory such as String theory or M-theory. Thirdly, in order to discuss possible formation of vortex lattice and distinguish between type I and II superconductors, one is tempted to relax the ansatz of planar symmetry. This will complicate the construction and analysis and we hope to report it in the near future. At last, this construction is a tractable model of strongly coupled system which may capture some physics of unconventional superconductors, in contrast to the conventional superconductors well described by GL theory macroscopically and BCS theory microscopically. Though we do not see fermionic degree of freedom from this macroscopic construction, the complex scalar, serving as order parameter, seems sufficient to explain such a critical phenomenon as good as the usual GL theory. In order to pursue a microscopic model along this line of reasoning, one may still need to understand better how to realize underlying fermionic degree of freedom in the context of AdS/CFT correspondence. FIG. 1 : 1The effective mass m 2 ef f evaluated at fixed temperature and boundary conditions at the horizon. From bottom up, the curves are with H = 0, 0.5 and 1 respectively. The dashing line indicates the Bretenlohner-Freedman bound, below which the AdS vacuum is unstable under perturbation of Ψ and condensation is expected. FIG. 2 : 2We plot order parameter O 2 as a function of temperature. The critical temperature T c decreases as applied magnetic field increases. HereH is the normalized H given by H 2/3 /T 0 , where T 0 = T c at H = 0. FIG. 3 : 3The phase diagram of T c against H c . The superconducting phase where O 2 = 0( O 1 = 0) exists in the lower left part below the solid (dashed) curve, while normal phase in the upper right part above the curve. 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[ "How Much is the Efficiency of Solar Cells Enhanced by Quantum Coherence?", "How Much is the Efficiency of Solar Cells Enhanced by Quantum Coherence?" ]	[ "Sangchul Oh \nQatar Environment and Energy Research Institute\nHamad Bin Khalifa University\nP.O. Box 5825Qatar Foundation, DohaQatar\n" ]	[ "Qatar Environment and Energy Research Institute\nHamad Bin Khalifa University\nP.O. Box 5825Qatar Foundation, DohaQatar" ]	[]	We study how much the efficiency of a solar cell as a quantum heat engine could be enhanced by quantum coherence. In contrast to the conventional approach that a quantum heat engine is in thermal equilibrium with both hot and cold reservoirs, we propose a new description that the quantum heat engine is in the cold reservoir and the thermal radiation from the hot reservoir is described by the pumping term in the master equation. This pumping term solves the problem of the incorrect mean photon number of the hot reservoir assumed by the previous studies. By solving the master equation, we obtain the current-voltage and the power-voltage curves of the photocell for different pumping rates. We find that, as the photon flux increases, the power output of the photocell increases linearly at first and then becomes saturated, but the efficiency decreases rapidly. It is demonstrated that while the power output is enhanced significantly by the quantum coherence via the dark state of the coupled donors, the improvement of the efficiency is not significant. PACS numbers: 42.50.Gy, 78.67-n, 82.39.Jn, 84.60.JtSolar cells and photosynthesis, which convert sunlight into electrical and chemical energies, respectively, may be regarded as heat engines. The maximum efficiency of a heat engine operating between hot and cold reservoirs is known as the Carnot efficiency, derived from the second law of thermodynamics. For a quantum heat engine, Scovil and Schulz-DuBois considered a three-level maser in thermal contact with two heat reservoirs, and showed its ultimate efficiency is that of a Carnot engine [1]. Shockley and Queisser obtained the upper limit of efficiency of a single p-n junction solar cell, based on the assumption that electron-hole pairs recombine only through the radiative process, i.e., the principle of the detailed balance [2]. The Shockley-Queisser limit, however, is far below the Carnot efficiency because of only one electron-hole pair generation per photon with energy larger than the band gap of the semiconductor generates.Recent studies have shown that quantum effects could play a key role in photosynthesis and solar cells. Engel and his co-workers observed the long-lived quantum coherence in exciton dynamics in the Fenna-Matthews-Olsen complex, using 2-dimensional electronic spectroscopy [3]. Following experimental and theoretical studies suggest that this quantum beat may be due to the interplay of electronic and vibronic quantum dynamics. Scully and his colleagues showed theoretically that quantum coherence could enhance the efficiency of a solar cell and a photosynthetic reaction center[4][5][6][7]. It has been argued that the quantum coherence could break the detailed balance, and thus the Shockley-Queisser limit of the efficiency of solar cells. Inspired by Scully et al.'s work, Creatore et al. [8] proposed a biologically inspired photocell model enhanced by a delocalized dark quantum state of two dipole-dipole coupled donors. Zhang et al. [9] showed that the delocalized dark state of three coupled donors could enhance more the efficiency of a photocell. Recently, Fruchtman et al. [11] showed that a photocell with asymmetric pair of coupled chromophores could outperform those with the symmetric dimer or with a pair of independent molecules.While theoretical studies on photocells mentioned above predict promising enhancement of the efficiency of a quantum heat engine, there is controversy, especially, raised by Kirk[12][13][14]. The claim of the role of quantum coherence in enhancing the efficiency needs to be more complete in the following sense. First, photocells as a quantum heat engine are assumed to be in thermal equilibrium with hot and cold reservoirs simultaneously. This assumption may give rise to a question on the temperature of a photocell. Second, the average photon number of the Sun with a temperature of 6000 K at the energy gap of donors was incorrectly used in the master equations in previous studies. Finally, while the previous studies have shown the power enhancement by quantum effects, they tells neither how much efficiency is enhanced nor whether the Shockley-Queisser limit is surpassed.In the paper, we present a realistic model of a photocell which is in thermal contact only with the cold reservoir. The pumping term in a master equation is introduced in order to take into account the photon flux from the hot reservoir. This resolves the issue of the incorrect mean photon number of the hot reservoir assumed by the previous studies, and makes it possible to calculate the efficiency. The power output of the photocell is obtained as a function of the strength of the pumping term, i.e., the photon flux. We show that the power increases linearly at first but becomes saturated as the pumping strength increases. We obtain the efficiency as a function of pumping strength and demonstrate that quantum coherence could enhance the efficiency, but not much.Solar Cell with Donor-Acceptor.-Let us start with a simple photovoltaic model, a four-level quantum sys-	null	[ "https://arxiv.org/pdf/1709.08337v1.pdf" ]	119,092,701	1709.08337	d728340120b2c01ae9aca192aae69351f3dc5b41	How Much is the Efficiency of Solar Cells Enhanced by Quantum Coherence? 25 Sep 2017 Sangchul Oh Qatar Environment and Energy Research Institute Hamad Bin Khalifa University P.O. Box 5825Qatar Foundation, DohaQatar How Much is the Efficiency of Solar Cells Enhanced by Quantum Coherence? 25 Sep 2017(Dated: September 26, 2017) We study how much the efficiency of a solar cell as a quantum heat engine could be enhanced by quantum coherence. In contrast to the conventional approach that a quantum heat engine is in thermal equilibrium with both hot and cold reservoirs, we propose a new description that the quantum heat engine is in the cold reservoir and the thermal radiation from the hot reservoir is described by the pumping term in the master equation. This pumping term solves the problem of the incorrect mean photon number of the hot reservoir assumed by the previous studies. By solving the master equation, we obtain the current-voltage and the power-voltage curves of the photocell for different pumping rates. We find that, as the photon flux increases, the power output of the photocell increases linearly at first and then becomes saturated, but the efficiency decreases rapidly. It is demonstrated that while the power output is enhanced significantly by the quantum coherence via the dark state of the coupled donors, the improvement of the efficiency is not significant. PACS numbers: 42.50.Gy, 78.67-n, 82.39.Jn, 84.60.JtSolar cells and photosynthesis, which convert sunlight into electrical and chemical energies, respectively, may be regarded as heat engines. The maximum efficiency of a heat engine operating between hot and cold reservoirs is known as the Carnot efficiency, derived from the second law of thermodynamics. For a quantum heat engine, Scovil and Schulz-DuBois considered a three-level maser in thermal contact with two heat reservoirs, and showed its ultimate efficiency is that of a Carnot engine [1]. Shockley and Queisser obtained the upper limit of efficiency of a single p-n junction solar cell, based on the assumption that electron-hole pairs recombine only through the radiative process, i.e., the principle of the detailed balance [2]. The Shockley-Queisser limit, however, is far below the Carnot efficiency because of only one electron-hole pair generation per photon with energy larger than the band gap of the semiconductor generates.Recent studies have shown that quantum effects could play a key role in photosynthesis and solar cells. Engel and his co-workers observed the long-lived quantum coherence in exciton dynamics in the Fenna-Matthews-Olsen complex, using 2-dimensional electronic spectroscopy [3]. Following experimental and theoretical studies suggest that this quantum beat may be due to the interplay of electronic and vibronic quantum dynamics. Scully and his colleagues showed theoretically that quantum coherence could enhance the efficiency of a solar cell and a photosynthetic reaction center[4][5][6][7]. It has been argued that the quantum coherence could break the detailed balance, and thus the Shockley-Queisser limit of the efficiency of solar cells. Inspired by Scully et al.'s work, Creatore et al. [8] proposed a biologically inspired photocell model enhanced by a delocalized dark quantum state of two dipole-dipole coupled donors. Zhang et al. [9] showed that the delocalized dark state of three coupled donors could enhance more the efficiency of a photocell. Recently, Fruchtman et al. [11] showed that a photocell with asymmetric pair of coupled chromophores could outperform those with the symmetric dimer or with a pair of independent molecules.While theoretical studies on photocells mentioned above predict promising enhancement of the efficiency of a quantum heat engine, there is controversy, especially, raised by Kirk[12][13][14]. The claim of the role of quantum coherence in enhancing the efficiency needs to be more complete in the following sense. First, photocells as a quantum heat engine are assumed to be in thermal equilibrium with hot and cold reservoirs simultaneously. This assumption may give rise to a question on the temperature of a photocell. Second, the average photon number of the Sun with a temperature of 6000 K at the energy gap of donors was incorrectly used in the master equations in previous studies. Finally, while the previous studies have shown the power enhancement by quantum effects, they tells neither how much efficiency is enhanced nor whether the Shockley-Queisser limit is surpassed.In the paper, we present a realistic model of a photocell which is in thermal contact only with the cold reservoir. The pumping term in a master equation is introduced in order to take into account the photon flux from the hot reservoir. This resolves the issue of the incorrect mean photon number of the hot reservoir assumed by the previous studies, and makes it possible to calculate the efficiency. The power output of the photocell is obtained as a function of the strength of the pumping term, i.e., the photon flux. We show that the power increases linearly at first but becomes saturated as the pumping strength increases. We obtain the efficiency as a function of pumping strength and demonstrate that quantum coherence could enhance the efficiency, but not much.Solar Cell with Donor-Acceptor.-Let us start with a simple photovoltaic model, a four-level quantum sys- We study how much the efficiency of a solar cell as a quantum heat engine could be enhanced by quantum coherence. In contrast to the conventional approach that a quantum heat engine is in thermal equilibrium with both hot and cold reservoirs, we propose a new description that the quantum heat engine is in the cold reservoir and the thermal radiation from the hot reservoir is described by the pumping term in the master equation. This pumping term solves the problem of the incorrect mean photon number of the hot reservoir assumed by the previous studies. By solving the master equation, we obtain the current-voltage and the power-voltage curves of the photocell for different pumping rates. We find that, as the photon flux increases, the power output of the photocell increases linearly at first and then becomes saturated, but the efficiency decreases rapidly. It is demonstrated that while the power output is enhanced significantly by the quantum coherence via the dark state of the coupled donors, the improvement of the efficiency is not significant. Solar cells and photosynthesis, which convert sunlight into electrical and chemical energies, respectively, may be regarded as heat engines. The maximum efficiency of a heat engine operating between hot and cold reservoirs is known as the Carnot efficiency, derived from the second law of thermodynamics. For a quantum heat engine, Scovil and Schulz-DuBois considered a three-level maser in thermal contact with two heat reservoirs, and showed its ultimate efficiency is that of a Carnot engine [1]. Shockley and Queisser obtained the upper limit of efficiency of a single p-n junction solar cell, based on the assumption that electron-hole pairs recombine only through the radiative process, i.e., the principle of the detailed balance [2]. The Shockley-Queisser limit, however, is far below the Carnot efficiency because of only one electron-hole pair generation per photon with energy larger than the band gap of the semiconductor generates. Recent studies have shown that quantum effects could play a key role in photosynthesis and solar cells. Engel and his co-workers observed the long-lived quantum coherence in exciton dynamics in the Fenna-Matthews-Olsen complex, using 2-dimensional electronic spectroscopy [3]. Following experimental and theoretical studies suggest that this quantum beat may be due to the interplay of electronic and vibronic quantum dynamics. Scully and his colleagues showed theoretically that quantum coherence could enhance the efficiency of a solar cell and a photosynthetic reaction center [4][5][6][7]. It has been argued that the quantum coherence could break the detailed balance, and thus the Shockley-Queisser limit of the efficiency of solar cells. Inspired by Scully et al.'s work, Creatore et al. [8] proposed a biologically inspired photocell model enhanced by a delocalized dark quantum state of two dipole-dipole coupled donors. Zhang et al. [9] showed that the delocalized dark state of three coupled donors could enhance more the efficiency of a photocell. Recently, Fruchtman et al. [11] showed that a photocell with asymmetric pair of coupled chromophores could outperform those with the symmetric dimer or with a pair of independent molecules. While theoretical studies on photocells mentioned above predict promising enhancement of the efficiency of a quantum heat engine, there is controversy, especially, raised by Kirk [12][13][14]. The claim of the role of quantum coherence in enhancing the efficiency needs to be more complete in the following sense. First, photocells as a quantum heat engine are assumed to be in thermal equilibrium with hot and cold reservoirs simultaneously. This assumption may give rise to a question on the temperature of a photocell. Second, the average photon number of the Sun with a temperature of 6000 K at the energy gap of donors was incorrectly used in the master equations in previous studies. Finally, while the previous studies have shown the power enhancement by quantum effects, they tells neither how much efficiency is enhanced nor whether the Shockley-Queisser limit is surpassed. In the paper, we present a realistic model of a photocell which is in thermal contact only with the cold reservoir. The pumping term in a master equation is introduced in order to take into account the photon flux from the hot reservoir. This resolves the issue of the incorrect mean photon number of the hot reservoir assumed by the previous studies, and makes it possible to calculate the efficiency. The power output of the photocell is obtained as a function of the strength of the pumping term, i.e., the photon flux. We show that the power increases linearly at first but becomes saturated as the pumping strength increases. We obtain the efficiency as a function of pumping strength and demonstrate that quantum coherence could enhance the efficiency, but not much. Solar Cell with Donor-Acceptor.-Let us start with a simple photovoltaic model, a four-level quantum sys-tem composed of a donor and a acceptor, as shown in Figs. 1 (a) and (b). We present the issue of the previous photocell models and solve it by introducing the pumping term in our model. Fig. 1 (a) depicts a photocell model of previous studies that is in the thermal equilibrium with both hot and cold reservoirs at the same time. The total Hamiltonian is written formally as H = H S + H H + H C + H SH + H SC ,(1) where H S is the Hamiltonian of the photocell with donor and acceptor, and H H (H C ) is the Hamiltonian of the hot (cold) reservoir represented by the collection of the harmonic oscillators. Typically, it is assumed that the interactions, H SH and H SC , between the system and the reservoirs are assumed to be time-independent. Using the Born and the Markov approximations, one can obtain the master equation for the system dynamics. As shown in the previous studies [5][6][7][8][9][10][11], the probabilities P i of occupation of energy levels E i obey the Pauli master equationṡ P 0 = γ 01 (n h 01 + 1) P 1 − n h 01 P 0 + χΓ P α + γ 0β (n c 0β + 1) P β − n c 0β P 0 ,(2a)P 1 = γ 01 n h 01 P 0 − (n h 01 + 1) P 1 + γ α1 n c α1 P α − (n c α1 + 1) P 1 ,(2b)P α = γ α1 (n c α1 + 1) P 1 − n c α1 P α − (1 + χ)Γ P α , (2c) P β = γ 0β n c 0β P 0 − (n c 0β + 1) P β + Γ P β .(2d) Here γ ij are the transition rates between level E i to level E j . The mean photon number n h ij (n c ij ) of the hot (cold) reservoir at temperature T h (T c ) for a given frequency ∆E ij = E j − E i is written as n h ij = 1 e ∆Eij/kB T h − 1 .(3) The parameters are taken as follows: E 1 − E 0 = 1.8 eV, E 1 − E α = E β − E 0 = 0.2 eV, γ 01 = 1. 24 µeV, γ α1 = 12 meV, and γ 0β = 24 meV [5][6][7][8][9][10][11]. These imply 1/γ 01 ≃ 0.5 ns, 1/γ α1 ≃ 0.55 fs, 1/γ 0α ≃ 0.26 fs, and χ = 0. So, the typical time to reach the steady state is the order of femotosecond. The temperates of the hot and cold reservoirs are T h = 6000 K and T c = 300 K, respectively. If the parameters are plugged into Eq. (3), the mean photon number of the hot reservoir at energy ∆E 01 = 1.8 eV is given by n h 01 ≃ 0.0317, and the mean photon number of the cold reservoir at energy ∆E 1α = ∆E 0β = 0.2 eV by n c 1α = n c 0β ≃ 4.368 × 10 −4 [15]. However, the previous papers [5][6][7][8][9][10][11] assumed n h 01 = 60000 that does not coincide with the value given by Eq. (3). In order to solve the pitfall of the previous studies depicted in Fig. 1 (a), we propose a new photocell model as shown in Fig. 1 (b). The donor of the new photocell is assumed to be in thermal equilibrium only with the cold reservoir, but not with the hot reservoir. The photon flux from the hot reservoir is described by the pumping term [16]. So the strength of the pumping term may correspond to the solar irradiance incident on the photocell. It is straightforward to obtain the Pauli master equations with the pumping term for the population dynamics of the new photocell model P 0 = γ 01 (n c 01 + 1) P 1 − n c 01 P 0 + χΓ P α + γ 0β (n c 0β + 1) P β − n c 0β P 0 + W p (P 0 − P 1 ) , P 1 = γ 01 n c 01 P 0 − (n c 01 + 1) P 1 + γ α1 n c α1 P α − (n c α1 + 1) P 1 + W p (P 1 − P 0 ) ,(4a)P α = γ α1 (n c α1 + 1) P 1 − n c α1 P α − (1 + χ)Γ P α , (4c) P β = γ 0β n c 0β P 0 − (n c 0β + 1) P β + Γ P β .(4b) Note that the mean photon number n h 01 of the hot reservoir in Eq. (2) is replaced by n c 01 of the cold reservoir and the pumping term W p in Eq. (4). The mean photon number n h 01 = 60, 000 of the previous studies [5][6][7][8][9][10][11] corresponds to W p /γ 01 ≃ 60, 000 and W p ≃ 1.1 × 10 15 s −1 . It is instructive to compare this pumping rate with the number of photons incident per unit area per unit time for the black-body radiation of the Sun at temperature T s = 6000 K, using the Planck distribution. The number of photons with energy greater than the energy gap E g = hν g absorbed by the donor per unit area per unit time is given by Q s (ν g , T s ) = 2π c 2 ∞ νg ν 2 e hν/kB Ts − 1 dν .(5) For E g = 1.8 eV, one obtains Q s ≃ 9.0 × 10 25 m −2 s −1 . So the pumping rate W p = 10 15 s −1 corresponds to the photon flux incident on a photocell with area 0.1 µm 2 . Eq. (4) is solved numerically using the Runge-Kutta method. After the populations reach the steady state, the current is calculated as I = eΓP α , and the voltage as V = E α − E β + k B T ln(P α /P β ). By changing the resistance Γ of the external load from zero to infinity, one obtains the current-voltage curve of the photocell for various pumping rates, as shown in Fig. 2. The magnitude of current I is readily estimated as follows. The generation rate of the excited electrons is propotional to the pumping rate, for example W p = 10 12 s −1 . The transfer rate of the excited electrons to the acceptor is fast, i.e., the order of femtosecond. Thus, the current is just given by the product of electron charge and the generation rate, I ∼ 1.6 × 10 −19 C × 10 12 s −1 = 0.16 µA, i.e., the order of microampere. We investigate how the efficiency and the maximum power change as a function of the pumping rate. The efficiency η of the photocell is calculated as η = P out P in = P m [µeV] 1.8 [eV] · W p [s −1 ] .(6) It would be expected that the more power the photocell generates, the higher solar irradiance it receives. However, as shown in Fig. 3, the maximum power increases linearly at the beginning but becomes saturated above a certain value of the pumping rate. This implies that there is a bottleneck in population dynamics. A saturation curve like Fig. 3 can be found in photosynthesis, which is well known as the photosynthesisirradiance curve [17][18][19]. Note that the maximum power as a function of pumping rate W p can be fitted by P (W p ) = a · W p /(W p + b) with a = 1.1 and b = 7. We find that the efficiency of the photocell decreases as the pumping rate W p increases. Photocell with coupled donors and an acceptor.-Let us turn to the main question how much the efficiency of a photocell is enhanced by quantum coherence. Similar to the previous studies [8,9], we consider the photocell model composed of two coupled donors and an acceptor, as shown in Fig. (4). Unlike the previous studies, the photocell is in thermal equilibrium only with the cold reservoir. The dark state formed by the coupled donors plays a key role in enhancing the power of the photocell in compared with a photocell with uncoupled donors. The dynamics for occupation probabilities of energy level E i is readily given by the Pauli master equatioṅ P 0 = γ 01 (1 + n c 01 ) P 1 − n c 01 P 0 + χΓ P α + γ 0β (1 + n c 0β ) P β − n c 0β P 0 + W p (P 1 − P 0 ) ,(7a)P 1 = γ 01 n c 01 P 0 − (1 + n c 01 ) P 1 + γ 12 n c 12 P 2 − (1 + n c 12 ) P 1 + W p (P 0 − P 1 ) ,(7b)P 2 = γ 12 (1 + n c 12 ) P 1 − n c 12 P 2 + γ α2 n c α2 P α − (1 + n c α2 ) P 2 ,(7c)P α = γ α2 [(1 + n c α2 ) P 2 − n c α2 P α ] − Γ(1 + χ) P α , (7d) P β = ΓP α + γ β0 n c β0 P 0 − (1 + n c β0 ) P β .(7e) We solve Eq. (7) with the parameters γ ij given by Refs. [8,9]. We obtain the current-voltage curve and the power-voltage curve for different pumping rates and for photocells with uncoupled and coupled donors to see the effect of quantum coherence, as shown in Fig. 5. It is interesting that at low pumping rate W p = 10 12 s −1 , the short-circuit current and the power are not enhanced by the quantum coherence. However, at high pumping rate W p = 10 15 s −1 , the quantum coherence gives rise to the strong enhancement of the short-circuit current and the power, agreeing with the previous studies [5][6][7][8][9][10][11]. Fig. 6 depicts the maximum power P m and the efficiency as a function of pumping rate W p for coupled and uncoupled donors. For both uncoupled and coupled donors, the maximum power P m increases at first and becomes saturated as the pumping rate increases, but the efficiency decreases. The photocell with coupled donors generates more power than that with uncoupled donors as the pumping rate increases, but the enhancement in efficiency due to the quantum coherence is not the case. In contrast to the claim of the previous studies [5][6][7][8][9][10][11], the enhancement of the efficiency due to the quantum coherence, via dark states or noise-induced quantum coherence, is very small at W p = 10 15 s −1 which corresponds to n h 02 = 60, 000. In conclusion, we proposed a new photocell model where the system is in thermal equilibrium only with the cold reservoir and the photon flux from the hot reservoir is described by the pumping term in the master equation. The pumping term resolves the problem of the incorrect mean photon number of the hot reservoir used by the previous studies. The maximum power and the efficiency were obtained as a function of the pumping rate. It is found that as the pumping rate increases, the power increases linearly at first but becomes saturated, and the efficiency decreases rapidly. It is shown that the quantum coherence via the dark state of the coupled donors clearly enhance the power significantly, but the efficiency tiny. Further study is needed to see whether quantum coher- ence could enhance the efficiency significantly and break the Shockley-Queisser limit of a single-junction solar cell. PACS numbers: 42.50.Gy, 78.67-n, 82.39.Jn, 84.60.Jt Figure 1 . 1(a) A photocell is in thermal equilibrium with both hot and cold reservoirs. (b) A photocell is in thermal equilibrium only with the cold reservoir. The hot reservoir excites the donor with pumping rate Wp. Figure 2 . 2(a) Current I and (b) power P are plotted as a function of voltage V for different pumping rates Wp. Figure 3 . 3Efficiency (blue solid line) and the maximum power Pmax (red dashed line) are plotted as a function of the pumping rate Wp. Figure 4 . 4A photocell with coupled donors and an acceptor is in thermal equilibrium only with the cold reservoir. The coupled donors form the bright state \|2 and the dark state \|1 . Figure 5 .Figure 6 . 56(a) Current I and power P are plotted as a function of voltage V for different pumping rates Wp, and for the photocells with coupled donors (solid lines) and uncoupled donors (dashed and dash-dotted lines). (a) Maximum power Pm and (b) efficiency as a function of the pumping rate Wp for uncoupled donors (dashed lines) and coupled donors (solid lines). The maximum power curves for uncoupled and coupled donors are fitted by two functions 1.08Wp/(Wp + 5) and 1.37Wp/(Wp + 6.5), respectively. . H E D Scovil, E O Schulz-Dubois, Phys. Rev. Lett. 2262H.E.D. Scovil and E.O. Schulz-DuBois, Phys. Rev. Lett. 2, 262 (1959). . W Shockley, H J Queisser, J. Appl. Phys. 32510W. Shockley and H.J. Queisser, J. Appl. Phys. 32, 510 (1961). . G S Engel, T R Calhoun, E L T K Read, T Ahn, Y.-C Mancal, R E Cheng, G R Blankenship, Flemming, Nature. 446782G.S. Engel, T.R. Calhoun, E.L. Read. T.K. Ahn, T. Mancal, Y.-C. Cheng, R. E. Blankenship, and G.R. Flem- ming, Nature 446, 782 (2007). . M O Scully, Phys. Rev. Lett. 104207701M.O. Scully, Phys. Rev. Lett. 104, 207701 (2010). M O Scully, K R Chapin, K E Dorfman, M B Kim, A Svidzinsky, Proc. Natl. Acad. Sci. USA. Natl. Acad. Sci. USA10815097M.O. Scully, K.R. 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[ "Astronomy & Astrophysics A modular set of synthetic spectral energy distributions for young stellar objects", "Astronomy & Astrophysics A modular set of synthetic spectral energy distributions for young stellar objects" ]	[ "T P Robitaille [email protected] \nMax Planck Institute for Astronomy\nKönigstuhl 1769117HeidelbergGermany\n\nHeadingley Enterprise and Arts Centre\nBennett RoadLS6 3HNLeedsUK\n" ]	[ "Max Planck Institute for Astronomy\nKönigstuhl 1769117HeidelbergGermany", "Headingley Enterprise and Arts Centre\nBennett RoadLS6 3HNLeedsUK" ]	[ "A&A" ]	In this paper, I present a new set of synthetic spectral energy distributions (SEDs) for young stellar objects (YSOs) spanning a wide range of evolutionary stages, from the youngest deeply embedded protostars to pre-main-sequence stars with few or no disks. These models include significant improvements on the previous generation of published models: in particular, the new models cover a much wider and more uniform region of parameter space, do not include highly model-dependent parameters, and include a number of improvements that make them more suited to modeling far-infrared and sub-mm observations of forming stars. Rather than all being part of a single monolithic set of models, the new models are split up into sets of varying complexity. The aim of the new set of models is not to provide the most physically realistic models for young stars, but rather to provide deliberately simplified models for initial modeling, which allows a wide range of parameter space to be explored. I present the design of the model set, and show examples of fitting these models to real observations to show how the new grid design can help us better understand what can be determined from limited unresolved observations. The models, as well as a Python-based fitting tool are publicly available to the community.	10.1051/0004-6361/201425486	[ "https://www.aanda.org/articles/aa/pdf/2017/04/aa25486-14.pdf" ]	119,374,535	1703.05765	c0f8398588d3fd5b1fb9f9c5e6f3994597f1621f	Astronomy & Astrophysics A modular set of synthetic spectral energy distributions for young stellar objects 2017 T P Robitaille [email protected] Max Planck Institute for Astronomy Königstuhl 1769117HeidelbergGermany Headingley Enterprise and Arts Centre Bennett RoadLS6 3HNLeedsUK Astronomy & Astrophysics A modular set of synthetic spectral energy distributions for young stellar objects A&A 60011201710.1051/0004-6361/201425486Received 8 December 2014 / Accepted 3 August 2016astronomical databases: miscellaneous -radiative transfer -stars: formation -stars: protostars In this paper, I present a new set of synthetic spectral energy distributions (SEDs) for young stellar objects (YSOs) spanning a wide range of evolutionary stages, from the youngest deeply embedded protostars to pre-main-sequence stars with few or no disks. These models include significant improvements on the previous generation of published models: in particular, the new models cover a much wider and more uniform region of parameter space, do not include highly model-dependent parameters, and include a number of improvements that make them more suited to modeling far-infrared and sub-mm observations of forming stars. Rather than all being part of a single monolithic set of models, the new models are split up into sets of varying complexity. The aim of the new set of models is not to provide the most physically realistic models for young stars, but rather to provide deliberately simplified models for initial modeling, which allows a wide range of parameter space to be explored. I present the design of the model set, and show examples of fitting these models to real observations to show how the new grid design can help us better understand what can be determined from limited unresolved observations. The models, as well as a Python-based fitting tool are publicly available to the community. Introduction Over the last two decades, increasingly sensitive and detailed surveys of our Galaxy at infrared wavelengths have resulted in a dramatic increase in the number of known young stellar objects (YSOs). The Spitzer/GLIMPSE survey at mid-infrared wavelengths contains tens of thousands of YSOs, with the sample of around 10 000 YSO candidates from Robitaille et al. (2008) representing only the brightest ones. Spitzer has also revealed thousands of YSOs in a number of individual regions, including over 8000 YSO candidates in the survey of Cygnus X North (Beerer et al. 2010), over 1000 YSO candidates in the c2d survey of nearby star-forming regions (Evans et al. 2009), and almost 3500 YSO in the Orion A and B molecular clouds (Megeath et al. 2012). The WISE survey (Wright et al. 2010), while less sensitive than Spitzer, covers the whole sky and has already unveiled thousands of YSOs (Koenig & Leisawitz 2014). Herschel has both provided long-wavelength data for known YSO candidates, and has also revealed new extremely embedded protostars (e.g., Stutz et al. 2013). While the Hubble Space Telescope (HST) and the Atacama Large Millimeter Array (ALMA) allow the study of a smaller sample of YSOs with exquisite resolution and sensitivity, the vast majority of YSOs in the Galaxy will remain unresolved for a while yet. Nevertheless, with such a large sample of known YSOs and YSO candidates, we can make significant progress in quantifying the formation of stars across a wide range of environments. Extracting physical properties from limited -often unresolved -observations however requires radiative transfer modeling. Visiting Researcher, School of Physics & Astronomy, The University of Leeds, Leeds LS2 9JT, UK In Robitaille et al. (2006, hereafter R06), we presented a set of approximately 20 000 radiative transfer models, each of which had spectral energy distributions (SEDs) computed for ten viewing angles and 50 apertures. The models were set up by first sampling the central source mass between 0.1 and 50 M , and the central source age between 10 3 and 10 7 yr (excluding postmain-sequence objects), and using evolutionary tracks to derive the source temperature and radius in each case. The parameters for the circumstellar environment, consisting of an accretion disk, infalling envelope, and bipolar cavities, were then sampled from ranges that were functions of the stellar mass and/or age. In total, 14 parameters were used to define the models. In Robitaille et al. (2007, hereafter R07), we presented a tool to rapidly fit model SEDs from R06 to observations, correctly taking into account the effects of extinction and apertures. The aim of the tool was not to identify the best-fitting model, but rather the ensemble of models that could provide a good fit, thereby identifying likely ranges in parameters space. The R06 models and the R07 fitting tool have been extensively used to model the SEDs of thousands of sources across many regions of star formation. These models have been invaluable in a number of studies to learn about the physical properties of young stars and regions. For example, Forbrich et al. (2010) carefully modeled the SEDs of a number of protostars in the IC 348 and NGC 2264 star-forming regions, demonstrating that it was crucial to include the intra-cluster extinction in order to reliably determine evolutionary stages of young sources in embedded clusters. Another example is the study by Mottram et al. (2011), which made use of the R06 models in order to determine bolometric fluxes and luminosities for over a thousand embedded massive stars in the Red MSX Source (RMS) survey. In this paper, I present a new set of model SEDs for YSOs that significantly improves on the R06 models. Despite the success of the R06 models, a number of issues with those models remain, and these are described in Sect. 2. I then present an overview of the setup and methods for the new set of models in Sects. 3 and 4 respectively, and present some initial results derived from the new models in Sect. 5. I discuss remaining caveats relating to the models in Sect. 6, and limitations inherent to SED modeling in Sect. 7. Limitations of the previous models In this section, I review the main limitations of the R06 models in order to justify the choices made for the new set of models. For a discussion of more general issues inherent in any SED modeling, the reader is referred to Sect. 7. Model-dependent parameters Of the 14 parameters defining the R06 models, some required specific assumptions to be made in order to be used in the analytical description of the model. For instance, the stellar mass and age had no direct impact on the SED, but instead, evolutionary tracks were needed to transform these into the stellar temperature and radius, which are the properties that actually have an effect on the SED. When fitting these models to observed SEDs, the parameters we can directly determine in a best-case scenario are the stellar temperature and luminosity (and therefore radius). The stellar mass and age are then only determined with the assumption of the evolutionary tracks. Similarly, the envelope infall rate was converted to an envelope density structure via the assumption of the Ulrich (1976) collapse model. Therefore, when one derives stellar masses, ages, and envelope infall rates from the R06 models, one is implicitly assuming a specific set of evolutionary tracks and a collapse model. However, users of the models may not necessarily realize the inherent assumptions, and most importantly cannot easily change these assumptions since they are built into the models. Unevenly sampled parameter space The R06 models are defined by sampling circumstellar environment parameters from ranges that are functions of the stellar mass and age. For example, the envelope infall rate is sampled from ranges that decrease over time, while the cavity opening angle is sampled from ranges that increase with time. The motivation for these was to restrict the parameter space coverage to regions that were thought to be realistic in order to minimize the computing time required, but the downside was that this led to correlations between parameters in the models before they were even applied to observations. Examples include a correlation between the envelope infall rateṀ env and stellar mass M (since the envelope infall rate is sampled from a constant range inṀ env /M ), between the envelope infall rate and the disk accretion rate (since both are sampled from ranges that decrease for larger ages), and an anticorrelation between infall rate and cavity opening angle (since, as mentioned above, the latter is sampled from a range that increases with time). These correlations can be seen in Fig. 1 of Robitaille (2008). These built-in correlations mean that it is very difficult to look for such trends in samples of objects, since the built-in trends would mask any real trend. Fixed model complexity The R06 models, defined by 14 parameters, are often used without testing simpler models. For example, all of the models with envelopes in the R06 set also have disks, and furthermore the higher the envelope density, the higher the minimum disk mass found in the models. This means that there are, in fact, no deeply embedded protostellar models that do not have a disk. This in turn implies that it is not possible to use the models to try and find evidence for disks in embedded YSOs because there are no models to test the hypothesis that there is no disk. Lack of cold dust at long wavelengths The outer radii of the envelopes in the R06 models were defined as the radius where the temperature would drop to 30 K if the envelope was optically thin to radiation. While for more embedded YSOs, the temperature may drop below 30 K at the outer radius, those models only guarantee that they include all dust hotter than 30 K. This in turn means that the models may not include enough cold material compared to observed sources, and may therefore show a deficit of long-wavelength emission. The reason for this limitation was that the models were primarily designed with the Spitzer IRAC/MIPS and shorter wavelength data in mind, and at the time there were no surveys equivalent to the Herschel data, which now provides high-resolution long-wavelength fluxes for thousands of YSOs. As a result, investigations using Herschel data have predictably found that the models are not always able to fit at the longest wavelengths. For example, Sewiło et al. (2010) found that some of the Herschel sources in the Large Magellanic Cloud were not well fit by the R06 models beyond 100 µm. Because the R06 models do not include much dust below 30 K, none of the model SEDs peak at wavelengths longer than 100 µm, but the coldest protostars observed with Herschel typically peak around 200 or 300 µm. While the R06 models were well-suited to Spitzer observations, new models are required now that Herschel observations of YSOs are common. Poor signal-to-noise at long wavelengths The R06 models were computed using the Monte-Carlo radiative transfer code developed by Whitney et al. (2003aWhitney et al. ( ,b, 2004, which at the time computed the long-wavelength SED by sampling equal energy photon packets in the same way as for shorter wavelengths. However, traditional Monte-Carlo sampling done in this way results in the number of photons emitted being highest where the SED is brightest, and conversely lower where the SED is fainter, such as in the far-infrared and at mm wavelengths. As a result, most SEDs in the R06 set had poor signal-tonoise (S/N) at 1 mm and beyond, and some of the less embedded models even had poor S/N beyond 100 µm. While in many cases this has not been too problematic, in the sense that observations also have poor S/N where for the faintest sources, we ideally need models with high S/N at all wavelengths to make the most efficient use of available observations. Model components and set-up In this section, I present an overview of the new set of models, which addresses the issues described in Sect. 2, and includes a number of further improvements. Design philosophy The new models were not computed in a single monolithic set, but rather as several sets of models with increasing complexity. A11, page 2 of 16 For instance, one of the sets consists of models with only a star with a surrounding disk, another set includes a disk and an envelope, but no bipolar cavities, and yet another set includes a disk, envelope, and bipolar cavities. This modularity allows us to ask which model offers the best representation of the data, before even looking at the actual values of the parameters (see an example in Sect. 5.4). For example, a source might be fit by a complex model with a central source, a disk, and an envelope with bipolar cavities, but it might be equally well fit by a model with only a disk, indicating that there is no strong evidence one way or another for the presence of an envelope. It is important to assess not only the goodness of fit but also the simplicity or complexity of the model, since it is easy but not always meaningful to fit any set of data with an arbitrarily complex model. This design can also allow the available sets of models to grow over time. The components used to generate the model sets are described in Sect. 3.2. For components containing free parameters, the free parameters were uniformly randomly sampled between a minimum and maximum value. The ranges of values used for each parameter are given in Table 1. By randomly sampling in uniform ranges, we can avoid correlations between parameters which were present in the R06 models. On the other hand, some of the combinations of parameters may be unphysical -since the definition of what might be considered physical will change over time (for example with stellar evolutionary tracks) the models presented here span a broad parameter space, and it is left to the user of the models to decide which models to ignore as being unphysical. In general, models can be divided into two main categories. The first are models that aim to be as realistic as possible -for example in the case of radiative transfer, models with a realistic 3d distribution of dust and with the best available dust model (with variable dust properties depending on location and temperature). The second category of models are models that are simpler but aim to provide insight into the effect of various physical processes, components, and so on. The collection of models presented in this paper is firmly in the second category: the aim is not to provide the most realistic model, but rather simple models that can be used to explore large regions of parameter space. This influences some of the decisions outlined in the following sections. I encourage users to use these models as a starting point for modeling observational data of young stars, but to follow this up with more detailed modeling if the observations cannot be reproduced with the simple models, or if spatially resolved data or spectra are available. Components As mentioned in Sect. 3.1 the new models consist of a combination of components which I describe in the following sections. Central source The central source, present in all models, was set to be a spherical source. Unlike the R06 models, the central source was not assigned a mass, and the stellar properties were not derived from evolutionary tracks. Instead, the central source was defined directly using a stellar radius R and effective temperature T . While this does result in some of the models having unphysical combinations of R and T , it allows the models to be independent of stellar evolutionary tracks. The idea is that users of the models can then -if needed -assume a specific set of tracks, decide which models are physical according to those tracks, and assign masses and ages to the stars. The effective temperature of the source was used to select appropriate stellar photosphere models. For temperatures including and above 4000 K, the photosphere models from Castelli & Kurucz (2004) were used, while for temperatures below 4000 K, models computed with the PHOENIX code (Brott & Hauschildt 2005) and intended for the Gaia mission 1 were used instead. Since the central source is not defined in terms of mass, we cannot calculate a surface gravity log [g]. However, log [g] does not have a large impact on stellar photospheres in the range 3000 K to 20 000 K, while above and below these temperatures, the largest difference depending on the choice of stellar photosphere model is generally of the order of 10% relative to the log [g] = 4.0 models. Therefore, the log [g] = 4.0 models were always used, with the caveat that these may be wrong by up to 10% for low and high temperatures, which affects mostly the near-infrared fluxes for non-embedded models. Disk In this set of models, only passive disks are included. For embedded YSOs, the effect of not explicitly including accretion is likely to be minimal because the heating from viscous dissipation in the disk is typically not very important, and the biggest effect is the increase in luminosity from the star. Since the star is embedded, the radiation from the star gets reprocessed and therefore the shape of the stellar spectrum is not important. When modeling disks at near-infrared and longer wavelengths, the passive disks can still be used to model accreting disks for a similar reason: while we expect a little extra heating in the disk from the viscous dissipation, most of the accretion luminosity comes from the stellar surface, and again the increase in luminosity of the central source is the most important effect, so a model of a passive disk with a higher central luminosity star will likely be adequate in most cases. Of course, the lack of accretion does mean that UV and optical fluxes for non-embedded sources with strong accretion cannot reliably be modeled, since none of the passive disk models will produce the adequate excess UV and optical emission typically observed toward accreting sources. Disks in hydrostatic equilibrium are expected to be flared (Shakura & Sunyaev 1973), and at earlier times, when the dust is well coupled to the gas, the dust in the disk follows the same structure. Over time, as the dust settles to the mid-plane, the effective flaring for the disk may change. The flaring and the scaleheight of the disk is therefore parametrized such that it covers the range of flaring from hydrostatic disks to flat disks. The density distribution of the passive flared disks is given in cylindrical polar coordinates (R, z, φ) by ρ(R, z, φ) = ρ disk 0 R 0 R β−p exp        − 1 2 z h(R) 2        ,(1) where ρ disk 0 is defined by the disk dust mass M disk , p is the surface density radial exponent, β is the disk flaring power, and the disk scaleheight h(R) is given by h(R) = h 0 R R 0 β · (2) The disk is truncated at the inner radius R disk min and the outer radius R disk max . The free parameters varied for all models with a disk are the disk dust mass M disk , outer radius R disk max , flaring exponent β, surface density exponent p, and scaleheight h 0 . For some of the models, the disk inner radius R disk min was also varied (as described in Sect. 3.3). Envelope Two types of envelopes were included in the models -the first are spherically-symmetric power-law envelopes, and the second are rotationally flattened envelopes. The reason for including both types of envelopes is because this will allow users to investigate the constraints on the envelope structure from observations: while a model with a more complex rotationally flattened envelope may fit well, it is important to also check whether a model with a simpler spherically symmetric envelope can also fit. The spherically symmetric models also have more flexibility as to what power-law to use for the radial dependence of the density. The power-law envelope density structure is given by ρ(r) = ρ env 0 r r 0 γ ,(3) where ρ env 0 is the density of the envelope at the radius r 0 , and serves as the scaling for the envelope density, and γ is the radial power of the density. The other type of envelope included in the models is (as in R06) the rotationally flattened envelope structure defined by Ulrich (1976). This envelope structure has a radial power-law dependence of around 3/2 outside the centrifugal radius R c , and 1/2 inside. The density structure has a singularity in the midplane at the radius R c , which is due (in the analytical model) to the pile-up of material due to conservation of angular momentum. The density structure is given by ρ(r, θ) = ρ env 0 r R c −3/2 1 + µ µ 0 −1/2       µ µ 0 + 2µ 2 0 R c r       −1 ,(4) where ρ env 0 is the scaling of the density structure which can be related (in the analytical model) to the infall rateṀ env : ρ env 0 =Ṁ env 4π GM R 3 c 1/2 ·(5) The µ 0 value is given by the equation for the streamline: µ 3 0 + µ 0 r R c − 1 − µ r R c = 0.(6) The envelope dust density is defined by specifying ρ env 0 rather thanṀ env since the former had a direct impact on the SED and is not dependent on the analytical model of infall (beyond the shape of the density structure itself). The R06 models used this envelope by specified the envelope infall rateṀ env , but this sometimes led to confusion since this infall is only used to set the density and does not result in an increase in accretion luminosity. Users of the new models will still be able to compute their ownṀ env values for the models, but by doing so manually, the assumptions will be clearer. For both envelope components, the envelope is truncated at the inner radius R env min , and extends all the way to the edge of the grid, which is discussed in Sect. 4. The free parameters are the envelope density scaling ρ env 0 , as well as the radial exponent γ in the case of the power-law envelope, and the centrifugal radius R c in the case of the Ulrich envelope. Bipolar cavities Observations of embedded YSOs almost always show signs of bipolar outflows. The primary effect of these outflows on the circumstellar envelopes is to carve out bipolar cavities by entraining the envelope material into the outflow. This leads to a region of lower density where radiation can escape more easily, and can sometimes be observed in scattered light (since the shorter wavelength radiation finds a much easier escape route via the cavities). Bipolar cavities can only be present in a model if an envelope component is also present. The power-law cavity is defined as a region with boundary given by z(R) = z 0 R R 0 c ·(7) A11, page 4 of 16 For \|z\| > z(R), the dust density is set to ρ cav 0 or the envelope density, whichever is lowest. The value of R 0 and z 0 are set by defining the half-opening angle of the cavity θ 0 at a spherical polar radius of r 0 = 10 000 au: z 0 = r 0 cos θ 0 , (8) r 0 = r 0 sin θ 0 .(9) The cavities are not empty, but instead contain dust with a constant density ρ cav 0 , except in regions where the envelope density would be lower. The free parameters are the power-law exponent of the cavity opening c, the opening angle θ 0 , and the density inside the cavity ρ cav 0 . Ambient medium The last model component is the ambient interstellar medium. Forming stars, especially embedded YSOs, are not completely isolated objects, and the effect of the interstellar medium can sometimes be important. As shown in Whitney et al. (2013) and Koepferl et al. (2015), the interstellar medium can be heated by the central stars and provide additional thermal emission at the longest wavelengths. For the models without an envelope, both models with and without an ambient medium are included, while for models with envelopes the ambient medium is included in all cases. The reason for always including it for envelopes is to avoid including a sharp cutoff in the envelope density, since the analytical expressions for the envelope densities never go to zero. By adding an ambient medium, the radiative transfer can be carried out to a radius where both the density and the temperature reach a constant value. The ambient medium, when included, is not simply a density that is added, but instead it is a lower limit to the density (so that densities above it remain unchanged). For the models presented here, the dust density was set to ρ amb = 10 −23 g/cm 3 , and the lower limit of the temperature in the whole model was set to T amb = 10 K. As discussed in Sect. 4.2.3, we need to ensure that we correctly subtract the background emission from this ambient medium once we extract the SEDs. Parameter space sampling As mentioned in Sect. 3.1, Table 1 lists the ranges of parameters sampled for each parameter. Table 2 lists all the model sets computed for this initial model release. The model set name is indicated, along with the components that the model set includes. The model set name is composed of several characters that indicate which component is present. The characters, in order, are s (star), p (passive disk), p (power-law envelope) or u (Ulrich envelope), b (bipolar cavities), h (inner hole), m (ambient medium), and i (interstellar dust). If a component is absent, a "-" is given instead. Since these model names are not immediately intuitive, the second column shows an icon representing the basic layout of the model. In the figures shown in later sections, these icons are then used in figure panels to specify which model set is being shown. All models include a star with the radius R and temperature T sampled from the ranges given in Table 1. For the passive disks, power-law and Ulrich envelopes, and bipolar cavity components, the ranges sampled from are also given in Table 1. For models where the ambient medium is included, the values are fixed to those given in Sect. 3.2.5. For all models that contain at least a disk, envelope, or ambient medium, the inner radius is set to the same value for all components, and is either set to R sub (the dust sublimation radius) or is sampled from the range given in Table 1. The number of models in each set should ideally be related to the number of variable parameters in that set, but the simplest set has two parameters, while the most complex has 12, so the number of models is set using n models = 10 000 × 2 max[n var −9,0] . This resulted in 10 000 models for models with nine or fewer variable parameters. The reason for enforcing the lower limit on the number of models is to ensure that there were enough representative models for each set. Dust properties For this initial collection of models sets, the dust properties are taken to be the same for all models, specifically the dust properties from Draine (2003a,b) using the Weingartner & Draine (2001) Milky Way grain size distribution A for R V = 5.5 and C/H = 30 ppm renormalized to C/H = 42.6 ppm. The full Mie scattering properties of this dust model were computed using the bhmie routine from (Bohren & Huffman 1983), modified by B. Draine 2 , and wrapped in a package that makes it easy to compute the properties for various size distributions 3 . As described in Sect. 3.1, the aim of the set of models presented here is not to provide the most physically realistic models, but rather to provide a consistent set of models spanning a large region of parameter space, which can then be used to explore the behavior of various physical components in the models. While it might be desirable for example to vary the dust properties in disks to account for grain growth toward the mid-plane, the models provided here are a first step to ensure that in the first place a model with uniform dust properties cannot fit the observations. In many instances, one may be tempted to make models more complex rather than explore more of the existing parameter space. The set of models could of course be expanded in future to include models with different or even variable dust properties, and include for example the effects of polycyclic aromatic hydrocarbons (PAHs), but this initial set deliberately includes much simpler dust properties. Methods Radiative transfer code The code used to compute the models presented here is Hyperion, an open-source 3d Monte-Carlo dust continuum radiative transfer code, described in detail in Robitaille (2011). The models presented here were computed with version v0.9.5 of the code. The radiative transfer was computed on a spherical polar grid. For models with spherical symmetry, a 1d grid with 400 radial cells was used, while for more complex axisymmetric models, a 2d grid with 400 radial cells and 300 polar cells was used. The radial cells were distributed so as to ensure that the width of a cell at the inner edge of the disk was small enough to resolve the temperature gradient well. This was done by picking values equally spaced in log space between δR and R disk max − R disk min , where δR is the width at the inner edge of the disk corresponding to τ = 0.1 at the peak wavelength of the stellar spectrum, then adding these values to R disk min (this is the default mode in Hyperion when computing YSO models). The outer radius of the grid was set to √ 2 times the radius of the largest aperture in which the SEDs were measured. The aperture sizes as well as the origin of the √ 2 factor are discussed in Sect. 4.2.3. Rather than using dust temperatures, the code carries out all calculations in terms of the absorbed energy per unit mass of dust (the absorbed specific energy). In the first few iterations, the code computes the absorbed specific energy in each cell using the algorithm outlined by Lucy (1999). Once the code detects that the calculation has converged, it then proceeds to computing the SEDs. The convergence detection is described in detail in Robitaille (2011) -for the models presented here, the specific energy is considered to be converged once the 99th percentile value of the energy difference is less than a factor of two, and the change in this value changes less than 10% from one iteration to the next. The modified random walk (MRW) algorithm (Min et al. 2009;Robitaille 2010) was used in order to speed up the computation of the Monte-Carlo propagation in optically thick regions. The raytracing algorithm described in Robitaille (2011) was also used in order to obtain excellent S/N at long wavelengths, to avoid issues such as those in the R06 models (cf. Sect. 2.5). Spectral energy distributions Overview The SEDs were computed for 250 wavelength bins logarithmically spaced between 0.01 µm and 5 mm. The SEDs also contain information about the origin of the emission, for example whether it comes directly from the star, whether it was scattered, or whether it was emitted by dust. Each SED was computed for all four Stokes parameters I, Q, U, and V, which means that in addition to the SED of the total flux, polarization spectra are also available. The linear polarization is given by p lin = Q 2 + U 2 I 2 ,(11) and the circular polarization by p circ = V I ·(12) Viewing angles The SEDs were computed for nine viewing angles for each model. In contrast to the R06 models, where the SEDs were produced using escaping photons binned into viewing angle ranges, and thus effectively producing SEDs averaged over a range of viewing angles, the models presented here were computed using peeling-off and raytracing (cf. Robitaille 2011), which allows one to compute SEDs at exact viewing angles. However, selecting the same viewing angles for all models is not ideal, since it does not adequately sample the full range of viewing angles and favors specific ones. For example, if SEDs are computed from 0 • to 90 • in steps of 10 • , then model SEDs for disks will vary dramatically between 80 • and 90 • (edge-on), and there will be no SEDs for viewing angles in between these values, so a source seen at a viewing angle of 85 • will not have any adequate models. Therefore, each model was instead computed for 9 viewing angles randomly sampled between 0 • to 90 • . However, since when looking at a model it is useful to see how the SED changes with viewing angle, and we do not want models to randomly have all angles close to pole-on, I used stratified sampling, which means randomly sampling one angle between 0 • and 10 • , one between 10 • and 20 • , and so on, which still produces a distribution that is random for our purposes. Apertures and extent along the line of sight The SEDs were computed in 20 circular apertures, with radii logarithmically spaced between the dust sublimation radius and an outer radius that depended on the type of model. For models with no ambient medium but with a disk, the largest aperture was simply set to the radius of the disk. For the star-only models with no ambient medium, all apertures were set to include all the flux from the star. For models with an ambient medium, the situation required a more careful treatment of the outer aperture as well as the emission region in three dimensions. Let us first consider the example of a constant density optically thin ambient medium with no central source: in this case, if the ambient density is set up in a spherical polar grid, then an observer would not observe a constant surface brightness, but rather would see a circularly symmetric region of emission peaking at the center and falling off toward the edges (essentially the projection of an optically thin sphere onto the sky). However, this is not desirable, since we need a model with a constant density and temperature to produce a constant surface brightness on the sky. Hyperion allows images and SEDs to be computed by including only photons emitted in a fixed range of distances along the line of sight -thus, one can in effect set up a slab outside which photons are ignored. This slab is set up to have a halfwidth d that is the same as the radius of the largest aperture a max . Thus, the region of emission included in the SEDs is therefore a cylinder rather than a sphere, which avoids the issues mentioned above. In order to make sure that the cylinder is completely filled, the spherical grid itself thus needs to extend to √ 2 times the radius of the largest aperture (also the half-width of the slab). A diagram of this set-up is shown in Fig. 1. With this set-up, as the apertures become larger, the surface brightness should asymptotically tend to the surface brightness of the ambient medium. The transition point at which the surface brightness reaches this level is r(ρ = ρ amb ) or r(T = T amb ), whichever is largest. While r(ρ = ρ amb ) is known from the start, it is impossible to know r(T = T amb ) in advance of running a model, since the temperature structure is required. One could therefore choose the radius at which the temperature would reach the ambient level in the optically thin limit. However, for optically thick models, since the temperature could in reality approach the ambient level at a much smaller radius, we would be wasting many photon packets to the constant density ambient region. Therefore, the radiative transfer was first run to determine the temperature structure, and only then were the final apertures to use for the SED decided. Postprocessing Once the SEDs were computed, a background surface brightness was subtracted from each aperture. This background surface brightness was determined by computing the modified blackbody emission that would arise from a medium with density A11, page 7 of 16 ρ amb , temperature T amb , thickness 2d, and assuming the dust properties described in Sect. 3.4. To avoid S/N issues, any flux densities for smaller apertures that are below 0.1% of the SED in the largest aperture are masked (ignored), for wavelengths larger than 10 µm -this was done because as the apertures get smaller, the long-wavelength emission becomes dominated by a smaller number of photons, and under this threshold the flux densities start to become unreliable. Finally, the model flux densities were all interpolated to a common set of 20 apertures logarithmically spaced between 10 2 au and 10 6 au. SED fitting code In R07, we presented a fitting algorithm for the R06 models, which was implemented as a Fortran fitting code that could be called from a public web interface. The underlying Fortran code was, however, not publicly available. While the web interface allowed for easy modeling of low numbers of sources, it was not suitable for modeling many forming stars across entire regions. I have now written a new Python-based fitting code that allows users to carry out modeling of many sources using the new models (as well as the R06 models). This will also allow users to customize the fitting workflow, and for example to eliminate unphysical models, or fit models from the different sets to the same data. Results Computational details The models were computed on a cluster with 120 cores over a period of several months. The total CPU time to compute the models was approximately 640 000 h. A small fraction (just below 2%) of models were terminated during the temperature calculations since they did not complete within the allocated time due to extremely high optical depths, and they are therefore missing from the initial v1.1 release of the models. However, they will be added to future releases. In addition, for the two sets of models with spherically symmetric power-law envelopes, the temperature calculations for some of the models did complete, but the calculation of the SEDs did not. For the v1.1 release of the models, the SEDs for these models were therefore recomputed without including the contribution from scattered stellar and scattered dust emission. Since this concerns the models with the highest optical depth, not including the scattered stellar emission is a valid approximation. On the other hand, for a subset of the models, not including the scattered dust emission may have an effect. By extrapolating the relationship between the contribution of the scattered dust emission to the total SED as a function of wavelength, it is possible to infer that between 2 and 14 µm, for 90% of the models, the effect of not including the scattered dust emission will result in a less than 10% effect. Outside this wavelength range, the effect will not be noticeable for any of the models. In future, Hyperion will include a much more efficient method for computing the scattered light. Figure 2 shows an overview of the SEDs in the model sets by selecting 2000 SEDs for each model set. These are normalized to the total luminosity of the SED to make it easier to compare them to one another. The aim of this figure is not to show each individual SED but to show trends over the different model sets. For example, as expected, models with envelopes have a much wider range of near-and mid-infrared fluxes compared to models with a disk alone. Models Model availability The models are publicly available 5 . Given the number of models, these are primarily delivered using a few large files, rather than millions of small files. Each model set consists of the following main files: a FITS file containing the fluxes and associated uncertainties for all models, apertures, and wavelengths; a FITS file containing the parameters for all the models. The motivation for providing the models as a few large files is that it allows the data to be more easily sliced, for example to return the flux at a specific wavelength for all models, for the largest aperture, or the fluxes for all apertures and wavelengths for a given model, and so on. Several FITS file readers such as that included in Astropy (Astropy Collaboration et al. 2013) are able to read subsets of data from FITS files as needed, meaning that the data can still be accessed if the FITS files are larger than the available memory. In addition to these main files, I provide the model fluxes obtained by convolving the SEDs with a number of common transmission curves, for example for instruments on Spitzer or Herschel, and I also provide a utility in the SED fitting tool (cf. Sect. 4.3) that allows users to convolve the models with any other transmission curve. Finally, I am also making available the individual model files, which can be used to extract information relating to the density and temperature grids, as well as the polarization spectra. Rather than consider the models a static and final version, the models are versioned and may be updated in future if bugs or other issues are found at any stage of the pipeline to generate the models. The version used in this paper is v1.1, and this will be updated if any issues are fixed. Fitting strategy General approach In R07 and subsequent studies, the approach taken to modeling observations with the R06 models was to simply fit all the models to the observations and directly look at the parameter values in order to place constraints on the physical properties. The model sets presented here allow us to better separate the problem into determining constraints related to the presence or absence of components (e.g., "Is there any evidence for a disk in this source?"), and determining constraints related to the physical parameter values (e.g., "What is the disk mass?"). As mentioned in Sect. 3.1, one of the consequences of the uniform sampling of parameter space is that not all models are necessarily physically realistic, so when fitting real objects, the user of the models is responsible for deciding which models are realistic. However, for the example presented in this section, all models are assumed to be equally realistic. Data and fitting parameters In this section, I show an example of modeling observational data using the new models. The data used is that for a source in the NGC 2264 star formation region (Forbrich et al. 2010). The source, referred to as Source 20 in Forbrich et al., is the 2MASS source J06404862+0935578 at a position of α = 06 h 40 m 48.62 s δ = 09 • 35 57.8 (ICRS). The fluxes extend from optical to sub-millimeter wavelengths, and were carefully checked as part of that study. In cases where the source was undetected, upper limits are provided. As in Forbrich et al. (2010), the distance range was assumed to be 869 to 961 pc (within ∼5% of d = 913 pc), and A V range to be from 0 to 40 mag. The extinction law used was the same as that used in Forbrich et al. Results In Fig. 3 I show for each model set presented here all the models that satisfy χ 2 − χ 2 best < 3 n data , where χ 2 best is the χ 2 of the best model for each model set (see R07 for details on the χ 2 calculation). The panels in the figure are ordered roughly in order of increasing complexity from left to right, then from top to bottom, so that the top left panel shows models that include only a star, and the bottom right panel shows models that include a star, disk, infalling envelope, bipolar cavity, and an ambient medium. At a first glance, a substantial fraction of the model sets provide a good fit to the data, with the exception of the models with neither disk or envelope, and the models with only an Ulrich envelope. All model sets with only a disk or only a power-law envelope, or with an envelope and a disk provide a good fit. It thus initially appears that one cannot place strong constraints on the nature of the object. However, I now show that one can use information about not only the quality of the fit, but also the fraction of models that provide a good fit in order to try and determine which model is most likely (even though many of the models remain possible). Bayesian model comparison Using a Bayesian approach, we can compare how well two models M 1 and M 2 explain a set of data D using the Bayes factor K ≡ P(D\|M 1 ) P(D\|M 2 ) ·(13) In this equation, the P(D\|M i ) terms are the likelihoods, which represent the probability that the data D are produced if the model M i is correct. Thus, the Bayes factor gives the ratio of the probability that one model produces the data compared to the other. If K = 1, then the two models are equally likely, whereas if K > 1 then model M 1 is more likely than M 2 (and vice versa). Assuming that a model M can be parametrized by a parameter vector θ, we can expand the likelihood as an integral over parameter space: P(D\|M) = p(θ\|M) p(D\|θ, M) dθ.(14) In our case, a model M corresponds to one of our model sets with a fixed number of parameters. For each model M we have a finite number of pre-existing samples of parameter space. We can therefore approximate P(D\|M) by a sum over the existing model samples θ j : P(D\|M) = N j=1 P(θ j \|M) P(D\|θ j , M),(15) where N is the number of samples. Assuming that the models have been sampled such that all models samples are equally likely, the term P(θ j \|M) is then equal to 1/N, and we can rewrite the likelihood as P(D\|M) = 1 N N j=1 P(D\|θ j , M).(16) In other words, the likelihood is the sum of the probabilities of that each model sample will produce the data, divided by the number of samples, or more simply, the likelihood is simply the probability of reproducing the data averaged over the model samples. Comparing which model is most likely is then simply a matter of comparing the values of P(D\|M) for different models M. We note that the assumption about all model samples being equally likely does mean that we are treating the initial sampling of the models (i.e., logarithmic or linear) as a prior. It is also important to note that uniform (linear or logarithmic) sampling does increase the risk of missing small areas of high probability, and the models presented here should be considered as a starting point for any modeling effort rather than a set that includes the optimal model for any specific dataset. The implication of Eq. (16) is that given two different model sets, even if both can fit the data equally well, the one where the highest fraction of models fit well should be preferred. If one A11, page 10 of 16 , where each panel shows all the fits for χ 2 − χ 2 best < 3 n data for a specific model set (and where χ 2 best is determined for each model set individually). A11, page 11 of 16 considers a simple model set, for example a model set with only a star and a disk, and then compares this to a model set with an envelope added, the region of parameter space providing a good fit in the model with the envelope will represent a smaller fraction of the total parameter space. Therefore, if both model sets have the same total number of samples, the region of parameter space providing a good fit will be sampled by fewer models, and therefore the fraction of models providing a good fit will be lower. Thus, the Bayes factor is simply a formal mathematical way of thinking about Occam's razor, which states that all things considered, the simpler model should be favored because more complex models require more fine tuning to reproduce the data. The challenge in our case is to estimate the probability P(D\|θ j , M). Formally, under the assumption of normal errors, we can write this as P(D\|θ j , M) ∝ exp − χ 2 2 ·(17) However, as I will discuss in Sect. 7.1, this is often too stringent a definition, because in the case of observed SEDs, there are a number of systematic sources of errors to consider, such as variability or the fact that the models are only simplistic representations of a more complex 3d reality. Therefore, I suggest here (as was done in R07) that we should adopt a less stringent definition of probability. For example, in R07, the approach was to treat all models with χ 2 − χ 2 best < 3 n data ,(18) as being "good", and remaining models as being "bad". Of course, this is a very binary definition that also does not reflect the continuum of good to bad fits, but it corresponds roughly to what one would consider reasonable vs. unreasonable models by eye. If we adopted this definition, we would be treating P(D\|θ j , M) as P(D\|θ j , M) ∝        1 for χ 2 − χ 2 best ≤ 3 n data 0 for χ 2 − χ 2 best > 3 n data .(19) Yet another possibility would be to correctly model all the systematic effects in SED modeling, and determine a real P(D\|θ j , M), although in many cases this is not possible without significant effort. The aim here is not to debate which definition of P(D\|θ j , M) should be used, and this will depend on the problem being solved. We can now consider the example from Fig. 3 and try to apply this formalism in order to determine which model set is the most likely. We adopt a definition of the probability from Eq. (19), but in order for the comparison to be fair between different models, we need to use the same value of χ 2 best for all models. Otherwise, a model set that has many poor models, where the best fit is poor, may rank the same as a model set that has many good models, where the best fit is good. We therefore define χ 2 best as being the χ 2 for the single best-fitting model over all model sets. For the purpose of this example, a criterion of χ 2 − χ 2 best ≤ 9 n data ,(20) provided a sensible separation between good and bad fits, so models that satisfy this are assigned a probability of one, while the remaining ones have probability of zero. If we write the number of "good" models as N good for the model set M, we can then re-write Eq. (16) as P(D\|M) ∝ N good N ·(21) In other words, we simply look at which model set has the highest fraction of good models. In this way, even if a specific model set contains the best fit by χ 2 value, if only one model provides a good fit in that set, it means that the parameters need to be finetuned to reproduce the data, while a model set where a larger fraction of models can reproduce the data is more likely because it requires less fine-tuning. Figure 4 shows all the models considered good when using the same χ 2 best for all models. This shows that already some model sets cannot provide any good fits to the data according to this definition. We now look at the values of P(D\|M) for the remaining model sets. Table 3 lists for each model set the fraction of models providing a good fit, the best-fit χ 2 value, and the relative score, which is given by the ratio of P(D\|M) to the mean of the P(D\|M) values for all model sets. A higher relative score means a more likely model. Using this, we can see that the most likely model is that of a simple disk with no additional inner hole, no envelope, and no ambient medium. If we look at the model set that also includes an ambient medium, the χ 2 value for this model set is lower (2.28 compared to 2.71), but because a smaller fraction of models fit, the model is overall less likely (the score is 1.947 compared to 7.600 for the model with no ambient medium). Similarly, the model with a flattened envelope, bipolar cavities, and embedded in an ambient medium has a best-fit χ 2 value only slightly larger (3.00 instead of 2.71), but the score is much lower (0.867 vs. 7.600). This is because the model with the envelope requires more fine-tuned conditions in order to reproduce the data. Of course, the model with the envelope may actually be correct, but the Bayes factor tells us that without any other information, it is easier for the simpler model to reproduce the data, and we should therefore consider it more likely. Parameter space Having identified a most likely model set, we can take a look at some of the parameters from the models that fit well. However, since other model sets provided reasonable fits, we examine the parameters only under the assumption that this most likely model set is correct. Figure 5 shows a 2d projection of parameter space, showing the stellar temperature against the total disk mass. The figure compares the results obtained with the R06 grid versus the model parameters for the sp s-i set in this paper (that is, the model set that includes only disks around a central star, with no variable inner radius). Also shown for reference are the results for the spubhmi set (the most complex one presented in this paper). The figure shows the significant difference between the original parameter space coverage, which is non-uniform and has a complex shape, and the new parameter space coverage, which is much more uniform in both model sets. The models that provide a good fit lie in a similar region of parameter space, but show more structure in the results from the R06 models, which could be due to biases in the parameter space coverage that may be in other dimensions that those shown here. In contrast, the new models that provide a good fit have a more uniform and extended distribution in the sp s-i set, and show that the temperature and total disk mass are both very uncertain, although certain ranges are ruled out. The lack of strong constraints is expected -without a detection in the sub-mm, the disk mass cannot be well constrained, and the mid-infrared fluxes can only provide a lower limit to the mass. Indeed, as shown in R06, midinfrared fluxes are only able to tell us once the disk has become optically thick, but above this they provide no constraints. The Fig. 3, but showing all the fits for χ 2 − χ 2 0,best < 9 n data where χ 2 0,best is the same for all panels, and set to the best fit over all model sets. 5. Comparison of parameter space coverage of all the models (gray points) and best fits to the source shown in Fig. 4 (black) for the R06 model grid (left), the subset of models presented in this paper that are in the sp s-i set (center), and the spubhmi set (right). The good fits are defined using Eq. (20), and we apply the same criterion to the R06 models. The limits are adjusted to those covered by the new model sets, and the left figure would include more points on the left and higher temperatures if the plot limits were expanded. While the disk masses sampled for the new models were dust masses, these have been converted to gas masses for the above figures assuming a gas-to-dust ratio of 100. stellar temperature is uncertain because of the large extinctionto some extent, the temperature and extinction are in fact correlated. Caveats While this new set of models addresses the limitations of the R06 models described in Sect. 2, there are nevertheless a number of caveats with the current models. Some of these were already mentioned above, and a summary is provided here. First, the caveats relating to parameter space coverage are the following: 1. A single dust model is assumed throughout the density structures (cf. Sect. 3.4). 2. The dust properties do not take into account emission from transiently heated very small grains and polycyclic aromatic hydrocarbons (PAHs). 3. All the models have a single source of emission, which is the central star, and the effects of the interstellar radiation field are not taken into account. 4. Accretion is not explicitly included. As mentioned in Sect. 3.2.2, this does not mean that the models cannot be used to study objects with accretion -it mainly means that the models cannot reproduce the UV and optical fluxes for non-embedded objects. 5. Not all models are necessarily equally physically plausible. It is up to the user to decide whether certain models should be ignored for not being physically realistic. Future versions of the models can of course address these limitations, for example by adding models with different dust properties and even variable dust properties, but it is important to also have basic models that have simple dust properties and no interstellar radiation field for comparison, in order to test whether there is actually evidence that these simple properties are not sufficient. Other important caveats include: 6. The shape of the photosphere models for the central source may be accurate only to 10% in some cases (as discussed in Sect. 3.2.1). This will only matter when modeling fluxes in the optical and near-infrared for non-embedded objects. 7. Parameters derived from the model parameters may still suffer from correlations, even though the main parameters are sampled uniformly inside ranges. For example, the distribution of radii and temperatures is uniform in log space, but the distribution of luminosities is not so. This is not an issue with the model sets presented here as such, but the limitation is that only the directly sampled model parameters are uniform (it would indeed be impossible to sample radii, temperatures, and luminosities evenly in the same model set). 8. The models should not be treated as final: similarly to software, the models can include issues that will be resolved in subsequent versions. For example, in the v1.1 release of the models described in this paper, a small fraction (2%) of the models are missing, and for two of the model sets, a small fraction of models will be missing scattered dust emission in the mid-infrared (see Sect. 5.1 for more details). When publishing results based on these models, it is therefore important to mention the version used. General limitations In Sect. 2, I outlined limitations with the R06 set of models, which are addressed with the new models. However, there are a number of limitations of SED modeling in general that no set of models can overcome, and are instead intrinsic to this type of modeling. Misspecification of models YSOs, especially accreting protostars, are complex 3d objects, not nicely axisymmetric density structures. Even disks may contain 3d structures such as spiral arms. This means that from the onset, we know that the models are wrong compared to the real density distributions, but we can still hope that they are similar enough to reality to provide interesting insight into those objects. In the language of statistics, our models are misspecified. Most models in Astrophysics are misspecified, but to a different degree: for instance, models for the orbits of binary stars, or for parallax motions, while still approximate, are much better approximations of reality than models of the density distribution around YSOs. In practice, the fact the models are misspecified means that one cannot fit these models to data using χ 2 minimization and expect that the χ 2 value will be convertible into a probability A11, page 14 of 16 using the classical P ∝ exp (−χ 2 /2). Instead, the χ 2 can only be used as a relative measure of goodness of fit -the absolute value is not meaningful. Blending and confusion With the exception of the nearest and low-density star-forming regions, more crowded regions and more distant star-forming regions suffer from source blending, and many objects that appear as single YSOs will often in fact be two or more objects. If this is the case, then some of the bulk properties such as the luminosity or dust mass may be close enough to the real values (although the latter depends on the temperature distribution, which will be different if multiple sources are present), but for the more detailed parameters the values will not be meaningful. For example, when modeling two blended sources, properties such as the disk flaring angle or scaleheight are no longer meaningful. Of course, it is not always possible to know if a source is suffering from blending, especially for distant star forming regions, but one should always bear in mind that this is likely, and therefore that the more distant regions one looks at, the less one should trust the details of the individual parameters, and for these regions, it may be more useful to carry out population synthesis modeling than to fit single SEDs to photometry which most likely contains contributions from several sources. Variability Even in the case of an isolated YSO, a further complication is (wavelength-dependent) variability. Many YSOs are known to be variable (e.g., Morales-Calderón et al. 2011;Günther et al. 2014), and this would not be a problem in itself if all the multiwavelength photometry was taken at the same time, but this is rarely the case. Most SEDs that we construct from archival data contain contributions at different wavelengths that were measured years apart in some cases. This can result in discontinuities in the SED from one wavelength to the next, making it challenging to fit the fluxes with a static model, but often not providing sufficient data to allow a detailed time-dependent model to be used. Summary I have presented a new set of model SEDs for young stellar objects, spanning a wide range of evolutionary stages, from the youngest deeply embedded protostars to pre-main-sequence stars with little or no disks. In Sect. 2, I discussed the limitations with the models that were originally published in R06, and in Sect. 3 I discussed how the new model set was designed in order to resolve these limitations. To summarize, the most significant changes compared to the previous generation of models are: -The new models do not depend on highly model-dependent values, such as the stellar age and mass, which depend on stellar evolutionary tracks. Instead, the new models are defined using parameters that have a direct impact on the radiative transfer. -The parameter space is sampled uniformly (in linear or logarithmic space depending on the parameter) which does not introduce correlations between the parameters used to define the models. -The models are split into sets with increasing complexity, with the simplest model set having two parameters, and the most complex having 12. A11, page 15 of 16 A&A 600, A11 (2017) -The envelope outer radius for envelope models should now be large enough to include 10-20 K dust that is essential for modeling Herschel observations. -The models now have a high S/N over the entire wavelength range. The aim of the new set of models is not to provide the most physically realistic models for young stars, but rather to provide deliberately simplified models for initial modeling, which allows us to cover a wider range of parameter space. In addition, the design of the new set of models allows us to separate the problem into determining constraints related to the presence or absence of components, and determining constraints related to the physical parameter values assuming a given model. The fitting example in Sect. 5.4 showed how these models can be used to analyze real observations of a YSO, and what we could determine from this. While the modeling results were highly degenerate, I showed how one can use a Bayesian approach to assign relative probabilities to the various models of increasing complexity. The first version of the models (v1.1) as well as a new Python-based fitting tool are publicly available. Fig. 1 . 1Schematic representation of a typical model. The z axis of the model is shown as the black vertical arrow. In this example, the line of sight is rotated by an inclination of i = 35 • relative to the z axis. The dotted lines show the extent of the largest aperture with radius a max , while the dashed lines show the extent of the slab with half-width d inside which photons are taken into account for the SED. To ensure that the models account for all the emission in the intersection of the largest aperture and the slab (the dark shaded area), the edge of the grid (the thick solid circle) needs to have a radius of √ 2d. Fig. 2 . 2A subset of 2000 SEDs for each model set, normalized to the total luminosity of each SED. Fig. 3 . 3Model fits to observations of source 20 in NGC 2264 (using the source nomenclature fromForbrich et al. 2010) Fig. 4 . 4As in Table 1 . 1Parameter ranges sampled.Parameter Symbol Minimum Maximum Sampling Stellar radius R 0.1 R 100 R Log Stellar temperature T 2000 K 30 000 K Log Disk mass [dust] M disk 10 −8 M 0.1 M Log Disk inner radius R disk min R sub 1000 R sub Log Disk outer radius R disk max 50 AU 5000 AU Log Disk flaring power β 1 1.3 Linear Disk surface density power p −2 0 Linear Disk scaleheight h 100AU 1 AU 20 AU Log Envelope density [dust] ρ env 0 10 −24 g/cm 3 10 −16 g/cm 3 Log Envelope density power γ −2 −1 Linear Envelope centrifugal radius R c 50 AU 5000 AU Log Cavity density [dust] ρ cav 0 10 −23 g/cm 3 10 −20 g/cm 3 Log Cavity opening angle θ 0 0 • 60 • Linear Cavity power c 1 2 Linear Table 2 . 2Model sets presented in this paper.Model set Icon Star Disk Envelope Cavity Ambient Inner radius Variables Models s s-i yes . . . . . . . . . . . . . . . 2 10 000 sp s-i yes passive . . . . . . . . . R sub 7 10 000 sp h-i yes passive . . . . . . . . . variable 8 10 000 s smi yes . . . . . . . . . yes R sub 2 10 000 sp smi yes passive . . . . . . yes R sub 7 10 000 sp hmi yes passive . . . . . . yes variable 8 10 000 s-p-smi yes . . . power-law . . . yes R sub 4 10 000 s-p-hmi yes . . . power-law . . . yes variable 5 10 000 s-pbsmi yes . . . power-law yes yes R sub 7 10 000 s-pbhmi yes . . . power-law yes yes variable 8 10 000 s-u-smi yes . . . Ulrich . . . yes R sub 4 10 000 s-u-hmi yes . . . Ulrich . . . yes variable 5 10 000 s-ubsmi yes . . . Ulrich yes yes R sub 7 10 000 s-ubhmi yes . . . Ulrich yes yes variable 8 10 000 spu-smi yes passive Ulrich . . . yes R sub 8 10 000 spu-hmi yes passive Ulrich . . . yes variable 9 10 000 spubsmi yes passive Ulrich yes yes R sub 11 40 000 spubhmi yes passive Ulrich yes yes variable 12 80 000 A11, page 6 of 16 The new code is available 4 under an open-source BSD license, and was used for the results shown in Sect. 5.4.4 http://doi.org/10.5281/zenodo.235786 A11, page 13 of 16 1 10 Disk (gas) mass [solar masses]10 4 Stellar temperature [K] 10 -6 10 -5 10 -4 0.001 0.01 0.1 1 10 Disk (gas) mass [solar masses] 10 4 Stellar temperature [K] 10 -6 10 -5 10 -4 0.001 0.01 0.1 1 10 Disk (gas) mass [solar masses] 10 4 Stellar temperature [K] Fig. Table 3 . 3Relative likelihoods of the model sets for the example source modeled.P(D\|M) Score Model set Icon (relative) χ 2 best (relative) s s-i 0.0000% 384.60 0.000 sp s-i 0.3167% 2.71 7.600 sp h-i 0.1156% 2.71 2.773 s smi 0.0000% 713.92 0.000 sp smi 0.0811% 2.28 1.947 sp hmi 0.0267% 3.38 0.640 s-p-smi 0.0700% 3.99 1.680 s-p-hmi 0.0100% 11.12 0.240 s-pbsmi 0.0178% 3.50 0.427 s-pbhmi 0.0011% 10.42 0.027 s-u-smi 0.0000% 40.80 0.000 s-u-hmi 0.0000% 39.24 0.000 s-ubsmi 0.0000% 46.59 0.000 s-ubhmi 0.0000% 56.08 0.000 spu-smi 0.0489% 4.09 1.173 spu-hmi 0.0056% 8.51 0.133 spubsmi 0.0361% 3.00 0.867 spubhmi 0.0206% 3.64 0.493 http://www.hs.uni-hamburg.de/EN/For/ThA/phoenix/ gaia_info.html A11, page 3 of 16 A&A 600, A11 (2017) http://www.astro.princeton.edu/~draine/scattering. html 3 https://github.com/hyperion-rt/bhmie http://doi.org/10.5281/zenodo.166732 A11, page 8 of 16 A11, page 16 of 16 Acknowledgements. I wish to thank Barbara Whitney and Kenny Wood for helpful discussions that helped shape this work, as well as the referee for suggestions that helped improve this paper. This research made use of Astropy, a communitydeveloped core Python package for Astronomy (Astropy Collaboration et al. 2013). All the scripts used to produce the figures in this paper and to carry out the fitting example in Sect. 5.4 will be made available alongside the models. Since I am no longer working as a researcher, the model sets presented here will not be expanded in future to include additional parameters (although I may release updated versions of the models to fix the issues described in Sect. 5.1), but all the materials needed to reproduce this work will be freely available so that anyone interested in expanding this work will be able to do so. . 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E L Wright, P R M Eisenhardt, A K Mainzer, http:/linker.aanda.org/10.1051/0004-6361/201425486/33AJ. 1401868Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868	[ "https://github.com/hyperion-rt/bhmie" ]
[ "TACTIC and MACE gamma-ray telescopes", "TACTIC and MACE gamma-ray telescopes" ]	[ "K K Yadav \nHIGRO collaboration) Astrophysical Sciences Division\nBhabha Atomic Research Centre\n400 085TrombayMumbaiIndia\n" ]	[ "HIGRO collaboration) Astrophysical Sciences Division\nBhabha Atomic Research Centre\n400 085TrombayMumbaiIndia" ]	[]	The TACTIC gamma-ray telescope, equipped with a tracking light collector of ∼9.5m 2 area and a 349-pixel imaging camera has been in operation at Mount Abu in Western India since 2001. Having a sensitivity of detecting the Crab Nebula above 1.2 TeV at 5.0σ significance level in 25h of observations, this telescope has detected gamma-ray emissions from Mrk501 and Mrk421 and is presently being deployed for monitoring of AGNs. As a new Indian initiative in γ-ray astronomy we are setting up the 21-m diameter MACE γ-ray telescope at the high altitude (4200m asl) astronomical site at Hanle in North India. This telescope will deploy a 1408-pixels integrated camera at its focal plane. Designed to operate at a trigger threshold of ∼30 GeV, this telescope is expected to be operational in 2011. Some of the salient features of the TACTIC telescope along with the results of its recent observations and the design details of the MACE telescope are presented in this paper.	null	[ "https://arxiv.org/pdf/0904.3647v1.pdf" ]	16,037,963	0904.3647	0c0674089320abe4da52462e01ae4ddc43b050a5	TACTIC and MACE gamma-ray telescopes 23 Apr 2009 K K Yadav HIGRO collaboration) Astrophysical Sciences Division Bhabha Atomic Research Centre 400 085TrombayMumbaiIndia TACTIC and MACE gamma-ray telescopes 23 Apr 2009 The TACTIC gamma-ray telescope, equipped with a tracking light collector of ∼9.5m 2 area and a 349-pixel imaging camera has been in operation at Mount Abu in Western India since 2001. Having a sensitivity of detecting the Crab Nebula above 1.2 TeV at 5.0σ significance level in 25h of observations, this telescope has detected gamma-ray emissions from Mrk501 and Mrk421 and is presently being deployed for monitoring of AGNs. As a new Indian initiative in γ-ray astronomy we are setting up the 21-m diameter MACE γ-ray telescope at the high altitude (4200m asl) astronomical site at Hanle in North India. This telescope will deploy a 1408-pixels integrated camera at its focal plane. Designed to operate at a trigger threshold of ∼30 GeV, this telescope is expected to be operational in 2011. Some of the salient features of the TACTIC telescope along with the results of its recent observations and the design details of the MACE telescope are presented in this paper. TACTIC telescope The TACTIC (TeV Atmospheric Cherenkov Telescope with Imaging Camera) γ-ray telescope located at Mt. Abu (24.6 • N, 72.7 • E, 1300m asl), is being used to study potential TeV γray sources. The telescope deploys a F/1 type tracking light collector of ∼9.5 m 2 area, made up of 34×0.6 m diameter, front-coated spherical glass facets which have been prealigned to produce an on-axis spot of ∼ 0.3 • diameter at the focal plane. The telescope uses a 349-pixel photomultiplier tube (ETL 9083UVB) -based imaging camera with a uniform pixel resolution of ∼0.3 • and a field-of-view of ∼6 • ×6 • to record images of atmospheric Cherenkov events. The innermost 121 pixels (11×11 matrix) are used for generating the event trigger, based on the NNP (Nearest Neighbour Pairs)/3NCT (Nearest Neighbour Non-Collinear Triplets) topological logic 1 , by demanding a signal ≥ 25/8 pe for the 2/3 pixels which participate in the trigger generation. Whenever the single channel rate of any two or more pixels in the trigger region goes outside the preset operational band, it is automatically restored to within the prescribed range by appropriately adjusting the high voltage of the pixels 2 . The resulting change in the photomultiplier (PMT) gain is monitored by repeatedly flashing a blue LED, placed at a distance of ∼1.5m from the camera. The advantages of using such a scheme are that in addition to providing control over chance coincidence triggers, it also ensures safe operation of PMTs with typical anode currents of ≤ 3 µA. The back-end signal processing hardware of the telescope is based on medium channel density NIM and CAMAC modules developed inhouse. The data acquisition and control system of the telescope 3 has been designed around a network of PCs running the QNX (version 4.25) real-time operating system. The triggered events are digitized by CAMAC based 12-bit Charge to Digital Converters (CDC) which have a full scale range of 600 pC. The telescope has a pointing and tracking accuracy of better than ±3 arc-minutes. The tracking accuracy is checked on a regular basis with so called "point runs", where a bright star whose declination is close to that of the candidate γ-ray source is tracked continuously for about 5 hours. The point run calibration data (corrected zenith and azimuth angle of the telescope when the star image is centered) are then incorporated in the telescope drive system software so that appropriate corrections can be applied directly in real time while tracking a candidate γ-ray source 4 . Recent TACTIC results In order to evaluate the performance of the TACTIC telescope the Crab Nebula "standard candle" has been observed repeatedly since 2001. Operating at a γ-ray threshold energy of ∼1.2 TeV, the telescope records a cosmic ray event rate of ∼2.0 Hz at a typical zenith angle of 15 • . The telescope has a 5σ sensitivity of detecting Crab Nebula in 25 hours of observation time. The consistent detection of a steady signal from the Crab Nebula along with excellent matching of its energy spectrum with that obtained by other groups, reassures that the performance of the TACTIC telescope is quite stable and reliable. The telescope has detected strong γ-ray signals from two active galactic nuclei (AGN) Mrk501 (2006 observations) 5 and Mrk421 ( 2005-06 observations) 6 while other two AGNs 1ES2344+514 7 and H1426+428 observed during 2004-05 and 2004-07 respectevely have been found to be in the quiescent state. Some of the recent results obtained on various candidate γ-ray sources are listed in Table 1. We believe that there is a considerable scope for the TACTIC telescope to monitor TeV γ-ray emission from other AGNs on a long-term basis. MACE telescope Exploring the γ-ray sky in the energy range ≥ 10GeV with low energy threshold ground based atmospheric Cherenkov telescopes is expected to lead to a potentially rich harvest of astrophysical discoveries, as has been already demonstrated by the HESS and MAGIC telescopes at γ-ray energies ≥ 100GeV . The low threshold energy can be attained by increasing the light collector area of the telescopes and installing them at higher altitudes where the photon density of the atmospheric Cherenkov events is higher 8 . As a new Indian initiative in gamma-ray astronomy, the Himalayan Gamma Ray Observatory (HIGRO) is being set up at Hanle (32.8 • N, 78.9 • E, 4200m asl) in the Ladakh region of North India. The site offers an average of about 260 uniformly distributed spectroscopic nights per year which is a major advantage in terms of sky coverage for source observations. Located closer to the shower maximum the Cherenkov photon The MACE (Major Atmospheric Cherenkov Experiment) telescope with high resolution imaging camera is designed to operate in the sub-TeV energy range as part of the HIGRO collaboration. As depicted in Figure 1 the altitude-azimuth mounted telescope will deploy a 21-m diameter parabolic light collector made of 356 panels of 984 mm × 984 mm size with each panel consisting of 4 spherical mirror facets of 488 mm × 488 mm size. Each facet is diamond turned to a mirror finish yielding a reflectivity of ≥ 85% in the visible band. The telescope will use the graded focal length (increases towards the periphery) mirrors in order to reduce the D 80 spot size (defined as the diameter of the circle within which 80% of the reflected rays lie) of the light collector to ∼15 mm for on-axis incidence. Each mirror panel will be equipped with motorized orientation controllers for aligning them to form a single parabolic light collector. The focal plane instrumentation will have a photomultiplier tube based imaging camera covering a field of view of 4 • ×4 • . The imaging camera will comprise of 1408 pixels arranged in a square matrix with uniform pixel resolution of 0.1 • . The inner 576 pixels with field of view of 2.4 • ×2.4 • will be used for generating the event trigger. The PMTs will be provided with acrylic front-aluminized light cones for enhancing the light collection efficiency of the camera. The signal processing instrumentation will also be housed within the camera and the acquired data will be sent to the control room over the computer network for processing and archiving. Detailed Monte Carlo simulation studies have been carried out using CORSIKA 10 code and the results suggest that using a pixel threshold of ≥4pe and a 4 nearest neighbour pixel trigger, gamma-ray energy threshold of ∼30 GeV is achievable by the MACE telescope. Figure 2 shows the differential trigger rates of γ-rays for the two different types of spectra. The energy thresholds are determined to be 44GeV for the Crab spectrum and 31GeV for the pure power law spectrum with a diffential index of 2.59 for the above mentioned configuration. : Gamma-ray differential rates for the two types of primary spectra calculated for the 4 nearest neighbour pixel, 4pe trigger configuration. For Crab, it gives an energy threshold of ∼ (44 ± 2)GeV and for the power law the threshold energy is ∼ (31 ± 2)GeV . Status of MACE telescope The detailed engineering and structural design of the MACE telescope has been completed. Fabrication of the mechanical structure has started and the telescope is likely to be installed at Hanle by 2011. Figure 1 : 121-m diameter MACE telescope density at Hanle is substantially high as compared to the sea level 9 . The higher photon density along with the low background light level at this site helps in lowering the energy threshold of the Cherenkov telescope being setup there. Figure 2 2Figure 2: Gamma-ray differential rates for the two types of primary spectra calculated for the 4 nearest neighbour pixel, 4pe trigger configuration. For Crab, it gives an energy threshold of ∼ (44 ± 2)GeV and for the power law the threshold energy is ∼ (31 ± 2)GeV . Table 1 : 1Observations on gamma-ray sources with TACTIC telescopeSr. Source Observation period Observation(h) Significance/UL 1 Crab Nebula Dec 2003 -Feb 2004 104.28 10.30σ Nov 2005 -Feb 2006 101.04 9.40σ Nov 2007 -Mar 2008 105.15 11.05σ 2 Mrk421 Dec 2005 -Apr 2006 201.72 11.5σ Jan 2007 -Mar 2007 83.5 ≤ 0.92 × 10 −12 ph cm −2 s −1 Jan 2008 -May 2008 149.70 9.60σ 3 Mrk501 Mar 2005 -May 2005 46.00 ≤ 4.62 × 10 −12 ph cm −2 s −1 Feb 2006 -May 2006 66.80 7.5σ 4 1ES2344+514 Oct 2004 -Dec 2005 60.15 ≤ 3.84 × 10 −12 ph cm −2 s −1 5 H1426+428 Mar 2004 -Jun 2007 165.70 ≤ 1.18 × 10 −12 ph cm −2 s −1 AcknowledgmentsWe would like to thank the organizers of the "44th Rencontres de Moriond" for providing financial support to attend the conference. We would also like to thank colleagues of the Astrophysical Sciences Division for their contribution towards the operation of the TACTIC telescope and the design of the MACE telescope. . S R Kaul, Nucl. Instrum. Methods A. 496400S.R. Kaul et al, Nucl. Instrum. Methods A 496, 400 (2003). . N Bhatt, Meas. Sci. Technol. 12167N. Bhatt et al, Meas. Sci. Technol 12, 167 (2001). . K K Yadav, Nucl. Instrum. Methods A. 527411K.K. Yadav et al, Nucl. Instrum. Methods A 527, 411 (2004). . R , Nucl. Instrum. Methods A. 578548R. Koul et al, Nucl. Instrum. Methods A 578, 548 (2007). . S V Godambe, J. Phys. G: Nucl. Part. Phys. 341683S. V. Godambe et al, J. Phys. G: Nucl. Part. Phys. 34, 1683 (2007). . K K Yadav, Astropart. Phys. 27447K. K. Yadav et al, Astropart. Phys. 27, 447 (2007). . S V Godambe, J. Phys. G: Nucl. Part. Phys. 3565202S. V. Godambe et al, J. Phys. G: Nucl. Part. Phys. 35, 065202 (2008). . F Aharonian, Astropart. Phys. 15335F. Aharonian et al, Astropart. Phys. 15, 335 (2001). R Cowsik, OG2.0527th ICRC. Hamburg2769R. Cowsik et al, 27th ICRC, Hamburg, OG2.05, 2769 (2001). D Heck, FZKA 6019 Forschungszentrum Karlsruhe. D. Heck et al, FZKA 6019 Forschungszentrum Karlsruhe,(1998).	[]
[ "The macroscopic monopolization in diagonal magnetoelectrics", "The macroscopic monopolization in diagonal magnetoelectrics" ]	[ "Nicola A Spaldin \nMaterials Theory\nETH Zurich\nWolfgang-Pauli-Strasse 278093ZurichSwitzerland\n", "Michael Fechner \nMaterials Theory\nETH Zurich\nWolfgang-Pauli-Strasse 278093ZurichSwitzerland\n", "Eric Bousquet \nMaterials Theory\nETH Zurich\nWolfgang-Pauli-Strasse 278093ZurichSwitzerland\n\nPhysique Théorique des Matériaux\nUniversité de Liège\nB-4000Sart TilmanBelgium\n", "Alexander Balatsky \nNORDITA\nKTH Royal Institute of Technology and Stockholm University\n23 106 91Roslagstullsbacken, StockholmSweden\n\nTheoretical Division\nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n\nCenter for Integrated Nanotechnologies\nLos Alamos National Laboratory\n87545Los AlamosNMUSA\n", "Lars Nordström \nDepartment of Physics and Astronomy\nUppsala University\nP.O. Box 516SE-75120UppsalaSweden\n" ]	[ "Materials Theory\nETH Zurich\nWolfgang-Pauli-Strasse 278093ZurichSwitzerland", "Materials Theory\nETH Zurich\nWolfgang-Pauli-Strasse 278093ZurichSwitzerland", "Materials Theory\nETH Zurich\nWolfgang-Pauli-Strasse 278093ZurichSwitzerland", "Physique Théorique des Matériaux\nUniversité de Liège\nB-4000Sart TilmanBelgium", "NORDITA\nKTH Royal Institute of Technology and Stockholm University\n23 106 91Roslagstullsbacken, StockholmSweden", "Theoretical Division\nLos Alamos National Laboratory\n87545Los AlamosNMUSA", "Center for Integrated Nanotechnologies\nLos Alamos National Laboratory\n87545Los AlamosNMUSA", "Department of Physics and Astronomy\nUppsala University\nP.O. Box 516SE-75120UppsalaSweden" ]	[]	We develop the formalism of the macroscopic monopolization -that is the monopole moment per unit volume -in periodic solids, and discuss its relationship to the diagonal magnetoelectric effect. For the series of lithium transition metal phosphate compounds we use first-principles density functional theory to calculate the contributions to the macroscopic monopolization from the global distribution of magnetic moments within the unit cell, as well as from the distribution of magnetization around the atomic sites. We find one example within the series (LiMnPO 4 ) that shows a macroscopic monopolization corresponding to a ferromonopolar ordering consistent with its diagonal magnetoelectric response. The other members of the series (LiMPO 4 , with M = Co, Fe and Ni) have zero net monopolization but have antiferromonopolar orderings that should lead to q-dependent diagonal magnetoelectric effects.	10.1103/physrevb.88.094429	[ "https://arxiv.org/pdf/1306.5396v2.pdf" ]	118,465,874	1306.5396	8e9a6de8cf0fd6125b5e683ea89c0ae68be3281b	The macroscopic monopolization in diagonal magnetoelectrics 29 Jun 2013 Nicola A Spaldin Materials Theory ETH Zurich Wolfgang-Pauli-Strasse 278093ZurichSwitzerland Michael Fechner Materials Theory ETH Zurich Wolfgang-Pauli-Strasse 278093ZurichSwitzerland Eric Bousquet Materials Theory ETH Zurich Wolfgang-Pauli-Strasse 278093ZurichSwitzerland Physique Théorique des Matériaux Université de Liège B-4000Sart TilmanBelgium Alexander Balatsky NORDITA KTH Royal Institute of Technology and Stockholm University 23 106 91Roslagstullsbacken, StockholmSweden Theoretical Division Los Alamos National Laboratory 87545Los AlamosNMUSA Center for Integrated Nanotechnologies Los Alamos National Laboratory 87545Los AlamosNMUSA Lars Nordström Department of Physics and Astronomy Uppsala University P.O. Box 516SE-75120UppsalaSweden The macroscopic monopolization in diagonal magnetoelectrics 29 Jun 2013(Dated: May 7, 2014)arXiv:1306.5396v2 [cond-mat.str-el] We develop the formalism of the macroscopic monopolization -that is the monopole moment per unit volume -in periodic solids, and discuss its relationship to the diagonal magnetoelectric effect. For the series of lithium transition metal phosphate compounds we use first-principles density functional theory to calculate the contributions to the macroscopic monopolization from the global distribution of magnetic moments within the unit cell, as well as from the distribution of magnetization around the atomic sites. We find one example within the series (LiMnPO 4 ) that shows a macroscopic monopolization corresponding to a ferromonopolar ordering consistent with its diagonal magnetoelectric response. The other members of the series (LiMPO 4 , with M = Co, Fe and Ni) have zero net monopolization but have antiferromonopolar orderings that should lead to q-dependent diagonal magnetoelectric effects. I. INTRODUCTION The linear magnetoelectric response of a solid is the linear order magnetization induced by an electric field or equivalently the linear order electric polarization induced by a magnetic field. It is described by a second-rank tensor, α, which can be non-zero when both time-reversal and space-inversion symmetries are broken, and may have diagonal or off-diagonal components, corresponding to a response parallel or perpendicular to the applied field respectively. Materials with anti-symmetric off-diagonal linear magnetoelectric responses have the same symmetry as the toroidal component of the second-order term in the magnetic multipole expansion, and so there has been much recent discussion in the literature of whether the toroidal moment, t, is a relevant and useful concept for describing such magnetoelectric effects. In particular, the term ferrotoroidics has been introduced to describe materials in which the toroidal moments are aligned cooperatively, and such materials have been considered to complete the group of primary ferroics. 1-3 Motivated by this suggestion, a theory of toroidization -defined to be the toroidal moment per unit volume -in bulk crystalline solids has been developed, which appropriately treats the multi-valuedness caused by the periodic boundary conditions 4 . Ferrotoroidic switching has been reported 3 , and attempts to demonstrate that the toroidal moment can act as a primary order parameter are ongoing. In addition, the local toroidal moments associated with the atomic V sites in V 2 O 3 and the atomic Cu sites in CuO have been detected directly using resonant xray diffraction [5][6][7] . Such local toroidal moments could be of tremendous importance, as it has been proposed that they are candidates for the order parameter in the pseudo-gap phase of cuprate superconductors 8 . The second-order term in the magnetic multipole expansion contains two additional contributions beyond the toroidal term, which describe in turn magnetic quadrupolar and magnetic monopolar components that couple respectively to the gradient and divergence of the magnetic field (see detailed derivation below). While the latter has not been extensively discussed on the grounds that Maxwell's equations tell us formally that B does not diverge, it is in fact non-zero in materials with a diagonal linear magnetoelectric response. Indeed, it could appropriately be described as a magnetoelectric monopole to distinguish it from the zeroth order term in the multipole expansion of the magnetic field which is the magnetic analogue to the electrical charge and indeed is formally zero. We emphasize also that the magnetoelectric monopole discussed here is a ground state property of the system, and so is distinct from those recently proposed and verified in spin ice, in which nonlocal magnetic monopoles exist as excited states 9,10 . The origin of the relationship between the monopolar contribution to the multipole expansion and the diagonal magnetoelectric response is illustrated in Fig. 1 (a) and (b) where we follow the discussion from Ref. 11. The monopolar magnetic vortex in panel (a) consists of local spin magnetic moments (black solid arrows) oriented outwards from a point -note that Maxwell's equations are not violated; while M diverges, it is compensated for by H and so B does not diverge. Since the spin moments s i are never parallel, it is known from the theory of multiferroics that there is a local radial electric polarization ∝ s i ×s j (unfilled grey arrows) associated with each pair of spins 12,13 . However these local radial polarizations are uniform around the vortex and the net electric polarization is zero. On application of a magnetic field, however, the spin moments reorient to align themselves more closely parallel to the field (panel (b)). The local contributions to the electric polarization no longer average to zero and a net polarization parallel to the magnetic field direction results. For completeness, we show in Figure 1 (c) and (d) the analogous relationship between a toroidal vortex and the offdiagonal magnetoelectric response. In this case an applied magnetic field modifies the spin orientations so that a net magnetic moment is induced perpendicular to the direction of applied field. The remainder of this paper is organized as follows: In the next section we review the definition of the magnetoelectric monopole starting from a multipole expansion of the magnetic field and show that it couples to the divergence thereof. In Section III we describe how the monopole can be calculated from first-principles electronic structure methods, as well as how it could be directly measured experimentally. We introduce the term monopolization to describe the monopole per unit volume in periodic solids, and show that it is natural both theoretically and experimentally to divide the total monopolization into two contributions: That arising from the local monopoles around individual ions, and that arising from the global distribution of magnetic moments within the solid. We discuss also the problems associated with defining the monopolization for an infinite periodic solid, and propose a practical solution. In Section IV we present results of the calculated monopolizations for the family of lithium transition metal phosphates, LiMPO 4 , M = Mn, Fe, Co, Ni. All members of this family have the same structure and overall magnetic order, but they differ in their local magnetic anisotropy and hence their magnetic symmetry. We find that the different magnetic symmetries lead to different monopolar orderings: In one case there is ferromonopolar ordering with a net macroscopic monopolization, and the remaining three cases have zero net monopolization, but with hidden "anti-ferromonopolar" orderings that have not previously been identified. In section V we develop the Ginzburg-Landau theory describing the coupling of the monopolization to homogeneous external magnetic and electric fields. In the final section we discuss the possible relevance of these concepts. II. THE MULTIPOLE EXPANSION Following the derivation in Ref. 14, we consider a magnetization density µ(r), that may arise from both spin and orbital contributions, in an inhomogeneous magnetic field H (r) that varies slowly on the scale of the system size. Then the interaction energy, H int , of the magnetization density with the magnetic field H int = − µ(r) · H (r) d 3 r(1) can be expanded in powers of field gradients calculated at some arbitrary reference point r = 0: H int = − µ(r) · H (0) d 3 r − r i µ j (r)∂ i H j (0) d 3 r − . . . . (2) where i, j are Cartesian directions. The first term is the interaction of the field with the magnetic moment of the system m = µ(r)d 3 r .(3) In the second term, the tensor M i j = r i µ j (r)d 3 r with nine components can be decomposed into three parts (summation over repeated indices is implied): i) the pseudoscalar from the trace of the tensor, a = 1 3 M ii = 1 3 r· µ(r)d 3 r ,(4) ii) the toroidal moment vector dual to the antisymmetric part of the tensor, t i = 1 2 ε i jk M jk , t = 1 2 r× µ(r)d 3 r ,(5) and iii) the traceless symmetric tensor q i j describing the quadrupole magnetic moment of the system, q i j = 1 2 M i j + M ji − 2 3 δ i j M kk = 1 2 r i µ j + r j µ i − 2 3 δ i j r· µ(r) d 3 r . (6) The expansion of Eqn. (2) can then be written in the form H int = −m · H (0) −a (∇ · H) r=0 −t · [∇ × H] r=0 −q i j (∂ i H j + ∂ j H i ) r=0 − . . ..(7) We see that the toroidal moment t couples to the curl of the magnetic field, and the quadrupole moment q i j couples to the field gradient, while the pseudoscalar a is coupled to the divergence of magnetic field, and so represents a monopolar component. III. CALCULATION AND MEASUREMENT OF THE MAGNETOELECTRIC MONOPOLE IN BULK, PERIODIC SOLIDS In this section we discuss the difficulties associated with the definition of the monopole in bulk, periodic solids, and propose solutions that allow a correspondence between calculated monopole moments and possible experimental measurements. First we note a simplification: Since the orbital contribution to the magnetization density, µ orb (r) is proportional to r × p(r), where p is the momentum, and r · r × p is zero, the orbital contribution to the monopole is always formally zero, and only the spin contribution need be considered. For systems of finite size, such as molecules or molecular clusters, that have zero net magnetic moment, the value of the monopole can be evaluated directly from the spin part of the magnetization density through the integral in Eqn. 4. Eqn. 4 is not directly applicable to extended systems where periodic boundary conditions are employed, however, because the integral contains the position operator, r. Therefore for a general continuous magnetization density µ(r) it will lead to arbitrary values, depending on the choice of unit cell used in the calculation. A. Decomposition of the monopole moment into atomic site and local moment contributions In anticipation of treating the bulk, periodic case, we rewrite Eqn. 4 by decomposing the position operator r into the positions of the constituent atoms, r α , relative to some arbitrary origin, plus the distance from each atomic center, [r − r α ]. The integral over all space then separates into a sum over the atomic sites, ∑ α and an integral around each atomic site, as , and Eqn. 4 can be rewritten as a = 1 3 r· µ(r)d 3 r = 1 3 ∑ α as (r α + [r − r α ])· µ(r)d 3 r = 1 3 ∑ α r α · as µ(r)d 3 r + as [r − r α ] · µ(r)d 3 r = 1 3 ∑ α r α · m α + as [r − r α ] · µ(r)d 3 r(8) where the summation, ∑ α is over all of the atoms α in the system, and m α is the local magnetic moment on the αth atom. We see then that the monopole can be decomposed into two components: The first, comes from the local monopoles at the atomic sites, which arise from the same current distribution around the site that simultaneously gives rise to the local dipole moment. We call this contribution a as for "atomic site", and at each site, α, it is given by a as α = 1 3 as [r − r α ].µ(r)d 3 r(9) where the atomic nucleus is at position r α and the integral is over some localized region around the atomic nucleus; in an electronic-structure calculation this can be chosen to be the "atomic sphere" or the "pseudo-atomic orbital" depending on the details of the implementation and the integral can in principle be evaluated over this finite region. In practice, we calculate the atomic site contributions to the monopole through expectation values of spherical tensors using a generalization of the method used previously to obtain inversion-even tensor moments in studies of correlated d or f electron materials 15,16 . For each atomic site α a local density matrix γ α inside a site-centered sphere is obtained from the electronic structure and expanded in spherical harmonics and spinors. In the present work we use the augmented plane wave plus local orbital (APW+lo) method and these spheres are naturally chosen to be the muffin-tin spheres. The density matrices are then further expanded with respect to their behavior (either even or odd) under space inversion i and time inversion θ : γ α = 1 ∑ ν=0 1 ∑ η=0 γ νη α θ γ νη α = (−1) ν γ νη α iγ νη α = (−1) η γ νη α .(10) For magnetoelectrically active multipole moments such as monopoles, only the component that is odd in both space inversion and time reversal that is γ 11 α , is relevant. In addition, for convenience we expand the density matrices in the Pauli matrices and the identity matrix in spin space, γ νη = 1 2 3 ∑ β =0 σ β γ νηβ α γ νηβ α = Sp σ β γ νη(11) where Sp is the trace over the spin degree of freedom. Now the monopole moment can be written in the form a α = 1 2 3 ∑ β =1 Tr Γ (110) σ β γ 11β α . Here the operator Γ (110) describes the coupling of two rank one tensors, r α and m α , to a rank zero a α , and and Tr is the trace over the orbital degree of freedom. In Figure 2 we show the generic magnetization textures for positive and negative atomic site monopoles, as well as for completeness the z component of a toroidal moment and the z 2 component of the quadrupolar tensor. The arrows represent the magnetization orientation on a sphere surrounding an atomic site and the color indicates whether the magnetization points outwards (green) or inwards (red). Note that these atomic site monopoles can in principle be measured by resonant x-ray spectroscopy 17 , which has been used successfully to detect an atomic site toroidal moment 7,18 . No unambiguous measurement of atomic monopoles has been made to date, however, because a material has not yet been identified that meets the stringent conditions required to achieve an observation in the resonant x-ray measurement. We point out also that, provided that the local magnetic site is not an inversion center, the atomic monopoles can be nonzero even in a system with overall zero monopole moment; we will explore some examples in Section IV. Such systems might be described as "anti-monopolar" and should show a q-dependent magnetoelectric effect. The second contribution to the monopole, which we write a lm for "local moment", arises from representing the magnetization density by a distribution of localized magnetic moments m α at the atomic sites: a lm = 1 3 ∑ α r α .m α .(13) In systems such as insulating 3d transition metal oxides, which have large localized magnetic moments that are spatially separated by distances of a fewÅ we expect this contribution to be the dominant contribution to the total monopole. Using Eq. (13) we can straightforwardly evaluate the monopoles of the arrangements of magnetic moments shown in Fig. 3. Taking the ±y-oriented magnetic moments to be spaced a distance d apart along the y direction, and the ±x-oriented moments a distance d apart along x, then the monopoles of arrangements a) and b) in Fig. 3 are a = − 2 3 dm and + 2 3 dm respectively, where m is the magnitude of each local magnetic dipole moment. Applying Eq. 13 to the arrangement show in c) yields the value + 1 3 dm; this can also be obtained by inspection by recognizing that c) consists of a monopole with magnetic moments at the same position of as in b) but of half the magnitude, plus a quadrupole, as shown in the lower panel of Fig. 3. The total monopole resulting from these two contributions is then a = a lm + ∑ α a as α(14) where the sum is over all the atomic sites. In all the cases shown in Fig. 3, the net magnetization is zero. There exists a complication, however, in the case where the region over which the monpole is to be evaluated has a net magnetic dipole. The complication is that all multipoles in systems with non-zero lower-order multipoles (the magnetic dipole in the case of the magnetoelectric monopole) are dependent on the choice of origin used to evaluate them. It is straightforward to see that for systems with nonvanishing magnetic dipole moment, for a change of origin defined by r → r ′ = r + R 0(15) the monopole changes as a → a ′ = a + 1 3 R 0 . µ(r)d 3 r .(16) It remains an open question in general, which we do not address here, whether such origin dependence of the multipoles is physically meaningful (see for example Ref. 19). One practical approach is to always choose as the origin the position of the average magnetic moment,R, defined so that µ(r −R)d 3 r = 0. This is equivalent to neglecting any uncompensated part of the magnetization and retaining only the compensated part in the calculation of the monopole 4 . Care must be taken, however, in situations where a change in net magnetic dipole moment, or a structural rearrangement occurs, to ensure that a consistent choice of origin is maintained. B. Bulk systems with periodic boundary conditions; the problem of multi-valuedness Next we turn to the case of a system with periodic boundary conditions. It is often convenient to describe the properties of a bulk crystalline solid in terms of a small repeat unit -the unit cell -which is then replicated using periodic boundary conditions to generate the infinite solid. Many intensive quantities such as the magnetization, which is defined to be the magnetic moment per unit volume, can then be simply obtained as the value of the quantity in a single unit cell divided by the unit cell volume. For the case of the macroscopic monopole per unit volume -which we propose to call the monopolization by analogy with magnetization, polarization, etc. -Eqn. 4 is not directly applicable to extended systems with periodic boundary conditions, because for a general continuous magnetization density µ(r), Eq. (4) evaluated over one unit cell will lead to arbitrary values, depending on the particular choice of unit cell used in the calculation. We note that this behavior is distinct from the origin dependence discussed in Section III A, and persists even in the case when the net magnetization is zero. In fact the difficulties are exactly analogous to those encountered in defining a macroscopic bulk toroidization, and indeed reflect those involved in defining a macroscopic bulk ferroelectric polarization, which were solved through the introduction of the modern theory of polarization [20][21][22] . A proposed solution in the case of the toroidization was described in detail in Ref. 4. In this section we extend the description to the case of the monopole and address the following questions: 1. How should the monopole density -the monopolization -of a bulk periodic solid be formally defined? 2. What are the consequences of the periodic boundary conditions within a bulk crystalline solid? For simplicity we develop the formalism for the case of the monopolization coming from the local moment contribution. First we note that, as we shall see later, the formalism requires that each local moment, m α , is equal to an integer number of Bohr magnetons. Since we consider only the spin part of the magnetic moment (the orbital part does not contribute to the monopole), a magnetic moment that is an integer number of Bohr magnetons corresponds to the moment of an integer number of electrons. In general, however, an integer number is not obtained from integrating the magnetization density over a sphere around an atomic site in a solid; in fact this number is not uniquely defined as it depends on the choice of integration radius. Rather, the spin moment of the corresponding spin-polarized Wannier function should be used; since a Wannier function in an insulating system contains an integer number of electrons its spin is always an integer number of Bohr magnetons. We then define the local moment monopolization, A lm = a lm /V , where V is the volume of the system with local moment monopole a lm . Then, for a large finite system containing N identical unit cells each of volume Ω: A lm = 1 3NΩ ∑ α r α .m α (17) = 1 3NΩ ∑ n,i (r i + R n ).m i .(18) Here, r i are the positions of the magnetic moments m i relative to the same (arbitrary) point within each unit cell, R n is a lattice vector with index n, and we have used the fact that the orientation of the magnetic moments is the same in each unit cell. The summation over i indicates the summation over all moments within a unit cell, and that over n indicates the summation over all unit cells. Expanding the scalar product, we obtain: A lm = 1 3Ω ∑ i r i .m i + 1 3NΩ ∑ n R n . ∑ i m i = 1 3Ω ∑ i r i .m i ,(19) using the fact that the sum over all lattice vectors contains both R n and −R n , so that ∑ n R n = 0. Thus, the local moment monopole of a system of N unit cells is just N times the monopole evaluated for one unit cell, and the corresponding monopolizations are identical. In an infinite periodic solid, we have a freedom in choosing the basis corresponding to the primitive unit cell of the crystal. In particular, we can translate any spin of the basis by a lattice vector R n without changing the overall periodic arrangement. However, such a translation of a spin by R n leads to a change in the local moment monopolization as follows: ∆A lm ni = 1 3Ω R n ·m i µ B ,(20) wherem i is a unit vector oriented in the direction of magnetic moment m i . The freedom in choosing the basis corresponding to the primitive unit cell thus leads to a multivaluedness of the monopolization with respect to certain "increments" (defined by Eq. (20)) for each magnetic sub-lattice i and lattice vector R n . This multivaluedness of the monopolization is reminiscent of the modern theory of electric polarization, [21][22][23] where the polarization changes by eR n /Ω when an elementary charge e is translated by a lattice vector R n . The resulting multivaluedness has led to the concept of the "polarization lattice" corresponding to a bulk periodic solid, 23 with eR n /Ω called the "polarization quantum" if R n is one of the three primitive lattice vectors. An even closer analogy is provided by the toroidization, which is multivalued with values spaced by the toroidization increment 1 2Ω R n × m, corresponding to translation of an elementary magnetic moment, m by a lattice vector 4 . Eq. (20) suggests the existence of an analogous "monopolization lattice", with monopolization increments 1 3Ω µ B R n ·m i , where R n is any primitive lattice vector andm i are the unit vectors indicating the orientations of the magnetic moments. Note that the monopolization, and hence the monopolization increments are scalar quantities. As a result the corresponding monopolization lattice can become rather dense, particularly in cases where the three lattice vectors are unequal but close in size, and the spin moments are noncollinear and canted away from the lattice vector directions. We illustrate the behavior and implications of the monopolization lattice next with a simple model one-dimensional example. C. A one-dimensional example a. The periodic non-monopolar state. To illustrate some consequences of the multivaluedness of the monopolization in periodic systems described in the previous section, we now consider the example of a one-dimensional antiferromagnetic chain of equally spaced magnetic moments as shown in Fig. 4a. The moments, with magnitude m = µ B , are spaced a distance d apart from each other along the x axis, and are alternating in orientation along ±x. Thus, the unit cell length is 2d and there are two oppositely oriented magnetic moments in each unit cell. Since this configuration does not possess a macroscopic magnetic dipole moment, the corresponding monopole moment is origin independent. The arrangement of magnetic moments in Fig. 4a is spaceinversion symmetric with respect to each moment site and thus cannot exhibit a macroscopic monopole moment. The local moment monopole of the single unit cell highlighted 4: Calculation of the monopolization for two different onedimensional antiferromagnetic periodic arrangements of magnetic moments. Our choice of unit cell is indicated by the shaded area in each case. a) shows a non-monopolar state, which is space-inversion symmetric with respect to each moment site. b) is a monopolar state. 2d m -m x a) b) d (d-λ) x m -m λ FIG. in Fig. 4a, calculated using Eq. (13), however, is identical to that calculated for the finite moment configuration in Fig. 3c, i.e. a lm = 1 3 dm, and the corresponding monopolization, A lm = a lm /Ω = 1 3 dm 2d = 1 3 m 2 (since the "volume" Ω of the one-dimensional unit cell is just its length, 2d). Since the moments of magnitude µ B are oriented exactly parallel to the x axis, the elementary monopolization increment in this case is ∆A lm = ± 1 3 µ B , which means that the monopolization of the unit cell is exactly equal to one half of the monopolization increment, and the allowed monopolization values for the periodic arrangement are A n = ( 1 2 + n) 1 3 µ B , where n can be any integer number. We see that in our example the allowed local moment monopolization values form a one-dimensional lattice of values, centrosymmetric around the origin. This is analogous to the cases of the electric polarization and the toroidization, where the polarization and toroidization lattices are invariant under all symmetry transformations of the underlying crystal structure. In particular, the polarization and toroidization lattices corresponding to centrosymmetric crystal structures are inversion symmetric, which is achieved in lattices that include either the zero or the half quantum/increment. We see that the same holds true for the local moment monopolization of our one-dimensional example, and that a centrosymmetric set of monopolization values can be understood as representing a non-monopolar state of the corresponding system. We also note that the formalism is only consistent for the case of local magnetic moments corresponding to integer numbers of Bohr magnetons, which in turn correspond to the spin contribution from integer numbers of electrons. In the case of the electric polarization, it is now widely recognized that only differences in the polarization lattices between different configurations, such as between a centrosymmetric non-polar reference structure and a ferroelectric polar crystal, are in fact measurable quantities. Since these differences are the same for each point of the polarization lattice they are well-defined quantities. Likewise in the case of the toroidization, only differences in toroidization lattices between for example different arrangements of magnetic moments or different ionic positions are measurable 4 toroidization and electric polarization, only differences in local moment monopolization, corresponding to two different bulk configurations, are measurable quantities and correspond to physical observables such as the difference in monopolization between a ferromonopolar state and its non-monopolar paraphase. Such quantities can be obtained by monitoring the change in monopolization on one arbitrarily chosen branch within the allowed set of values, when transforming the system from the initial to the final state along a well-defined path. b. Monopolar state and changes in monopolization. In order to obtain a nontrivial macroscopic monopolization the system has to break both space and time inversion symmetry. In the case of the one-dimensional antiferromagnetic chain this can be achieved by "moment pairing", i.e. if the distances between neighboring magnetic moments alternate as shown in Fig. 4b. Here the magnetic moments of magnitude m = µ B are spaced alternately a distance of (1 − λ )d and (1 + λ )d apart from each other along the x axis (−1 < λ < 1). The nonmonopolar example above corresponds to λ = 0. Since the unit cell size is the same as in the non-monopolar case, the elementary monopolization increment is again ∆A lm = ± 1 3 µ B . The monopolization of the unit cell indicated in Fig. 4b is A lm = 1 3 (− λ d + 1) µ B 2 , so that the allowed values of A lm for the full periodic arrangement are: A lm = 1 2 + n 1 3 (− λ d + 1)µ B .(21) Fig . 5 shows the allowed monopolization values as a function of the displacement λ of the moments from their positions in the centrosymmetric, non-monopolar state. The change in monopolization between two configurations with λ = λ 1 and λ = λ 2 for a certain branch n is given by: A lm n (λ 2 ) − A lm n (λ 1 ) = 1 3 λ 2 − λ 1 d µ B 2 ,(22) i.e. it is independent of the branch index n. In particular, if the non-centrosymmetric distortion is inverted (λ 2 = λ 0 , λ 1 = −λ 0 ), the change in monopolization is 2A lm s = 1 3 λ 0 µ B d so that A lm s = 1 3 λ 0 µ B 2d can be interpreted as the spontaneous monopolization, again in analogy to the case of the electric polarization, where the spontaneous polarization is given by the branch-independent change in polarization compared to a centrosymmetric reference structure. Another possible way to alter the monopolization is by changing the orientation of the magnetic moments instead of changing their positions. In particular, we expect that a full 180 • rotation of all magnetic moments, which is equivalent to the operation of time reversal, should invert the macroscopic "spontaneous monopolization", and should therefore lead to the same change 2A lm s as discussed above. If we allow the magnetic moments to rotate out of the x direction, while preserving the antiparallel alignment of the two basis moments, the monopolization is given by A lm n (λ , α) = 1 2 + n 1 3 (− λ d + 1)µ B cos α ,(23) where α is the angle between the magnetic moments and the x direction. Note here a difference from the case of the toroidization -since the monopolization is a scalar, rotation of the magnetic moments away from perfect alignment reduces the absolute magnitude of the monopolization. In contrast, in the toroidal case a rotation could reduce the toroidization along one axis while simultaneously increasing it along another. Interestingly, in this example, the magnetic moment rotation which reduces the monopolization induces a toroidization, effectively converting the monopolar response into a toroidal one through the moment reorientation. The change in monopolization for a full 180 • rotation of the moments is thus: A lm n (λ 0 , 180 • ) − A lm n (λ 0 , 0 • ) = 2 1 2 + n 1 3 (− λ 0 d + 1)µ B ,(24) and apparently depends on the branch index n. However, if one calculates the same change in monopolization for the nonmonopolar state with d = 0, one obtains: λ . Here, we use the terminology "proper" and "improper" in analogy to the case of the proper and improper piezoelectric response, 24 where a similar branch dependence is caused by volume changes of the unit cell, and the improper piezoelectric response has to be subtracted appropriately. Fig. 6 shows the initial and final states for the two cases where either the atoms carrying the moments are displaced, or the magnetic moment directions are inverted. The two final states are equivalent except for a translation of all moments by half a unit cell along y, which, due to Neumann's principle, is irrelevant for the macroscopic properties. The spontaneous monopolization of the upper state in Fig. 6 is therefore the same as for the lower state in the Figure. A lm n (0, 180 • ) − A lm n (0 • ) = −2 1 2 + n 1 3 µ B .(25) IV. MONOPOLIZATIONS IN REAL MATERIALS -THE LI TRANSITION-METAL PHOSPHATES We now turn to a real materials example, and choose the family of lithium transition-metal phosphates, LiMPO 4 , M = Mn, Fe, Co, Ni, as our model system. All of the LiMPO 4 compounds crystallize in the olivine structure with the orthorhombic space group Pnma and the crystallographic point group D 2h . [25][26][27][28][29] The lattice parameters and atomic coordinates, obtained from first-principles calculations in this work and Refs. 30 and 31, are given in Table I. The transition metal cations occupy the sites with Wyckoff positions 4c; these are surrounded by strongly distorted oxygen octahedra and have local C s = {e, i 2y } symmetry. All compounds have a transition to an antiferromagnetic state at some tens of kelvin. The resulting magnetic order breaks the inversion symmetry in all cases and hence allows for the linear magnetoelectric effect. Across the series, however, three distinct antiferromagnetic orderings emerge 28,[32][33][34][35] , summarized in Table II. These different antiferromagnetic orderings lead in turn to different magnetic symmetries and different allowed monopolar contributions. For simplicity we neglect small cantings of the magnetic moments away from the easy axis that are reported or known for many of the compounds. We also list the localmoment spin magnetic moment for each transition metal ion. A. Symmetry analysis In Table III we show the character table of the D 2h symmetry group and indicate which irreducible representations are adopted by each possible collinear ordering of the transition metal magnetic moments, m, along the cartesian axes, as well as the symmetries of the possible monopolar a, toroidal t and quadrupolar q orderings on the transition metal sites. In LiMnPO 4 the easy axis is the a axis, and the magnetic moments adopt a C-type antiferromagnetic ordering with order parameter m 1 − m 2 − m 3 + m 4 34 , this combination belongs to the A u irreducible representation of the the D 2h symmetry group. (This ordering allows for a simultaneous A-type antiferromagnetic canting along the c axis which is negligible in our DFT calculations and we neglect here. Note that a weak ferromagentic canting has also been reported, which is not compatible with the Pnma symmetry analysis 36 ; this we also neglect.) We see from the line corresponding to the A u irreducible representation in Table III that the ordering of local M-site monopole moments all with the same sign also has A u symmetry, therefore LiMnPO 4 is ferromonopolar and supports a macroscopic monopolization. Conversely there is no net toroidal moment, with only an anti-ferrotorodial ordering along the b direction allowed on the Mn sites. This is consistent with the experimental observation that the magnetoelectric response has only diagonal components 37 . We note also that the z 2 and x 2 − y 2 quadrupolar components have the same symmetry as the monopole; these quadrupolar contributions are responsible for the inequality between the magnitudes of the diagonal elements of the magnetoelectric tensor. LiCoPO 4 has been of particular recent interest because the observation of ferrotoroidic domains using nonlinear optical techniques has been reported. 3 Both LiCoPO 4 and LiFePO 4 also adopt a C-type antiferromagnetic ordering, but in contrast to LiMnPO 4 , both have their easy axis primarily along the b axis 33,38 . This corresponds to the B 1u irreducible representation which we see from Table III disallows both a macroscopic monopolization and any local monopolar contribution on the transition metal sites. This symmetry allows, however, a toroidal moment parallel to the c axis. As a result the magnetoelectric responses of both compounds are entirely off-diagonal 37,39 , although α xy is not exactly equal to -α yx (which would be the case for a purely toroidal response) because a ferroquadrupolar q xy component is allowed with the same symmetry as t z . (We note that recently it was found that the magnetic moments in LiCoPO 4 and LiFePO 4 are rotated slightly away from the b direction 32,40 . Such a symmetry lowering is not compatible with the Pnma space group and requires an additional structural distortion that has not yet been identified. We do not treat these further symmetry lowerings here.) Finally we turn to the case of LiNiPO 4 , which again has Ctype AFM ordering, but this time with easy axis along the c direction 35 , so that the Ni sublattice has magnetic point group mm ′ m and transforms according to the B 2u representation. (This symmetry also allows a small A-type AFM canting of the magnetic moments along the a direction which has been reported 35 and which we neglect here). While this sym-point group 4c (M) 4a (Li) 8d (O 3 ) D 2h e c 2z c 2y c 2x i i 2z i 2y i 2x a, q z 2 /x 2 −y 2 t x , q yz t y , q zx t z , q xy m x m y m z a a A g 1 1 1 1 1 1 1 1 0 1g 1 1 -1 -1 1 1 -1 -1 2g 1 -1 1 -1 1 -1 1 -1 0 3g 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1u 1 1 -1 -1 -1 -1 1 1 0 2u 1 -1 1 -1 -1 1 -1 3u 1 -1 -1 1 -1 1 1 -1 0 metry does not allow a net macroscopic monopolization, local monopoles are allowed on the Ni ions and must order with an antimonopolar arrangement. A macroscopic toroidal moment is again allowed, this time along the b direction, consistent with the corresponding off-diagonal magnetoelectric effect 30,35,41 . + − −+ 0 + + −− 0 + − +− 0 0 + + + + − − −− B+ + −− 0 + − −+ 0 + − +− 0 + + ++ 0 + + − − − − ++ B+ + −− 0 + − −+ 0 + + ++ 0 0 + − + − − + −+ B1 + − −+ 0 + + −− 0 + + ++ 0 + − +− 0 + − − + − + +− A u+ + ++ 0 + − +− 0 + − −+ 0 + + −− + + ++ + + + + + + ++ B+ − +− 0 + + ++ 0 + − −+ 0 + + −− + + − − + + −− B1 + − +− 0 + + ++ 0 + + −− 0 + − −+ + − +− + − + − + − +− B+ + ++ 0 + − +− 0 + + −− 0 + − −+ + − − + + − −+A u Mn C x , A z   α xx α yy α zz   (0, 0, 0) ∅ B 1u Co, Fe C y   α xy α yx   (0, 0, T z ) 0 B 2u Ni C z , A x   α xz α zx   (0, T y , 0) 0 In this series, therefore, we find one example -LiMnPO 4 -of a material with a net monopolization, in which the local monopole moments on the transition metal sites are aligned in a ferromonopolar arrangement. We also find an example -LiNiPO 4 -which has no macroscopic monopolization, but has a finite-q antimonopolar ordering on the transition metal sites. In the remaining two compounds -LiCoPO 4 and LiFePO 4 -the macroscopic monopolization and the local monopoles on the transition metal sites are both zero by symmetry. We summarize our symmetry analysis in Table IV. While it is at first sight tempting to describe LiCoPO 4 and LiFePO 4 as non-monopolar, this is not strictly correct, as we discuss next. First, we note that in the LiMPO 4 family, the P atom and the O 1 and O 2 atoms also occupy 4c sites, and so follow the same symmetry transformations as the transition metal ions. This means that for LiMnPO 4 and LiNiPO 4 local monopoles are allowed on these atoms. Of the remaining sites, the 4a of Li have only ı as a symmetry operation, and the 8d sites of the O 3 have no site symmetry. In Table III we also list the symmetries and possible monopole orderings of the 4a and 8d sites. We find that for the A 1u irreducible representation of LiMnPO 4 , the monopoles on Li and O 3 have the same ferromonopolar ordering as the Mn sites. Likewise, for LiNiPO 4 , in which the Ni sites have antiferromonopolar ordering, an antiferromonopolar ordering of the Li and O 3 monopoles is also found. Most notably, for LiFePO 4 and LiCoPO 4 , which have non-monopolar transition metal 4b sites, antiferromagnetically ordered monopoles are allowed on the 4a and 8d sites. In the next section we use first-principles density functional theory to calculate the magnitudes of these various contributions. B. Density functional calculations of atomic site monopoles and macroscopic monopolizations Our calculations were done using the local spin density approximation with an additional Hubbard U correction on the transition metal sites (the LSDA+U method). We took values of U=5eV and J=0.75eV for all systems; these values correctly reproduce the experimentally reported magnetic orderings and anisotropies. For structural optimizations we used the Vienna ab initio simulation package (VASP) 42 with a plane-wave basis set and projector augmented wave 43 potentials. Our energy cutoff and k-point grid were 500 eV and 2 × 2 × 4 respectively. We used default VASP PAW potentials with the following electrons in the valence: Li (1s, 2s), O (2s, 2p), P (3s, 3p), Co (3d, 4s), Mn, Fe and Ni (3p, 3d, 4s). Structural relaxations were performed in the absence of spin-orbit coupling. For the monopole calculations we used the structures obtained form the VASP code, then used the linearized augmented plane wave (LAPW) method as implemented in the ELK code 44 with spin-orbit coupling included to calculate the charge and spin density. We used a basis set of l max(apw) = 10, a 9 × 5 × 5 k-point sampling of the Brillouin zone and took the product of the muffin tin radius and the maximum reciprocal lattice vector to be 7.5. To calculate the atomic site monopoles (a as ) we decomposed the the density matrix into tensor moments as described in Section III 16 and evaluated the d − p matrix elements for the transition metal atoms and the p − s matrix elements for the Li, P and O atoms. In Table V we report our calculated local atomic site monopoles a as , for the series of transition metal phosphates, as well as the local moment contribution, a lm . Note that the orbital component makes no contribution by symmetry to the atomic site monopoles, and its magnitude is negligible in the local moment monopole of the ferromonopolar LiMnPO 4 because of the half-filled Mn 2+ d shell. We also report the total macroscopic monopolizations, normalized to the unit volume, A. The first thing to note is that, in the ferromonopolar case of LiMnPO 4 , the local moment monopole is as expected considerably larger -by around three orders of magnitudethan the atomic site monopoles. The value of the local moment monopole in one four-formula unit unit cell is 2.09 µ BÅ , whereas the local atomic site monopoles are all around 10 −3 µ BÅ . Even when summed over all the atomic sites, the contribution from the atomic site monopoles is still only 8.52 × 10 −3 µ BÅ ; it is so small in part because of cancellations between site monopoles of different sign. The macroscopic monopolization, A, which is the total monopole per unit volume, then derives almost entirely from the local moment contribution. We obtain a value of A = 6.95 × 10 −3 µ B /Å 2 modulo the monopolization increment of 11.54 × 10 −3 µ B /Å 2 µ B /Å 2 . Note that, since we treat the magnetic moments as collinear along a lattice vector there is just one monopolization increment. For the other compounds a net monopolization is forbidden by symmetry, and so the local moment monopole and the total monopolization are both formally zero. We find, however, non-zero values for those atomic site monopoles that are allowed by symmetry, always with the appropriate symmetryallowed antiferromagnetic ordering. Particularly interestingly, we find that when atomic site monopoles are symmetry allowed on the P and O atoms, they are comparable to or larger than the values on the transition metals. The relative sizes of the atomic site monopoles can be understood from inspection of the magnetization density: In Fig. 8 we show the isosurface of our calculated magnetization density at 0.00125 µ B /Å 3 for LiNiPO 4 , with blue and red surfaces indicating positive and negative density, as well as a slice through the magnetization density coinciding with the Ni site positions. The small deviation from a perfectly spherical distribution around the Ni atom is indicative of the monopolar and other non-dipolar multipolar contributions. It is clear that the magnetization density around the oxygen atoms, while smaller in magnitude, is more non-spherical than that around Ni. In particular, the magnetization density changes sign at the O 3 sites, indicating a highly non-spherical magnetization density which is consistent with their having the largest atomic site monopoles, The atomic site monopole on Li, although non-zero by symmetry for every case, is always small, consistent with the highly ionic nature of the Li + ion; since the charge density around the Li ions is close to zero, the magnetization density is too (Fig. 8). Finally we note that the atomic site monopole on Ni in LiNiPO 4 is one order of magnitude smaller than that on Mn in LiMnPO 4 , even though its local magnetic dipole moments is only ∼2.5 times smaller. Our initial computer experiments suggest that this is partly a result of the different magnetic anisotropy in the two cases, as a calculation with the Ni moments constrained to have the same orientation as those of Mn in LiMnPO 4 yields increased atomic site monopoles. A detailed study of the factors that determine the magnitudes of atomic site monopoles will be the subject of future work. V. MULTIFERROIC FREE ENERGY WITH MONOPOLE CONTRIBUTIONS As stated above, from a macroscopic symmetry point of view, the symmetries which allow for a macroscopic monopolization are identical with that allowing for a diagonal component of the linear magnetoelectric effect tensor. In this section, we develop the relationship between these two quantities by analyzing the following free energy expression: U = 1 2ε P 2 − P · E + 1 2χ M 2 − M · H + 1 2 β A 2 + 1 4 γA 4 + cAP · M ,(26) where ε and χ are the electric and magnetic susceptibilities, β and γ are temperature-dependent coefficients, and c determines the strength of the magnetoelectric coupling. This is the simplest possible free energy expression that can simultaneously describe (i) a phase transition from a para-monopolar (A = 0) into a ferromonopolar phase (A = 0), (ii) the coupling of the electric polarization P and the magnetization M to the electric field E and the magnetic field H, respectively, and (iii) a coupling between the electric polarization, the magnetization, and the monopolization. Note that only the magnetization and the polarization couple to H and E, the monopolization in general does not couple to any homogeneous external fields, in agreement with the fundamental definitions discussed in Sec. II. The trilinear form of the coupling term in Eq. (26) is the lowest possible order that is compatible with the overall space and time reversal symmetries. Since our purpose here is to discuss the new features arising from this trilinear coupling, we leave for future work the analysis of gradient terms in the free energy that would be required to describe for example variations in monopolization, magnetization or polarization at domain walls. The equilibrium values for P and M can be obtained by minimizing Eq. (26). This leads to: P = ε(E − cAM )(27) and M = χ(H − cAP ) .(28) If one inserts Eq. (28) into Eq. (27) one obtains (to leading order in A): P = ε(E − χcAH) .(29) The last term in Eq. (29) is a symmetric linear magnetoelectric effect proportional to the monopolization. Thus, the presence of the trilinear coupling term between monopolization, magnetization, and polarization in Eq. (26) gives rise to a diagonal magnetoelectric effect P = αH in the ferromonopolar phase, with α ii = α j j = α kk = ε χcA .(30) (Note that an off-diagonal magnetoelectric effect is obtained from a trilinear coupling between toroidization, magnetization and polarization, as discussed in Ref. 4). Conversely, the presence of a monopolar contribution can be inferred from the existence of a diagonal linear magnetoelectric response, the magnitude of which is determined by the product of the dielectric susceptibility, magnetic permeability, monopolization and the strength of the coupling between A, P and M . If the linear magnetoelectric response is diagonal and isotropic, then there can be no quadrupolar contributions and the response arises entirely from monopolar contributions. We see also from Eqn. 29 that in the case of antiferromonopolar ordering, a homogeneous magnetic field will induce a finite-q polarization. Such a relationship could be used in the case of q = π/a, to provide a more fundamental definition of an antiferroelectric in simultaneously antiferromonopolar systems, than the current unsatisfactory working definition based on the observation of double-loop hysteresis. Finally we mention that an additional interesting consequence of the relationship between the monopolization and the diagonal magnetoelectric effect is the induction of monopoles by electric charge. This has been discussed previously in the context of axion electrodynamics 45 , and is currently being revisited in the context of topological insulators 46 . VI. SUMMARY, CONCLUSIONS, AND OUTLOOK In summary we have presented a theoretical analysis of magnetoelectric monopoles in bulk periodic solids. We introduced the term "monopolization" to describe the monopole moment per unit volume, and considered two contributions, one arising from the local variation in magnetization density around the atom and the second from the distribution of localized magnetic dipole moments throughout the unit cell. We found that the latter dominates the total monopolization in transition metal compounds with ferromonopolar ordering. We showed that, for ferromonopolar materials, periodic boundary conditions lead to a multivaluedness of the monopolization, suggesting that only differences in monopolization are well-defined observable macroscopic quantities. We found also that care must be taken in evaluating such monopolization differences: For example in the example of the distorted one-dimensional antiferromagnetic chain discussed in Sec.III C, the change in monopolization due to a structural distortion can be calculated straightforwardly, whereas in the case of a magnetic moment reversal one has to subtract the improper monopolization change that is caused by the corresponding change in the monopolization increment. Quantitative measurements of monopolizations are challenging. The atomic site monopolization can in principle be detected using resonant x-ray scattering, although the experimental constraints are rather rigorous and a suitable material for such an experiment has not yet been identified. In particular, for most space group symmetries the sites that allow an atomic site monopole also allow an atomic site quadrupolar component, and disentangling the two contributions is not straightforward 18 . This problem can be circumvented by selecting materials with an isotropic diagonal magnetoelectric response 47 , however few such materials have been identified to date. Even more problematic is the question of how to measure the macroscopic local moment monopolization. According to the fundamental definition of the monopole moment, this is in principle possible by measuring the effect on a sample of a diverging magnetic field, however such a field is not accessible. It is possible that earlier observations of a quadrupolar magnetic field around a spherical sample of the prototypical diagonal magnetoelectric Cr 2 O 3 48,49 also incorporate a monopolar contribution; the theory underlying these measurements will be revisited in future work 50 . It has also been recently proposed that signatures of monopolar behavior will manifest in the transport properties of diagonal magnetoelectrics 51 An open question, both for ferrotoroidic and ferromonopolar materials is whether the toroidal moment or monopole moment respectively can be a primary order parameter, or is always secondary to an antiferromagnetic or structural ordering. Currently no case has been identified even theoretically in which the monopolization is non-zero while there is no magnetic ordering, although it is possible that some "hiddenorder parameter" materials that are of current interest might prove to fall into this class 8 . The fact that the monopole order parameter is a scalar might be helpful in distinguishing responses that arise from the antiferromagnetism from those of the monopole, in cases where the antiferromagnetic order parameter is a vector. Within the class of secondary ferromonopolar materials, it is also an open question whether there is a fundamental difference between the case in which the primary order parameter is the AFM ordering, and that where it is a structural phase transition from a centrosymmetric antiferromagnet (which does not allow monopolization) to a non-centrosymmetric monopolar state. Finally, we mention that it has been argued that ferrotoroidicity is a key concept for fitting all forms of ferroic order in a simple fundamental scheme based on the different transformation properties of the corresponding order parameters with respect to time and space inversion (see Refs. 1-3, in particular Fig. 2 in Ref. 3). It is clear that from a symme-try point of view, that the monopolization could play a similar role, since a ferromonopolar material also breaks both space-inversion and time-reversal symmetry. As a result the nonlinear optical techniques used in Ref. 3 to identify ferrotoroidic ordering are sensitive also to the monopolar symmetry breaking, and could provide indirect evidence for the presence of monopolization. In addition, the four fundamental forms of ferroic order, with order parameters transforming according to the four different representations of the "parity group" generated by the two operations of time and space reversal 52 could be chosen to be ferroelasticity, ferroelectricity, ferromagnetism, and ferromonopolicity (rather than ferrotoroidicity). Whether the scalar nature of the monopole, compared with the vector nature of the toroidal moment, makes this choice more or less appropriate is an open question. FIG. 1 : 1Diagonal ((a) and (b)) and off-diagonal ((c) and (d)) magnetoelectric responses of monopolar and toroidal spin arrangements. From Ref. 11. FIG. 2 :FIG. 3 : 23Representation of (left to right) positive and negative monopoles, the z component of the toroidal moment and the z 2 component of the quadrupole moment. Representative arrangements of local magnetic moments (shown by arrows) that have monopolar contributions. The arrangements in a) and b) are purely monopolar, and have equal and opposite monopoles. c) consists of the sum of a monopolar contribution (of size half that of b) and a quadrupolar contribution; the decomposition is shown in the lower panel. FIG. 5 : 5. In the next section we show that, in analogy with the cases of the Allowed values of the monopolization for the antiferromagnetic chain ofFig. 4as a function of displacement λ from the nontoroidal case (λ = 0). The cartoons at the bottom indicate the corresponding positions of the magnetic moments within the unit cell. FIG. 6 : 6Obviously, in this case the corresponding change in macroscopic monopolization should be zero, since both the initial and final states (and all intermediate states) correspond to a non-monopolar configuration and thus A lm s = 0. If one subtracts the improper change in A lm , Eq. (25), from the change in monopolization calculated in Eq. (24), one obtains the proper change in monopolization 2A lm s = 1 3 λ 0 µ B d , which is identical to that obtained by inverting the non-centrosymmetric distortion Effect on the magnetic moment configuration of Fig. 4b (middle panel) of a reversal of all magnetic moments (lower panel) and of a reversal of the non-centrosymmetric distortion d (upper panel). Note that the upper and lower final states are identical, with the moments in the upper and lower panels translated by half a unit cell relative to each other. FIG. 7 : 7Structure of the lithium transition metal phosphates. The 1 -4 labeling of the transition metal atoms is consistent with their labeling inTables II and III.: a, b and c lattice parameters and Wyckoff positions for the lithium transition metal phosphates, LiMPO 4 , M = Mn, Fe, Co and Ni. All values were obtained by structural relaxation using density functional theory within the LSDA+U method as described in the FIG. 8 : 8Calculated magnetization density isosurface for LiNiPO 4 . The blue and red surfaces correspond to positive or negative density, respectively. TABLE II : IIExperimentally determined magnetic orderings for the lithium transition metal phosphates. TABLE III : IIICharacter table of the D 2h point group, and symmetry analyses for the 4c site (dipole, monopole, toroidal and quadrupole ordering) and the 4a and 8d sites (monopole ordering only) of the Pnma space group. label M magnetic order ME Toroidal Monopole TABLE IV : IVSummary of the measured primary (C-type) magnetic or- dering, and the resulting additional magnetic orderings, toroidal and monopole moments, and components of the magnetolectric tensor (ME), obtained by symmetry analysis for the LiMPO 4 series. TABLE V : VCalculated atomic site monopoles, local moment monopoles, and macroscopic monopolizations for the Li transition metal phosphates. H. Schmid, in Introduction to Complex Mediums for Optics and Electromagnetics, edited by W. S. Weiglhoger and A. 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[ "Nucleon Momentum Decomposition in QCD", "Nucleon Momentum Decomposition in QCD" ]	[ "Y M Cho \nSchool of Electrical and Computer Engineering\nUlsan National Institute of Science and Technology\n689-798UlsanKorea\n\nSchool of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea\n", "Mo-Lin Ge \nTheoretical Physics Division\nChern Institute of Mathematics Nankai University\n300071TianjinChina\n", "D G Pak \nBogoliubov Laboratory of Theoretical Physics\nJoint Institute for Nuclear Research Dubna\nMoscow region141980Russia\n", "Pengming Zhang \nInstitute of Modern Physics\nChinese Academy of Science\n730000LanzhouChina\n" ]	[ "School of Electrical and Computer Engineering\nUlsan National Institute of Science and Technology\n689-798UlsanKorea", "School of Physics and Astronomy\nSeoul National University\n151-747SeoulKorea", "Theoretical Physics Division\nChern Institute of Mathematics Nankai University\n300071TianjinChina", "Bogoliubov Laboratory of Theoretical Physics\nJoint Institute for Nuclear Research Dubna\nMoscow region141980Russia", "Institute of Modern Physics\nChinese Academy of Science\n730000LanzhouChina" ]	[]	Based on the gauge invariant quark canonical momentum we construct two theoretically possible decompositions of nucleon momentum to those of quarks and gluons. We predict that either 6% or 21% of nucleon momentum is carried by gluons, depending on what type of gluons are in nucleons. We clarify the existing confusions on this problem and discuss the physical implications of our result on the proton spin crisis problem. 14.20.Dh, Keywords: canonical and kinematic quark momentum, gauge invariant canonical quark momentum, gauge invariant decomposition of nucleon momentum, gluon momentum in nucleon	null	[ "https://arxiv.org/pdf/1102.1130v2.pdf" ]	119,277,755	1102.1130	dac29685c249b98c03e11bdccbb2042b3d61ece3	Nucleon Momentum Decomposition in QCD 29 Apr 2011 Y M Cho School of Electrical and Computer Engineering Ulsan National Institute of Science and Technology 689-798UlsanKorea School of Physics and Astronomy Seoul National University 151-747SeoulKorea Mo-Lin Ge Theoretical Physics Division Chern Institute of Mathematics Nankai University 300071TianjinChina D G Pak Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research Dubna Moscow region141980Russia Pengming Zhang Institute of Modern Physics Chinese Academy of Science 730000LanzhouChina Nucleon Momentum Decomposition in QCD 29 Apr 2011arXiv:1102.1130v2 [nucl-th] Based on the gauge invariant quark canonical momentum we construct two theoretically possible decompositions of nucleon momentum to those of quarks and gluons. We predict that either 6% or 21% of nucleon momentum is carried by gluons, depending on what type of gluons are in nucleons. We clarify the existing confusions on this problem and discuss the physical implications of our result on the proton spin crisis problem. 14.20.Dh, Keywords: canonical and kinematic quark momentum, gauge invariant canonical quark momentum, gauge invariant decomposition of nucleon momentum, gluon momentum in nucleon An important problem in nuclear physics is to find out how much fraction of nucleon momentum is carried by the gluons. It has generally been believed that gluons carry about a half of nucleon momentum [1]. But recently there has been a new assertion that only about one-fifth of the nucleon momentum should be carried by the gluons [2]. This has created considerable controversy and confusion in the literature [3,4]. To resolve this problem one has to know how to decompose the momentum of nucleon to those of its constituents. At the first glance this problem seems to be simple enough. But in gauge theories it is very difficult to obtain a gauge invariant decomposition of momentum or spin to those of the constituents. In fact it has long been suggested that this is impossible in gauge theories. The reason is that the gauge interaction makes a gauge invariant decomposition of the total momentum (and spin) to those of the constituents very difficult [5,6]. The purpose of this Letter is to clarify the confusion on this problem and provide new nucleon momentum decompositions to predict the fraction of gluon momentum in nucleons. To understand the problem, consider the canonical decomposition of the momentum of positronium in QED P (qed) µ = P e µ + P γ µ = i ψ γ 0 ∂ µ ψd 3 x + [(∂ µ A α )F α0 + 1 4 δ 0 µ F 2 αβ ]d 3 x.(1) This does provide a decomposition of momentum to those * Electronic address: [email protected] of the constituents, but is not gauge invariant. We can change it to the popular gauge invariant decomposition adding a surface term [2,4] P (qed) µ =P e µ +P γ µ = i ψ γ 0 D µ ψd 3 x + (F µα F α0 + 1 4 δ 0 µ F 2 αβ )d 3 x.(2) But this also may not be the desired decomposition because the first term involves both electron and photon. The problem stems from the fact that charged particles have two momentums, the "canonical" one given by −i∂ µ and the "kinematic" one given by −iD µ , but neither is suitable for the momentum decomposition of composite particles [5,6]. This is because the canonical momentum is not gauge invariant, and the kinematic momentum contains the gauge field. Moreover, there are actually two different issues in this problem. The first is theoretical: How to make a gauge invariant decomposition of the total momentum. The second is experimental: How to make a measurable (and gauge invariant) decomposition of the total momentum. This is more subtle because here we must figure out what are the measurable momentums of the constituents. To obtain a gauge invariant decomposition of the positronium momentum, we first decompose the photon field to the vacuum part Ω µ and the physical part X µ [7], A µ = Ω µ + X µ , Ω µ = ∂ µ θ, ∂ µ X µ = 0.(3) Notice that this decomposition is gauge independent. Moreover, the gauge transformation affects only the pure gauge part, so that the physical part remains gauge invariant. In particular, the physical part here becomes a Lorentz covariant four-vector, so that we can make X 0 = 0 choosing a proper Lorentz frame (as far as X µ is space-like). Now, adding a surface term to (2), we can easily change it to [4] P µ = i ψ γ 0D µ ψd 3 x + [(∂ µ X α )F α0 + 1 4 δ 0 µ F 2 αβ ]d 3 x,D µ = ∂ µ − ieΩ µ .(4) Unlike (1) or (2), each term now is gauge invariant and at the same time involves only one constituent. So, theoretically it does describe a gauge invariant decomposition of the total momentum. It has generally been believed that the kinematic momentum is what experiments measure, because this is gauge invariant. This has made (2) a popular decomposition. But here we have shown that the canonical momentum can also be expressed by a covariant derivative. So there are actually two gauge invariant momentums that we can construct and thus can possibly measure. If so, which momentum is measurable and why is that so? Classically it appears that the conserved momentum of a charged particle moving in an electromagnetic field is the sum of the kinematic momentum of the particle and the electromagnetic momentum (the Poynting vector) [6]. This favors the kinematic momentum. But quantum mechanically the kinematic momentum operators do not satisfy the canonical momentum commutation relation, since they do not commute. Moreover, the canonical momentumD µ defined by the vacuum potential is gauge invariant and at the same time satisfies the canonical commutation relation. This strongly implies that (4) is the correct momentum decomposition. In QCD the conserved momentum obtained by Noether's theorem is given by P (qcd) µ = i ψ γ 0 ∂ µ ψd 3 x + [(∂ µ A α ) · F α0 + 1 4 δ 0 µ F 2 αβ ]d 3 x.(5) Adding a surface term we can change it to the popular gauge invariant decomposition P (qcd) µ = i ψ γ 0 D µ ψd 3 x + [ F µα · F α0 + 1 4 δ 0 µ F 2 αβ ]d 3 x.(6) But again the first term contains quarks and gluons. To cure this defect we first have to find out the gauge covariant canonical momentum operator which does not include gluons. To construct such momentum operator we must have a gauge independent decomposition of the non-Abelian gauge potential to the vacuum partΩ µ and the physical part Z µ similar to (3). Consider SU(2) QCD for simplicity, and letn i (i = 1, 2, 3) be a gauge covariant right-handed orthonormal basis in SU(2) space. Then imposing the vacuum condition to the potential ∀ i D µni = (∂ µ + g A µ ×)n i = 0. (n 2 i = 1)(7) we obtain the most general vacuum [10], A µ →Ω µ = 1 2g ǫ ijk (n i · ∂ µnj )n k .(8) Next, we make the decomposition A µ =Ω µ + Z µ ,(9) and find that under the gauge transformation we have δΩ µ = 1 gD µ α, δ Z µ = − α × Z µ , D µ = ∂ µ + gΩ µ × .(10) where α is the (infinitesimal) gauge parameter. Notice that Z µ becomes a Lorentz covariant (as well as gauge covariant) four-vector. Finally, we impose the transversality condition to Z µ to make it physical D µ Z µ = 0.(11) Notice that this is not a gauge condition, because it applies to any gauge. Obviously this is the generalization of (3) to QCD which provides the desired decomposition. Now, we can modify (5) to P (qcd) µ = i ψ γ 0D µ ψd 3 x + [(D µ Z α ) · F α0 + 1 4 δ 0 µ F 2 αβ ]d 3 x,(12) adding a surface term − (∂ αΩµ · F α0 )d 3 x.(13) Clearly this provides a gauge invariant decomposition of total momentum to those of the quarks and gluons. But this may not be the desired decomposition that we are looking for. The reason is that QCD has two types of gluons, so that we have to figure out which become the constituents of nucleons [7][8][9]. To see this one has to understand that QCD allows the Abelian decomposition which separates the gluons to the colorless binding gluons and the colored valence gluons gauge independently. Because of this we have two types of QCD, the restricted QCD (RCD) made of the binding gluons and the standard QCD made of all gluons. Moreover, QCD can be viewed as RCD which has the valence gluons as the colored source [8,9]. So the valence gluons (just like the quarks) become another colored source which has to be confined. This means that RCD plays the crucual role in confinement, which is known as the Abelian dominance in QCD [11,12]. This interpretation has been confirmed numerically in a series of lattice QCD calculations [13,14]. Most importantly, the quark model of hadrons tells that nucleons (in particular the low-lying nucleons) are made of three quarks which are colored, not quarks and colored gluons [15]. The colored gluons make up glueballs. This implies that valence gluons have no place in nucleons. If so, only quarks and binding gluons should contribute to the nucleon momentum. But so far this important point has completely been ignored. To exclude the contribution of the valence gluons in (6) we have to separate the valence gluons from the binding gluons. This can be done by the Abelian decomposition [8,9]. Letn =n 3 be the unit isotriplet which selects the color direction at each space-time point, and make the Abelian projection imposing the condition, D µn = (∂ µ + g A µ ×)n = 0. (n 2 = 1)(14) This selects the restricted potential A µ = A µn − 1 gn × ∂ µn . (A µ =n · A µ )(15) With this we have the Abelian decomposition [8,9], A µ =Â µ + X µ , (n · X µ = 0)(16) where X µ is the valence potential. Notice thatÂ µ by itself forms a connection space, so that under the (infinitesimal) gauge transformation we have [8,9] δÂ µ = 1 gD µ α, δ X µ = − α × X µ ,(17)whereD µ = ∂ µ + gÂ µ ×. What is important about this decomposition is that it is gauge independent. Oncen is chosen, the decomposition follows automatically, independent of the choice of a gauge. SinceÂ µ still contains the pure gauge degrees, we need to decompose it to the vacuum and physical parts, A µ =Ω µ + B µ ,D µ B µ = 0, B µ = B µn , B µ = A µ − 1 gn 1 · ∂ µn2 .(18) Notice that B µ (just like X µ ) is gauge and Lorentz covariant. This is because bothÂ µ andΩ µ form a connection space which is closed under the gauge transformation. Now, it is straightforward to obtain the desired decomposition of nucleon momentum. All we have to do is to replace Z µ by B µ and F µν toF µν in (6), P (rcd) µ = i ψ γ 0D µ ψd 3 x + [(D µ B α ) ·F α0 + 1 4 δ 0 µF 2 αβ ]d 3 x,(19) where B µ is the transverse binding gluon. Notice that (12) is physically very similar to the QED expression (4). Clearly we can derive (12) from RCD. In fact RCD has the conserved momentum P (rcd) µ = i ψ γ 0 ∂ µ ψd 3 x + [(∂ µÂα ) ·F α0 + 1 4 δ 0 µF 2 αβ ]d 3 x.(20) From this we can obtain (12) adding a surface term. Now, we come back to the difficult question: What are the quark and gluon momentums in nucleon? In QED we have two gauge invariant electron momentums, the canonical −iD µ and the kinematic −iD µ . But in QCD we have three. To see this notice that we can express (12) by P (rcd) µ = i ψ γ 0D µ ψd 3 x + [F µα ·F α0 + 1 4 δ 0 µF 2 αβ )d 3 x,(21) adding a surface term. Notice that the first term represents the quark kinematic momentum, but this contains only the binding gluons. This tells that there are two gauge invariant quark kinematic momentums, −iD µ and −iD µ , on top of the gauge invariant canonical momentum −iD µ . So here we can not just say that it is the kinematic momentum that experiments measure. The above analysis tells us the followings. First, in gauge theories there is a gauge invariant decomposition of total momentum (and spin) of a composite particle to those of the constituents, one in QED and two in QCD. But these decompositions involve the canonical momentum which may or may not be measurable by experiment. Second, if the canonical momentum is not measurable, there is no gauge invariant decomposition of total momentum (and spin) to those of constituents in the strict sense. But we still have "partial" decompositions which involve the kinematic momentum, again one in QED and two in QCD. The reason why we have two competing decompositions in QCD is because we have two types of gluons. If nucleons contain only the binding gluons, (12) or (21) must be the correct one. But if they contain all gluons, we must have (6) or (19). To find which decomposition is correct, suppose only the kinematic momentum is mesurable. In this case we have in the asymptotic limit [1] P g µ = 2n g 2n g + 3n f P tot µ . Now, the difference between (21) and the popular (6) is that (21) includes only the binding gluons (n g = 2) but (6) includes all gluons (n g = 8). So (6) gives the wellknown prediction (with n f = 5 as usual) that about 51% of nucleon momentum is carried by gluons [1]. In contrast (21) tells that only about 21% of nucleon momentum must be carried by gluons. Now, suppose only the canonical momentum is measurable. In this case (22) must change, and it has been proposed that (22) be replaced by [2] P g µ = n g n g + 6n f P tot µ . This should be confirmed by an idependent calculation, but suppose this is true. Then (12) which assumes that nucleons contain all gluons predicts that gluons carry about 21% of nucleon momentum [2]. Notice the strange coincidence between this prediction and that of (21) based on (22 Exactly the same argument applies to the nucleon spin crisis problem [7,16,17]. Here again there are three (one canonical and two kinematic) quark orbital angular momentums. Moreover, assuming that only the canonical angular momentum is measurable, we have two nucleon spin decompositions depending on which gluons are in nucleons [7]. And only one of them can describe the correct nucleon spin decomposition. Independent of the details the essence of our analysis can be summarized as follows. First, there exist more than one logically acceptable gauge invariant quark and gluon momentums in QCD. Indeed quarks have three and gluons have four such momentums, as we have shown in (6), (12), (19), and (21). This is because QCD potential allows the vacuum and Abelian decompositions (9) and (16), so that quarks have one canonical and two kinematic gauge invariant momentums [8,9]. Clearly this is against the common wisdom [2-4, 16, 17]. Second, we must know which gluons are in nucleons to have a correct momentum (and spin) decomposition. So far this point has completely been ignored, because it has always been believed that all gluons are in nucleons [2-4, 16, 17]. But the Abelian decomposition tells that QCD has two types of gluons, and the quark model implies that only the binding gluons are in nucleons [8,9,15]. Certainly this is a very interesting new idea which is totally different from the standard belief, and it is important to find out which gluons are in nucleons. In this Letter we showed that we can tell which view is correct by measuring the gluon momentum in nucleons. ). They have the same prediction, but totally different physics. If nucleons have only binding gluons, however, (19) tells that only about 6% of momentum must be carried by gluons. Notice that the fraction of gluon momentum becomes less if nucleons has only binding gluons, for obvious reason. We hope that our analysis will help to settle the current controversies on nucleon momentum and spin decomposition [2-4, 16, 17]. A detailed discussion on this and related issues will be presented elsewhere[7,18]. . H Georgi, H Politzer, Phys. Rev. 9416H. Georgi and H. Politzer, Phys. Rev. D9, 416 (1974); . D Gross, F Wilczek, 980D. Gross and F. Wilczek, ibid. 980 (1974); . H Politzer, Phys. Rep. 14129H. Politzer, Phys. Rep. 14, 129 (1974). . X S Chen, X F Lu, W M Sun, F Wang, T Goldman, Phys. Rev. Lett. 10362001X. S. Chen, X. F. Lu, W. M. Sun, F. Wang, and T. Goldman, Phys. Rev. Lett. 103, 062001 (2009). . X Ji, Phys. Rev. Lett. 10439101X. Ji, Phys. Rev. Lett. 104, 039101 (2010); . X S Chen, X F Lu, W M Sun, F Wang, T Goldman, hep-ph/08124336X. S. Chen, X. F. 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[ "Tenth-order lepton g−2: Contribution from diagrams containing a sixth-order light-by-light-scattering subdiagram internally", "Tenth-order lepton g−2: Contribution from diagrams containing a sixth-order light-by-light-scattering subdiagram internally" ]	[ "Tatsumi Aoyama \nFaculty of Science\nNagoya University\n464-8602NagoyaAichiJapan\n", "Katsuyuki Asano \nDepartment of Physics\nNagoya University\n464-8602NagoyaAichiJapan\n", "Masashi Hayakawa \nDepartment of Physics\nNagoya University\n464-8602NagoyaAichiJapan\n", "Toichiro Kinoshita \nLaboratory for Elementary-Particle Physics\nCornell University\n14853IthacaNew YorkUSA\n", "Makiko Nio \nTheoretical Physics Laboratory\nNishina Center\nRIKEN\n351-0198WakoJapan\n", "Noriaki Watanabe \nDepartment of Physics\nNagoya University\n464-8602NagoyaAichiJapan\n" ]	[ "Faculty of Science\nNagoya University\n464-8602NagoyaAichiJapan", "Department of Physics\nNagoya University\n464-8602NagoyaAichiJapan", "Department of Physics\nNagoya University\n464-8602NagoyaAichiJapan", "Laboratory for Elementary-Particle Physics\nCornell University\n14853IthacaNew YorkUSA", "Theoretical Physics Laboratory\nNishina Center\nRIKEN\n351-0198WakoJapan", "Department of Physics\nNagoya University\n464-8602NagoyaAichiJapan" ]	[]	This paper reports the result of our evaluation of the tenth-order QED correction to the lepton g −2 from Feynman diagrams which have sixth-order light-by-light-scattering subdiagrams, none of whose vertices couple to the external magnetic field. The gauge-invariant set of these diagrams, called Set II(e), consists of 180 vertex diagrams. In the case of the electron g − 2 (a e ), where the light-by-light subdiagram consists of the electron loop, the contribution to a e is found to be −1.344 9 (10) (α/π) 5 . The contribution of the muon loop to a e is −0.000 465 (4) (α/π) 5 . The contribution of the tau-lepton loop is about two orders of magnitudes smaller than that of the muon loop and hence negligible. The sum of all of these contributions to a e is −1.345 (1) (α/π) 5 .We have also evaluated the contribution of Set II(e) to the muon g − 2 (a µ ). The contribution to a µ from the electron loop is 3.265 (12) (α/π) 5 , while the contribution of the tau-lepton loop is −0.038 06 (13) (α/π) 5 . The total contribution to a µ , which is the sum of these two contributions and the mass-independent part of a e , is 1.882 (13) (α/π) 5 .	10.1103/physrevd.81.053009	[ "https://arxiv.org/pdf/1001.3704v2.pdf" ]	118,680,535	1001.3704	716d84e609bab19834f9075526d138360358e59e	Tenth-order lepton g−2: Contribution from diagrams containing a sixth-order light-by-light-scattering subdiagram internally 11 Mar 2010 Tatsumi Aoyama Faculty of Science Nagoya University 464-8602NagoyaAichiJapan Katsuyuki Asano Department of Physics Nagoya University 464-8602NagoyaAichiJapan Masashi Hayakawa Department of Physics Nagoya University 464-8602NagoyaAichiJapan Toichiro Kinoshita Laboratory for Elementary-Particle Physics Cornell University 14853IthacaNew YorkUSA Makiko Nio Theoretical Physics Laboratory Nishina Center RIKEN 351-0198WakoJapan Noriaki Watanabe Department of Physics Nagoya University 464-8602NagoyaAichiJapan Tenth-order lepton g−2: Contribution from diagrams containing a sixth-order light-by-light-scattering subdiagram internally 11 Mar 2010(Dated: March 12, 2010)RIKEN-TH 182numbers: 1340Em1460Cd1460Ef1220Ds This paper reports the result of our evaluation of the tenth-order QED correction to the lepton g −2 from Feynman diagrams which have sixth-order light-by-light-scattering subdiagrams, none of whose vertices couple to the external magnetic field. The gauge-invariant set of these diagrams, called Set II(e), consists of 180 vertex diagrams. In the case of the electron g − 2 (a e ), where the light-by-light subdiagram consists of the electron loop, the contribution to a e is found to be −1.344 9 (10) (α/π) 5 . The contribution of the muon loop to a e is −0.000 465 (4) (α/π) 5 . The contribution of the tau-lepton loop is about two orders of magnitudes smaller than that of the muon loop and hence negligible. The sum of all of these contributions to a e is −1.345 (1) (α/π) 5 .We have also evaluated the contribution of Set II(e) to the muon g − 2 (a µ ). The contribution to a µ from the electron loop is 3.265 (12) (α/π) 5 , while the contribution of the tau-lepton loop is −0.038 06 (13) (α/π) 5 . The total contribution to a µ , which is the sum of these two contributions and the mass-independent part of a e , is 1.882 (13) (α/π) 5 . I. INTRODUCTION The anomalous magnetic moment g − 2 of the electron has provided the most stringent test of the validity of quantum electrodynamics, QED. The experimental value with the least uncertainty is the one obtained by the Harvard group in 2008 (a ≡ g−2 2 ) [1] a e (HV08) = 1 159 652 180.73 (28) × 10 −12 . To confront the prediction of the standard model with this measurement the hadronic contribution up to the order α 3 [2][3][4][5][6][7][8][9], the electroweak contribution up to the two-loop order [10][11][12], and the QED radiative correction up to the eighth order must be taken into account [13][14][15]. In order to match or exceed further improvement in the accuracy of the experimental value, it is necessary to evaluate the tenth-order QED radiative correction to g − 2. To meet this challenge we launched several years ago the project to compute all 12672 Feynman diagrams that contribute to the tenth-order a e [16,17]. The most difficult to evaluate is the gauge-invariant set, called Set V, which consists of 6354 diagrams that have no virtual lepton loop. To deal with this set systematically we have developed an automatic code-generating algorithm gencodeN [16,18]. We now have fortran codes for all diagrams of Set V generated by gencodeN. Numerical evaluation of these integrals is in progress at present. Meanwhile, we have also made steady progress in the evaluation of other types of tenthorder diagrams, and have published some of the results [17,19,20]. At the tenth order there appear five gauge-invariant sets of diagrams, called Set I(j), Set II(e), Set II(f), Set III(c), and Set VI(j), which contain light-by-light-scattering subdiagram(s) internally, i.e., none of whose vertices is the external vertex [16,17]. (See Fig. 1.) Feynman diagrams containing a light-by-light-scattering subdiagram internally appear for the first time in the eighth-order QED correction to the lepton g −2. The paper is organized as follows. Section II describes the strategy we have adopted for the numerical study. The renormalization is set up so that ultraviolet divergences can be subtracted away without introducing spurious infrared divergence. Section III gives the results of our numerical work which covers the contributions of all diagrams of Set II(e) to a e and a µ . Section IV is devoted to the discussion and summary. where the lower case "a" includes the factor α The charge conjugation and time reversal symmetries of QED help us to reduce the number of independent amplitudes. For instance, A67 gives the same contribution to g −2 as A47. Recall also that the diagram in which the lepton loop runs in the opposite direction gives the same contribution as the original one. In this way, we obtain a complete set of independent where the number in the brackets accounts for the symmetry factor for each diagram as well as the doubling due to two directions that a lepton loop takes. Thus far no one has succeeded in evaluating the diagrams of Set II(e) analytically. We resort to the numerical means utilizing the parametric integral formulation [16,18,22,23]. The evaluation of g−2 can be simplified significantly by focusing on the quantity associated with the self-energy diagram Gij, such as the magnetic moment amplitude M Gij , using the Ward-Takahashi identity which relates the regularized self-energy function Σ Gij (p) of the diagram Gij to the sum Λ Gij (p, q) of the contributions from the regularized vertex diagrams obtained by inserting a QED vertex into Gij in all possible ways [24]. The next step is to renormalize the integrals on the computer, which we carry out by subtractive renormalization. Since the bare amplitudes of individual diagrams have different structures of UV singularities, the numerical subtraction of UV singularities must be carried out for each diagram separately. Our aim is to construct subtraction terms that (i) share the same UV singularity as the integrand of the bare amplitude in the common Feynman parameter space, and (ii) do not introduce spurious IR singularities. The second point is not a trivial requirement. For instance, the usual on-shell secondorder vertex renormalization constant contains an IR divergence. In general the subtraction term constructed under the usual on-shell renormalization condition introduces an IR singularity that is not present in the bare amplitude. To avoid this problem we perform the renormalization in two steps. The first step is to construct the UV-finite amplitude ∆M Gij in which only the UV-divergent part of the corresponding on-shell vertex (or self-energy) term is subtracted, leaving out the UV-finite piece unsubtracted. We call this step an intermediate renormalization. The second step is to carry out the finite residual renormalization to account for the difference between the intermediate renormalization and the usual on-shell renormalization. The IR-divergent parts of the usual on-shell renormalization constants appear in the second step but cancel out when summed over the entire gauge-invariant set. The subtraction terms of ∆M Gij are constructed as follows. The UV singularities associated with the second-order vertex and self-energy subdiagrams are subtracted via Koperation, retaining the Feynman cut-off until UV divergences cancel out by renormalization [23]. The UV singularities of the light-by-light-scattering (l-l) loops are subtracted while maintaining the Pauli-Villars regularization in order to avoid dealing with divergent hence undefined quantities. The Pauli-Villars mass is sent to infinity only after renormalization is carried out. Note that not only the usual on-shell renormalization but also the intermediate renormalization are defined on the mass-shell insofar as it is IR-safe. To avoid confusion let us call the usual on-shell renormalization as the full renormalization henceforth. Let us illustrate our renormalization procedure taking A47 of Fig. 3 as an example. A47 has four UV-divergent subdiagrams which can be identified by the sets of lepton lines involved: S 1 = {1, 2, 4, 5, 6, 7, 8, 9} , S 2 = {2, 3, 4, 5, 6, 7, 8, 9} , S 3 = {4, 5, 6, 7, 8, 9} , S 4 = {8, 9} .(4) Both subdiagrams S 1 and S 2 are the eighth-order vertex subdiagrams, S 3 is the sixth-order l-l subdiagram, and S 4 is the second-order vertex subdiagram. Each UV subtraction term of ∆M Gij is associated with a Zimmermann's forest. A47 has 11 normal forests. Let C S denote the operator which extracts the full renormalization constant of the subset S from M Gij , and let K S denote the operator which extracts the UV singularity of the subset S by the intermediate renormalization defined by the K operation, respectively. Then the UV-finite quantity ∆M A47 is defined by ∆M A47 = M A47 − C S 1 M A47 − C S 2 M A47 − C S 3 M A47 − K S 4 M A47 +K S 4 C S 3 M A47 + K S 4 C S 1 M A47 + K S 4 C S 2 M A47 +C S 3 C S 1 M A47 + C S 3 C S 2 M A47 −K S 4 C S 3 C S 1 M A47 − K S 4 C S 3 C S 2 M A47 .(5) Expression of ∆M Gij for other Gij in Eq. (3) can be written down similarly. It is by definition that all subtraction terms on the right-hand side of Eq. (5) are factorizable. For instance, the operator C S 1 acting on M A47 produces the product of L S 1 and M 2 : C S 1 M A47 = L S 1 M 2 ,(6) where L S 1 is the full eighth-order vertex renormalization constant of the diagram that contains the sixth-order light-by-light-scattering subdiagram, and M 2 = a 2 = 1 2 is the secondorder lepton g−2. Of course this equation is meaningless unless it is regularized. L S 1 and M 2 can be expressed as regularized integrals in the parametric integral formulation [22] on two separate Feynman parameter spaces with constraints i : all lines∈S 1 x i = 1, k=3, d y k = 1, where y d is the Feynman parameter associated with the photon d. The K-operator, K S 4 , acts on the regularized integrand J(z) of M A47 directly and produces a function J S 4 (z) that possesses the same UV singularity associated with the subdiagram S 4 as J(z). By definition K-operation also has the factorization property. For instance, the operator K S 4 acting on the regularized M A47 produces the factorized result K S 4 M A47 = L UV 2 M 8A ,(7) where L UV 2 is the UV-divergent part of the regularized second-order on-shell vertex renormalization constant L 2 and does not include the IR-divergent part of L 2 [23]. M 8A denotes the amplitude of the magnetic moment from the eighth-order diagram A of Fig. 2. The regularization mass must be sent to infinity after K S 4 M A47 is combined with M A47 . The difference of L UV 2 and L 2 is accounted for at the stage of the residual renormalization. The subtraction term C S 3 M A47 can be written (somewhat) symbolically as C S 3 M A47 = Π(0, 0, 0, 0)M 4 ,(8) Here Π is a short-hand form of the sixth-order l-l subdiagram defined by Π κλµν (k a , k b , k c , k d )\| ka=k b =kc=k d =0 ,(9) where k a , etc., are the momenta carried by the photon line a, etc., and M 4 is obtained from M A47 by shrinking the l-l loop of S 3 to a point. The UV divergence of M A47 arising from the subdiagram S 3 is cancelled by the term C S 3 M A47 of Eq. (5), which results in full renormalization of the S 3 divergence. Actually, M A47 contains another UV divergence arising from S 4 which we subtract by the operator K S 4 . The complete removal of UV divergences arising from S 3 and S 4 is achieved by the combination M A47 − C S 3 M A47 − K S 4 M A47 + K S 4 C S 3 M A47 .(10)∆M Gij − 4 L 2 − L UV 2 + B 2 − B UV 2 A l (LL8, l ′ ) = Gij∈ Eq.(3) ∆M Gij − 4∆B 2 × A l (LL8, l ′ ) , ∆B 2 ≡ L 2 − L UV 2 + B 2 − B UV 2 = 3 4 .(11) Now, at last, we can send the regulator mass to infinity. B 2 is the full second-order wave function renormalization constant and B UV 2 is the UV-divergent part of B 2 defined by the K operation. The IR divergence of (L 2 − L UV 2 ) cancels that of (B 2 − B UV 2 ) leaving a finite term as expected. (Note that B + L = 0 while B UV 2 + L UV 2 is finite but not zero. See Ref. [24] for the exact definitions of B UV 2 and L UV 2 .) Eq. (11) shows that the residual renormalization term is proportional to the eighth-order contribution A l (LL8, l ′ ) to the anomalous magnetic moment of the lepton l from the diagrams of Fig. 2, in which loops are given by lepton l ′ . The numerical study in Ref. [25] has provided an accurate value for the mass-independent contribution A e (LL8, e) A e (LL8, e) = −0.990 72 (10) . In addition the paper [25] reports the electron-loop contribution to a µ A µ (LL8, e) = −4.432 43 (58) . We have also evaluated the muon loop contribution A e (LL8, µ) to the electron g −2, and the tau-lepton loop contribution A µ (LL8, τ ) to the muon g−2 needed for this work A e (LL8, µ) = −0.000 177 8 (12),(14) A µ (LL8, τ ) = −0.015 868 (37). The remaining task is to evaluate every ∆M Gij in various combination of the external and internal leptons. III. NUMERICAL RESULTS OF ∆M Gij fortran codes for ∆M Gij are rather complicated and not easy to obtain. In order to facilitate this problem we adapted the automating code gencodeN specifically for the Set II(e) which generates the integrands of ∆M Gij as fortran-formatted source programs. (See Refs. [16,18] for the details of automation.) Two independent sets of automating codes together with another set of manually-produced codes were constructed to confirm their validity. The integral for the diagram Gii, i.e, the one containing the second-order self-energy subdiagram, was found to exhibit worse convergence than the others. In order to alleviate this problem, we modify the integrand in the following way. For instance, in the diagram A44 in Fig. 3, which contains a second-order self-energy subdiagram, the integrand of ∆M Gij depends on the Feynman parameters z 4 , z 8 only through the combination z 48 ≡ (z 4 + z 8 ). Thus, the number of independent variables is reduced from 12 to 11. This seems to improve somewhat the convergence of iteration procedure. The results of numerical integration of ∆M Gij for the mass-independent contribution to the lepton g−2 are presented in Table I. Following Eq. (11) the last line of Table I together with the value (12) for A e (LL8, e) yields the mass-independent contribution to g −2 from The integration of ∆M Set II(e) diagrams where l = e, µ, or τ . Recall that the actual contribution to g−2, a e (II(e), e), is A e (II(e), e) times the factor α π 5 . The electron g −2 also receives the Set II(e) contribution induced by the virtual muon loop. To see its numerical significance, the computation of ∆M Gij for the muon loop is explicitly performed. The results are shown in Table II The size of this contribution is less than the numerical uncertainty of the electron-loop contribution A e (II(e), e) given in Eq. (16). Since the tau-lepton loop contribution is expected to be about two-orders of magnitudes smaller than the muon loop contribution and hence negligible, we present the sum of Eqs. (16) and (17) The main contribution of Set II(e) to the muon g −2 arises from the diagrams each of which is induced by an electron loop. We present the numerical result of ∆M Gij for the electron-loop contribution in Table III. This table shows that the sum of ∆M Gij is an order of magnitude larger than that of the mass-independent ∆M Gij in Table I Thus the electron-loop contribution A µ (II(e), e) is not much larger than the muon loop contribution A µ (II(e), µ) of Eq. (16). Since the sign of A µ (II(e), e) is opposite to that of A µ (II(e), µ), we are curious about the role that the tau-lepton contribution A µ (II(e), τ ) might play. Table IV shows the result of ∆M Gij for the Set II(e) contribution to the muon g−2 induced by the tau-lepton loop. Equation (11), together with the value (15) of A µ (LL8, τ ), gives A µ (II(e), τ ) = −0.038 06 (13) , which is two orders of magnitude smaller than A µ (II(e), e) or A µ (II(e), µ). Summing up Eqs. (19), (16) and (20), the Set II(e) contribution to the muon g−2 is found to be a µ (II(e)) = 1.882 (13) α π 5 . (21) IV. DISCUSSION AND SUMMARY In this paper, we computed the contribution to the lepton g − 2 from the tenth-order QED diagrams of Set II(e) that contain the sixth-order light-by-light-scattering subdiagram internally. The use of Ward-Takahashi identity, as well as the symmetries of QED, reduces the computation of 180 Feynman diagrams to that of 20 integrals ∆M Gij . The intermediate renormalization to define ∆M Gij is chosen so that the UV divergence associated with the second-order self-energy or vertex subdiagram is subtracted away by K-operation. Meanwhile the UV divergence arising from the l-l loop is subtracted by full renormalization. This leads to simplification of the final result as is seen in Eq. (11). The Set II(e) contribution to the electron g − 2 is obtained by evaluating the electron and muon virtual effects. The result is given in Eq. (18). The size is of the typical order of magnitude for the tenth-order. The numerical computation was carried out as accurately as possible with the available computer resources. The contribution to the muon g − 2 is obtained by evaluating the virtual effects of all leptons. The result is given in Eq. (21). The contribution of the electron loop to the muon g−2 is not much larger than the muon loop contribution. We found that A µ (II(e), e) in Eq. (19) involves partial cancellation between the sum of ∆M Gij over all Gij in Set II(e) and the residual renormalization term in Eq. (11). In spite of these problems we were able to obtain the result for a µ (II(e)) with the uncertainty less than 1% using the high performance computer system, RICC. Figure 2 .FIG. 1 : 21-order self-energy diagrams with the fourth-order internal light-by-light-scattering subdiagrams. Vertex diagrams relevant to lepton g−2 can be obtained by inserting a single external QED vertex into one of open lepton lines labeled 1, 2, 3 of individual diagrams of The diagrams of Set I(j) are those involving two fourth-order light-by-light-scattering subdiagrams, both internal, which have been evaluated and published recently [19]. The diagrams of other subsets are obtained by adding O(α) correction to those of Fig. Representative diagrams of gauge-invariant sets which contain an internal light-by-lightscattering subdiagram. Set II(f) consists of diagrams obtained by inserting a second-order vacuum-polarization loop into one of internal photon lines of the diagrams of Fig. 2 in all possible ways. They have been evaluated by a simple modification of the fortran codes developed previously for the eighth-order work. The result was published in Ref. [17]. The diagrams of Set III(c) are obtained by attaching a virtual photon line to the open lepton path of the individual diagrams of Fig. 2 in all possible ways. Evaluation of this set is in progress. The diagrams of Set VI(j) contain two light-by-light-scattering subdiagrams, one of which is internal, while the other is external. The numerical result of the Set VI(j) was published in Ref. [17]. The diagrams of Set II(e) are obtained by attaching both ends of a virtual photon line to the lepton loop of the individual diagrams of Fig. 2 in all possible ways, forming the sixth-order internal light-by-light-scattering subdiagram. This paper reports the result of our work on Set II(e), which consists of 180 Feynman vertex diagrams. II. COMPUTATIONAL PROCEDUREThis section describes the strategy for computing the diagrams of Set II(e). We denote the contribution of Set II(e) to the magnetic moment of the lepton l induced by the virtual loop of lepton l ′ as a l (II(e), l ′ ) = A l (II(e), l ′ ) α π 5 , FIG. 2 : 2Eighth-order self-energy Feynman diagrams LL8 that contain the fourth-order light-bylight-scattering subdiagram internally. In Ref.[27], A, B and C are called "LLJ", "LLL" and "LLK", respectively. The vertex diagrams are obtained by inserting a single QED vertex into one of the lepton lines 1, 2 or 3 of G (G = A, B, C). π 5 5while the upper case "A" does not. Recall that a l (II(e), l ′ = l) and A l (II(e), l ′ = l) are independent of l and called mass-independent contributions. A l (II(e), l ′ = l) depends only on the mass ratio m l ′ /m l . We use the values found in Ref. [21] for lepton masses. As explained in Sec. I, the diagrams of Set II(e) are obtained by attaching an internal photon line to the lepton loop of the individual diagrams of Fig. 2 in all possible ways. Let us denote the diagrams of Set II(e) as Gij by specifying (i) the base eighth-order diagram G, where G is one of A, B, or C of Fig. 2, and (ii) a pair (i, j) of lepton lines of the closed loop (4 ≤ i ≤ j ≤ 7) to which an additional internal photon line is attached. For instance, the insertion of two QED vertices into the middle of the lines 4 and 7 of the diagram A ofFig. 2and the introduction of a virtual photon line which connects these vertices produces the tenth-order diagram called A47. Representative diagrams of Set II(e) are shown inFigure 3. FIG. 3 : 3Representative diagrams of Set II(e). A47 and A44 involve a second-order vertex subdiagram and a second-order self-energy subdiagram, respectively. A manipulation similar to that of Sec. III D of Ref.[16] expresses L S 1 M 2 as an integral over the single Feynman parameter space with j∈A47 z j = 1. With this form of integrand of L S 1 M 2 the pointwise subtraction of the overall UV divergence of M A47 residing in the subdiagram S 1 can be achieved. [Actually, C S 1 M A47 still has divergences from the subdiagrams S 3 and S 4 which must be subtracted by other terms of Eq.(5).] When the contributions of all diagrams of Set II(e) listed in Eq. (3) are put together, Π(0, 0, 0, 0) from all diagrams cancel out and we obtain a simple result A l (II(e), l ′ ) = Gij∈ Eq.(3) Gij is carried out with the help of the adaptive-iterative Monte-Carlo integration routine VEGAS [26] on the massively parallel computer, RIKEN Integrated Cluster of Clusters (RICC). The number of sampling points for each iteration is 10 8 for all diagrams with the second-order self-energy subdiagrams and 2 × 10 8 for all others. A l (II(e), l) = −1.344 86 (99), . Putting together the last line of this table and the value (14) of A e (LL8, µ) we obtain the muon loop contribution to the electron g−2 from Set II(e) diagrams A e (II(e), µ) = −0.000 465 (4). as our current best value for the contribution to the electron g−2 from Set II(e) diagrams a e (II(e)) = −1.345 TABLE I : INumerical results for mass-independent ∆M Gij from diagrams Gij in Set II(e). Full symmetry factors are included for the individual values. The number of sampling points for eachiteration is 10 8 for all diagrams with the second-order self-energy subdiagrams and 2 × 10 8 for all others. The first, second, third and fourth columns show the name of the diagram, the value of integrand and its uncertainty, the χ 2 value of VEGAS integration, and the number of iterations of VEGAS integration. If χ 2 is very close to 1, then the numerical integration by VEGAS is reliable. Gij ∆M Gij (uncertainty) χ 2 iterations A44 5.397 41 (33) 1.04 2420 A55 2.796 88 (23) 1.09 1250 A77 2.422 84 (20) 1.08 1210 A47 0.100 26 (15) 1.02 1120 A45 −1.559 22 (16) 0.98 1150 A46 1.180 76 (12) 1.10 1140 A57 1.653 245 (93) 1.09 800 B44 −4.440 95 (34) 1.06 2660 B55 −4.741 06 (33) 0.99 2500 B47 1.725 96 (17) 1.05 990 B45 2.521 96 (17) 1.07 1070 B46 −0.349 57 (11) 1.04 1040 B57 −2.254 206 (97) 1.10 790 C44 −5.054 64 (34) 1.03 2570 C55 −2.398 68 (24) 1.09 1670 C77 −2.431 20 (22) 1.06 1460 C47 1.574 62 (16) 1.06 990 C45 1.821 84 (17) 1.00 1200 C46 −1.881 49 (12) 1.03 1280 C57 −0.401 777 (91) 1.03 710 sum −4.317 02 (94) A. electron g−2 TABLE II : IINumerical results for ∆M Gij of electron g−2 from diagrams Gij in set II(e) in each of which muon induces light-by-light scattering. Full symmetry factors are included in the individual values. The number of sampling points for each iteration is 10 8 for all diagrams with the second-order self-energy subdiagrams and 2 × 10 8 for all others. Gij ∆M Gij (uncertainty) ×10 3 χ 2 iterations A44 11.424 89 (49) 1.19 1360 A55 5.822 60 (36) 1.29 640 A77 6.167 18 (33) 1.23 640 A47 3.674 56 (28) 1.06 320 A45 0.715 82 (275) 1.14 320 A46 3.296 85 (25) 1.18 400 A57 3.686 56 (20) 1.55 240 B44 −6.068 57 (45) 1.02 1280 B55 −7.947 942 (44) 0.98 1280 B47 −1.252 29 (29) 1.16 240 B45 0.775 54 (25) 1.04 320 B46 −0.389 63 (26) 0.91 240 B57 −3.590 13 (20) 1.54 240 C44 −7.361 42 (46) 1.14 1120 C55 −3.125 44 (31) 1.17 720 C77 −2.927 29 (34) 1.33 480 C47 0.213 13 (28) 0.95 240 C45 −0.306 03 (30) 0.91 240 C46 −3.423 30 (26) 1.29 400 C57 −0.383 70 (17) 1.25 240 sum −0.998 6 (14) TABLE III : IIINumerical results for ∆M Gij of muon g −2 from diagrams Gij in set II(e) in each of which electron induces light-by-light scattering. Full symmetry factors included in the individual values. The number of sampling points for each iteration is 10 8 for all diagrams with the secondorder self-energy subdiagrams and 2 × 10 8 for all others.Gij ∆M Gij (uncertainty) χ 2 iterations A44 21.914 7 (35) 1.02 2180 A55 10.747 4 (21) 0.99 1540 A77 10.438 3 (20) 0.99 1300 A47 6.166 8 (27) 1.01 1370 A45 7.843 7 (28) 1.04 1370 A46 −13.679 5 (14) 0.98 1050 A57 −14.181 8 (11) 1.09 810 B44 −25.919 5 (40) 1.02 3050 B55 −25.634 7 (37) 1.03 3050 B47 39.794 5 (28) 1.08 1760 B45 41.011 0 (28) 1.11 1810 B46 −16.936 4 (15) 0.99 1130 B57 −22.029 9 (11) 1.23 840 C44 −41.123 9 (42) 0.99 3010 C55 −20.500 5 (29) 1.02 1770 C77 −20.929 8 (29) 1.05 1610 C47 46.365 7 (30) 1.05 1810 C45 47.994 0 (30) 1.04 1890 C46 −22.236 3 (15) 1.06 1130 C57 −19.136 5 (12) 1.08 810 sum −10.032 (12) TABLE IV : IVNumerical results for ∆M Gij of muon g − 2 from diagrams Gij in set II(e) in each of which tau-lepton induces light-by-light scattering. Full symmetry factors are included in the individual values. The number of sampling points for each iteration is 10 8 for all diagrams with the second-order self-energy subdiagrams and 2 × 10 8 for all others.Gij ∆M Gij (uncertainty) χ 2 iterations A44 0.422 177 (18) 1.09 1600 A55 0.215 057 (18) 1.19 480 A77 0.219 086 (19) 1.18 320 A47 0.112 235 (14) 0.96 240 A45 −0.008 963 (17) 1.07 160 A46 0.119 699 (16) 1.37 160 A57 0.138 883 (10) 1.63 160 B44 −0.239 377 (18) 1.02 1400 B55 −0.298 375 (18) 0.99 1280 B47 −0.017 671 (16) 1.11 160 B45 0.053 945 (16) 1.16 160 B46 −0.014 195 (13) 0.96 160 B57 −0.140 250 5 (80) 1.43 240 C44 −0.288 614 (18) 1.04 1280 C55 −0.125 456 (18) 1.14 400 C77 −0.120 731 (18) 1.18 320 C47 0.019 880 (15) 1.13 160 C45 0.011 773 (16) 0.89 160 C46 −0.129 852 (14) 1.09 240 C57 −0.014 913 9 (86) 1.17 160 sum −0.085 66 (7) AcknowledgmentsThis work is supported in part by JSPS Grant-in-Aid of Scientific Research (C) The work of T. K. was supported by the U. S. National Science Foundation under Grant No. NSF-PHY-0757868. 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[ "Understanding and Mitigating the Effect of Outliers in Fair Ranking", "Understanding and Mitigating the Effect of Outliers in Fair Ranking" ]	[ "Fatemeh Sarvi [email protected] ", "Maria Heuss [email protected] ", "Mohammad Aliannejadi [email protected] ", "Sebastian Schelter [email protected] ", "Maarten De Rijke [email protected] ", "Fatemeh Sarvi ", "Maria Heuss ", "Mohammad Aliannejadi ", "Sebastian Schelter ", "Maarten De ", "Rijke ", "\nAIRLab\nUniversity of Amsterdam Amsterdam\nThe Netherlands\n", "\nUniversity of Amsterdam\nAmsterdamThe Netherlands\n", "\nUniversity of Amsterdam\nAmsterdamThe Netherlands\n", "\nUniversity of Amsterdam\nAmsterdamThe Netherlands\n", "\nUniversity of Amsterdam\nAmsterdamThe Netherlands\n" ]	[ "AIRLab\nUniversity of Amsterdam Amsterdam\nThe Netherlands", "University of Amsterdam\nAmsterdamThe Netherlands", "University of Amsterdam\nAmsterdamThe Netherlands", "University of Amsterdam\nAmsterdamThe Netherlands", "University of Amsterdam\nAmsterdamThe Netherlands" ]	[ "Proceedings of the Fifteenth ACM International Conference on Web Search and Data Mining (WSDM '22)" ]	Traditional ranking systems are expected to sort items in the order of their relevance and thereby maximize their utility. In fair ranking, utility is complemented with fairness as an optimization goal. Recent work on fair ranking focuses on developing algorithms to optimize for fairness, given position-based exposure. In contrast, we identify the potential of outliers in a ranking to influence exposure and thereby negatively impact fairness. An outlier in a list of items can alter the examination probabilities, which can lead to different distributions of attention, compared to position-based exposure. We formalize outlierness in a ranking, show that outliers are present in realistic datasets, and present the results of an eye-tracking study, showing that users' scanning order and the exposure of items are influenced by the presence of outliers. We then introduce OMIT, a method for fair ranking in the presence of outliers. Given an outlier detection method, OMIT improves fair allocation of exposure by suppressing outliers in the top-ranking. Using an academic search dataset, we show that outlierness optimization leads to a fairer policy that displays fewer outliers in the top-, while maintaining a reasonable trade-off between fairness and utility.CCS CONCEPTS• Information systems → Learning to rank.	10.1145/3488560.3498441	[ "https://arxiv.org/pdf/2112.11251v2.pdf" ]	245,353,305	2112.11251	4b7639c564c665f177eaf7be975a992829731fdb	Understanding and Mitigating the Effect of Outliers in Fair Ranking 2022. February 21-25, 2022. February 21-25, 2022 Fatemeh Sarvi [email protected] Maria Heuss [email protected] Mohammad Aliannejadi [email protected] Sebastian Schelter [email protected] Maarten De Rijke [email protected] Fatemeh Sarvi Maria Heuss Mohammad Aliannejadi Sebastian Schelter Maarten De Rijke AIRLab University of Amsterdam Amsterdam The Netherlands University of Amsterdam AmsterdamThe Netherlands University of Amsterdam AmsterdamThe Netherlands University of Amsterdam AmsterdamThe Netherlands University of Amsterdam AmsterdamThe Netherlands Understanding and Mitigating the Effect of Outliers in Fair Ranking Proceedings of the Fifteenth ACM International Conference on Web Search and Data Mining (WSDM '22) the Fifteenth ACM International Conference on Web Search and Data Mining (WSDM '22)Tempe, AZ, USA WSDM '22; Tempe, AZ, USA2022. February 21-25, 2022. February 21-25, 202210.1145/3488560.3498441ACM Reference Format:. ACM, New York, NY, USA, 9 pages. https://doi.org/Fair ranking; Outliers Traditional ranking systems are expected to sort items in the order of their relevance and thereby maximize their utility. In fair ranking, utility is complemented with fairness as an optimization goal. Recent work on fair ranking focuses on developing algorithms to optimize for fairness, given position-based exposure. In contrast, we identify the potential of outliers in a ranking to influence exposure and thereby negatively impact fairness. An outlier in a list of items can alter the examination probabilities, which can lead to different distributions of attention, compared to position-based exposure. We formalize outlierness in a ranking, show that outliers are present in realistic datasets, and present the results of an eye-tracking study, showing that users' scanning order and the exposure of items are influenced by the presence of outliers. We then introduce OMIT, a method for fair ranking in the presence of outliers. Given an outlier detection method, OMIT improves fair allocation of exposure by suppressing outliers in the top-ranking. Using an academic search dataset, we show that outlierness optimization leads to a fairer policy that displays fewer outliers in the top-, while maintaining a reasonable trade-off between fairness and utility.CCS CONCEPTS• Information systems → Learning to rank. INTRODUCTION The primary goal of a ranker as used in a search engine or recommender system is to optimize the list in order to satisfy user needs by sorting items in their order of relevance to the query [22]. Recently there has been a growing concern about the unfairness towards minority groups caused by this simplistic assumption [5,6]. Several studies have proposed approaches to achieve fair ranking policies. The goal is to ensure that the protected groups receive a predefined share of visibility. Exposure-based methods [6,24,26,[31][32][33] quantify the expected amount of attention each individual or group of items receives from users, where attention is typically related to the item's position and based on the observation that users are more likely to click on items presented at higher positions [5,19]. But item position is not the only factor that affects exposure [13]. Inter-item dependencies also play a key role [8]. E.g., consider a user who is trying to buy a phone. When searching on an e-commerce platform, if an item in the list is on promotion and has a "Best Seller" badge, this can be distracting so that it gets more attention from the user, irrespective of its position in the list; the item would stand out even more if it is the only one with this feature. We hypothesize that inter-item dependencies have an effect on examination probability and exposure of items. We focus on the case of having an outlier in the ranking and aim to understand and address its effect on user behavior. We hypothesize that exposure received by an item is influenced by the existence of an outlier in the list, and assume that this effect should be considered while allocating exposure to protected groups in a fair ranking approach. We define outliers as items that observably deviate from the rest. The properties and method with which we identify outliers in a set of items are dependent on the task. The properties are observable item features that can be presentational in nature or correspond to ranking features used to produce the ranked list. E.g., in the e-commerce search example, if only one item in a result page has a "Best Seller" tag, it is an outlier based on this presentational feature. To begin, we perform an exploratory analysis on the TREC Fair Ranking dataset. We observe that a large number of outliers exist in the rankings, where we use multiple outlier detection techniques to identify outliers based on the papers' citations as they can make an item more attractive and catchy than others. Next, we perform an eye tracking study, where we measure the attention that each item in a ranked list gets through eye tracking, so as to show that users can actually perceive outliers in rankings. We find that attention is more focused on outlier items. The scanning order and exposure received by each item may be influenced by the existence of outliers. Unlike other types of bias studied in search and recommendation [19,27,28,37,40], our eye-tracking study reveals that outlierness comes from inter-item dependencies. It affects the examination propensities for items around the outlier in a way that is less dependent on the position and based on relationships between items presented together. However, it is translated to bolded keyword matches in the title and abstracts, which can be calculated for each item separately, independent of its neighbors. Attractiveness does not alter the examination model based on position bias, and only results in relatively more clicks on items when they are presented with more bolded matched keywords [40]. While allocating fair exposure to protected items or groups in a fair ranking solution, we should account for the effect of outliers. We propose an approach to account for the existence of outliers in rankings without damaging the utility or fairness of the ranking by mitigating outlierness, called OMIT. OMIT jointly optimizes (1) user utility, (2) item fairness, and (3) fewer outliers in top-positions as a convex optimization problem that can be solved optimally through linear programming. Via its solution, we derive a stochastic ranking policy using Birkhoff-von Neumann (BvN) decomposition [7]. OMIT reduces the number of outliers at top-10 positions on the TREC 2020 dataset by 80.66%, while maintaining the NDCG@10, compared to a state-of-the-art ranking baseline. The main contributions of this work are as follows: (1) we introduce, study and formalize the problem of outlierness in ranking and its effects on exposure distribution and fairness; (2) we run an extensive eye-tracking user study in two search domains to support our hypothesis about the existence of an effect of outliers on items' exposure; (3) inspired by our analysis, we propose OMIT, an efficient approach that mitigates the outlierness effect on fairness; (4) we compare OMIT to competitive baselines on two TREC datasets in terms of fairness, outlierness, and utility; OMIT is able to remove outliers while balancing utility and fairness; and (5) we make the data from our eye-tracking study plus the code that implements our baselines and OMIT publicly available. BACKGROUND Exposure and utility. Consider a single query , that we will often leave out for notational simplicity, for which we want to rank a set of documents D = { 1 , 2 , . . . , }. Suppose we are given document utilities u ∈ R , where is a proxy for the relevance of document for . Let v ∈ R , be the attention vector, where denotes how much attention a document gets at position , and which is decreasing with the position. This vector encodes the assumed position bias, e.g., = 1/log(1 + ). We require a probabilistic ranking in the form of a doubly stochastic document-position matrix P ∈ [0, 1] × where denotes the probability of putting document at position . Such a matrix can be decomposed into a convex combination of permutation matrices, which allows us to sample a concrete ranking [32]. The exposure of a document under ranking P denotes the expected attention that this document will get. Using the position based attention vector , this can be modeled as a function of the ranking and position bias: Exposure( \|P) = =1 . The expected utility of a ranking P is the sum of the documents' utilities weighted by the exposure given to them by P: (P) = ∑︁ =1 Exposure( \|P) = ∑︁ =1 ∑︁ =1 = u Pv. (1) Without fairness considerations, a utility-maximizing ranking can be found by sorting the documents in descending order of utility. Group fairness. Suppose now that the documents D can be partitioned into two disjoint sets D and D , where documents in D belong to a historically disadvantaged group (e.g., publications from not so well established institutes), and those in D belong to the privileged group (e.g., publications from well-established institutes). We want to ensure a certain notion of fairness in the ranking. We want to avoid disparate treatment of the different groups. We use the disparate treatment ratio [32], which measures how unequal the exposure given to the disadvantaged group (in relation to the corresponding utility of the disadvantaged group) is compared to the corresponding ratio of the privileged group, as: dTR(D , D \|P) = ∈D Exposure( \|P)/ ∈D ∈D Exposure( \|P)/ ∈D .(2) Note that dTR is 1 if the groups are treated fairly and smaller than 1 if the ranking is unfair towards the disadvantaged group. We often encounter disparate treatment when only optimizing for the expected utility of a ranking [5,6,32]. We can find a utility maximizing ranking P that avoids disparate treatment by solving the following optimization problem [32]: P = arg max P u ⊤ Pv (expected utility) such that 1 ⊤ P = 1 ⊤ (row stochasticity) P1 = 1 (column stochasticity) 0 ≤ P ≤ 1 (valid probabilities) f ⊤ Pv = 0, (dTR constraint)(3) where 1 denotes a vector and f is the vector constructed to encode the avoidance of disparate treatment, with = 1 ∈D \|D \| ∈D − 1 ∈D \|D \| ∈D ,(4) where 1 ∈D = 1 if document is in the disadvantaged group and 0 otherwise (and analogously for 1 ∈D ) [32]. Degrees of outlierness. Outliers are items that deviate from the rest of the data [18]. They can be interesting observations or suspicious anomalies. Either way, they are considered noise that can affect the statistical analysis. We describe three outlier detection methods that we will use later in the paper. Let = { 1 , . . . , \| ∈ R} be a set of values for which we want to identify outliers. Median Absolute Deviation (MAD). Although it is common practice to use Z-scores to identify possible outliers, this can be misleading (particularly for small sample sizes) due to the fact that the maximum Z-score is at most ( − 1)/ √ . Iglewicz and Hoaglin [17] recommend using the modified Z-score: = 0.6745( −˜)/MAD, where MAD is the median absolute deviation and˜is the median of . These authors recommend that modified Z-scores with an absolute value of greater than 3.5 be labeled as potential outliers. Median K-Nearest Neighbor (MedKNN). This model [3] uses the K-Nearest Neighbor algorithm to define a distance-based outlier detection method. For each point we have a value ( ) as the weight calculated from the nearest neighbors; outliers are the points with the largest values of . We use Median K-Nearest Neighbor, which computes ( ) as the median distance of to its neighbors. To find the nearest neighbors, the method linearizes the search space and uses Hilbert space-filling curve to search efficiently; the method scales linearly in the dimensionality and the size of the data. Copula-Based Outlier Detection (COPOD). COPOD [21] is a novel outlier detection method based on estimating tail probabilities using empirical copula. COPOD uses empirical cumulative distribution functions (ECDFs) to compute tail probabilities. These tail probabilities estimate the probability of observing a point at least as extreme as for each data point. If is an outlier, the probability of observing a point as extreme as should be small. It means that this point has a rare occurrence. This method is deterministic and efficient, and scalable to high dimensionality data. OUTLIERS IN RANKING A common assumption is that exposure is a function of position [6,24,26,32,36]. We argue that this assumption holds only if ranked items can be deemed similar, meaning that no item is perceived as an outlier. Below we introduce outliers in the context of ranking. We then determine that outliers are present in rankings in realistic datasets. We also report on an eye-tracking study that shows that the presence of outliers in ranked lists impacts user behavior. A definition of outliers in ranking. For a ranked list, we define outliers as items that stand out among the window of items that the user can see at once, drawing the user's attention. Outlier items often have (visible) characteristics that distinguish them from their neighbors. E.g., consider Fig. 1, which shows a result page for the query "smart phone". The result page view consists of 6 products, each presented with characteristics such as title, image, and price. The item at position three deviates from the other items in terms of several visual characteristics; it has more details, some promotive tags, and bold keywords. Other features, such as more positive reviews, or a higher price, may also influence the user's perceived relevancy. In this example, the third item can be considered as an outlier according to such visual characteristics. Formally, we define outliers in ranking as follows. Consider a ranked list of items in D = { 1 , 2 , . . . , }, that has been produced in response to a query. We call an observable characteristic of an item in a ranked list an observable item feature. These features can be purely presentational in nature, like the bold keywords in Fig. 1, or correspond to ranking features used by the search engine to produce the ranked list, e.g., the average user rating. Note that detecting an item as an outlier in a ranking depends on the context in which we see the item. Throughout the paper, we consider the full ranked list of items as the context in which we detect outliers. In Section 6 we study varying sizes for the context. Moreover, it is possible to use multiple observable features to detect the outliers. For example, we can consider image size as 1 and price as 2 , and then use any combination of these two feature values to present item . Below, when we refer to an item being an outlier in a given ranked list, we assume that it is clear from the context what outlier detection method M and observable item feature are being used. Do outliers in ranking exist? To determine whether outliers are present in rankings in datasets, we take a retrieval test collection, compute feature values for one of the (potentially observable) rankings features appropriate for the collection, and determine whether there are outliers among the top-20 documents returned for the test queries (using ListNet as the ranker, see Section 5). For the experiments in Section 6 we use the academic search dataset provided by the TREC Fair Ranking track. 1 It contains information about papers and authors extracted from the Semantic Scholar Open Corpus. 2 It comes with queries and relevance judgments; see Table 1 for some descriptive statistics. We used the number of citations as observable feature for this dataset as they can make an item more attractive than others (when reported). In the remainder of the section, we report the analysis only on the TREC 2020 data, as we observed similar trends in both datasets. Fig. 2 shows the mean, maximum, and minimum of papers' citation counts for all search sessions in the data. There is a high variance between mean and maximum citations, which implies that the data is outlier-prone based on this feature. We plot outlier counts for each position in the top-20. Fig. 3 depicts the number of relevant and non-relevant outliers detected by the outlier detection methods introduced in Section 2 at different positions. The stacked bars show that in spite of the attractiveness of outliers, most of these items are irrelevant, judging by the click data. In total, 88.5%, 89.8%, and 90.1% of the outliers are irrelevant when MedKNN, COPOD, and MAD are employed as the outlier detection method, respectively. The average percentage of irrelevant documents in the top 20 positions in the dataset is 83.3%. This suggests that by pushing these items to lower positions we can improve the degree of outlierness without jeopardizing utility. Do users perceive outliers in the ranking? Outliers are present in realistic datasets, but do they impact user behavior? Prior studies stress the importance of relationships between ranked items [30], but it is unknown how an outlier in a ranking affects the examination probability. To address this gap, we conduct an eye-tracking study. We ask participants to interact with search engine result pages, as they normally would, and find the items that they prefer and think are relevant. We track their eye movements via an online webcam-based eye tracking service. 3 We use two scenarios, e-commerce, and scholarly search. We focus on a list view; in both scenarios, participants are able to see all items in one page. For each scenario, we include two result pages, one without an outlier item and one with (as in Fig. 1). In the absence of outliers we expect participants to follow the position bias examination assumption [19]; in the presence of outliers, we expect that users' attention is diverted towards them. We recruit 40 university students and staff for both scenarios. In the instructions, we describe the overall goal of the research and ask participants to read the instructions carefully. We describe what webcam-based eye tracking is and that the eye-tracking service will ask them for access to their webcam. We instruct participants to first read and understand the query and then start scanning the result page as if they submitted the query themselves. For reporting, we consider four eye-tracking measures based on participants' eye fixations: (1) fixation count (the number of fixations within an area; more fixation means more visual attention); (2) time spent (shows the amount of time that participants spent on average looking at an area); (3) Time To First Fixation (TTFF; the amount of time that it takes participants on average to look at one 3 RealEye, https://realeye.io. area for the first time); and (4) revisit count (indicates how many times on average participants looked back at the area) [15]. Outliers in e-commerce. For this scenario, we mimick the Amazon Marketplace search result page. Fig. 1 depicts our example ranked list with an outlier. The third item in the list stands out from other items for different reasons, including price and sales-related tags (e.g., being on sale), as well as other information that is available for this item. Comparing Fig. 4a and 4b, we see that in the presence of an outlier, items at the top of the list receive less attention, contradicting the position bias assumption. Fig. 5(a) reports the eye-tracking measures for the e-commerce scenario. We highlight the outlier in each list with an asterisk. In the no-outlier condition, participants exhibit linear behavior in terms of scanning the items. The highest number of fixations, time spent, revisits belongs to the items on the top of the list, and it decreases as we go down the list. TTFF demonstrates a linear behavior of the first fixation time, that is, most of the participants started scanning the results from top to bottom. The ranked list with an outlier exhibits very different measurements. Attention is more focused on the outlier item. Also, we see that from the TTFF values, the average time for the first fixation is the lowest for this item, suggesting that the scanning order and exposure are influenced by the existence of the outlier. This is also evident by comparing the heatmaps in Fig. 4a and 4b. Outliers in scholarly search. In the second scenario, we study the effect of outliers on scholarly search result pages. To this end, we mimick the result page from PapersWithCode. 4 To save space, we omit the eye fixation heatmap for the result page without outliers; it shows the familiar F-shape. Fig. 4c shows the eye fixation heatmap for the result page with an outlier item; the fourth item has a different thumbnail and a large number of GitHub stars, making this item stand out in the list. Similar to the e-commerce scenario, the eye fixations are very different from the F-shape typical for the no-outlier case. Fixations and time spent are the highest for the outlier item, suggesting that it draws lots of attention, and contradicts the cascade examination assumption. However, different from the e-commerce case, we do not observe as big a difference in the TTFF values between the conditions with and without outliers, suggesting that the order of item scans is not affected as much as in the e-commerce example. MITIGATING OUTLIERNESS IN FAIR LEARNING TO RANK We now present a ranking algorithm for mitigating outlierness for fairness in ranking, called OMIT, that simultaneously accounts for item fairness and outlierness effect requirements. Based on the observations reported in the previous section we know that outliers can influence exposure and examination order, in a way that can be considered as a type of bias. We take a first step towards mitigating the outlierness phenomenon by proposing a remarkably simple, yet effective solution that removes outliers from the top positions where the distribution of exposure is most critical. Our solution aims at decreasing outlierness in the top-positions, while retaining the ranking's utility and fairness with position-based assumptions. OMIT is based on the linear programming method described in Section 2. In addition to optimizing for user utility while staying within the fairness constraints, our goal is to reduce the number of outliers in the top-of all rankings. OMIT has two steps. In the first, we search for the marginal rank probability matrix that satisfies item group fairness by solving a linear program that optimizes both for user utility and fewer outliers in the top-items with linear constraints for fairness. In the second, we derive a stochastic ranking policy from the solution of the linear programming method 4 https://paperswithcode.com using the Birkhoff-von Neumann decomposition [7]; cf. also [32]. Step 1: Computing MRP matrix. Let D = { 1 , . . . , } be a set of items that we aim to rank for a given query . Each ranking (·\| ) corresponds to some permutation matrix that re-orders the elements of D . 5 As discussed in Section 3, to determine which items are outliers, we use the domain specific observable item features ( 1 ), . . . , ( ) that potentially impact the user's perception of an item. Using these characteristics as features, one can use any outlier detection method to determine which items should be considered outliers. The majority of outlier detection methods, including the ones we use (Section 3), find outliers by calculating scores that indicate the degree of outlierness. This results in a list of outlierness values M( ( 1 )\|D), . . . , M( ( )\|D), where ≤ is the size of the outlierness context that the algorithm takes into account while detecting the outliers. We define a vector o D , that contains, for each document, the information whether it is an outlier with respect to the full ranked list: o D = { } with = M( ( )\|D), if is outlier in D 0, else.(5) We use o and o D interchangeably. Note that we are considering outliers in the context of the full list, = , i.e., the outlier detection algorithm takes the whole ranked list as input. We assume that items that are perceived as outliers in this context are likely to be perceived as outliers when seen in the smaller context of the topitems. Below, we show that this heuristic works well in practice. OMIT works by pushing outliers away from the top-. Let be the permutation matrix corresponding to a ranking . The amount of outlierness that a ranking places in the top-are equal to Outlierness D ( \| M) = ∑︁ =0 (o ) = o h,(6) where h = (1, . . . , 1, 0, . . . , 0) is a vector containing 1 for the first positions and 0 for all positions after that. Similarly, the expected outlierness, that is placed in the top-by P is given by Outlierness D (P\| M) = o Ph.(7) While optimizing for user utility we can use the expected outlierness 5 For simplicity, we interchangeably use ( ·) and ( · \| ), as well as D and D . Algorithm 1 Outlier mitigation for fair ranking (OMIT) Input: D , M, , Output: 1: Create o D as Eq. 5 using D and M 2: h ← (ℎ 1 , . . . , ℎ ) such that ℎ = 1 if ≤ else 0 3: P ← arg max P u Pv − o Ph such that P is doubly-stochastic and fair (Eq. 8) 4: ← BvN-Decomposition(P) 5: Return to add an objective that will function as a regularization term, penalizing ranking policies with outliers in the top-. We extend the linear programming approach described in Section 2 to solve: P = arg max P u Pv − o Ph such that P is doubly-stochastic P is fair. (8) For item fairness, we adopt the disparate treatment constraints as described in Section 2. Both terms of the optimization objective, and the constraints for fairness and finding a doubly stochastic matrix are linear in 2 variables. Hence, the resulting linear program can be solved efficiently using standard optimization algorithms [32]. Step 2: Constructing a stochastic ranking policy. The solution to the linear program P is a matrix indicating the probability of showing each item at any position. To generate actual rankings, we need to derive a stochastic ranking policy from P and sample rankings to present to users. We follow [32] and use Birkhoff-von Neumann decomposition to compute , which decomposes the doubly stochastic matrix P into the convex sum of permutation matrices = 1 1 + · · · + , with 0 ≤ ≤ 1, = 1 [7]. This results in at most ( − 1) 2 + 1 rankings [23], corresponding to , that are shown to the user with probability , respectively. 6 OMIT model summary. Algorithm 1 presents an overview of OMIT. OMIT takes as input the initial ranking (optimized for utility), outlier detection method M, outlierness context size , and the number of top items that we aim to remove outliers . In line 1, o D is created for a given outlier detection technique and outlierness context size, followed by line 2 where we create the h list that takes into account the top of the list that we aim to mitigate outlierness. In line 3, we solve the linear program that jointly solves the fairness and outlierness problem and pass the stochastic ranking in line 4 to the BvN decomposition algorithm. Finally, we return the output of the BvN method as the output. EXPERIMENTAL SETUP We target the following research questions: (RQ1) How do different outlier detection methods affect OMIT's performance? (RQ2) How does our OMIT trade-off between utility, fairness, and outlierness, compared to baselines? (RQ3) We adopt the constraints proposed in [32] (called FOE) to optimize a ranked list for fairness and utility through linear programming, as described in Section 4. Given that OMIT adds additional constraints, is the overall linear program more effective when we treat the doubly stochastic matrix constraints as hard or soft constraints? (RQ4) How does changing the context of detecting outliers affect OMIT's outlierness improvement and utility? (RQ5) How does changing affect OMIT's outlierness improvement and utility in the top-positions? Data. We use data from the TREC Fair Ranking 2019 and 2020 track (see Section 3). We make the group definitions over the two datasets consistent. As for the TREC 2019 data, we bin the original article groups into two classes. For the TREC 2020 data, we adopt the group definitions from the original data, that is, documents are assigned to two groups based on their authors' h-index. Moreover, we follow the TREC setup to generate query sequences, leading to multiple occurrences of the same query, using the provided frequencies. Specifically, we evaluate on a query sequence of size 10, 000, including all the queries in the evaluation data. Evaluation metrics. We evaluate methods for fair learning to rank in the presence of outliers in terms of utility, item fairness, and outlierness. For utility and fairness, we use NDCG and dTR (see Eq. 2), respectively, as our metrics and report their expected values. To measure the expected outlierness of the policy P up to position in the ranking, we use Outlierness D (P\| M) as defined in Eq. 7. Similarly we define the expected number of outliers up to position in ranking for policy P as #Outliers D (P\| M) = o Ph,(9) where o = 1 >0 (o ) is the binarized version of o where each outlier item is assigned 1, and all the rest are assigned 0. Compared methods. To evaluate OMIT, we build several baseline methods, combining different options for each component of our model (initial ranking, fairness of exposure, outlier mitigation): • Initial ranking: The initial ranking of all compared methods is generated using ListNet [9]. ListNet is a learning to rank model, optimizing for utility. We use it in our experiments to create initial ranked lists, D , using the click data provided in the training set, with 30 maximum epoch, and a validation ratio of 0.3. • Fairness of exposure: We use two variants of FOE [32] based on hard vs. soft doubly stochastic matrix constraints, and call them FOE and FOE , respectively. 7 • Outlier mitigation: We employ two outlier mitigation techniques, namely, RO and OMIT. RO removes all the outlier items detected by M from the ranking, while OMIT is our proposed outlier mitigation method as described in Section 4. We specify methods as combinations of the three components mentioned above. E.g., "ListNet + FOE + OMIT" uses the initial ranked list produced by ListNet, applying FOE fairness post-processing with hard constraints and the OMIT outlier mitigation model. EMPIRICAL RESULTS Effect of outlier detection method. We address RQ1 by changing the outlier detection method, while keeping the other parts of the model fixed. Table 2 reports the results of using three different outlier detection methods in OMIT. For comparison, we also report the results of ListNet without outlier mitigation (row a) and report the relative improvements. All three outlier detection methods effectively reduce outlierness compared to the baselines. COPOD achieves the best results by reducing outlierness by 80.3% and 80.6% in terms of Outlierness@10 on the TREC 2019 and 2020 data, respectively. In terms of dTR, COPOD outperforms MAD and MedKNN Table 2: Comparing loss in fairness and utility, with gains in outlierness for different outlier detection methods on the TREC 2019 and 2020 Fair Ranking data. Models used: (a) ListNet and (b) ListNet + FOE + OMIT. Δ values denote the percentage of relative improvement compared to (a). * refers to statistically significant improvements compared to (a) using a two-tailed paired t-test ( < 0.05). on both datasets where it increases dTR by 4.5% on TREC 2020. We see no significant difference in the utility achieved by the methods. Given that COPOD is parameter-free and scalable, we prefer it over the other two methods. For the remaining experiments, we choose COPOD as the outlier detection method. NDCG ↑ Fairness ↑ # Outliers ↓ Outlierness ↓ Model Outl. @5 @10 dTR @10 Δ(%) @10 Δ(%) Utility, fairness, and outlierness trade-offs. To answer RQ2, we turn to Table 2, which shows the results for OMIT and the baseline methods in terms of utility, fairness, and outlierness. Although ListNet is purely optimized for utility, it does not achieve the highest NDCG in all cases. This suggests that optimizing for fairness and outlierness could even improve the utility. As shown in Fig. 3 there is a high density of outliers among top positions that are mostly irrelevant. Therefore, when OMIT pushes these items to lower positions, it improves outlierness and utility measures simultaneously. Moreover, we see that mitigating outlierness does not cause any significant effect on dTR, showing that OMIT is capable of retaining position-based item fairness. 2020 data, respectively. We return the original ranking as the output when FOE does not find an optimum solution. To fix this problem we implemented the constraint for doubly stochastic matrix as a soft constraint and we check for validity of the permutation matrices generated by the decomposition algorithm. Table 3 demonstrates the effectiveness of the soft variant of FOE (row g vs. f). Effect of . To answer RQ4, we examine the effect of the parameter, which determines the outlierness context size. We change from 10 to 40 items while keeping other parameters fixed, and observing the behavior of the model. This is important since the outlierness of an item depends on its context, e.g., an item can be considered as an outlier in the top-10 items, but may not be an outlier in the top-20 items. The two left plots in Fig. 6 show the outlierness improvements (compared to ListNet) and utility in terms of NDCG@10 for different values of on the TREC 2019 and 2020 data. We see that Outlierness@10 improves for larger values of , suggesting that determining outlierness of items in a bigger pool of items is more accurate and allows OMIT to mitigate the outliers in the top-10 positions more effectively. Effect of . To answer RQ5, we study the effect of changing when optimizing for mitigating outlierness in top-positions. We are interested in finding out how utility and outlierness are influenced when optimizing for mitigating outlierness for a longer list of top ranks ranging from 10 to 40. This mimics the cases where more items can be shown to a user, hence outliers in longer lists can be observed by the user. The left plots in Fig. 6 depict the results for both datasets. We observe that outlierness improvement drops for greater values of since it is more challenging for the algorithm to push all outliers to lower positions. Fig. 3 shows that most outliers are located at top positions, so greater values of do not necessarily translate to more outliers. The changes in utility scores of the lists are marginal, with a 0.5% increase and 1.6% decrease for larger values of for the TREC 2019 and 2020 data, respectively. The difference in utility of the two datasets is due to the fact that there are more relevant outliers in TREC 2019 than in TREC 2020. RELATED WORK Bias in implicit feedback. Users' implicit feedback, such as clicks, can be a great source of relevance judgment that has been shown to help improve search quality [2]. clicks may be misleading due to different types of bias, which causes the probability of a click to differ from the probability of relevance. Recent work on discovering and correcting for different types of bias in logged click data concerns position bias [19,20], presentation bias [40], selection bias [29], trust bias [2,35], popularity bias [1], and recency bias [12]. Inter-item dependencies affect the perceived relevance of items [13]. We introduce a phenomenon that is anchored in inter-item dependencies and may result in biased clicks. Our work differs from previously discussed types of bias by considering interitem relationships. Showing items as outliers can make them more attractive to users, influencing their perceived relevance. Presentation bias [40] concerns a related effect; bold keywords in titles make some items appear more attractive. However, this definition of attractiveness is independent of other items in the list. Metrikov et al. [25] show that click-through rates can be manipulated by adding more images to top positions next to the ad slots on the search result page; they did not study the effect on item exposure or biased clicks. We focus on the effect of outliers, as an inter-item dependency on the examination probability and item exposure, and introduce it as a potential source of bias in click data. Fair ranking. Following [42], we distinguish two ways of measuring fairness of rankings. Work on probability-based fairness determines the probability that a ranking is the result of a fair process [4,10,11,16,34,39,41]. Exposure-based methods determine the expected exposure for each item in the ranking and aim to ensure that this exposure is fairly distributed [6,14,24,26,[31][32][33]38]. Our work belongs to the second category. To estimate the expected exposure of each item or group, we need to take into account different types of bias that the user might have when observing system outputs. Previous work has mostly focused on position bias [6,32,38]. We emphasize the role of inter-item dependencies in the exposure that an item receives, which can be a source of unfairness when not considered in computing the expected exposure. We extend the re-ranking approach introduced in [32], to not only produce fair rankings but also avoid showing the outliers in the top-. An important effort for developing a benchmark for evaluating retrieval systems in terms of fairness is the TREC Fair Ranking track (see footnote 1). We expand the use of the track's resources to include the study of outliers in fair ranking. CONCLUSION & FUTURE WORK We introduced and studied a phenomenon related to fair ranking, that is, outlierness. We analyzed data from the TREC Fair Ranking track and found a significant number of outliers in the rankings. We hypothesized that the presence of outliers in a ranking impacts the way users scan the result page. We confirmed this hypothesis with an eye-tracking study in two scenarios: e-commerce and scholarly search. We proposed OMIT, an approach to mitigate the existence and effect of outliers in a ranked list. With OMIT, we introduced a ranking constraint based on the outlierness of items in a ranking and combined it with fairness constraints. We formulated the problem of outlier mitigation as a linear programming problem and produced stochastic rankings, given an initial ranking. Using OMIT one can reduce outliers in rankings without compromising user utility or position-based item fairness. We analyzed the effects of different outlier detection approaches and compared their results. Our experiments also showed that there is a trade-off between the depth of outlier detection and user utility. Now that we have established the impact of outliers in rankings, future work on fair ranking should consider the presence of outliers by default. One limitation of our work is our focus on removing the bold outliers defined in the context of the whole list from the toppositions. We plan to improve our model to mitigate the outlierness of all sliding windows of size . In addition, we want to improve the performance for cases where outliers are relevant items, e.g., by considering alternative methods of outlier mitigation. Finally, a natural extension of our work could be to quantify the effect that an outlier in the presentation of the ranking can have on the examination probability distribution; this could open the door to unbiased learning to rank approaches that counter the outlier bias in logged user data. DATA AND CODE To facilitate reproducibility of our work, all code and parameters are shared at https://github.com/arezooSarvi/OMIT_Fair_ranking. Definition 3 . 1 ( 31Degree of outlierness). Let be an observable item feature, and M be one of the outlier detection methods discussed in Section 2. The degree of outlierness of an item in the ranked list[ 1 , . . . , ] is the value calculated by M for ( ) in the context of { ( 1 ), . . . ( )}, that determines how much ( ) differs from the other elements of the set. We write M( ( )\|D))) for this value. Definition 3.2 (Outliers in ranking). We say that according to M, item is an outlier in the ranked list [ 1 , . . . , ] for feature , if M Figure 1: E-commerce example used in our eye-tracking user study. A result page with one outlier at position 3, identified by more descriptive fields, higher price, and colored tags. clicks in product pages 0.169 0.170 identifies ( ) as an outlier in the set { ( 1 ), . . . , ( )}. Figure 2 : 2Distribution of the number of citations of top-20 papers returned for test queries in the TREC Fair Ranking track 2020 data. Figure 3 :Figure 4 : 34Number of outliers at each position w.r.t. different outlier detection methods, considering the number of citations as the observable feature. Each stacked bar shows the number of irrelevant and relevant outliers. Examples used in the eye-tracking study. (a) Heatmap for a result list without outlier for the query "smart phone". Items at top ranks receive more attention, following the position bias assumption. (b) Heatmap for a similar page (the same list as in Fig. 1) but with one outlier, at position 3. Participants exhibit increased attention towards and around the outlier item. (c) Heatmap for a scholarly search example, with an outlier (at position 4). Eye tracking measurements for scholarly search. Figure 5 : 5Eye tracking measurements for each position, based on participants' eye fixations. The positions where outliers were shown are marked with an asterisk. Figure 6 : 6NDCG@10 and outlierness@10 improvement percentage for different values of and different sizes of the window in which we detect the outliers. Table 1 : 1Descriptive statistics of TREC Fair Ranking Track 2020 data. Table 3 3shows that OMIT effectively decreases the number of outliers in top-10 positions by at most 83.49% (and 82.93%) when used with FOE (row g in the table) on the TREC 2019 (and 2020) data. For the outlierness metrics, these values are 80.29% and 80.66%. Referring back to our data analysis, we observed a high density of non-relevant outlier items at the top of the list, indicating the high possibility of user distraction towards these non-relevant items, as suggested by our eye-tracking study. Hard vs. soft constraint. We turn to RQ3 and experiment with two variants of the FOE model, which differ in the constraint for generating a doubly stochastic matrix (see Eq. 8). This constraint is important since the BvN algorithm can guarantee to generate valid permutations only if the input is doubly stochastic. We observed that forcing the convex optimization to output such matrices can make the constraints too hard to satisfy even when only optimizing for fairness and utility. Hence, the algorithm cannot find an opti- mum solution for many of the queries. For example, FOE cannot find solutions for 47%, and 46% of the queries on TREC 2019 and Table 3 : 3Comparing loss in fairness and utility, with gains in outlierness for COPOD on the TREC 2019 and TREC 2020 Fair Ranking data. Models used: (a) ListNet; (b) ListNet+FOE ; (c) ListNet+FOE ; (d) ListNet+FOE + ; (e) ListNet+FOE + ; (f) ListNet + FOE + OMIT; (g) ListNet + FOE + OMIT. Δ values denote the percentage of relative improvement compared to (a). Other conventions are the same as in Table 2. NDCG ↑ Fairness ↑ # Outliers ↓ Outlierness ↓ Model @5 @10 dTR @10 Δ(%) @10 Δ(%) TREC 2019 (a) 0.671 0.757 0.982 1.260 − 0.873 − (b) 0.670 0.756 0.991 1.235 − 0.870 − (c) 0.673 0.758 0.982 1.225 − 0.852 − (d) 0.663 0.697 0.996* 1.114 11.58 0.861 1.37 (e) 0.667 0.700 0.965 1.072* 14.92 0.834 4.47 (f) 0.672 0.757 0.996* 1.080* 14.28 0.753* 13.74 (g) 0.667 0.753 0.995* 0.208* 83.49 0.172* 80.29 TREC 2020 (a) 0.240 0.356 0.267 1.043 − 0.755 − (b) 0.237 0.355 0.249 1.073 − 0.780 − (c) 0.241 0.357 0.262 1.046 − 0.758 − (d) 0.242 0.362 0.293* 1.143 −9.58 0.811 −7.41 (e) 0.242 0.362 0.269 1.148 −10.06 0.817 −8.21 (f) 0.237 0.359 0.282* 0.885* 15.15 0.645* 14.56 (g) 0.240 0.366* 0.279* 0.178* 82.93 0.146* 80.66 https://fair-trec.github.io/ 2 http://api.semanticscholar.org/corpus/ We used the implementation from https://github.com/jfinkels/birkhoff. We used the implementation from https://github.com/MilkaLichtblau/BA_Laura. 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[]	[ "Francesco Vaccarino " ]	[]	[]	Let R be a commutative ring and let n, m be two positive integers. Let A R (n, m) be the polynomial ring in the commuting independent variables x i (j) with i = 1, . . . , m ; j = 1, . . . , n and coefficients in R. The symmetric group on n letters Sn acts on A R (n, m) by means of σ(x i (j)) = x i (σ(j)) for all σ ∈ Sn and i = 1, . . . , m ; j = 1, . . . , n. Let us denote by A R (n, m) Sn the ring of invariants for this action: its elements are usually called multisymmetric functions and they are the usual symmetric functions when m = 1. In this paper we give a presentation of A R (n, m) Sn in terms of generators and relations that holds for any R and any n, m, thereby answering a classical question.2000 Mathematics Subject Classification. 05E05, 13A50, 20C30.	null	[ "https://export.arxiv.org/pdf/math/0405490v2.pdf" ]	119,583,733	math/0405490	1d9c04fbcf6fcb842becdf49fc46fc1fa306b178	24 Mar 2005 Francesco Vaccarino 24 Mar 2005arXiv:math/0405490v2 [math.CO] THE RING OF MULTISYMMETRIC FUNCTIONS Typeset by A M S-T E X 1and phrases Characteristic-free invariant theorysymmetric functionsrepre- sentations of symmetric groups Let R be a commutative ring and let n, m be two positive integers. Let A R (n, m) be the polynomial ring in the commuting independent variables x i (j) with i = 1, . . . , m ; j = 1, . . . , n and coefficients in R. The symmetric group on n letters Sn acts on A R (n, m) by means of σ(x i (j)) = x i (σ(j)) for all σ ∈ Sn and i = 1, . . . , m ; j = 1, . . . , n. Let us denote by A R (n, m) Sn the ring of invariants for this action: its elements are usually called multisymmetric functions and they are the usual symmetric functions when m = 1. In this paper we give a presentation of A R (n, m) Sn in terms of generators and relations that holds for any R and any n, m, thereby answering a classical question.2000 Mathematics Subject Classification. 05E05, 13A50, 20C30. Introduction Let R be a commutative ring and let n, m be two positive integers. Let A R (n, m) be the polynomial ring in the commuting independent variables x i (j) with i = 1, . . . , m ; j = 1, . . . , n and coefficients in R. The symmetric group on n letters S n acts on A R (n, m) by means of σ(x i (j)) = x i (σ(j)) for all σ ∈ S n and i = 1, . . . , m ; j = 1, . . . , n. Let us denote by A R (n, m) Sn the rings of invariants for this action: its elements are usually called multisymmetric functions and they are the usual symmetric functions when m = 1. In this case, A R (n, 1) ∼ = R[x 1 , x 2 , . . . , x n ], and R[x 1 , x 2 , . . . , x n ] Sn is freely generated by the elementary symmetric functions e 1 , . . . , e n given by the equality (1 + tx i ). (0.1) Here e 0 = 1 and t is a commuting independent variable (see [M]). Furthermore one has e k (x 1 , . . . , x n ) = i1<i2<···<i k ≤n x i1 x i2 · · · x i k (0.2) Unless otherwise stated, we now assume that m > 1. We first obtain generators of the ring A R (n, m) Sn . Let A R (m) := R[y 1 , . . . , y m ], where y 1 , . . . , y m are commuting independent variables, let f = f (y 1 , . . . , y m ) ∈ A R (m) and define f (j) := f (x 1 (j), . . . , x m (j)) for 1 ≤ j ≤ n. (0.3) Notice that f (j) ∈ A R (n, m) for all 1 ≤ j ≤ n and that σ(f (j)) = f (σ(j)), for all σ ∈ S n and j = 1, . . . , n. Define e k (f ) := e k (f (1), f (2), . . . , f (n)) i.e. (1 + tf (i)), (0.4) where t is a commuting independent variable. Then e k (f ) ∈ A R (n, m) Sn . One may think about the y i as diagonal matrices in the following sense: let M n (A R (n, m)) be the full ring of n × n matrices with coefficients in A R (n, m). Then there is an embedding ρ n : A R (m) ֒→ M n (A R (n, m)) (0.5) given by ρ n (y i ) :=   x i (1) 0 . . . 0 0 x i (2) . . . 0 0 0 . . . x i (n)   for i = 1, . . . , m. (0.6) Now (0.4) gives n k=0 t k e k (f ) = n j=1 (1 + tρ n (f ) jj ) = det(1 + tρ n (f )), (0.7) where det(−) is the usual determinant of n × n matrices. Let M m be the set of monomials in A R (m). For µ ∈ M m let ∂ i (µ) denote the degree of µ in y i , for all i = 1, . . . , m. We set ∂(µ) := (∂ 1 (µ), . . . , ∂ m (µ)) (0.8) for its multidegree. The total degree of µ is i ∂ i (µ). Let M + m be the set of monomials of positive degree. A monomial µ ∈ M + m is called primitive it is not a power of another one. We denote by M + m the set of primitive monomials. We define an S n invariant multidegree on A R (n, m) by setting ∂(x i (j)) = ∂(y i ) ∈ N m for all 1 ≤ j ≤ n and 1 ≤ i ≤ m. If f ∈ A R (m) is homogeneous of total degree l, then e k (f ) has total degree kl (for all k and n). We are now in a position to state the first part of our result (recall that m > 1). Theorem 1 (generators). The ring of multisymmetric functions A R (n, m) Sn is generated by the e k (µ), where µ ∈ M + m , k = 1, . . . n and the total degree of e k (µ) is less or equal than n(m − 1). If n = p s is a power of a prime and R = Z or p · 1 R = 0, then at least one generator has degree equal to n(m − 1). If R ⊃ Q then A R (n, m) Sn is generated by the e 1 (µ), where µ ∈ M + m and the degree of µ is less or equal than n. To obtain the relations between these generators, we need more notation on (multi)symmetric functions. The action of S n on A R (n, 1) ∼ = R[x 1 , x 2 , . . . , x n ] preserves the usual degree. We denote by Λ k R,n the R-submodule of invariants of degree k. Let q n : R[x 1 , x 2 , . . . , x n ] − → R[x 1 , x 2 , . . . , x n−1 ] be given by x n → 0 and x i → x i , for i = 1, . . . , n − 1. This map sends Λ k n,R to Λ k n−1,R and it is easy to see that Λ k n,R ∼ = Λ k k,R for all n ≥ k. Denote by Λ k R the limit of the inverse system obtained in this way. The ring Λ R := k≥0 Λ k R is called the ring of symmetric functions (over R). It can be shown [M] that Λ R is a polynomial ring, freely generated by the (limits of the) e k , that are given by ∞ k=0 t k e k := ∞ i=1 (1 + tx i ). (0.9) Furthermore the kernel of the natural projection π n : Λ R − → Λ n,R is generated by the e n+k , where k ≥ 1. In a similar way we build a limit of multisymmetric functions. For any a ∈ N m we set A R (n, m, a) for the linear span of the monomials of multidegree a. One has where the projective limit is taken with respect to n over the projective system (A R (n, m, a) Sn , π n ). Set A R (n, m) =A R (∞, m) := a∈N m A R (∞, m, a). (0.14) We set, by abuse of notation, e k (f ) := lim ← e k (f ) ∈ A R (∞, m) (0.15) with k ∈ N and f ∈ A(m) + , the augmentation ideal, i.e. (1 + tf (j)). where α := (α µ ) µ∈M + m is such that α µ ∈ N, µ∈M + m α µ ≤ k and λ α := µ∈M + m λ αµ . We can now state the second part of our main result. Theorem 2 (relations). (1) The ring A R (∞, m) is a polynomial ring, freely generated by the (limits of ) the e k (µ), where µ ∈ M + m and k ∈ N. The kernel of the natural projection A R (∞, m) − → A R (n, m) Sn is generated as R-module by the coefficients e α of the elements e n+k (f ), where k ≥ 1 and f ∈ A R (m) + . (2) If R ⊃ Q then A R (∞, m) is freely generated by the e 1 (µ), where µ ∈ M + m . The kernel of the natural projection is generated as an ideal by the e n+1 (f ), where f ∈ A R (m) + . In Dalbec's paper [D] generators and relations are found in the case where R ⊃ Q. The relations found there are actually the same we find: indeed what Dalbec calls monomial multisymmetric functions are exactly those e α we introduced in (0.17), so that his Proposition 1.9 is a special case of our Proposition 3.1(1) when R ⊃ Q. Another paper on this theme, giving a minimal presentation when the base ring is a characteristic 2 field, is [A]. Again, its main results on multisymmetric functions are a corollary of ours when R is a characteristic 2 field. The results of this paper were presented in 1997 at a congress on algebraic groups representations in Ascona (CH) organized by H.P.Kraft. They are published only now for personal reasons. Notations and basic facts The monomials of A R (n, m) form a R-basis, permuted by the action of S n . Thus, the sums of monomials over the orbits form a R-basis of the ring of multisymmetric functions. We now introduce some notation and preliminary results concerning these functions and orbit sums. Let k ∈ N, we denote by f the sequence (f 1 . . . , f k ) in A R (m) and by α the element (α 1 , . . . , α k ) ∈ N k , where α j ≤ n . Let t 1 , . . . , t k be commuting independent variables, we set as usual t α : = i t αi i . We define elements e α (f ) ∈ A R (n, m) Sn by α t α e α (f ) := det(1 + h t h ρ n (f h )) = n i=1 (1 + h t h f h (i)). (1.1) Example 1.1. Let n = 3 and f, g ∈ A R (m) then e (2,1) (f, g) = f (1)f (2)g(3) + f (1)g(2)f (3) + g(1)f (2)f (3). If n = 4 then (4) Let k = m and f j = y j for j = 1, . . . , m, then the e α (y) = e (α1,...,αm) (y 1 , . . . , y m ) where α j ≤ n are the well-known elementary multisymmetric functions. These generate A R (n, m) Sn when R ⊃ Q (see [G] or [W]), and satisfy e (2,1) (f, g) =f (1)f (2)g(3) + f (1)g(2)f (3) + g(1)f (2)f (3)+ f (1)f (2)g(4) + f (1)g(2)f (4) + g(1)f (2)f (4)+ f (1)f (3)g(4) + f (1)g(3)f (4) + g(1)f (3)f (4)+ f (2)f (3)g(4) + f (2)g(3)f (4) + g(2)f (3)fα t α e α (y) = det(1 + j t j ρ n (y j )) = n i=1 (1 + m j=1 t j x j (i)). (1.2) Lemma 1.2. The multisymmetric function e (α1,...,α k ) (f 1 , . . . , f k ) is the orbit sum (under the considered action of S n ) of f 1 (1)f 1 (2) · · · f 1 (α 1 )f 2 (α 1 + 1) · · · f 2 (α 1 + α 2 ) · · · f k ( h α h ). Proof. Let E be the set of mappings φ : {1, . . . , n} → {1, . . . , k + 1}. We define a mapping φ → φ * of E into N k+1 by putting φ * (i) equal to the cardinality of φ −1 (i). For two elements φ 1 , φ 2 of E, to satisfy φ * 1 = φ * 2 it is necessary and sufficient that there should exist σ ∈ S n such that φ 2 = φ 1 • σ. Set f k+1 := 1 R and E(α) := {φ ∈ E \| φ * = (α 1 , . . . , α k , n − i α i )}, then we have e α (f ) = φ∈E(α) f φ(1) (1)f φ(2) (2) · · · f φ(n) (n) (1.3) and the lemma is proved. It is clear that e (α1,...,α k ) (f 1 , . . . , f k ) = e (α τ (1) ,...,α τ (k) ) (f τ (1) , . . . , f τ (k) ) for all τ ∈ S k . If two entries are equal, say f 1 = f 2 , then, by (1.1) e (α1,...,α k ) (f 1 , . . . , f k ) = (α 1 + α 2 )! α 1 !α 2 ! e (α1+α2,...,α k ) (f 1 , f 3 . . . , f k ). (1.4) Let N (M + m ) be the set of functions M + m − → N with finite support. We set \| α \|:= µ∈M + m α(µ) (1.5) Let α ∈ N (M + m ) , then there exist k ∈ N and µ 1 , . . . , µ k ∈ M + m such that α(µ i ) = α i = 0 for i = 1, . . . , k and α(µ) = 0 when µ = µ 1 , . . . , µ k . We set e α := e (α1,...,α k ) (µ 1 , . . . , µ k ), ( 1.6) i.e. we substitute (µ 1 , . . . , µ k ) to variables in the elementary multisymmetric function e (α1,...,α k ) (y 1 , . . . , y k ). Then \|α\|≤n t α e α = n i=1 (1 + µ∈M + m t µ µ(i)),(1.7) where t µ are commuting independent variables indexed by monomials and t α := µ∈M + m t α(µ) µ (1.8) for all α ∈ N (M + m ) . If α ∈ N (M + m ) is such that α(µ) = k for some µ ∈ M + m and α(ν) = 0 for all ν ∈ M + m with ν = µ, we see that e α = e k (µ), the k-th elementary symmetric function evaluated at (µ(1), µ(2), . . . , µ(n)). Lemma 1.3. Given a monomial µ ∈ A R (n, m), there exist µ 1 , . . . , µ n ∈ A R (m) such that µ = µ 1 (1) · · · µ n (n). Proof. Let µ = ij x i (j) aij then µ j = i y aij i for j = 1, . . . , n. Proof. By Lemma 1.2 and (1.6), the e α are a complete system of representatives (for the action of S n ) of the orbit sums of the products {µ 1 (1)µ 2 (2) · · · µ n (n) : µ i ∈ M m , i = 1, . . . , n}. So the first statement follows by Lemma 1.3. Notice that ∂(e α ) = µ∈M + m α µ ∂(µ) to prove the second statement. Generators Let us calculate the product between two elements e α , e β ∈ B n,m,R of the basis B n,m,R . Theorem 2.1 -Product Formula. Let k, h ∈ N, f 1 . . . , f k , g 1 , . . . , g h ∈ A R (m) and t 1 , . . . , t k , s 1 , . . . , s h be commuting independent variables. Set as in (1.1) e α (f ) := e (α1,...,α k ) (f 1 , . . . , f k )and e β (g) := e (β1,...,β h ) (g 1 , . . . , g h ). Then e α (f )e β (g) = γ e γ (f , g, fg), where fg := (f 1 g 1 , f 1 g 2 , . . . , f 1 g h , f 2 g 1 , . . . , f 2 g h , . . . , f k g h ) and γ := (γ 10 , . . . , γ k0 , γ 01 , . . . , γ 0h , γ 11 , γ 12 , . . . , γ kh ) are such that            γ ij ∈ N \| γ \|≤ n h j=0 γ ij = α i for i = 1, . . . , k k i=0 γ ij = β j for j = 1, . . . , h. Proof. The result follows from ( αj ≤n k j=1 t αj j e α (f ))( β l ≤n h l=1 s β l l e β (g)) = ( α t α e α (f ))( β s β e β (g)) = n i=1 (1 + k j=1 t j f j (i)) n i=1 (1 + h l=1 s l g l (i)) = n i=1 (1 + k j=1 t j f j (i) + h l=1 s l g l (i) + j,l t j s l f j (i)g l (i)). Introduce the new variables u jl with j = 1, . . . , k and l = 1, . . . , h, then n i=1 (1 + k j=1 t j f j (i) + h l=1 s l g l (i) + j,l t j s l f j (i)g l (i)) = n i=1 (1 + k j=1 t j f j (i) + h l=1 s l g l (i) + j,l u jl (i)g l (i)) = γ v γ e γ (f , g, fg) where v is the cumulative variable t, s, u. Then substitute u jl = t j s l to obtain γ v γ e γ (f , g, fg) = γ ( k a=1 t γa0 a h b=1 s γ 0b b k a=1 h b=1 (t a s b ) γ ab e γ (f , g, fg)), where fg = (f 1 g 1 , f 1 g 2 , . . . , f k g 1 , . . . , f k g h ) and γ satisfy the condition of the theorem. Example 2.2. Let us calculate in A R (2, 3) S2 e (1,1) (a, b)e 2 (c) = 0≤k,h≤1 e (1−k,1−h,2−k−h,h,k) (a, b , c, ac, bc) = e (1,1) (ac, bc), since 1 − k + 1 − h + 2 − k − h + h + k = 4 − k − h ≤ 2. Corollary 2.3. Let k ∈ N, a 1 , . . . , a k ∈ A R (m), α = (α 1 , . . . , α k ) ∈ N k with α j ≤ n. Then e (α1,...,α k ) (a 1 , . . . , a k ) belongs to the subring of A R (n, m) Sn generated by the e i (µ), where i = 1, . . . , n and µ is a monomial in the a 1 , . . . , a k . Proof. We prove the claim by induction on j α j (notice that 1 ≤ k ≤ j α j ) assuming that α i > 0 for all i. If j α j = 1 then k = 1 and e (α1,...,α k ) (a 1 , . . . , a k ) = e 1 (a 1 ). Suppose the claim true for all e (β1,...,β h ) (b 1 , . . . , b h ) with b 1 , . . . , b h ∈ A R (m) and i β i < j α j . Let k, a 1 , . . . , a k , α be as in the statement, then we have by Theorem 2.1 e α1 (a 1 )e (α2,...,α k ) (a 2 , . . . , a k ) = = e (α1,...,α k ) (a 1 , . . . , a k ) + e γ (a 1 , . . . , a k , a 1 a 2 , . . . , a 1 a k ), where γ = (γ 10 , γ 01 , . . . , γ 0h , γ 11 , γ 12 , . . . , γ 1h ) with h = k − 1, h j=0 γ 1j = α 1 with h j=1 γ 1j > 0, and γ 0j + γ 1j = α j for j = 1, . . . , h. Thus γ 10 + γ 01 + · · · + γ 0h + γ 11 + · · · + γ 1h = j α j − h j=1 γ 1j < j α j . Hence e (α1,...,α k ) (a 1 , . . . , a k ) = e α1 (a 1 )e (α2,...,α k ) (a 2 , . . . , a k ) − e γ (a 1 , . . . , a k , a 1 a 2 , a 1 a 3 , . . . , a 1 a k ), where r,s γ rs < j α j . So the claim follows by induction hypothesis. We now recall some basic facts about classical symmetric functions, for further reading on this topic see [M]. We have another distinguished kind of functions in Λ R beside the elementary symmetric ones: the power sums. For any r ∈ N the r-th power sum is p r := i≥1 x r i . Let g ∈ Λ R , set g · p r = g(x r 1 , x r 2 , . . . , x r k , . . . ), this is again a symmetric function. Since the e i generate Λ R we have that g · p r can be expressed as a polynomial in the e i . In particular, P h,k := e h · p k is a polynomial in the e i . Proposition 2.5. For all f ∈ A R (m), and k, h ∈ N, e h (f k ) belongs to the subring of A R (n, m) Sn generated by the e j (f ). Proof. Let f ∈ A R (m) and consider e h (f k ) ∈ A R (n, m) Sn , we have (see Introduction) e h (f k ) = e h (f (1) k , . . . , f (n) k ) = P h,k (e 1 (f (1), . . . , f (n)), . . . , e n (f (1), . . . , f (n))) and the result is proved. We are now ready to prove Theorem 1 stated in the introduction. Proof of Theorem 1. Recall that a monomial µ ∈ M + m is called primitive if it is not a power of another one and we denote by M + m the set of primitive monomials. The elements e α ∈ B n,m,R , that form a R-basis by Prop.1.4, can be expressed as polynomials in e i (µ) with i = 1, . . . , n and µ ∈ M + m , by Cor.2.3. If µ = ν k with ν ∈ M + m , then e i (µ) can be expressed as a polynomial in the e j (ν), by Prop.2.5. Since for all µ ∈ M + m there exist k ∈ N and ν ∈ M + m such that µ = ν k , we have that A(n, m) Sn is generated as a commutative ring by the e j (ν), where ν ∈ M + m and j = 1, . . . , n. The theorem then follows by the following result due to Fleischmann [F]: the ring A R (n, m) Sn is generated by elements of total degree ℓ ≤ n(m − 1), for any commutative ring R, with sharp bound if n = p s a power of a prime and R = Z or p · 1 R = 0. If R ⊃ Q then the result follows from Newton's Formulas and a well-known result of H.Weyl (see [G], [W]). Relations We write a generating series for the orbits of monomials G(t) := n i=1 (1 + M + m t µ µ(i)) = α,\|α\|≤n t α e α ,(3.1) where α ∈ N (M + m ) and t α e α (n) = 0 when α = 0. Recall the map π n : A R (n, m) − → A R (n − 1, m) defined by π n (x i (j)) = 0 if j = n x i (j) if j ≤ n − 1 for all i. (3.2) Then we have of course that π n (G n (t)) = G n−1 (t), so that π n ((e α )) = e α if \| α \|≤ n 0 otherwise. (4) The R-module A R (∞, m) is free with basis {e α : α ∈ N (M + m ) }. Proof. (1) By (3.3) and (3.5), for all a ∈ N m , the following is a split exact sequence of R-modules 0 − → ker π n,a − → A(n, m, a) Sn πn,a − −− → A(n − 1, m, a) Sn−1 − → 0, and the claim follows. (2) If m j=1 a j < n, then kerπ n,a = 0, indeed ∂(e α ) = µ∈M + m α µ ∂(µ) = a =⇒\| α \|≤ m j=1 a j < n. Hence A(h, m, a) S h ∼ = A(b, m, a) S b where b := m j=1 a j , for all h ≥ m j=1 a j and the claim follows by (3.5). (3) follows from (1) and (2). (4) follows from (3) and (3.8) Remark 3.2. Notice that A R (m) ⊗n ∼ = A R (n, m) as multigraded S n -algebras by means of f 1 ⊗ · · · ⊗ f n ↔ f 1 (1)f 2 (2) · · · f n (n) (3.10) for all f 1 , . . . , f n ∈ A R (m). Hence A R (n, m) Sn ∼ = T S n (A R (m)), where T S n ( − ) denotes the symmetric tensors functor. Since T S n (A R (m)) ∼ = R T S n (A Z (m)) (see [B]), we have A R (n, m) Sn ∼ = R ⊗ A Z (n, m) Sn (3.11) for any commutative ring R. We then work with R = Z and we suppress the Z subscript for the sake of simplicity. Remark 3.3. The Z-module A(∞, m) can be endowed with a structure of N mgraded ring such that the π n are N m -graded ring homomorphisms: the product e α e β , where α, β ∈ N (M + m ) , is defined by using the product formula of Theorem 2.1 with no upper bound on \| γ \|, where γ appears in the summation. σ m : Z[e i,µ ] i∈N,µ∈M + m − → A(∞, m) given by σ m : e i,µ → e i (µ), for all i ∈ N, µ ∈ M + m is an isomorphism, i.e. A(∞, m) is freely generated as a commutative ring by the e i (µ), where i ∈ N and µ ∈ M + m . Proof. Since we defined the product in A(∞, m) as in Theorem 2.1, it is easy to verify, repeating the reasoning of the previous section, that A(∞, m) is generated as a commutative ring by the e i (µ), where i ∈ N, µ ∈ M + m . Hence σ m is onto for all m ∈ N. Let a ∈ N m and consider the restriction σ m,a : C(m, a) − → A(∞, m, a). It is onto as we have just seen. A Z-basis of C(m, a) is Proof of Theorem 2. (1) As before we set R = Z and the result follows by Remark 3.2, Prop.3.4. and Prop.3.1. for all µ∈M + m λ µ µ ∈ A(m) + . Hence g(e α ) = 0 for all e α with \| α \|= k ; thus g = 0. If R ⊃ Q the result then follows from Newton's formulas and Cor.3.5. Aknowledgement I would like to thank M.Brion, C.De Concini and C.Procesi, in alphabetical order, for useful discussions. I would also like to thank the referee for its valuable suggestions. A R (n, m, a). (0.10) Let π n : A R (n, m) − → A R (n − 1, m) be given byπ n (x i (j)) = 0 if j = n x i (j) if j ≤ n − 1 for all i.(0.11) Then (see (3.5)) we prove that, for all a ∈ N m π n (A R (n, m, a) Sn ) = A R (n − 1, m, a) Sn−1 . (0.12) For any a ∈ N m set A R (∞, m, a) := lim ← A R (n, m, a) Sn , (0.13) k is a homogeneous polynomial of degree k. Now, if f = µ∈M + m λ µ µ, we set e k (f ) := α λ α e α (0.16) := {e α : \| α \|≤ n} is a R-basis of A R (n, m) Sn .The set B n,m,a,R := {e α : \| α \|≤ n and ∂(e α ) = a} is a R-basis of A R (n, m, a) Sn , for all a ∈ N m . Example 2. 4 . 4Consider e (2,1) (a, b) in A R (3, m) as in Example 1.2, then e (2,1) (a, b) = e 2 (a)e 1 (b) − e (1,1) (a, ab) = e 2 (a)e 1 (b) − e 1 (a)e 1 (ab) + e 1 (a 2 b). all a ∈ N m the restriction π n,a : A R (n, m, a) → A R (n − 1, that π n,a (A R (n, m, a) Sn ) = A R (n − 1, m, a) Sn−1 (3.5) and then (A R (n, m, a) Sn , π n,a ) is a projective sytem. For any a ∈ N m setA R (∞, m, a) := lim ← A R (n, m, a) Sn , (3.6)where the projective limit is taken with respect to n over the above projective system and setπn,a : A R (∞, m, a) − → A R (n, m, a) the classical case (m = 1) and recalling (3.1), (3.3) we make an abuse of notation and set e α := lim ← e α (n), for any α ∈ N (M + m ) . In the same way we set e j (f ) := lim ← e j (f ) with j ∈ N, where f ∈ A R (m) + is homogeneous of positive multidegree, so that j ∂(f ) = a. Proposition 3.1. Let a ∈ N m . (1) The R-module kerπ n,a is the linear span of {e α ∈ A R (∞, m, a) : \| α \|> n}. (2) The R-module homomorphismsπ n,a : A R (∞, m, a) − → A R (n, m, a) Sn are onto for all n ∈ N and A R (∞, m, a) ∼ = A R (n, m, a) Sn for all n ≥\| a \| . (3) The R-module A R (∞, m, a) is free with basis {e α : ∂(e α ) = a}, m, a) := Z[e i,µ ] i∈N,µ∈M + m with multidegree given by ∂(e i,µ ) = ∂(µ)i. Then the multigraded ring homomorphism On the other hand, a Z-basis of A(∞, m, a) is{e α : αµ∈N,µ∈M + m α µ ∂(µ) = a}.Let µ ∈ M + m , then there are an unique k ∈ N and an unique ν ∈ M + m such that µ = ν k . Henceαµ∈N,µ∈M + m α µ ∂(µ) = k∈N,αµ∈N,ν∈M + m α µ k ∂(ν),so that C(m, a) and A(∞, m, a) have the same (finite) Z-rank and thus are isomorphic via σ m,a . Corollary 3 . 5 . 35Let R ⊃ Q then A R (∞, m) is a polynomial ring freely generated by the e 1 (µ), where µ ∈ M + m . Proof. By Prop.3.4 and Theorem 1. ( 2 ) 2By Prop.3.1 the kernel ofA(∞, m)π n −→ A(n, m) Sn has basis {e α : \| α \|> n}. Let V k be the submodule of A(∞, m) with basis {e α : \| α \|= k}. Let A k be the sub-Z-module of Q ⊗ V k generated by the e k (f ) with f ∈ A(m) + . Let g : Q ⊗ V k − → Q be a linear form identically zero on A k The mod 2 cohomology rings of the symmetric groups and invariants. A M Feschbach, Topology. A. M.Feschbach, The mod 2 cohomology rings of the symmetric groups and invariants, Topology (2002), 57-84. Elements of mathematics -Algebra II Chapters 4-7. B N Bourbaki, Springer-VerlagBerlinB. N.Bourbaki, Elements of mathematics -Algebra II Chapters 4-7, Springer-Verlag, Berlin, 1988. Multisymmetric functions. D J Dalbec, Beiträge Algebra Geom. 401D. J.Dalbec, Multisymmetric functions, Beiträge Algebra Geom. 40(1) (1999), 27-51. A new degree bound for vector invariants of symmetric groups. F P Fleischmann, Trans. Am. Math. Soc. 350F. P.Fleischmann, A new degree bound for vector invariants of symmetric groups, Trans. Am. Math. Soc. 350 (1998), 1703-1712. Discriminants, resultants and multidimensional determinants. G I Gelfand, M Kapranov, A Zelevinsky, Birkahuser, BostonG. I.Gelfand,M.Kapranov, A.Zelevinsky, Discriminants, resultants and multidimensional deter- minants, Birkahuser, Boston, 1994. Symmetric Functions and Hall Polynomials -second edition, Oxford mathematical monograph. M I G Macdonald, M. I.G.Macdonald, Symmetric Functions and Hall Polynomials -second edition, Oxford math- ematical monograph, 1995. Dipartimento di Matematica -Politecnico di Torino -Corso Duca degli Abruzzi 24 -10129 -Torino -Italy E-mail address: vaccarino@syzygie. H Weyl ; Princeton, N J , Princeton University PressThe classical groups. itH.Weyl, The classical groups, Princeton University Press, Princeton N.J., 1946. Dipartimento di Matematica -Politecnico di Torino -Corso Duca degli Abruzzi 24 -10129 -Torino -Italy E-mail address: [email protected]	[]
[ "Two-site dynamical mean field theory for the dynamic Hubbard model", "Two-site dynamical mean field theory for the dynamic Hubbard model" ]	[ "G H Bach \nDepartment of Physics\nDepartment of Physics\nUniversity of Alberta\nT6G 2G7EdmontonABCanada\n", "J E Hirsch \nDepartment of Physics\nUniversity of California\nSan Diego, La Jolla92093-0319CA\n", "F Marsiglio \nUniversity of Alberta\nT6G 2G7EdmontonABCanada\n" ]	[ "Department of Physics\nDepartment of Physics\nUniversity of Alberta\nT6G 2G7EdmontonABCanada", "Department of Physics\nUniversity of California\nSan Diego, La Jolla92093-0319CA", "University of Alberta\nT6G 2G7EdmontonABCanada" ]	[]	At zero temperature, two-site dynamical mean field theory is applied to the Dynamic Hubbard model. The Dynamic Hubbard model describes the orbital relaxation that occurs when two electrons occupy the same site, by using a two-level boson field at each site. At finite boson frequency, the appearance of a Mott gap is found to be enhanced even though it shows a metallic phase with the same bare on-site interaction U in the conventional Hubbard model. The lack of electronhole symmetry is highlighted through the quasi-particle weight and the single particle density of states at different fillings, which qualitatively differentiates the dynamic Hubbard model from other conventional Hubbard-like models.PACS numbers:	10.1103/physrevb.82.155122	[ "https://arxiv.org/pdf/1008.3905v1.pdf" ]	118,339,113	1008.3905	95e8e7bc5e826364dbb4252825b82c41f4d77c8d	Two-site dynamical mean field theory for the dynamic Hubbard model 23 Aug 2010 G H Bach Department of Physics Department of Physics University of Alberta T6G 2G7EdmontonABCanada J E Hirsch Department of Physics University of California San Diego, La Jolla92093-0319CA F Marsiglio University of Alberta T6G 2G7EdmontonABCanada Two-site dynamical mean field theory for the dynamic Hubbard model 23 Aug 2010(Dated: August 25, 2010)arXiv:1008.3905v1 [cond-mat.str-el]PACS numbers: At zero temperature, two-site dynamical mean field theory is applied to the Dynamic Hubbard model. The Dynamic Hubbard model describes the orbital relaxation that occurs when two electrons occupy the same site, by using a two-level boson field at each site. At finite boson frequency, the appearance of a Mott gap is found to be enhanced even though it shows a metallic phase with the same bare on-site interaction U in the conventional Hubbard model. The lack of electronhole symmetry is highlighted through the quasi-particle weight and the single particle density of states at different fillings, which qualitatively differentiates the dynamic Hubbard model from other conventional Hubbard-like models.PACS numbers: I. INTRODUCTION Dynamic Hubbard models 1 represent a new class of model Hamiltonians that describe the modification of the electronic wavefunction that occurs when an atomic orbital becomes occupied by more than one electron. The key difference between dynamic and standard Hubbardlike models can be understood by considering a Helium atom. Helium has two electrons in the 1s shell and the strength of the interaction between them is comparable to the electron-ion interaction. As a consequence, the two-electron wave function cannot be simply represented by a single Slater determinant formed by the electronic wavefunction of the singly occupied orbital (He + ) for each electron, since the wavefunction of one electron is modified by the presence of the other electron. The effect is more pronounced in the negative ion H − because of the weaker attraction between the electrons and the nucleus. Conventional tight binding models like the Hubbard model completely neglect this effect since they assume that the wavefunction for two electrons on a site is a simple product of the wavefunctions for a singly occupied site 1 . The Dynamic Hubbard model tries to include the physics of orbital relaxation by modulating the on-site interaction term (Hubbard U ) with an auxiliary boson (or spin) degree of freedom. 1 An effective low-energy Hamiltonian for this model modifies the hopping term so that the hopping amplitude depends on the electronic occupation of the sites involved in the hopping process; this is known as the correlated hopping model 2 . In this paper, we focus on the dynamic Hubbard model with an auxiliary spin-1/2 degree of freedom; the two states of this pseudospin can be viewed as describing the modification of the electronic wavefunction upon double occupancy. This kind of model was first suggested two decades ago. 3 The model was studied further through world line quantum Monte Carlo (QMC) methods 4 , exact diagonalization (ED) 5 , and an approximate perturbative analysis starting from a generalized Lang-Firsov transformation. 6 The exact treatments suffer from small size effects, but more recently, Bouadim et al. 7 also studied this model using determinant QMC on somewhat larger clusters (N = 6 × 6); however, due to the fermion sign problem, they encountered difficulties, especially at low temperatures. In recent years it has become clear that dynamical mean field theory (DMFT) is a valuable way to treat the local aspects of both quasiparticles and incoherent high energy excitations on the same footing in strongly correlated electron systems. 8 DMFT is practical because, instead of considering a large lattice model whose Hilbert space is exponentially large, one needs to solve merely a single impurity model; for this problem various algorithms for an exact solution already exist. 9 Even though the single impurity model can be treated numerically by QMC, it is still a computationally expensive problem. An approximate but effective alternative was proposed by Potthoff, 10 who proposed the so-called "two-site DMFT"; this method uses two sites, one for the impurity and one for the 'bath' of conduction electrons, which is readily solved exactly. Using only one bath site renders the mapping to the single impurity Anderson model (SIAM) approximate, but the self-consistency is now easily controlled through the bath parameters. In Ref. 10 the validity of the two-site DMFT simplification was established for the Hubbard model; for example, it predicts qualitatively correctly the existence of a Mott transition critical point, and even though it is not a conserving theory in the sense of Baym and Kadanoff, 11 the violation of the Luttinger sum rule, for example, is fairly small. Given our interest in establishing qualitative trends for the Dynamic Hubbard model, and the existence of QMC results (albeit for small lattices) as a benchmark, we will adopt the two-site DMFT approximation to study the dynamic Hubbard model. There have been some studies of correlated hopping models using DMFT, in particular applied to the Falicov-Kimball model 12,13 . Because the interaction term involves two sites rather than one, the self-energy is nonlocal and the formalism becomes considerably more complicated. Instead with the dynamic Hubbard model considered here we can describe the physics of correlated hopping in a much simpler way with a single-site selfenergy, by simply considering the model in the limiting case where the interaction becomes non-retarded (large ω 0 in the Hamiltonian Eq. (1)). As we illustrate below, the two-site DMFT treatment of the Dynamic Hubbard model gives semi-quantitative agreement with the QMC results. For half-filling and below, the properties of the Dynamic Hubbard model mimic those of the Hubbard model. For example a Mott insulating phase appears for strong enough on-site interaction, and, with an attractive on-site interaction, pairing is enhanced. In addition, however, electron-hole asymmetry naturally arises; this is evident, for example, in the dependence of the quasi-particle weight on the electron/hole number density. Thus this model captures the essential physics that a few electrons in a nearly empty electronic energy band can behave very differently from a few holes in a nearly filled band. The paper is organized as follows: The next section will briefly describe the Dynamic Hubbard model with an auxiliary spin 1/2 degree of freedom and will provide a synopsis of the two-site DMFT approximation. In section III we present some numerical results and discuss some of the characteristic properties of the dynamic Hubbard model, especially those that differentiate it from the simple Hubbard model. In addition, we show some comparisons with the QMC results, 7 which indicate that the two-site approximation works very well. The last section IV will summarize our results and suggest directions for further study. II. MODEL AND METHOD We consider here the Dynamic Hubbard Hamiltonian 1 with a spin-1/2 degree of freedom in the electron representation: H DHM = <i,j>σ t ij (c † iσ c jσ + c † jσ c iσ ) − µ i,σ n iσ + i (ω 0 σ x i + gω 0 σ z i ) + i (U − 2gω 0 σ z i )n i↑ n i↓ . (1) The first term is the electron hopping term; c † iσ (c iσ ) is an electron creation (annihilation) operator at site i with spin σ. Following Potthoff 10 we use a Bethe lattice with infinite connectivity with nearest neighbour hopping only, so that t ij = −t < 0 for nearest neighbours only. The parameter t = t * / √ q, with q the connectivity, and t * = 1 sets the energy scale. The second term is the usual chemical potential term which determines the electron filling. The auxiliary spin degree of freedom is given in the third term; the two levels have a spacing given by ω 0 . The fourth term describes interactions between two electrons. In addition to the onsite Hubbard U term, there is an additional coupling to the auxiliary spin degree of freedom. As explained in Ref. 1 and reviewed in the next section, this term varies the actual on-site repulsion, dependent on the state of the auxiliary degree of freedom. Dynamical Mean Field Theory has been widely studied in a number of correlated fermion systems, and this approach has been quite successful, as reviewed in Ref. 8. In particular, in models where the local dynamics is expected to play the most important role (as opposed to spatial correlations), the DMFT without the use of cluster methods 14,15 should be accurate. The Dynamic Hubbard model should be ideally suited for these conditions. DMFT maps a lattice model onto a quantum single impurity model through self-consistent conditions; in this paper, we consider particularly the single impurity Anderson model (SIAM): H imp = σ (ǫ d − µ)d † σ d σ + ns σ,k=2 (ǫ k − µ)a † kσ a kσ + ns σ,k=2 V k (d † σ a kσ + h.c.) + (ω 0 σ x + gω 0 σ z ) + (U − 2gω 0 σ z )n d↑ n d↓ (2) where d σ , d † σ are the impurity operators with spin σ which only act on the single impurity orbital with energy ǫ d . This impurity site is hybridized to a bath with n s − 1 degrees of freedom through the coupling parameter V k ; the Coulomb interaction (U −2gω 0 σ z ) only occurs when two electrons are on the impurity site. The on-site Green's function for the lattice model is given by G(ω) = ∞ −∞ dx ρ(x) ω + µ − Σ(ω) − x(3) where Σ(w) is the local self-energy in infinite dimensions (q → ∞) and ρ(x) is the free density of electron states for a Bethe lattice: ρ(x) = 1 2πt ⋆2 4t ⋆2 − x 2 .(4) Note that a 'momentum' dependent spectral Green function is implied by Eq. (3), and, for momentum corresponding to bare energy x, is given by G(x, ω) −1 ≡ ω + µ − Σ(ω) − x;(5) this implies a spectral function A(x, ω) ≡ − 1 π ImG(x, ω), to be used later in the optical conductivity. Because of the simplicity of the non-interacting density of states, Eq. (3) can be integrated analytically. The result is: G(ω) = 1 t ω + µ − Σ(ω) 2t − ω + µ − Σ(ω) 2t 2 − 1 ,(6) where the () is taken with a sign equal to sgn[Re(ω + µ − Σ(ω))]. For the SIAM, the impurity Green's function can be written as: G imp (ω) = 1 ω + µ − ǫ d − ∆(ω) − Σ imp (ω)(7) in which ∆(ω) = ns k=2 V 2 k /(ω + µ − ǫ k ) is the hybridization function and Σ imp is the impurity self-energy. The self-consistent process is based on the local nature of the quantum system in the limit of infinite dimensions, in which the on-site Green's function for the lattice model can be averaged over all momenta and only depends on the frequency as we obtain in Eq. (3). Instead of directly solving the dynamic Hubbard lattice model, we only need to solve the SIAM (with fewer degrees of freedom); from this we obtain the impurity Green's function which should be the same as the on-site Green's function for the lattice: G(ω) = G imp (ω) (8) Σ(ω) = Σ imp (ω)(9) Eqs. (8,9) are the self-consistency conditions. A solution is required for the impurity Green's function. While an exact solution is available through QMC techniques, 9 Potthoff suggested a much faster though approximate procedure known as the two-site DMFT, 10 which he benchmarked for the Hubbard model. In this approach, 10 the SIAM with only two sites is diagonalized; one site is for the impurity and one site represents the bath, so n s = 2. Therefore the bath parameters are single numbers, as ǫ k = E c and V k = V . Besides making the procedure significantly faster, the two-site DMFT is more transparent, as the self-consistency conditions are analytic. The self-consistency conditions (8,9) are replaced by conditions at high and low frequency, to give two new self-consistentcy conditions which relate directly to the bath parameters: n imp = n(10)V 2 = zM (0) 2(11) where n imp (n) is the filling for the impurity site (conduction band in the lattice model). The parameter z = 1/(1 − Re(dΣ(0))/dω) has the meaning of the quasiparticle weight in the metal phase and M (0) 2 = <i,j> t 2 ij is the second moment of the non-interacting density of states. For the model adopted here, Eq. (4), this becomes M (0) 2 = t * 2 ≡ 1. Therefore the right-hand-side of Eq. (11) reduces to the quasiparticle weight. In fact, the procedure follows closely that given in Ref. 10, and the reader is referred to that publication for full details. Once the impurity problem is solved, one can obtain the density of states for the original lattice through A(ω) = − 1 π ImG(ω + i0 + ) = ρ(ω + µ − Re(Σ(ω))) (12) where the second equality only follows because the self energy is given by a two-pole approximation; 10 the selfenergy is obtained from the self-consistent condition, Eq. (9) and Dyson's equation, Σ imp (ω + i0 + ) = G (0) imp (ω + i0 + ) −1 − G imp (ω + i0 + ) −1 . (13) We will also be interested in the behavior of the optical conductivity, which within the same local approximation is given by the expression 8 σ 1 (ω) = e 2 t 2 a 2 π 2 ν +∞ −∞ dǫ f (ǫ) − f (ǫ + ω) ω × +∞ −∞ dyρ(y)A(y, ǫ)A(y, ǫ + ω),(14) where a is the lattice constant and ν = a d is the volume of the unit cell in d dimensions. As stated above, the single particle spectral function, A(x, ω), is defined as A(x, ω) ≡ − 1 π Im 1 ω + µ − Σ(ω) − x .(15) This function is immediately known once the self-energy is determined. III. RESULTS AND DISCUSSION A. Site Hamiltonian The Dynamic Hubbard model Hamiltonian, Eq. (1), consists of electron degrees of freedom that can move throughout the lattice, along with pseudospin degrees of freedom that reside at each lattice site. The pseudospins model the ability of the ions to 'react' to the different electronic configurations by changing the orbitals when electrons are and are not present. Following Ref. 4, we focus on the on-site Hamiltonian for electrons: H (i) DHM = ω 0 σ i x + gω 0 σ i z + [U − 2gω 0 σ i z ]n i↑ n i↓ ,(16) which is easily solved, given the presence of 0, 1, or 2 electrons. Using the spin-1/2 σ z eigenstates, \|+ >,\|− >, as a basis, we find, that with n electrons present, the eigenstates are \|n > a = u(n)\|+ > +v(n)\|− > (17) \|n > b = v(n)\|+ > +u(n)\|− >(18) with eigenvalues: ǫ(n) a = δ n,2 U − ω 0 1 + g 2 ǫ(n) b = δ n,2 U + ω 0 1 + g 2(19) The eigenvector components are given by u(0) = u(1) = v(2) (20) v(0) = v(1) = u(2),(21) with u 2 (0) = 1 2 (1 − g 1 + g 2 ) (22) v 2 (0) = 1 2 (1 + g 1 + g 2 ).(23) The expectation value of the pseudospin, in the ground state, illustrates the relaxation of the orbital required, depending on the number of electrons present. For example, the expectation value of σ z , in the ground state, is given by < 0\|σ z \|0 >=< 1\|σ z \|1 >= u 2 (0) − v 2 (0) = −g/ 1 + g 2 ,(24) and < 2\|σ z \|2 >= u 2 (2) − v 2 (2) = +g/ 1 + g 2 .(25) Similarly, for the ground state, we obtain < 0\|σ x \|0 >=< 1\|σ x \|1 >=< 2\|σ x \|2 >= −1/ 1 + g 2 . (26) For large g the z-component of the pseudospin switches from close to −1 to a value close to +1 as the occupancy changes from one to two electrons, but does not change when the occupancy changes from zero to one electron. In contrast, the x-component remains constant as the occupancy changes. Note that this occurs independently of the value of the on-site Coulomb repulsion, U ; in particular, the excitation energy associated with an excited pseudospin state is given by the difference of Eqs. (19), Ω 0 = 2ω 0 1 + g 2 .(27) B. Mott transition In this subsection, we examine how the dynamics of the auxiliary boson field affects the Mott transition in the Dynamic Hubbard model. In the next subsection we 11)), and the vertical increase of Ec at half-filling is also a symptom of the Mott gap. Note, in the cases with non-zero g (dynamic Hubbard model), the asymmetry with respect to half-filling. consider effects related to electron-hole asymmetry and 'undressing'. The effect of the dynamic field on the Mott transition is best addressed near half-filling. One of the signatures of this transition, as approached from the Fermi Liquid side, is the disappearance of quasi-particle weight (QW). As noted in Eq. (11) and below, the self-consistent parameter V measures the quasi-particle spectral weight. In Fig. 1 we show results for both self-consistent parameters, E c and V for a number of parameters. Focusing on the static Hubbard model, it is clear that, within the DMFT approximation, a Mott transition takes place for a critical value of 4 < U c < 8, since, for U = 4 the parameter V remains non-zero over all electron fillings, while, for U = 8 the parameter V dips to zero at halffilling. Similarly, upon examining E c vs. n ( Fig. 1(a)), one sees a vertical jump at half-filling (present for U = 8 but absent for U = 4) as the characteristic signature of the Mott phase. The other two parameter choices illustrate that, for sufficiently large pseudospin coupling g, the Mott transition occurs, even for modest values of the bare Hubbard interaction, U . Thus the Mott transition is induced for U = 4 with g = 3.8, for example. This fact is further reinforced in Fig. 1(a), and redrawn in Fig. 2 using the relation between E c and µ that is implicit in Eq. (10), where now a plateau is present near half-filling for the parameter set U = 4 and g = 3.8, thus indicating the occurrence of a Mott transition. Further insight into the occurrence of the Mott insulator (and the inherent particle-hole asymmetry in this model described in the next subsection) can be gained by examining the behaviour of the expectation values of the pseudospin variables. We plot in Fig. 3 the expectation values σ x and σ z as a function of filling for two sets of parameters, one in which a metal-insulator transition does not take place (g = 2, shown in pink), and one in which it does (g = 3.8, shown in green). We should note that our values are in quantitative agreement with those of Ref. (7) (note that our values for σ x are negative while their's are positive). Fig. 3 makes clear that below half-filling the pseudospin expectation values are fairly constant as a function of electron filling. An expectation value of σ z = −1 maintains an effective U that is U + 2gω 0 , much higher than U itself, so that double occupancy is restricted (see Fig. 5 below). A simple way to view the metal-insulator transition is through a variational approach, where, on the one hand, one uses the product state in which the electrons form a Fermi sea, while the pseudospins remain in their ground state, σ zi = −1, for all sites, i. A simple calculation yields where n is the electron concentration, and we have used a simplifying assumption that the electron density of states is a constant, g(ǫ) = 1/(8t), appropriate to a 2D tightbinding model with nearest neighbour hopping t. This is essentially a Hartree calculation. The competing state is an insulator, with one electron per site (at least up to half-filling), no hopping, and a similar pseudospin state, σ zi = −1, for all sites, i. The energy per site for this state is −gω 0 . Therefore, restricting ourselves to half-filling, a metal-insulator transition will occur for U beyond U crit , where U crit = 8t − 2gω 0 . The critical value of U is clearly lower as g increases; this is because while the pseudospins remain in their ground state the effective value of on-site Coulomb repulsion is increased by the presence of the coupling to the pseudospin. Therefore U itself can be smaller and, in combination with the effect of the pseudospin, still instigate a metal-insulator transition. Note that the values of σ z , as given by Eq. (24) for low filling are −0.89 and −0.97 for g = 2 and g = 3.8, respectively; these are close to −1, as used in the variational calculation, and also agree very well with the numerical results shown in Fig. 3. This in turn leads to a more accurate estimate of the bare on-site repulsion, given by 4 H Fermi N = −2tn(2 − n) − gω 0 + (U + 2gω 0 )n 2 /4,(28)U bare = U + 2g 2 ω 0 1 + g 2 ,(29) which results when the background degree of freedom is not allowed to relax. As the filling increases above half-filling, the expectation values of the pseudospins change markedly. Electrons are no longer able to avoid double occupancy, so the pseudospin steadily changes from the \|− > state to the \|+ > state, to lower the on-site energy from U max = U + 2gω 0 to U min = U − 2gω 0 . As Fig. 3 indicates, there is essentially a linear increase of < σ z > from − g √ 1+g 2 ∼ −1 to + g √ 1+g 2 ∼ +1 as the band becomes completely full (n = 2), as indicated by Eqs. (24) and (25). The relaxation of the pseudospin degree of freedom results in a lower quasiparticle weight, as Fig. 1(b) indicates. In Fig. 2, the curve representing g = 3.8 (shown in green) approaches full occupation at µ ≈ 4. This point can be understood by the fact that U bare is approximately zero when n = 2 (see Eq. (29)), so that the chemical potential goes to the top of the bare band (2t * ) plus the energy shift due to the pseudospin ω0 √ g 2 +1 (g 2 −1) ≈ 1.7t * . As pointed out in Ref. (10), for the Hubbard model, the two-site DMFT approximation correctly produces three peaks in the single particle density of states corresponding to the lower and upper Hubbard bands, and a quasi-particle resonance peak at the Fermi energy, with quasi-particle weight z. For the dynamic Hubbard model, we show in Fig. 4 results for the parameters U = 4, g = 3.8 and ω 0 = 0.5, along with results for the Hubbard model with U = 8, for a number of different electron densities. Because of the pseudospin degree of freedom, the spectrum should contain at least four peaks, due to the appearance of more than two poles in the self energy; this is clearly the case in Fig. 4. Below half-filling ( Fig. 4(a)), the peak structure is very similar to that found in the Hubbard model, with U = 8. At half-filling ( Fig. 4(b), n = 1.0), the resonant peak at the Fermi level has all but disappeared, indicative of the Mott transition. Also shown are the results for much higher filling (Fig. 4(c), n = 1.5), where the results are clearly not symmetric with those at n = 0.5 ( Fig. 4(a)), and certainly no longer similar to the results for the Hubbard model with U = 8. As found in Ref. (7), the peak near the Fermi energy is considerably sharper at n = 1.5 compared with n = 0.5, indicating that this model is less free electron-like at high electron filling compared to low electron filling (as seen in Fig. 1(b) as well). Fig. 5 shows the double occupancy as a function of electron filling. Again, very little double occupancy is present below half filling, as expected, though it is clear that the presence of the electron-pseudospin coupling g suppresses the double occupancy near half-filling, and therefore enhances the Mott-like features of the Hubbard model. Above half-filling, the double occupancy quickly rises; though it is a more subtle effect here, the presence of g enhances the double occupancy since, as the filling increases the effective Hubbard U is decreasing due to the relaxation of the pseudospin degrees of freedom. In Fig. 6 we show the total energy as a function of electron filling, for a number of parameters, as indicated in the figure. For an empty or completely full band the numerical results agree with those given by the analytical results obtained for the site Hamiltonian, Eq. (16): for the dynamic model with U = 4, g = 3.8, and ω0 = 0.5, for three different fillings. The two models behave very similarly below halffilling; at half-filling both undergo a Mott transition with the disappearance of the width of the resonance peak at zero frequency. Above half-filling the two models differ markedly; in particular, the peak width at the Fermi level is considerably smaller for the dynamic Hubbard model compared with the static Hubbard model. E(n = 0) = −ω 0 1 + g 2(30)E(n = 2) = U − ω 0 1 + g 2 .(31) The results in Ref. (7) (their Fig. 14) are also in excellent agreement with the exact results given by the above equations. C. Electron-hole asymmetry and undressing phenomenology It was proposed in Ref. (5) that the dynamic Hubbard model describes superconductivity driven by "undressing": namely, that when the Fermi level is near the top of the band, pairing of hole carriers will lead to transfer of spectral weight from high to low frequencies and in particular in an increase of the quasiparticle weight and a decrease in the effective mass. These effects should appear both in the single particle spectral function and in Fig. 7, except that now g is increased to a value g = 2. Increasing g also increases the asymmetry with respect to half filling. two particle spectral functions such as the optical conductivity. In the present paper we do not consider pairing correlation functions, and hence we cannot ascertain from our results whether or not the model describes superconductivity. However, we can study properties of the model under hole doping. It is expected that the effects discussed in the previous paragraph should also occur both in the normal and the superconducting state for an almost filled electron band as a function of increased hole doping 5 . For the parameters considered in the previous subsec- Fig. 7, Mott physics is de-emphasized; instead, a decrease in spectral weight as electron filling increases arises primarily because of the role of the pseudospin degree of freedom. Note that the curves arise from data obtained through converging the parameter V through the DMFT iterative process. The symbols come from integrating the spectral function peak at the Fermi level (see Fig. 12 below), and the good agreement is clear. tion, it was found that doping the full band with holes led to a decrease rather than an increase in the quasiparticle weight, in contradiction to these expectations. Thus we also do not expect superconductivity driven by "undressing" for those parameters. However, we find that the expected "undressing" behavior does occur for lower values of the on-site repulsion U and/or larger values of the coupling g as well as for higher values of the boson frequency ω 0 . Figure 7 shows the behavior of the quasiparticle weight z (recall V = z 2 ) for U = 2, g = 1 and various values of ω 0 versus band filling. It can be seen that for ω 0 ≥ 2 the quasiparticle weight indeed increases when the full band is doped with holes (i.e. the quasiparticle weight decreases with electron filling as n → 2). Fig. 8 shows the same behavior occurring already for ω 0 ≥ 1 when we increase the value of the coupling to g = 2 with U = 2. Similarly, the same behaviour occurring for ω 0 ≥ 1 can be obtained by reducing the on-site repulsion to U = 1 while keeping g = 1 (Figure 9). For these parameter ranges a Mott transition at half-filling does not occur. As shown in Fig. 10, for large values of the frequency ω 0 , the behavior of the quasiparticle weight versus band filling n is described approximately by the expression z = [1 + (S − 1) n 2 ] 2(32) with Fig. 9, except that now we have increased ω0 to very large values; in this limit we should recover the correlated hopping model, and the result at low hole doping (near full electron occupation) should follow the analytical result as indicated. 5 This is clearly the case. as expected. 5 Also, as Fig. 10 shows, for smaller values of ω 0 the n−dependence of z is qualitatively similar but, as n → 2, the magnitude is larger than that given by Eq. (32). This dependence of quasiparticle weight on the boson frequency is consistent with the behavior found in Ref. 6 using a generalized Lang-Firsov transformation within an Eliashberg treatment. It was also found in that work that smaller boson frequency enhances the tendency to pairing. Figure 11 shows that the analytical form, Eq. (32) is indeed very accurate when U is not present. Results shown for increasing values of U indicate additional decreasing of quasiparticle weight that occurs due to well documented 'Hubbard physics'. Nonetheless, for all cases, the overall decreasing trend as a function of electron filling is clearly coming from 'quasiparticle dressing' due to the pseudospin degree of freedom (Eqs. (32) and (33)). S 2 = 1 1 + g 2(33) In Figure 12 we show the behavior of the single particle spectral function for one particular electron filling, n = 1.8, for a variety of 'momenta'. Each spectral function consists primarily of two peaks, separated by roughly the pseudospin excitation energy Ω 0 given by Eq. (27). The weight of the quasiparticle peak at the Fermi level, shown in the 5th panel, corresponds to the residue z plotted for all fillings in Fig. 9 (middle curve). As shown there, this quasiparticle weight decreases (increases) with increasing electron (hole) concentration. In Fig. 13 we show the behaviour of the optical conductivity obtained using Eq. (14), for several electron fillings. An exact calculation of the optical conductivity for a two-site model 16 shows that the optical conductivity can generally be divided into four contributions, as we now describe. There will be a Drude part centered at ω = 0 with a width 1/τ , normally due to elastic impurity scattering. In our calculations, this part appears as a δ-function at the origin, with artificially imposed broadening (see Fig. 13). This component involves transitions between the two coherent parts of the spectral functions (see Eq. (14)). For low electron fillings these transitions give rise to essentially the entire frequency dependent conductivity. Three more components contribute as the electron filling comes close to and exceeds half-filling. They have characteristic frequencies ω i , and weights C i , for i = L, M, H, corresponding to 'low', 'medium', and 'high'. These designations are relevant only when U is sufficiently small, as in Fig. 13. Then the 'low' frequency part is peaked near ω L ≈ U/2 + (U/2) 2 + 4t 2 S 2 , and involves transitions between the ground state and an excited state with a doubly occupied site (both states have pseudospins in their ground states). The quantity S ≡ 1\|2 gives an estimate of the polaronic effect of the pseudospin excitation required as an electron undergoes a hop. 16 The characteristic frequency for the 'medium' range is ω M ≈ Ω 0 = 2ω 0 1 + g 2 . This is the excitation energy for a pseudospin (see Eq. (27)), and transitions between states differing by a pseudospin excitation are responsible for this part of the conductivity. These transitions will play a more significant role as the electron-pseudospin coupling strength, g, increases. Finally, the high frequency characteristic frequency is ω H ≈ 2Ω 0 , and requires a transition in which two pseudospin excitations are created. This necessarily tends The spectral function for various energies, as shown, for n = 1.8. Here, we have used U = 1, g = 1, and ω0 = 2 (see pink curve in Fig. 9). The spectral functions consist primarily of two peaks, separated by an energy corresponding to the pseudospin excitation energy, Ω0. The 2nd last frame shows the spectral function at an energy corresponding to the Fermi level, and the weight under the peak at ω = 0 corresponds to the quasiparticle residue, z. to happen only when the electron concentration is very high, i.e. in the hole region. The weights of these various contributions are difficult to estimate in advance; they will depend on the hopping overlap integrals. Fig. 13 shows the optical conductivity at various fillings for U = 1, g = 1, and ω 0 = 2.0; then ω L ≈ 2.0, ω M ≈ 5.7 and ω H ≈ 11.3. For these same microscopic parameters, we also show the expectation values of the z and x components of the pseudospin degree of freedom as a function of electron density in Fig. 14. Their values are given analytically at n = 0 and n = 2: σ x = −1/ 1 + g 2 for both limits and σ z = −g/ 1 + g 2 (g/ 1 + g 2 ) for n = 0 (n = 2), and DMFT obviously gets these correctly. The steady increase of σ z with increasing electron density reflects the increased occurrence of the excited pseudospin state in the ground state as the lattice becomes more crowded. A more in-depth understanding of the optical conductivity comes from examining the spectral weight, partitioned into the various contributions, as described above. For concreteness we define the Drude conductivity to include contributions in the range, 0 < ω < ω L . The other contributions are defined as follows: low frequency in the range ω L < ω < ω M , midrange for ω M < ω < ω H , and the high frequency range for ω > ω H . This is shown in Fig. 15. First, note that the total spectral weight is generally asymmetric as a function of electron concentration. Note, moreover, that as the electron concentration approaches zero or full filling, the spectral weight approaches zero, since there are no carriers in either case. The Drude weight, defined as the area under the nearzero frequency portion of the conductivity, is also asymmetric; this is shown by the black triangular points. An integration over the Drude portion of Eq. (14) shows that the Drude spectral weight satisfies the sum rule 8 : ∞ 0 σ Dr (ω)dω = − πe 2 a 2 2d 2 ν < K >= ω * p 2 8π(34) and in the limit of infinite dimension d → ∞: ω * p 2 4π = 4πt 2 e 2 a 2 2 ν zρ(ǫ F )(35) Here ω * p is the renormalized plasma frequency. We can see from Eq. (35) that the Drude weight depends on the quasiparticle weight z and the DOS at ǫ F = µ − Re(Σ(0)). The result from Eq. (35) is also plotted in Fig. 15, and is even more asymmetric than the total spectral weight. In addition, other contributions are also plotted as a function of electron filling. Note that the total weight is Drude-like both for n → 0 and for n → 2, i.e. the contributions from the finite frequency portions fall off more quickly. Nonetheless, it is clear that spectral weight at high frequency is most intense in the hole-like region (n > 1) as compared with the electron-like region (n < 1). This can be understood by the following argument. First, we focus on the (minor) role of transitions involving the Hubbard U . These are described by the so-called 'low' frequency contribution, and they peak at half-filling. This is because that filling corresponds to the situation where more fluctuations are liable to occur. At low fillings U hardly plays any role, while at high filling U , by virtue of playing the same role for almost all electrons, again hardly plays any role. With excitations involving the pseudospin degree of freedom the situation is a little more subtle. At low to intermediate fillings, the pseudospins play almost no role because single electron transitions do not require a relaxation of the pseudospin degree of freedom. At high to intermediate fillings, however, transitions generally involve the pseudospin degree of freedom, because the overlap between a singly occupied state and a doubly occupied state, S ≡ 1\|2 , can be much less than unity (in contrast, for this model, the corresponding overlap between an empty and singly occupied site, T ≡ 0\|1 remains unity). Thus, there is an electron-hole asymmetry, with the pseudospin physics playing a large role in the electron density region 1 < n < 2. In fact, variation in the pseudospin expectation value, σ x , reflects this fact also. Note the resemblance between the electron density dependence of σ x in Fig. 14 (inverted) with the electron density dependence shown by the spectral weight in the medium frequency range shown by the blue asterisks in Fig. 15 (the parameter regimes used in Fig. 3 depict this variation of σ x in the hole-doped region even more pronouncedly). Hence, optical spectral weight in the medium frequency region plays a more important role for electron densities on the hole-like side of the phase diagram. This is consistent with the explanation in Ref. (7), that since σ x measures the fluctuations of σ z (in imaginary time), it should achieve a minimum (they obtained a maximum for reasons that are unclear) at the point where σ z is changing the most quickly (as a function of electron density). This tends to occur midway through the hole-like side of the phase diagram, i.e. near n ≈ 1.5. Such transfer of spectral weight from high to low frequencies as function of increasing hole concentration is seen in high T c cuprates, both in the single particle spectral function (photoemission experiments) 17 and in the two particle spectral function (optical conductivity) 18-21 . in the density of states. An important property of this model is its asymmetry with respect to half filling. The hole side (n > 1) is always considerably more dressed than the electron side, because of the relaxation of the pseudospin degree of freedom. This occurs because electrons minimize their Coulomb repulsion, at a cost of becoming somewhat more sluggish in their movements, i.e. they form polaronic-like states. We have identified a parameter regime where the electron-hole asymmetry is very clear (see , and the quasiparticle spectral weight increases linearly with hole doping away from n = 2, as described in an effective model with correlated hopping. 2 This quasiparticle undressing is a general phenomenon that occurs not only as a function of hole doping, but as a function of, for example, temperature changes and phase transitions. 22 Understanding the degree to which this undressing is robust as the auxiliary degree of freedom is reduced from the anti-adiabatic limit to a more physical regime is one of the goals of this paper. Fig. 9 in particular illustrates that the degree of dressing (at n = 2) is reduced as ω 0 is reduced, and therefore the degree of 'undressing' as the electron occupation is decreased from n = 2 is reduced. Future work will determine the impact of this frequency scale on superconductivity. We computed single particle spectral weights and the optical conductivity for this model. In both cases the frequency dependence is distinctly different for electronlike and hole-like doping levels. The asymmetric behavior for holes vs. electrons is clear as a function of doping; determining this asymmetry for a given doping level can be established through photoemission or tunneling experiments. Fig. 13), is plotted as a function of electron filling. As in Fig. 13, we have used the same parameter set U = 1, g = 1, ω0 = 2.0. FIG. 1 : 1Bath parameters (a) Ec and (b) V as a function of electron filling, for parameters with and without a Mott transition. For the static Hubbard model we use U = 8 and U = 4, as cases with and without a Mott gap, to illustrate the expected features in either case. The disappearance of V 2 at half-filling demonstrates the vanishing of quasiparticle spectral weight (see Eq. ( online) Electron filling as a function of chemical potential, for the cases indicated inFig. 1. Note the plateaus near half-filling, indicative of the Mott transition. online) The pseudospin expectation values < σx > and < σz > as a function of electron filling, for the dynamic Hubbard model, with g = 3.8 (with a Mott gap transition), and g = 2 (without a Mott transition). Note that the pseudospin plays very little role below half-filling, but undergoes a big change above half-filling. online) A comparison of the spectral function for the static model with U = 8 to the one online) The dependence of double occupancy vs. electron filling, for various parameters, for both the static and dynamic Hubbard models. For sufficiently large g in the dynamic Hubbard model, the double occupancy is driven to zero at half-filling. FIG . 6: (Color online) The total ground state energy vs. electron filling for the same sets of parameters as inFig. 5. The values at n = 0 and n = 2 agree with the analytically derived values, as explained in the text. FIGFIG. 8 : 8. 7: (Color online) The quasiparticle spectral weight, z vs. electron filling, for a variety of parameter values. For the dynamic Hubbard model, increasing the pseudospin frequency ω0 leads to a steady decrease in the spectral weight, particularly at high value of electron filling, thus making the model more asymmetric with respect to half filling. (Color online) Same as FIG. 9 : 9(Color online) Same asFig. 7, but with a smaller value of U (here U = 1), and we have added a higher frequency result (ω0 = 20). With respect to FIG. 10: (Color online) Same as Fig. 9, except that now we have increased ω0 to very large values; in this limit we should recover the correlated hopping model, and the result at low hole doping (near full electron occupation) should follow the analytical result as indicated. 5 This is clearly the case. FIG . 11: (Color online) To disentangle spectral weight reduction due to the Hubbard U vs. the dynamic pseudospin effect (g), we show the quasiparticle spectral weight z vs. electron filling n for several values of U , including U = 0. All numerical results shown are for a large value of pseudospin frequency, ω0 = 50. Agreement with the analytical result is excellent for U = 0. Discrepancies for other values of U shown come primarily from the quasiparticle reduction due to U , which is not accounted for in the analytical result. FIG. 12: (Color online) The spectral function for various energies, as shown, for n = 1.8. Here, we have used U = 1, g = 1, and ω0 = 2 (see pink curve in Fig. 9). The spectral functions consist primarily of two peaks, separated by an energy corresponding to the pseudospin excitation energy, Ω0. The 2nd last frame shows the spectral function at an energy corresponding to the Fermi level, and the weight under the peak at ω = 0 corresponds to the quasiparticle residue, z. FIG . 13: (Color online) The optical conductivity is shown at different filling for U = 1, g = 1, ω0 = 2. Though not so apparent in this figure, there are four primary components with characteristic frequencies, ω = 0 (Drude), ωL ≈ U/2 + (U/2) 2 + 4t 2 S 2 ≈ 2, ωM = Ω0 = 2ω0 1 + g 2 ≈ 5.7 and ωH = 2Ω0 ≈ 11.3. FIG . 14: (Color online) The pseudospin expectation values < σx > and < σz > as a function of electron filling with U = 1, g = 1,FIG. 15: (Color online) The spectral weight of the optical conductivity, as a function of electron filling. The total weight, along with the weight of each component (see figure caption in IV. CONCLUSIONSWe have investigated various properties of the dynamic Hubbard model by using dynamical mean field theory in the two-site approximation. 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[ "Brownian Motion on Spaces with Varying Dimension ", "Brownian Motion on Spaces with Varying Dimension " ]	[ "Zhen-Qing Chen ", "Shuwen Lou " ]	[]	[]	In this paper we introduce and study Brownian motion on state spaces with varying dimension. Starting with a concrete case of such state spaces that models a big square with a flag pole, we construct a Brownian motion on it and study how heat propagates on such a space. We derive sharp two-sided global estimates on its transition density functions (also called heat kernel). These two-sided estimates are of Gaussian type, but the measure on the underlying state space does not satisfy volume doubling property. Parabolic Harnack inequality fails for such a process. Nevertheless, we show Hölder regularity holds for its parabolic functions. We also derive the Green function estimates for this process on bounded smooth domains. Brownian motion on some other state spaces with varying dimension are also constructed and studied in this paper.Recently, analysis on non-smooth spaces has attracted lots of interest. In real world, there are many objects having varying dimension. It is natural to study Brownian motion and "Laplacian operator" on such spaces. A simple example of spaces with varying dimension is a large square with a thin flag pole. Mathematically, it is modeled by a plane with a vertical line installed on it:Here and in the sequel, we use := as a way of definition and denote [0, ∞) by R + . Spaces with varying dimension arise in many disciplines including statistics, physics and engineering (e.g. molecular dynamics, plasma dynamics). See, for example,[28,37]and the references therein. The goal of this paper is to construct and study Brownian motion and Laplacian on spaces of varying dimension, in particular, to investigate how heat propagates on such spaces. Intuitively, Brownian motion on space R 2 ∪R of (1.1) should behave like a two-dimensional Brownian motion when it is on the plane, and like a one-dimensional Brownian motion when it is on the vertical line (flag pole). However the space R 2 ∪R is quite singular in the sense that the base O of the flag pole where the plane and the vertical line meet is a singleton. A singleton would never be visited by a two-dimensional Brownian motion, which means Brownian motion starting from a point on the plane will never visit O. Hence there is no chance for such a process to climb up the flag pole. So the idea is to collapse or short (imagine putting an infinite conductance on) a small closed disk B(0, ε) ⊂ R 2 centered at the origin into a point a * and consider the resulting Brownian motion with darning on the collapsed plane, for which a * will be visited. The notion of Brownian motion with darning is coined in [12] and its potential theory has been studied in details in[11]and[13,. Through a * we put a vertical pole and construct Brownian motion with varying dimension on R 2 ∪ R + by joining together the Brownian motion with darning on the plane and the one-dimensional Brownian motion along the pole. It is possible to construct the process rigorously via Poisson point process of excursions. But we find that the most direct way to construct BMVD is by using a Dirichlet form approach, which will be carried out in Section 2.To be more precise, the state space of BMVD on E is defined as follows. Fix ε > 0 and p > 0. Denote by B ε the closed disk on R 2 centered at (0, 0) with radius ε. Let D 0 = R 2 \ B ε . By identifying B ε with a singleton denoted by a * , we can introduce a topological space E := D 0 ∪ {a * } ∪ R + , with the origin of R + identified with a * and a neighborhood of a * defined as {a * } ∪ (U 1 ∩ R + ) ∪ (U 2 ∩ D 0 ) for some neighborhood U 1 of 0 in R 1 and U 2 of B ε in R 2 . Let m p be the measure on E whose restriction on R + and D 0 is the Lebesgue measure multiplied by p and 1, respectively. In particular, we have m p ({a * }) = 0.Definition 1.1. Let ε > 0 and p > 0. A Brownian motion with varying dimensions (BMVD in abbreviation) with parameters (ε, p) on E is an m p -symmetric diffusion X on E such that (i) its part process in R or D 0 has the same law as standard Brownian motion in R or D 0 ;(ii) it admits no killings on a * .It follows from the m p -symmetry of X and the fact m p ({a * }) = 0 that BMVD X spends zero Lebesgue amount of time at a * .The following will be established in Section 2.	10.1214/18-aop1260	[ "https://arxiv.org/pdf/1604.07870v1.pdf" ]	25,757,228	1604.07870	cebe648aa4b47090ea23548ca15f906dc2297db0	Brownian Motion on Spaces with Varying Dimension * 26 Apr 2016 April 28, 2016 Zhen-Qing Chen Shuwen Lou Brownian Motion on Spaces with Varying Dimension * 26 Apr 2016 April 28, 2016arXiv:1604.07870v1 [math.PR]AMS 2010 Mathematics Subject Classification: Primary 60J6060J35; Secondary 31C2560H3060J45 Keywords and phrases: Space of varying dimensionBrownian motionLaplaciantransition density functionheat kernel estimatesHölder regularityGreen function In this paper we introduce and study Brownian motion on state spaces with varying dimension. Starting with a concrete case of such state spaces that models a big square with a flag pole, we construct a Brownian motion on it and study how heat propagates on such a space. We derive sharp two-sided global estimates on its transition density functions (also called heat kernel). These two-sided estimates are of Gaussian type, but the measure on the underlying state space does not satisfy volume doubling property. Parabolic Harnack inequality fails for such a process. Nevertheless, we show Hölder regularity holds for its parabolic functions. We also derive the Green function estimates for this process on bounded smooth domains. Brownian motion on some other state spaces with varying dimension are also constructed and studied in this paper.Recently, analysis on non-smooth spaces has attracted lots of interest. In real world, there are many objects having varying dimension. It is natural to study Brownian motion and "Laplacian operator" on such spaces. A simple example of spaces with varying dimension is a large square with a thin flag pole. Mathematically, it is modeled by a plane with a vertical line installed on it:Here and in the sequel, we use := as a way of definition and denote [0, ∞) by R + . Spaces with varying dimension arise in many disciplines including statistics, physics and engineering (e.g. molecular dynamics, plasma dynamics). See, for example,[28,37]and the references therein. The goal of this paper is to construct and study Brownian motion and Laplacian on spaces of varying dimension, in particular, to investigate how heat propagates on such spaces. Intuitively, Brownian motion on space R 2 ∪R of (1.1) should behave like a two-dimensional Brownian motion when it is on the plane, and like a one-dimensional Brownian motion when it is on the vertical line (flag pole). However the space R 2 ∪R is quite singular in the sense that the base O of the flag pole where the plane and the vertical line meet is a singleton. A singleton would never be visited by a two-dimensional Brownian motion, which means Brownian motion starting from a point on the plane will never visit O. Hence there is no chance for such a process to climb up the flag pole. So the idea is to collapse or short (imagine putting an infinite conductance on) a small closed disk B(0, ε) ⊂ R 2 centered at the origin into a point a * and consider the resulting Brownian motion with darning on the collapsed plane, for which a * will be visited. The notion of Brownian motion with darning is coined in [12] and its potential theory has been studied in details in[11]and[13,. Through a * we put a vertical pole and construct Brownian motion with varying dimension on R 2 ∪ R + by joining together the Brownian motion with darning on the plane and the one-dimensional Brownian motion along the pole. It is possible to construct the process rigorously via Poisson point process of excursions. But we find that the most direct way to construct BMVD is by using a Dirichlet form approach, which will be carried out in Section 2.To be more precise, the state space of BMVD on E is defined as follows. Fix ε > 0 and p > 0. Denote by B ε the closed disk on R 2 centered at (0, 0) with radius ε. Let D 0 = R 2 \ B ε . By identifying B ε with a singleton denoted by a * , we can introduce a topological space E := D 0 ∪ {a * } ∪ R + , with the origin of R + identified with a * and a neighborhood of a * defined as {a * } ∪ (U 1 ∩ R + ) ∪ (U 2 ∩ D 0 ) for some neighborhood U 1 of 0 in R 1 and U 2 of B ε in R 2 . Let m p be the measure on E whose restriction on R + and D 0 is the Lebesgue measure multiplied by p and 1, respectively. In particular, we have m p ({a * }) = 0.Definition 1.1. Let ε > 0 and p > 0. A Brownian motion with varying dimensions (BMVD in abbreviation) with parameters (ε, p) on E is an m p -symmetric diffusion X on E such that (i) its part process in R or D 0 has the same law as standard Brownian motion in R or D 0 ;(ii) it admits no killings on a * .It follows from the m p -symmetry of X and the fact m p ({a * }) = 0 that BMVD X spends zero Lebesgue amount of time at a * .The following will be established in Section 2. Introduction Brownian motion takes a central place in modern probability theory and its applications, and is a basic building block for modeling many random phenomena. Brownian motion has intimate connections to analysis since its infinitesimal generator is Laplacian operator. Brownian motion in Euclidean spaces has been studied by many authors in depth. Brownian motion on manifolds and on fractals has also been investigated vigorously, and is shown to have intrinsic interplay with the geometry of the underlying spaces. See [26,29,30,32] and the references therein. In most of these studies, the underlying metric measure spaces are assumed to satisfy volume doubling (VD) property. For Brownian motion on manifolds with walk dimension 2, a remarkable fundamental result obtained independently by Grigor'yan [22] and Saloff-Coste [33] asserts that the following are equivalent: (i) two-sided Aronson type Gaussian bounds for heat kernel, (ii) parabolic Harnack equality, and (iii) VD and Poincaré inequality. This result is then extended to strongly local Dirichlet forms on metric measure space in [9,34,35] and to graphs in [19]. For Brownian motion on fractals with walk dimension larger than 2, the above equivalence still holds but one needs to replace (iii) with (iii') VD, Poincaré inequality and a cut-off Sobolev inequality; see [3,4,1]. Theorem 1.2. For every ε > 0 and p > 0, BMVD with parameters (ε, p) exists and is unique in law. We point out that BMVD on E can start from every point in E. We further characterize the L 2 -infinitesimal generator L of BMVD X in Section 2, which can be viewed as the Laplacian operator on this singular space. We show that u ∈ L 2 (E; m p ) is in the domain of the generator L if and only if ∆u exists as an L 2 -integrable function in the distributional sense when restricted to D 0 and R + , and u satisfies zero-flux condition at a * ; see Theorem 2.3 for details. It is not difficult to see that BMVD X has a continuous transition density function p(t, x, y) with respect to the measure m p , which is also called the fundamental solution (or heat kernel) for L. Note that p(t, x, y) is symmetric in x and y. The main purpose of this paper is to investigate how the BMVD X propagates in E; that is, starting from x ∈ E, how likely X t travels to position y ∈ E at time t. This amounts to study the properties of p(t, x, y) of X. In this paper, we will establish the following sharp two-sided estimates on p(t, x, y) in Theorem 1.3 and Theorem 1.4. To state the results, we need first to introduce some notations. Throughout this paper, we will denote the geodesic metric on E by ρ. Namely, for x, y ∈ E, ρ(x, y) is the shortest path distance (induced from the Euclidean space) in E between x and y. For notational simplicity, we write \|x\| ρ for ρ(x, a * ). We use \| · \| to denote the usual Euclidean norm. For example, for x, y ∈ D 0 , \|x − y\| is the Euclidean distance between x and y in R 2 . Note that for x ∈ D 0 , \|x\| ρ = \|x\| − ε. Apparently, ρ(x, y) = \|x − y\| ∧ (\|x\| ρ + \|y\| ρ ) for x, y ∈ D 0 (1. 2) and ρ(x, y) = \|x\| + \|y\| − ε when x ∈ R + and y ∈ D 0 or vice versa. Here and in the rest of this paper, for a, b ∈ R, a ∧ b := min{a, b}. The following are the main results of this paper. Theorem 1.3. Let T > 0 be fixed. There exist positive constants C i , 1 ≤ i ≤ 14 so that the transition density p(t, x, y) of BMVD satisfies the following estimates when t ∈ (0, T ]: (i) For x ∈ R + and y ∈ E, C 1 √ t e −C 2 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 3 √ t e −C 4 ρ(x,y) 2 /t . (1.3) (ii) For x, y ∈ D 0 ∪ {a * }, when \|x\| ρ + \|y\| ρ < 1, C 5 √ t e −C 6 ρ(x,y) 2 /t + C 5 t 1 ∧ \|x\| ρ √ t 1 ∧ \|y\| ρ √ t e −C 7 \|x−y\| 2 /t (1.4) ≤ p(t, x, y) ≤ C 8 √ t e −C 9 ρ(x,y) 2 /t + C 8 t 1 ∧ \|x\| ρ √ t 1 ∧ \|y\| ρ √ t e −C 10 \|x−y\| 2 /t ; and when \|x\| ρ + \|y\| ρ ≥ 1, C 11 t e −C 12 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 13 t e −C 14 ρ(x,y) 2 /t . (1.5) Since p(t, x, y) is symmetric in (x, y), the above two cases cover all the cases for x, y ∈ E. Theorem 1.3 shows that the transition density function p(t, x, y) of BMVD on E has onedimensional character when at least one of x, y is in the pole (i.e. in R + ); it has two-dimensional character when both points are on the plane and at least one of them is away from the pole base a * . When both x and y are in the plane and both are close to a * , p(t, x, y) exhibits a mixture of one-dimensional and two-dimensional characters; see (1.4). Theorem 1.3 will be proved through Theorems 4.5-4.7. The large time heat kernel estimates for BMVD are given by the next theorem, which are very different from the small time estimates. Theorem 1.4. Let T > 0 be fixed. There exist positive constants C i , 15 ≤ i ≤ 32, so that the transition density p(t, x, y) of BMVD satisfies the following estimates when t ∈ [T, ∞): (i) For x, y ∈ D 0 ∪ {a * }, C 15 t e −C 16 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 17 t e −C 18 ρ(x,y) 2 /t . (ii) For x ∈ R + , y ∈ D 0 ∪ {a * }, C 19 t 1 + \|x\| log t √ t e −C 20 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 21 t 1 + \|x\| log t √ t e −C 22 ρ(x,y) 2 /t when \|y\| ρ ≤ 1, and 26 ρ(x,y) 2 /t when \|y\| ρ > 1. C 23 t 1 + \|x\| √ t log 1 + √ t \|y\| ρ e −C 24 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 25 t 1 + \|x\| √ t log 1 + √ t \|y\| ρ e −C (iii) For x, y ∈ R + , 28 \|x−y\| 2 /t + C 27 t 1 + (\|x\| + \|y\|) log t √ t e −C 29 (x 2 +y 2 )/t ≤ p(t, x, y) 31 \|x−y\| 2 /t + C 30 t 1 + (\|x\| + \|y\|) log t √ t e −C 32 (x 2 +y 2 )/t . Theorem 1.4 will be proved through Theorems 5.14, 5.15 and 5.17. Due to the singular nature of the space, the standard Nash inequality and Davies method for obtaining heat kernel upper bound do not give sharp bound for our BMVD. We can not employ either the methods in [22,33,9,34,35] on obtaining heat kernel estimates through volume doubling and Poincaré inequality or the approach through parabolic Harnack inequality. In fact, (E, m p ) does not have volume doubling property, and we will show the parabolic Harnack inequality fails for BMVD X; see Proposition 2.1 and Remark 4.8(iii). Hence a new approach is needed to study the heat kernel of BMVD. A key role is played by the "signed radial process" of BMVD, which we can analyze and derive its two-sided heat kernel estimates. From it, by exploring the rotational symmetry of BMVD, we can obtain short time sharp two-sided heat kernel estimates by the following observation. The sample paths of BMVD starting at x reach y at time t in two possible ways: with or without passing through a * . The probability of the first scenario is given exactly by the probability transition density function of killed Brownian motion in D 0 . The probability of the second scenario can be computed by employing the strong Markov property of BMVD at the first hitting time of the pole base a * and reducing it to the signed radial process of BMVD, exploring the symmetry of BMVD starting from a * . The large time heat kernel estimates are more delicate. For large time estimate, the key is to obtain the correct on-diagonal estimate. This is done through some delicate analysis of BMVD and Bessel process on the plane. As a corollary of the sharp two-sided heat kernel estimates, we find that the usual form of the parabolic Harnack inequality fails for parabolic functions of BMVD. Nevertheless, we will show in Section 6 that joint Hölder regularity holds for bounded parabolic functions of X. C 27 √ t 1 ∧ \|x\| √ t 1 ∧ \|y\| √ t e −C≤ C 30 √ t 1 ∧ \|x\| √ t 1 ∧ \|y\| √ t e −C Let X D be the part process of BMVD killed upon exiting a bounded connected C 1,1 open subset D of E. Denote by p D (t, x, y) its transition density function. Using the Green function estimates and boundary Harnack inequality for absorbing Brownian motion in Euclidean spaces, we can derive sharp two-sided estimates on Green function G D (x, y) := ∞ 0 p D (t, x, y)dt for BMVD in D. Recall that an open set D ⊂ R d is called to be C 1,1 if there exist a localization radius R 0 > 0 and a constant Λ 0 > 0 such that for every z ∈ ∂D, there exists a C 1, 1 −function φ = φ z : R d−1 → R satisfying φ(0) = 0, ∇φ(0) = (0, . . . , 0), ∇φ ∞ ≤ Λ 0 , \|∇φ(x) − ∇φ(z)\| ≤ Λ\|x − z\| and an orthonormal coordinate system CS z : y = (y 1 , . . . , y d ) := ( y, y d ) with its origin at z such that B(z, R 0 ) ∩ D = {y ∈ B(0, R 0 ) in CS z : y d > φ( y)}. For the state space E, an open set D ⊂ E will be called C 1,1 in E, if D ∩ (R + \ {a * }) is a C 1,1 open set in R + , and D ∩ D 0 is a C 1,1 open set in R 2 . Theorem 1.5. Suppose D is a bounded C 1,1 domain of E that contains a * . Let G D (x, y) be the Green function of BMVD X killed upon exiting D. Then for x = y in D, we have G D (x, y) ≍              δ D (x) ∧ δ D (y), x, y ∈ D ∩ R + ; (δ D (y) ∧ 1) (δ D ∧ 1) + ln 1 + δ D∩D 0 (x)δ D∩D 0 (y) \|x−y\| 2 , x, y ∈ D ∩ D 0 ; δ D (x)δ D (y), x ∈ D ∩ R + , y ∈ D ∩ D 0 . Here δ D (x) := dist ρ (x, ∂D) := inf{ρ(x, z) : z / ∈ D} and δ D∩D 0 (x) := inf{ρ(x, z) : z / ∈ D ∩ D 0 }. For two positive functions f and g, f ≍ g means that f /g is bounded between two positive constants. In the following, we will also use notation f g (respectively, f g) to mean that there is some constant c > 0 so that f ≤ cg (respectively, f ≥ cg). The above space E of varying dimension is special. It serves as a toy model for further study on Brownian motion on more general spaces of varying dimension. Another two examples of spaces of varying dimension and BMVD on them are given and studied in Section 8 of this paper. Even for this toy model, several interesting and non-trivial phenomena have arisen. The heat kernel estimates on spaces of varying dimension are quite delicate. They are of Gaussian type but they are not of the classical Aronson Gaussian type. The different dimensionality is also reflected in the heat kernel estimates for BMVD. Even when both points x and y are on the plane, the heat kernel p(t, x, y) exhibits both one-dimensional and two-dimensional characteristics depending on whether both points are close to the base point a * or not. In addition, both the Euclidean distance \|x − y\| and the geodesic distance ρ(x, y) between the two points x and y play a role in the kernel estimates. As far as we know, this is the first paper that is devoted to the detailed study of heat propagation on spaces of varying dimension and related potential theory. Our approach is mainly probabilistic. For other related work and approaches on Markov processes living on spaces with possibly different dimensions, we refer the reader to [20,24,25,27] and the references therein. The rest of the paper is organized as follows. Section 2 gives a Dirichlet form construction and characterization of BMVD, as well as its infinitesimal generator. Nash-type inequality for X is given in Section 3. In Section 4, we present small time heat kernel estimates for X, while the large time estimates are given in Section 5. Hölder continuity of parabolic functions of X is established in Section 6. Section 7 is devoted to the two-sided sharp estimates for Green function of X in bounded C 1,1 domains in E that contain the pole base a * . BMVD on a large square with multiple vertical flag poles or with an arch are studied in Sections 8. For notational convenience, in this paper we set p D (t, x, y) := p(t, x, y) − p D (t, x, y), (1.6) where D is a domain of E and p D (t, x, y) is the transition density of the part process killed upon exiting D. In other words, for any non-negative function f ≥ 0 on E, E p D (t, x, y)f (y)m p (dy) = E x [f (X t ); t ≥ τ D ] ,(1.7) where τ D := inf{t ≥ 0 : X t / ∈ D}. Thus while p D (t, x, y) gives the probability density that BMVD starting from x hits y at time t without exiting D, p D (t, x, y) is the probability density for BMVD starting from x leaves D before ending up at y at time t. We use C ∞ c (E) to denote the space of continuous functions with compact support in E so that there restriction to D 0 and R + are smooth on D 0 and on R + , respectively. We also follow the convention that in the statements of the theorems or propositions C, C 1 , · · · denote positive constants, whereas in their proofs c, c 1 , · · · denote positive constants whose exact value is unimportant and may change from line to line. Preliminaries Throughout this paper, we denote the Brownian motion with varying dimension by X and its state space by E. In this section, we will construct BMVD using Dirichlet form approach. For the definition and basic properties of Dirichlet forms, including the relationship between Dirichlet form, L 2 -infinitesimal generator, resolvents and semigroups, we refer the reader to [12] and [21]. For a connected open set D ⊂ R d , W 1,2 (D) is the collection of L 2 (D; dx)-integrable functions whose first order derivatives (in the sense of distribution) exist and are also in L 2 (D; dx). Define E 0 (f, g) = 1 2 D ∇f (x) · ∇g(x)dx, f, g ∈ W 1,2 (D). It is well known that when D is smooth, (E 0 , W 1,2 (D)) is a regular Dirichlet form on L 2 (D; dx) and its associated Hunt process is the (normally) reflected Brownian motion on D. Moreover, every function f in W 1,2 (D) admits a quasi-continuous version on D, which will still be denoted by f . A quasi-continuous function is defined quasi-everywhere (q.e. in abbreviation) on D. When d = 1, by Cauchy-Schwartz inequality, every function in W 1,2 (D) is 1/2-Hölder on D. Denote by W 1,2 0 (D) the E 0 1 -closure of C ∞ c (D), where E 0 1 (f, f ) := E 0 (u, u) + D u(x) 2 dx. It is known that for any open set D ⊂ R d , (E 0 , W 1,2 0 (D) ) is a regular Dirichlet form on L 2 (D; dx) associated with the absorbing Brownian motion in D. For a subset A ⊂ E, we define σ A := inf{t > 0, X t ∈ A} and τ A := inf{t ≥ 0 : X t / ∈ A}. Similar notations will be used for other stochastic processes. We will use B ρ (x, r) (resp. B e (x, r)) to denote the open ball in E under the path metric ρ (resp. in R + or R 2 under the Euclidean metric) centered at x ∈ E with radius r > 0. A measure µ on E is said to have volume doubling property if there exists a constant C > 0 so that m p (B ρ (x, 2r)) ≤ Cm p (B ρ (x, r)) for all x ∈ E and every r ∈ (0, 1]. Proof. Note that for small r > 0 and x 0 ∈ D 0 with \|x 0 \| ρ = r, m p (B ρ (x 0 , r)) = πr 2 while m p (B ρ (x 0 , 2r)) = 2εr+r 2 +r. Thus there does not exist a constant C > 0 so that m p (B ρ (x, 2r)) ≤ Cm p (B ρ (x, r)) for all x ∈ E and every r ∈ (0, 1]. The following is an extended version of Theorem 1.2. Theorem 2.2. For every ε > 0 and p > 0, BMVD X on E with parameter (ε, p) exists and is unique. Its associated Dirichlet form (E, F) on L 2 (E; m p ) is given by F = f : f \| D 0 ∈ W 1,2 (D 0 ), f \| R + ∈ W 1,2 (R + ), and f (x) = f (0) q.e. on ∂D 0 ,(2.1) E(f, g) = 1 2 D 0 ∇f (x) · ∇g(x)dx + p 2 R + f ′ (x)g ′ (x)dx. (2.2) Proof. Let E and F be defined as above. Let u 1 (x) = E x [e −σ Bε ] when x ∈ D 0 and u 1 (x) = E x e −σ D 0 when x ∈ R + . It is known that u 1 \| D 0 ∈ W 1,2 (D 0 ), u 1 \| R + ∈ W 1,2 (R + ), u 1 (x) = 1 for x ∈ ∂D 0 and u 1 (0) = 1. Hence u 1 ∈ F. Existence: Let F 0 = f : f \| R 2 ∈ W 1,2 0 (D 0 ), f \| R + ∈ W 1,2 0 (R + ) . Then F is the linear span of F 0 ∪ {u 1 }. It is easy to check that (E, F) is a strongly local regular Dirichlet form on L 2 (E; m). So there is an m p -symmetric diffusion process X on E associated with it. Using the Dirichlet form characterization, the part process of X killed upon hitting a * is an absorbing Brownian motion in D 0 or R + , depending on the starting point. So X is a BMVD on E. Moreover, X is conservative; that is, it has infinite lifetime. Uniqueness: Conversely, if X is a BMVD, it suffices to check from definition that its associated Dirichlet form (E * , F * ) in L 2 (E; m p ) has to be (E, F) given in (2.1)-(2.2). Indeed, since a * is non-polar for X, for all u ∈ F * , H 1 a * u(x) := E x e −σ a * u(X σ a * ) = u(a * )E x [e −σ a * ] ∈ F ∩ F * and u − H 1 a * u(x) ∈ F 0 . Thus F * ⊂ F. On the other hand, since the part process of X killed upon hitting a * has the same distribution as the absorbing Brownian motion on D 0 ∪ (0, ∞), which has Dirichlet form (E, F 0 ) on L 2 (E \ {a * }; m p ), we have F 0 ⊂ F * . It follows that F ⊂ F * Nash Inequality and Heat Kernel Upper Bound Estimate Recall that D 0 := R 2 \ B e (0, ε). In the following, if no measure is explicitly mentioned in the L p -space, it is understood as being with respect to the measure m p ; for instance, L p (E) means L p (E; m p ). Lemma 3.1. There exists C 1 > 0 so that f 2 L 2 (E) ≤ C 1 E(f, f ) 1/2 f L 1 (E) + E(f, f ) 1/3 f 4/3 L 1 (E) for every f ∈ F. Proof. Since D 0 ⊂ R 2 and R + are smooth domains, we have by the classical Nash's inequality, f 2 L 2 (D 0 ) ≤ c ∇f L 2 (D 0 ) f L 1 (D 0 ) for f ∈ W 1,2 (D 0 ) ∩ L 1 (D 0 ), and f 3 L 2 (R + ) ≤ C f 2 L 1 (R + ) f ′ L 2 (R + ) for f ∈ W 1,2 (R + ) ∩ L 1 (R + ) . The desired inequality now follows by combining these two inequalities. The Nash-type inequality in Lemma 3.1 immediately implies that BMVD X on E has a symmetric density function p(t, x, y) with respect to the measure m p and that the following on-diagonal estimate by [10, Corollary 2.12] holds. Proposition 3.2. There exists C 2 > 0 such that p(t, x, y) ≤ C 2 1 t + 1 t 1/2 for all t > 0 and x, y ∈ E. Since X moves like Brownian motion in Euclidean spaces before hitting a * , it is easy to verify that for each t > 0, (x, y) → p(t, x, y) is continuous in (E \ {a * }) × (E \ {a * }). For each t > 0 and fixed y ∈ E \ {a * }, E p(t/2, y, z) 2 m p (dz) = p(t, y, y) < ∞. So by the Dirichlet form theory, x → p(t, x, y) = E p(t/2, x, z)p(t/2, z, y)m p (dz) is E-quasicontinuous on E. Since a * is non-polar for X, x → p(t, x, y) is continuous at a * , and hence is continuous on E. By the symmetry and Chapman-Kolmogorov equation again, we conclude that x → p(t, x, a * ) = E p(t/2, x, z)p(t/2, z, a * )m p (dz) is continuous on E. Consequently, p(t, x, y) is well defined pointwisely on (0, ∞) × E × E so that for each fixed t > 0 and y ∈ E, p(t, x, y) is a continuous function in x ∈ E. We can use Davies method to get an off-diagonal upper bound estimate. Proposition 3.3. There exist C 3 , C 4 > 0 such that p(t, x, y) ≤ C 3 1 t + 1 t 1/2 e −C 4 ρ(x,y) 2 /t for all t > 0 and x, y ∈ E. Proof. Fix x 0 , y 0 ∈ E, t 0 > 0. Set a constant α := ρ(y 0 , x 0 )/4t 0 and ψ(x) := α\|x\| ρ . Then we define ψ n (x) = ψ(x) ∧ n. Note that for m p -a.e. x ∈ E, e −2ψn(x) \|∇e ψn(x) \| 2 = \|∇ψ n (x)\| 2 = \|α\| 2 1 {\|x\|ρ≤ n \|α\| } (x) ≤ α 2 . Similarly, e 2ψn(x) \|∇e −ψn(x) \| 2 ≤ α 2 . By [10,Corollary 3.28], p(t, x, y) ≤ c 1 t + 1 t 1/2 exp −\|ψ(y) − ψ(x)\| + 2t\|α\| 2 . (3.1) Taking t = t 0 , x = x 0 and y = y 0 in (3.1) completes the proof. As we have seen from Theorems 1.3 and 1.4, the upper bound estimate in Proposition 3.3 is not sharp. Signed Radial Process and Small Time Estimate In order to get the sharp two-sided heat kernel estimates, we consider the radial process of X. Namely, we project X to R by applying the following mapping from E to R: u(x) =    −\|x\|, x ∈ R + ; \|x\| ρ , x ∈ D 0 . (4.1) We call Y t := u(X t ) the signed radial process of X. Observe that u ∈ F loc , where F loc denotes the local Dirichlet space of (E, F), whose definition can be found, for instance, in [12,21]. By Fukushima decomposition [21,Chapter 5], Y t − Y 0 = u(X t ) − u(X 0 ) = M [u] t + N [u] t , P x -a.s. for q.e. x ∈ E, where M [u] t is a local martingale additive functional of X, and N [u] t is a continuous additive functional of X locally having zero energy. We can explicitly compute M [u] and N [u] . For any ψ ∈ C ∞ c (E), E(u, ψ) = 1 2 D 0 ∇\|x\| · ∇ψdx + p 2 R + (−1)ψ ′ dx = − 1 2 D 0 div x \|x\| ψdx − 1 2 ∂Be(0,ε) ψ(0) ∂\|x\| ∂ n σ(dx) + pψ(0) 2 = − 1 2 D 0 1 \|x\| ψdx − 2πε − p 2 ψ(0) = − E ψ(x)ν(dx), where n is the outward pointing unit vector normal of the surface ∂B e (0, ε), σ is the surface measure on ∂B e (0, ε) ⊂ R 2 , and ν(dx) := 1 \|2x\| 1 D 0 (x)dx + 2πε − p 2 δ {a * } . Recall that we identify 0 ∈ R + with a * . It follows from [21,Theorem 5.5.5] that dN [u] t = 1 2(u(X t ) + ε) 1 {Xt∈D 0 } dt + (2πε − p)dL 0 t (X), (4.2) where L 0 t (X) is the positive continuous additive functional of X having Revuz measure 1 2 δ {a * } . We call L 0 the local time of X at a * . Next we compute M [u] , the predictable quadratic variation process of local martingale M [u] ). Let u n = (−n) ∨ u ∧ n, and it immediately follows u n ∈ F. Let F b denote the space of bounded functions in F. By [21,Theorem 5.5.2], the Revuz measure µ un for M [un] can be calculated as follows. For any f ∈ F b ∩ C c (E), E f (x)µ un (dx) = 2E(u n f, u n ) − E(u 2 n , f ) = E f (x)\|∇u n (x)\| 2 m p (dx), which shows that µ un (dx) = \|∇u n (x)\| 2 m p (dx) = 1 Bρ(a * ,n) m p (dx). By the strong local property of (E, F), we have µ u = µ un on B ρ (a * , n). It follows that µ u (dx) = m p (dx). Thus by [12, Proposition 4.1.9], M [u] t = t for t ≥ 0 and so B t := M [u] t is a one-dimensional Brownian motion. Combining this with (4.2), we conclude dY t = dB t + 1 2(Y t + ε) 1 {Xt∈D 0 } dt + (2πε − p)dL 0 t (X) = dB t + 1 2(Y t + ε) 1 {Yt>0} dt + (2πε − p)dL 0 t (X). (4.3) We next find the SDE for the semimartingale Y . The semi-martingale local time of Y is denoted as L 0 t (Y ), that is, L 0 t (Y ) := lim δ↓0 1 δ t 0 1 [0,δ) (Y s )d Y s = lim δ↓0 1 δ t 0 1 [0,δ) (Y s )ds, (4.4) where Y t = t is the quadratic variation process of the semimartingale Y . Proposition 4.1. L 0 t (Y ) = 4πεL 0 t (X). Proof. By computation analogous to that for Y t = u(X t ), one can derive using Fukushima's decomposition for v(X t ) := \|X t \| ρ , that dv(X t ) = d B t + 1 2(\|X t \| ρ + ε) 1 {Xt∈D 0 } dt + (2πε + p)dL 0 t (X), where B is a one-dimensional Brownian motion. Observe that v(X t ) = \|Y t \|. Thus we have d\|Y t \| = d B t + 1 2(Y t + ε) 1 {Yt>0} dt + (2πε + p)dL 0 t (X). On the other hand, by Tanaka's formula, we have d\|Y t \| = sgn(Y t )dY t + dL 0 t (Y ) = sgn(Y t )dB t + sgn(Y t ) 1 2(Y t + ε) where sgn(x) := 1 if x > 0 and sgn(x) := −1 if x ≤ 0. Since the decomposition of a continuous semi-martingale as the sum of a continuous local martingale and a continuous process with finite variation is unique, one must have (2πε + p)L 0 t (X) = sgn(Y t )(2πε − p)L 0 t (X) + L 0 t (Y ). (4.5) The local time L 0 t (X) increases only when Y t = 0. Therefore (2πε + p)L 0 t (X) = −(2πε − p)L 0 t (X) + L 0 t (Y ), and so 4πεL 0 t (X) = L 0 t (Y ). The semi-martingale local time in (4.4) is non-symmetric in the sense that it only measures the occupation time of Y t in the one-sided interval [0, δ) instead of the symmetric interval (−δ, δ). One can always relate the non-symmetric semi-martingale local time L 0 (Y ) to the symmetric semi-martingale local time L 0 (Y ) defined by L 0 t (Y ) := lim δ↓0 1 2δ t 0 1 (−δ,δ) (Y s )d Y s = lim δ↓0 1 2δ t 0 1 (−δ,δ) (Y s )ds. Lemma 4.2. L 0 t (Y ) = 2πε+p 4πε L 0 t (Y ) = (2πε + p)L 0 t (X). Proof. Viewing \|Y t \| = \| − Y t \| and applying Tanaka's formula to the semimartingale −Y , we can derive in a way analogous to the computation leading to (4.5) that (2πε + p)L 0 t (X) = −sgn(−Y t )(2πε − p)L 0 t (X) + L 0 t (−Y ) = (2πε − p)L 0 t (X) + L 0 t (−Y ). Thus we get 2pL 0 t (X) = L 0 t (−Y ), which yields L 0 t (Y ) = 1 2 L 0 t (Y ) + L 0 t (−Y ) = 1 2 4πεL 0 t (X) + 2pL 0 t (X) = (2πε + p)L 0 t (X). Lemma 4.2 together with (4.3) gives the following SDE characterization for the signed radial process Y , which tells us precisely how X moves after hitting a * . Proposition 4.3. dY t = dB t + 1 2(Y t + ε) 1 {Yt>0} dt + 2πε − p 2πε + p d L 0 t (Y ). (4.6) Let β = 2πε−p 2πε+p . SDE (4.6) says that Y is a skew Brownian motion with drift on R with skew parameter β. It follows (see [32]) that starting from a * , the process Y (respectively, X) has probability (1 − β)/2 = p 2πε+p to enter (−∞, 0) (respectively, the pole) and probability (1 + β)/2 = 2πε 2πε+p to enter (0, ∞) (respectively, the plane). SDE (4.6) has a unique strong solution; see, e.g., [5]. So Y is a strong Markov process on R. The following is a key to get the two-sided sharp heat kernel estimate on p(t, x, y) for BMVD X. Proposition 4.4. The one-dimensional diffusion process Y has a jointly continuous transition density function P (Y ) (t, x, y) with respect to the Lebesgue measure on R. Moreover, for every T ≥ 1, there exist constants C i > 0, 1 ≤ i ≤ 4, such that the following estimate holds: C 1 √ t e −C 2 \|x−y\| 2 /t ≤ p (Y ) (t, x, y) ≤ C 3 √ t e −C 4 \|x−y\| 2 //t , (t, x, y) ∈ (0, T ] × R × R. (4.7) Proof. Let β := 2πε−p 2πε+p and Z be the skew Brownian motion dZ t = dB t + β L 0 t (Z), where L 0 t (Z) is the symmetric local time of Z at 0. The diffusion process Y can be obtained from Z through a drift perturbation (i.e. Girsanov transform). The transition density function p 0 (t, x, y) of Z is explicitly known and enjoys the two-sided Aronson-type Gaussian estimates (4.7); see, e.g., [32]. One can further verify that \|∇ x p 0 (t, x, y)\| ≤ c 1 t −1 exp(−c 2 \|x − y\| 2 /t), from which one can deduce (4.7) by using the same argument as that for Theorem A in Zhang [38, §4]. Proposition 4.4 immediately gives the two-sided estimates on the transition function p(t, x, y) of X when x, y ∈ R + since X t = −Y t when X t ∈ R + . Theorem 4.5. For every T ≥ 1, there exist C i > 0, 5 ≤ i ≤ 8, such that the following estimate holds: C 5 √ t e −C 6 \|x−y\| 2 /t ≤ p(t, x, y) ≤ C 7 √ t e −C 8 \|x−y\| 2 /t for t ∈ (0, T ] and x, y ∈ R + . Let A be any rotation of the plane around the pole. Using the fact that starting from a * , AX t has the same distribution as X t , we can derive estimates for p(t, x, y) for other x, y ∈ E. The next result gives the two-sided estimates on p(t, x, y) when x ∈ R and y ∈ D 0 . Theorem 4.6. For every T ≥ 1, there exist constants C i > 0, 9 ≤ i ≤ 12, such that for all x ∈ R + , y ∈ D 0 and t ∈ [0, T ], C 9 √ t e −C 10 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 11 √ t e −C 12 ρ(x,y) 2 /t . Proof. We first note that in this case by the symmetry of p(t, x, y), p(t, x, y) = p(t, y, x) = t 0 P y (σ a * ∈ ds)p(t − s, a * , x). By the rotational invariance of two-dimensional Brownian motion, P y (σ a * ∈ ds) only depends on \|y\| ρ , therefore so does y → p (X) (t, x, y). For x ∈ R + and y ∈ D 0 , set p(t, x, r) := p(t, x, y) for r = \|y\| ρ . For all a > b > 0 and x ∈ R + , b a p (Y ) (t, −\|x\|, y)dy = P −\|x\| (a ≤ Y t ≤ b) = P x (X t ∈ D 0 with a ≤ \|X t \| ρ ≤ b) = y∈D 0 :a≤\|y\|ρ≤b p(t, x, y)m p (dy) = y∈D 0 :a+ε≤\|y\|≤b+ε p(t, x, y)m p (dy) = b a 2π(r + ε) p(t, x, r)dr. This implies when x ∈ R + , y ∈ D 0 , p (Y ) (t, −\|x\|, \|y\| ρ ) = 2π(\|y\| ρ + ε) p(t, x, \|y\| ρ ) = 2π(\|y\| ρ + ε)p(t, x, y). (4.8) We thus have by Proposition 4.4 that c 1 √ t e −c 2 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ c 3 √ t e −c 4 ρ(x,y) 2 /t for x ∈ R + and y ∈ D 0 with \|y\| ρ < 1. (4.9) When \|y\| ρ > 1, we first have p(t, x, y) = 1 2π(\|y\| ρ + ε) p (Y ) (t, −\|x\|, \|y\| ρ ) 1 (\|y\| ρ + ε) √ t e −c 3 ρ(x,y) 2 /t ≤ 1 √ t e −c 3 ρ(x,y) 2 /t , while since ρ(x, y) ≥ \|y\| ρ > 1, p(t, x, y) = 1 2π(\|y\| ρ + ε) p (Y ) (t, −\|x\|, \|y\| ρ ) 1 (\|y\| ρ + ε) √ t e −c 4 ρ(x,y) 2 /t 1 √ t √ t √ T ρ(x, y) e −c 4 ρ(x,y) 2 /t 1 √ t e −(c 4 +1)ρ(x,y) 2 /t . (4.10) This completes the proof. Theorem 4.7. For every T ≥ 1, there exist constants C i > 0, 13 ≤ i ≤ 22, such that for all t ∈ [0, T ] and x, y ∈ D 0 , the following estimates hold. When max{\|x\| ρ , \|y\| ρ } ≤ 1, C 13 √ t e −C 14 ρ(x,y) 2 /t + C 13 t 1 ∧ \|x\| ρ √ t 1 ∧ \|y\| ρ √ t e −C 15 \|x−y\| 2 /t ≤ p(t, x, y) ≤ C 16 √ t e −C 17 ρ(x,y) 2 /t + C 16 t 1 ∧ \|x\| ρ √ t 1 ∧ \|y\| ρ √ t e −C 18 \|x−y\| 2 /t ; (4.11) and when max{\|x\| ρ , \|y\| ρ } > 1, C 19 t e −C 20 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 21 t e −C 22 ρ(x,y) 2 /t . (4.12) Here \| · \| and \| · \| ρ denote the Euclidean metric and the geodesic metric in D 0 , respectively. Proof. For x ∈ D 0 and t ∈ (0, T ], note that p(t, x, y) = p D 0 (t, x, y) + p D 0 (t, x, y), (4.13) where p D 0 (t, x, y) = t 0 p(t − s, a * , y)P x (σ {a * } ∈ ds). (4.14) As mentioned in the proof for Theorem 4.6, p(t − s, a * , y) is a function in y depending only on \|y\| ρ . Therefore so is y → p D 0 (t, x, y). Set p D 0 (t, x, r) := p D 0 (t, x, y) for r = \|y\| ρ . For any b > a > 0, P x (σ a * < t, X t ∈ D 0 with a ≤ \|X t \| ρ ≤ b) = a≤\|y\|ρ≤b p D 0 (t, x, y)m p (dy) = 2π b a (r + ε) p D 0 (t, x, r)dr. On the other hand, P x (σ a * < t, X t ∈ D 0 with a ≤ \|X t \| ρ ≤ b) = P (Y ) \|x\|ρ (σ a * < t, Y t > 0 with a ≤ \|Y t \| ρ ≤ b) = t 0 b a p (Y ) (t − s, 0, r)dr P (Y ) \|x\|ρ (σ 0 ∈ ds). It follows that 2π(r + ε) p D 0 (t, x, r) = t 0 p (Y ) (t − s, 0, r)P (Y ) \|x\|ρ (σ {0} ∈ ds) = p (Y ) (t, −\|x\| ρ , r). In other words, p D 0 (t, x, y) = 1 2(\|y\| ρ + ε) p (Y ) (t, −\|x\| ρ , \|y\| ρ ). (4.15) It is known that the Dirichlet heat kernel p D 0 (t, x, y) enjoys the following two-sided estimates: c 1 t 1 ∧ \|x\| ρ √ t 1 ∧ \|y\| ρ √ t e −c 2 \|x−y\| 2 /t ≤ p D 0 (t, x, y) ≤ c 3 t 1 ∧ \|x\| ρ √ t 1 ∧ \|y\| ρ √ t e −c 4 \|x−y\| 2 /t (4.16) for t ∈ (0, T ] and x, y ∈ D 0 . We now consider two different cases. Case (i): max{\|x\| ρ , \|y\| ρ } ≤ 1 and t ∈ (0, T ]. In this case, it follows from (4.13)-(4.16) and Proposition 4.4 that c 5 √ t e −c 6 (\|x\|ρ+\|y\|ρ) 2 /t + c 5 t 1 ∧ \|x\| ρ √ t 1 ∧ \|y\| ρ √ t e −c 7 \|x−y\| 2 /t ≤ p(t, x, y) ≤ c 8 √ t e −c 9 (\|x\|ρ+\|y\|ρ) 2 /t + c 8 t 1 ∧ \|x\| ρ √ t 1 ∧ \|y\| ρ √ t e −c 10 \|x−y\| 2 /t . (4.17) Observe that (\|x\| ρ + \|y\| ρ ) 2 /t ≍ ρ(x, y) 2 /t if \|x\| ρ ∧ \|y\| ρ ≤ √ t. (4.18) When \|x\| ρ ∧ \|y\| ρ > √ t, for a > 0, b > 0, 1 √ t e −a(\|x\|ρ+\|y\|ρ) 2 /t + 1 t 1 ∧ \|x\| ρ √ t 1 ∧ \|y\| ρ √ t e −b\|x−y\| 2 /t ≍ 1 √ t e −a(\|x\|ρ+\|y\|ρ) 2 /t + 1 √ t + 1 t e −b\|x−y\| 2 /t = 1 √ t e −a(\|x\|ρ+\|y\|ρ) 2 /t + e −b\|x−y\| 2 /t + 1 t e −b\|x−y\| 2 /t (4.19) The desired estimate (4.11) now follows from (4.17)-(4.19) and the fact (1.2). Case (ii): max{\|x\| ρ , \|y\| ρ } > 1 and t ∈ (0, T ]. By the symmetry of p(t, x, y) in x and y, in this case we may and do assume \|y\| ρ > 1 > t/T . It then follows from (4.15)-(4.16), Proposition 4.4 and (4.10) that c 11 t e −c 12 (\|x\|ρ+\|y\|ρ) 2 /t + c 11 t 1 ∧ \|x\| ρ √ t 1 ∧ \|y\| ρ √ t e −c 13 \|x−y\| 2 /t ≤ p(t, x, y) ≤ c 14 t e −c 15 (\|x\|ρ+\|y\|ρ) 2 /t + c 14 t 1 ∧ \|x\| ρ √ t 1 ∧ \|y\| ρ √ t e −c 16 \|x−y\| 2 /t . (4.20) When \|x\| ρ ∧ \|y\| ρ ≤ √ t, the lower bound estimate (4.12) follows from (4.20) and (4.18), while the upper bound estimate (4.12) follows from Proposition 3.3. Whereas when \|x\| ρ ∧ \|y\| ρ > √ t, the desired estimate (4.12) follows from (4.20) and (1.2). This completes the proof of the theorem. Remark 4.8. (i) One cannot expect to rewrite the estimate of (4.11) as t −1 e −cρ(x,y) 2 /t . A counterexample is that x = y = a * , in which case x and y can be viewed as either on R or on D 0 , therefore both Proposition 4.4 and Theorem 4.6 have already confirmed that p(t, x, y) ≍ t −1/2 , which is consistent with the (4.11). (ii) The Euclidean distance appearing in (4.11) cannot be replaced with the geodesic distance. To see this, take x = (ε + t −1/2 , 0) and y = (−ε − t −1/2 , 0) in D 0 . The estimate of (4.11) is comparable with t −1/2 + t −1 exp(−ε 2 /t), but if we replaced \|x − y\| with ρ(x, y), it would be comparable with t −1/2 + t −1 . For fixed ε, as t ↓ 0, t −1/2 + t −1 exp(−ε 2 /t) ∼ t −1/2 , but t −1/2 + t −1 ∼ t −1 . (iii) Theorem 1.3 also shows that the parabolic Harnack inequality fails for X. For a precise statement of the parabolic Harnack inequality, see, for example, [22,33,35]. For s ∈ (0, 1], take some y ∈ D 0 such that \|y\| ρ = √ s. Set Q + := (3s/2, 2s) × B ρ (y, 2 √ s) and Q − := (s/2, s) × B ρ (y, 2 √ s). Let u(t, x) := p(t, x, y). It follows from Theorem 1.3 that sup Q + u ≍ s −1/2 + s −1 ≍ s −1 and inf Q − u ≍ s −1 . Clearly there does not exist any positive constant C > 0 so that sup Q + u ≤ C inf Q − u holds for all s ∈ (0, 1]. This shows that parabolic Harnack inequality fails for X. Large time heat kernel estimates Recall that X denotes the BMVD process on E and its signed radial process defined by (4.3) is denoted by Y . In this section, unless otherwise stated, it is always assumed that T ≥ 8 and t ∈ [T, ∞). With loss of generality, we assume that the radius ε of the "hole" B(0, ε) satisfies ε ≤ 1/4. We begin with the following estimates for the distribution of hitting time of a disk by a two-dimensional Brownian motion, which follow directly from [23, §5.1, Case (a), α = 1] and a Brownian scaling. Proposition 5.1 (Grigor'yan and Saloff-Coste [23]). Let X be a Brownian motion on R 2 and K be the closed ball with radius ε centered at the origin. (i) If 0 < t < 2\|x\| 2 and \|x\| ≥ 1 + ε, then c log \|x\| exp −C\|x\| 2 /t ≤ P x (σ K ≤ t) ≤ C log \|x\| exp −c\|x\| 2 /t , (5.1) for some positive constants C > c > 0. (ii) If t ≥ 2\|x\| 2 and \|x\| ≥ 1 + ε, then P x (σ K ≤ t) ≍ log √ t − log \|x\| log √ t , (5.2) and ∂ t P x (σ K ≤ t) ≍ log \|x\| t(log t) 2 . (5.3) Our first goal is to establish an upper bound estimate on t 0 p(s, a * , a * )ds the Proposition 5.4. This will be done through two Propositions by using the above hitting time estimates. Proposition 5.2. p(t, a * , a * ) is decreasing in t ∈ (0, ∞). Proof. This follows from d dt p(t, a * , a * ) = d dt x p(t/2, a * , x) 2 m p (dx) = ∂ ∂t p(t/2, a * , x) p(t/2, a * , x)m p (dx) = L x p(t/2, a * , x)p(t/2, a * , x)m p (dx) = −E (p(t/2, a * , x), p(t/2, a * , x)) ≤ 0. Proposition 5.3. There exists some constant C 1 > 0 such that p(t, a * , a * ) ≤ C 1 log t t for t ∈ [8, ∞). Proof. For t ≥ 8 and x ∈ D 0 with 1 < \|x\| ρ < t/3, by Proposition 5.2, p(t, x, a * ) = t 0 P x (σ a * ∈ ds)p(t − s, a * , a * ) ≥ p(t, a * , a * )P x (σ a * ≤ t) ≍ p(t, a * , a * ) 1 − log \|x\| log √ t , where the " ≍ " is due to (5.3). Therefore, 1 ≥ P a * X t ∈ D 0 with 1 < \|X t \| ρ < t/4 = D 0 ∩{1<\|x\|ρ< √ t/4} p(t, a * , x)dx ≥ c 1 p(t, a * , a * ) √ t/4+ε 1+ε 1 − log r log √ t rdr ≥ c 2 p(t, a * , a * ) t log t . By selecting C 2 large enough, the above yields the desired estimate for p(t, a * , a * ) for t ≥ 8. Proposition 5.4. There exists some C 2 > 0 such that t 0 p(s, a * , a * )ds ≤ C 2 log t, for all t ≥ 4. Proof. For t ≥ 8 and x ∈ D 0 with 1 < \|x\| ρ < t/2, we have by (5.3), p(t, x, a * ) ≥ t t/2 p(t − s, a * , a * )P x (σ a * ∈ ds) ≍ log \|x\| t(log t) 2 t/2 0 p(s, a * , a * )ds. Thus by using polar coordinate, This yields the desired estimate. 1 ≥ P a * X t ∈ D 0 with 1 < \|X t \| ρ < t/2 = {x∈D 0 :1<\|x\|ρ< √ t/2} p(t, a * , x)m p (dx) √ t/2 1 log(r + ε) t(log t) 2 (r + ε)dr · t/2 0 p(s, a * , a * )ds = 1 t(log t) 2 √ t/2+ε The above proposition suggests that p(t, a * , a * ) ≤ c/t for t ∈ [4, ∞). However, in order to prove this rigorously, we first compute the upper bounds for p(t, a * , x) for different regions of x, and then use the identity p(t, a * , a * ) = E p(t/2, a * , x) 2 m p (dx) to obtain the sharp upper bound estimate for p(t, a * , a * ). Proposition 5.5. There exists C 3 > 0 such that for all t ≥ 8 and x ∈ D 0 with 1 < \|x\| ρ < t/2, p(t, a * , x) ≤ C 3 t log √ t \|x\| . Proof. By Proposition 5.3, (5.2) and Proposition 5.4, p(t, x, a * ) = t/2 0 p(t − s, a * , a * )P x (σ a * ∈ ds) + t t/2 p(t − s, a * , a * )P x (σ a * ∈ ds) ≤ c 1 log t t P x (σ a * ≤ t/2) + log \|x\| t(log t) 2 t/2 0 p(s, a * , a * )ds ≤ c 2 log t t log √ t/\|x\| log √ t + log \|x\| t log t ≤ c 3 1 t log √ t \|x\| + 1 t ≤ c 4 t log √ t \|x\| . The following asymptotic estimate for the distribution of Brownian hitting time of a disk from [36, Theorem 2] will be used in the next proposition. Lemma 5.6 (Uchiyama [36]). Let (B t ) t≥0 be the standard two-dimensional Brownian motion and σ r := inf{t > 0, \|B t \| ≤ r}. Denote by p r,x (t) the probability density function of σ r with B 0 = x. For every r 0 , uniformly for \|x\| ≥ r 0 , as t → ∞, p r 0 ,x (t) = log 1 2 e c 0 \|x\| 2 t (log t + c 0 ) 2 exp − \|x\| 2 2t +      O 1+(log(\|x\| 2 /t)) 2 \|x\| 2 (log t) 3 for \|x\| 2 ≥ t, 2γ log(t/\|x\| 2 ) t(log t) 3 + O 1 t(log t) 3 for \|x\| 2 < t, where c 0 is a positive constant only depending on r 0 and γ = − ∞ 0 e −u log udu is the Euler constant. Proposition 5.7. There exists C 4 , C 5 > 0 such that for all t ≥ 8 and all x ∈ D 0 , p(t, a * , x) ≤      C 4 1 t e −C 5 \|x\| 2 /t + (log(\|x\| 2 /t)) 2 \|x\| 2 (log t) 2 when \|x\| ρ > √ t, C 4 t e −C 5 \|x\| 2 /t + 1 (log t) 2 when √ t/2 ≤ \|x\| ρ ≤ √ t. Proof. Note that p(t, a * , x) = t 0 p(t − s, a * , a * )P x (σ a * ∈ ds) = t/2 0 p(t − s, a * , a * )P x (σ a * ∈ ds) + t t/2 p(t − s, a * , a * )P x (σ a * ∈ ds). (5.4) By the monotonicity of p(t, a * , a * ) established in Proposition 5.2, estimate (5.1) and Proposition 5.3, t/2 0 p(t − s, a * , a * )P x (σ a * ∈ ds) ≤ p(t/2, a * , a * )P x (σ a * ≤ t/2) (5.1) ≤ p(t/2, a * , a * ) c 1 log \|x\| e −c 2 \|x\| 2 /t ≤ c 1 log t t log \|x\| e −c 2 \|x\| 2 /t ≤ c 3 t e −c 2 \|x\| 2 /t . (5.5) On the other hand, by Proposition 5.4 and Lemma 5.6, for \|x\| ρ ≥ √ t, t t/2 p(t − s, a * , a * )P x (σ a * ∈ ds) ≤ sup s∈[t/2,t] p ε,x (s) · t t/2 p(t − s, a * , a * )ds ≤ c 4 log \|x\| t(log t) 2 e −\|x\| 2 /(2t) + log \|x\| 2 /t 2 \|x\| 2 (log t) 3 log t = c 4 log \|x\| t log t e −\|x\| 2 /(2t) + log \|x\| 2 /t 2 \|x\| 2 (log t) 2 ≤ c 4 t e −c 5 \|x\| 2 /t + c 4 log \|x\| 2 /t 2 \|x\| 2 (log t) 2 , (5.6) while for √ t/2 ≤ \|x\| ρ ≤ √ t, t t/2 p(t − s, a * , a * )P x (σ a * ∈ ds) ≤ sup s∈[t/2,t] p ε,x (s) · t t/2 p(t − s, a * , a * )ds ≤ c 6 log \|x\| t(log t) 2 e −\|x\| 2 /(2t) + 1 t(log t) 3 log t ≤ c 6 t e −c 5 \|x\| 2 /t + c 6 t(log t) 2 . (5.7) The desired estimate now follows from (5.4)-(5.7). Proposition 5.8. There exists C 6 > 0 such that p(t, a * , x) ≤ C 6 log t t e −x 2 /(2t) for all t ≥ 8 and x ∈ R + . Proof. Starting from x ∈ R + , BMVD X on E runs like a one-dimensional Brownian motion before hitting a * . Thus by the known formula for the first passage distribution for one-dimensional Brownian motion, P x (σ a * ∈ dt) = 1 √ 2πt 3 xe −x 2 /(2t) dt for x ∈ (0, ∞).p(t, a * , x) = t/2 0 p(t − s, a * , a * )P x (σ a * ∈ ds) + t t/2 p(t − s, a * , a * )P x (σ a * ∈ ds) ≤ p(t/2, a * , a * ) t/2 0 1 √ 2πs 3 xe −x 2 /(2s) ds + t t/2 p(t − s, a * , a * ) 1 √ 2πs 3 xe −x 2 /(2s) ds log t t ∞ 2x 2 /t 1 √ r e −r/2 dr + x √ t 3 e −x 2 /t t t/2 p(t − s, a * , a * )ds log t t e −x 2 /t + 1 t e −x 2 /(2t) · log t log t t e −x 2 /(2t) . We are now in a position to establish the following on-diagonal upper bound estimate at a * . Theorem 5.9. There exists C 7 > 0 such that p(t, a * , a * ) ≤ C 7 t −1/2 ∧ t −1 for all t ∈ (0, ∞). Proof. For t ≥ 8, we have p(t, a * , a * ) = E p(t/2, a * , x) 2 m p (dx) = R + + D 0 ∩{0<\|x\|ρ<1} + D 0 ∩{1<\|x\|ρ< √ t/2} + D 0 ∩{\|x\|ρ> √ t/2} p(t/2, a * , x) 2 m p (dx). (5.9) It follows from Proposition 5.8 that R + p(t/2, a * , x) 2 m p (dx) log t t 2 p ∞ 0 e −x 2 /(2t) dx ≍ c 1 (log t) 2 t 3/2 . (5.10) By Proposition 3.3, D 0 ∩{0<\|x\|ρ<1} p(t/2, a * , x) 2 m p (dx) ≤ sup x∈D 0 :0<\|x\|ρ<1 p(t/2, a * , x) 2 m p (D 0 ∩ {0 < \|x\| ρ < 1}) 1 √ t 2 = 1 t . (5.11) In view of Proposition 5.7, D 0 ∩{\|x\|ρ> √ t/2} p(t/2, a * , x) 2 m p (dx) ≤ D 0 ∩{\|x\|ρ> √ t/2} c 1 t e −c 2 \|x\| 2 /t 2 m p (dx) + D 0 ∩{\|x\|ρ> √ t} c 1 log \|x\| 2 /t 2 \|x\| 2 (log t) 2 2 m p (dx) + D 0 ∩{ √ t/2≤\|x\|ρ≤ √ t} c 1 1 t(log t) 2 2 m p (dx). (5.12) Using polar coordinate, D 0 ∩{\|x\|ρ> √ t/2} c 1 t e −c 2 \|x\| 2 /t 2 m p (dx) = c 3 ∞ √ t/2+ε r t 2 e −c 2 r 2 /t dr ≤ c 4 t , (5.13) while D 0 ∩{\|x\|ρ> √ t} log \|x\| 2 /t 2 \|x\| 2 (log t) 2 2 m p (dx) = 2π ∞ √ t+ε r log(r 2 /t) 4 r 4 (log t) 4 dr u=r/ √ t ≤ 2π ∞ 1 (log u) 4 u 3 t 3/2 (log t) 4 √ t du = 2π t(log t) 4 ∞ 1 (log u) 4 u 3 du = c 5 t(log t) 4 ,(5.14) and D 0 ∩{ √ t/2≤\|x\|ρ≤ √ t} 1 t(log t) 2 2 m p (dx) ≍ 1 t(log t) 4 1 t . (5.15) Hence it follows from (5.12)-(5.15) that D 0 ∩{\|x\|ρ> √ t/2} p(t/2, a * , x) 2 m p (dx) ≤ c 6 t . (5.16) By Proposition 5.5 and using polar coordinates, D 0 ∩{1<\|x\|ρ< √ t/2} p(t/2, a * , x) 2 m p (dx) 1 t 2 x∈D 0 ∩{1<\|x\|ρ< √ t/2} log √ t \|x\| 2 m p (dx) = 2π t 2 √ t/2+ε 1+ε r log √ t r 2 dr u=r/ √ t = 2π t ( √ t/2+ε)/ √ t (1+ε)/ √ t u(log u) 2 du ≤ c 7 t . (5.17) Combining (5.10),(5.11), (5.16) and (5.17), we conclude that p(t, a * , a * ) ≤ c 8 /t for t ≥ 8. On the other hand, taking x = y = 0 = a * in Theorem 4.5 yields that p(t, a * , a * ) ≤ c 9 t −1/2 for t ∈ (0, 8]. This completes the proof of the theorem. Theorem 5.10. There is a constant C 8 ≥ 1 so that C −1 8 t −1/2 ∧ t −1 ≤ p(t, a * , a * ) ≤ C 8 t −1/2 ∧ t −1 for all t ∈ (0, ∞). Proof. In view of Theorem 5.9, it remains to establish the lower bound estimate. By Cauchy-Schwartz inequality, for M ≥ 1 to be determined later, p(t, a * , a * ) = E p(t/2, a * , x) 2 m p (dx) ≥ {x∈E:\|x\|ρ≤M √ t} p(t/2, a * , x) 2 m p (dx) ≥ 1 m p ({x ∈ E : \|x\| ρ ≤ M √ t}) {x∈E:\|x\|ρ≤M √ t} p(t/2, a * , x)m p (dx) 2 t −1/2 ∧ t −1 P a * (\|X t \| ρ ≤ M √ t) 2 .(5.18) We claim that by taking M large enough, P a * (\|X t \| ρ ≤ M √ t) ≥ 1/2 for every t > 0, which will then give the desired lower bound estimate on p(t, a * , a * ). Recall the signed radial process Y = u(X) of X from (4.1) satisfies SDE (4.6). For any a > 0 and δ ∈ (0, ε), let Z δ,a and Z δ,−a be the pathwise unique solution of the following SDEs; see [5,Theorem 4.3]: Z δ,a t = a + B t + t 0 1 Z δ,a s + δ 1 {Z δ,a s >0} ds + L 0 t (Z δ,a ), (5.19) Z δ,−a t = −a + B t + t 0 1 Z δ,−a s − δ 1 { Z δ,−a s <0} ds − L 0 t ( Z δ,−a ),(5.20) where B is the Brownian motion in (4.6), and L 0 (Z δ,a ), L 0 ( Z δ,−a ) are the symmetric local times of Z δ,a and Z δ,−a at 0, respectively. Denote by Y a and Y −a the pathwise solutions of (4.6) with dZ a t = a + B t + t 0 1 Z a s ds, dZ −a t = −a + B t + t 0 1 Z −a s ds. In fact, Z a and −Z −a are both two-dimensional Bessel processes on (0, ∞) starting from a. They have infinite lifetimes and never hits 0. By [5, Theorem 4.6] again, diffusion processes Z δ,a is decreasing in δ, and Z δ,−a is increasing in δ. It is easy to see from the above facts that lim δ→0 Z δ,a t = Z a t and lim δ→0 Z δ,−a t = Z −a t . Consequently, with probability one, Y a t ≤ Z a t and Y −a t ≥ Z −a t for every t ≥ 0. In particular, we have for every t > 0, P(Y a t ≥ M √ t) ≤ P(Z a t ≥ M √ t) and P(Y −a t ≤ −M √ t) ≤ P(Z −a t ≤ −M √ t) = P(Z a t ≥ M √ t). Let W be two-dimensional Brownian motion. Then we have from the above that for every t > 0, P a (Y t ≥ M √ t) + P −a (Y t ≤ −M √ t) ≤ 2P (a,0) (\|W t \| ≥ M √ t). Passing a → 0 yields, by the Brownian scaling property, that P a * (\|X t \| ρ ≥ M √ t) = P 0 (\|Y t \| ≥ M √ t) ≤ 2P 0 (\|W t \| ≥ M √ t) = 2P 0 (\|W 1 \| ≥ M ), which is less than 1/2 by choosing M large. This completes the proof of the theorem. In the next two propositions, we use the two-sided estimate for p(t, a * , a * ) as well as Markov property of X to get two-sided bounds for p(t, a * , x) for different regions of x. We first record an elementary lemma that will be used later. Lemma 5.11. For every x > 0, 1 1 + x e −x 2 /2 ≤ ∞ x e −y 2 /2 dy ≤ eπ 1 + x e −x 2 /2 . Proof. Define φ(x) = ∞ x e −y 2 /2 dy − 1 1+x e −x 2 /2 . Then φ ′ (x) = − x (1+x) 2 e −x 2 /2 < 0. Since lim x→∞ φ(x) = 0, we have φ(x) > 0 for every x > 0. This establishes the lower bound estimate of the lemma. For the upper bound, note that for x ∈ (0, 1), ∞ x e −y 2 /2 dy ≤ 1 2 ∞ −∞ e −y 2 /2 dy = π 2 ≤ eπ/2 e −x 2 /2 , while for every x > 0, using a change of variable y = x + z, ∞ x e −y 2 /2 dy ≤ e −x 2 /2 ∞ 0 e −xz dz = x −1 e −x 2 /2 . This establishes the upper bound estimate of the lemma. Proposition 5.12. There exist constants C i > 0, 9 ≤ i ≤ 10, so that for all x ∈ R + and t ≥ 2, C 9 t 1 + \|x\| log t √ t e −2x 2 /t ≤ p(t, a * , x) ≤ C 10 t 1 + \|x\| log t √ t e −x 2 /2t . Proof. Observing that when x ∈ R + , we have by (5.8), p(t, a * , x) = p(t, x, a * ) = t 0 p(t − s, a * , a * )P x (σ a * ∈ ds) = t/2 0 p(t − s, a * , a * ) x √ 2πs 3 e −x 2 /2s ds + t t/2 p(t − s, a * , a * ) x √ 2πs 3 e −x 2 /2s ds. (5.21) By Theorem 5.10, Lemma 5.11 and a change of variable r = x/ √ s, t/2 0 p(t − s, a * , a * ) x √ 2πs 3 e −x 2 /2s ds ≍ 1 t t/2 0 x s 3/2 e −x 2 /2s ds = 2 t ∞ x √ 2/t e −r 2 /2 dr ≍ 1 t 1 1 + (x/ √ t) e −x 2 /t , while x √ 2πt 3 e −x 2 /t t/2 0 p(r, a * , a * )dr ≤ t t/2 p(t − s, a * , a * ) x √ 2πs 3 e −x 2 /2s ds ≤ 2x √ πt 3 e −x 2 /2t t/2 0 p(r, a * , a * )dr. This, Theorem 5.10 and (5.21) yields the desired result. Proposition 5.13. There exist C i > 0, 11 ≤ i ≤ 14 such that C 11 t e −C 12 \|x\| 2 ρ /t ≤ p(t, a * , x) ≤ C 13 t e −C 14 \|x\| 2 ρ /t for all t ≥ 1 and x ∈ D 0 . Proof. When t ∈ [1,8], the estimates follows from Theorem 4.7. So it remains to establish the estimates for t > 8. We do this by considering three cases. Note that p(t, a * , x) = p(t, x, a * ). Case 1. 1 ≤ \|x\| ρ < 2 √ t. We have by Theorem 5.10 and Proposition 5.1, p(t, x, a * ) = t/2 0 p(t − s, a * , a * )P x (σ a * ∈ ds) + t t/2 p(t − s, a * , a * )P x (σ a * ∈ ds) ≍ 1 t P x (σ a * ≤ t/2) + log \|x\| t(log t) 2 t/2 0 p(s, a * , a * )ds ≍ 1 t 1 − log \|x\| log √ t + log \|x\| t(log t) 2 t/2 0 1 √ s ∧ 1 s ds ≍ 1 t 1 − log \|x\| log √ t + log \|x\| t log t ≍ 1 t . Case 2. \|x\| ρ ≥ 2 √ t. In the following computation, B ρ (x, r) := {y ∈ E : ρ(x, y) < r}, and B e (x, r) := {y ∈ E : \|y − x\| < r}. We denote by {W ; P 0 x , x ∈ R 2 } two-dimensional Brownian motion and p 0 (t, x, y) = (2πt) −1 exp(−\|x − y\| 2 /2t) its transition density. Since p(t, x, a * ) = E x p(t − σ Bρ(a * ,2) , X σ Bρ (a * ,2) , a * ); σ Bρ(a * ,2) < t , it follows from Case 1, Theorem 4.5 and Theorem 4.7 that there is c 1 ≥ 1 so that p(t, x, a * ) E x t − σ Bρ(a * ,2) −1 e − (2+2ε) 2 2c 1 (t−σ Bρ(a * ,2) ) ; σ Bρ(a * ,2) < t = E 0 x t − σ Be(0,2+ε) −1 e − (2+2ε) 2 2c 1 (t−σ Be(0,2+ε) ) ; σ Be(0,2+ε) < t E 0 x c 1 t − σ Be(0,2+ε) −1 e − (2+ε) 2 2(c 1 t−σ Be(0,2+ε) ) ; σ Be(0,2+ε) < t ≤ E 0 x c 1 t − σ Be(0,2+ε) −1 e − (2+ε) 2 2(c 1 t−σ Be(0,2+ε) ) ; σ Be(0,2+ε) < c 1 t ≍ E 0 x p 0 (c 1 t − σ Be(0,2+ε) ), W σ Be(0,2+ε) , 0); σ Be (0,2+ε) < c 1 t = p 0 (c 1 t, x, 0) = (2πc 1 t) −1 exp(−\|x\| 2 /2c 1 t). Similarly, for the lower bound estimate, it follows from Case 1, Theorem 4.5 and Theorem 4.7 that there is c 2 ∈ (0, 1] so that p(t, x, a * ) E x t − σ Bρ(a * ,2) −1 e − 2 2 2c 2 (t−σ Bρ(a * ,2) ) ; σ Bρ(a * ,2) < t ≥ E 0 x t − σ Be(0,2+ε) −1 e − 2 2 2c 2 (t−σ Be(0,2+ε) ) ; σ Be(0,2+ε) < c 2 t E 0 x c 2 t − σ Be(0,2+ε) −1 e − (2+ε) 2 2(c 2 t−σ Be(0,2+ε) ) ; σ Be(0,2+ε) < c 2 t ≍ E 0 x p 0 (c 2 t − σ Be(0,2+ε) ), W σ Be(0,2+ε) , 0); σ Be (0,2+ε) < c 2 t = p 0 (c 2 t, x, 0) = (2πc 2 t) −1 exp(−\|x\| 2 /2c 2 t). Realizing that \|x\| ρ > 2 √ t > 4 √ 2 implies that \|x\| ρ ≍ \|x\|, we get the desired estimates in this case. Case 3. 0 < \|x\| ρ < 1. Note that by Proposition 3.2, y∈D 0 ∩Bρ(a * ,2) p(t/2, a * , y)p(t/2, y, x)m p (dy) ≤ D 0 ∩Bρ(a * ,2) c 1 √ t 2 m p (dy) ≍ 1 t , (5.22) while by Cases 1 and 2 above, D 0 ∩Bρ(a * ,2) c p(t/2, x, y)p(t/2, a * , y)m p (dy) ≤ sup y∈D 0 ∩Bρ(a * ,2) c p(t/2, a * , y) 1 t . (5.23) On the other hand, by Theorem 5.10 again, R + p(t/2, a * , y)p(t/2, y, x)m p (dy) = E x p(t/2, X t/2 , a * ); X t/2 ∈ R + = E x p(t/2, X t/2 , a * ); σ a * < t/2 and X t/2 ∈ R + = E x E a * [p(t/2, X t/2−s , a * )]\| s=σ a * ; σ a * < t/2 and X t/2 ∈ R + = E x p(t − σ a * , a * , a * ); σ a * < t/2 and X t/2 ∈ R + ≍ 1 t P x σ a * ≤ t/2 and X t/2 ∈ R + ≤ 1 t . (5.24) The estimates (5.22), (5.23) and (5.24) imply that 2) p(t/2, a * , y)p(t/2, y, x)m p (dy) + 2) p(t/2, a * , y)p(t/2, y, x)m p (dy) p(t, a * , x) = D 0 ∩Bρ(a * ,D 0 ∩B c ρ (a * ,+ R + p(t/2, a * , y)p(t/2, y, x)m p (dy) 1 t . On the other hand, there is a constant c 3 > 0 so that P x (σ a * ≤ 1) ≥ c 3 for all x ∈ D 0 with \|x\| ρ ≤ 1. Hence we have by Theorem 5.10 that for t ≥ 2 and x ∈ D 0 with \|x\| ρ ≤ 1, p(t, a * , x) = p(t, x, a * ) ≥ 1 0 p(t − s, a * , a * )P x (σ a * ∈ ds) 1 t P x (σ a * ≤ 1) 1 t . In conclusion, we have p(t, a * , x) ≍ 1 t for t ≥ 4 and x ∈ D 0 with \|x\| ρ ≤ 1. This completes the proof of the proposition. We are now in the position to derive estimates on p(t, x, y) for (x, y) ∈ D 0 ×D and t ∈ [8, ∞), by using the two-sided estimate of p(t, x, a * ) and the Markov property of X. Theorem 5.14. There exist constants C i > 0, 15 ≤ i ≤ 18, such that the following estimate holds: C 15 t e −C 16 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 17 t e −C 18 ρ(x,y) 2 /t , (t, x, y) ∈ [8, ∞) × D 0 × D 0 . Proof. As before, denote by {W ; P 0 x , x ∈ R 2 } two-dimensional Brownian motion and p 0 (t, x, y) = (2πt) −1 exp(−\|x − y\| 2 /2t) its transition density. We first note that, as a special case of [40, Theorem 1.1(a)], there are constants c 1 > c 2 > 0 so that for t ≥ 1 and x, y ∈ D 0 , (\|x\| ρ ∧ 1) (\|y\| ρ ∧ 1) t −1 e −c 1 \|x−y\| 2 /t p 0 D 0 (t, x, y) (\|x\| ρ ∧ 1) (\|y\| ρ ∧ 1) t −1 e −c 2 \|x−y\| 2 /t . It follows that there is c 3 ∈ (0, 1] so that p 0 D 0 (t, x, y) p 0 D 0 (c 3 t, x, y) for every t ≥ 1 and x, y ∈ D 0 . (5.25) We will prove the theorem by considering two different cases. Case 1. \|x\| ρ + \|y\| ρ > 2. Without loss of generality, we assume \|y\| ρ > 1. In this case, it is not hard to verify that ρ(x, y) ≍ \|x − y\|. Recall from (1.6)-(1.7) that for x, y ∈ D 0 , p D 0 (t, x, y) := p(t, x, y) − p D 0 (t, x, y) = E x [p(t − σ a * , a * , y); σ a * < t] . By Proposition 5.13 and the assumption that ε ∈ (0, 1/4], there are constants c 4 ≥ 1 so that for every x, y ∈ D 0 with \|y\| ρ > 1 p D 0 (t, x, y) t 0 1 t − s e − (\|y\|ρ+3ε) 2 2c 4 (t−s) P x (σ a * ∈ ds) ≤ c 1 t 0 1 c 4 t − s e − (\|y\|ρ+2ε) 2 2(c 4 t−s) P 0 x σ Be(0,ε) ∈ ds E 0 x p 0 (c 4 t − s, W σ Be(0,ε) , y); σ Be(0,ε) < c 4 t ≤ p 0 (c 4 t, x, y). (5.26) Similarly, there is a constant c 5 ∈ (0, c 3 ] so that p D 0 (t, x, y) t 0 1 t − s e − (\|y\|ρ−ε) 2 2c 5 (t−s) P x (σ a * ∈ ds) ≥ c 5 t 0 1 c 5 t − s e − \|y\| 2 ρ 2(c 5 t−s) P 0 x σ Be(0,ε) ∈ ds E 0 x p 0 (c 5 t − s, W σ Be(0,ε) , y); σ Be(0,ε) < c 5 t = p 0 D 0 (c 5 t, x, y). (5.27) Since p D 0 (t, x, y) = p 0 D 0 (t, x, y) and c 4 ≥ 1, we have from (5.26) that p(t, x, y) = p D 0 (t, x, y) + p D 0 (t, x, y) p 0 (t, x, y) + p 0 (c 4 t, x, y) p 0 (c 4 t, x, y) t −1 e −ρ(x,y) 2 /2c 4 t . On the other hand, we have by (5.25) and (5.27) that for every t ≥ 1 and x, y ∈ D 0 satisfying \|x\| ρ + \|y\| ρ > 2, p(t, x, y) p 0 D 0 (c 5 t, x, y) + p 0 D 0 (c 5 t, x, y) = p 0 (c 5 t, x, y) t −1 e −\|x−y\| 2 /2c 5 t t −1 e −ρ(x,y) 2 /c 5 t , where the last " " is due to the fact that \|x − y\|/ √ 2 ≤ ρ(x, y), which can be verified easily from the assumptions that \|x\| ρ + \|y\| ρ > 2 and ε ≤ 1/4. This establishes the desired two-sided estimates in this case. Case 2. \|x\| ρ + \|y ρ ≤ 2. In this case, it suffices to show that p(t, x, y) ≍ t −1 for t ≥ 8. The proof is similar to that of Case 3 of Proposition 5.13. By Proposition 3.2, for t ≥ 8, y∈D 0 ∩Bρ(a * ,2) p(t/2, x, z)p(t/2, z, y)m p (dy) ≤ D 0 ∩Bρ(a * ,2) c 6 √ t 2 m p (dy) ≍ 1 t , (5.28) while by Case 1, D 0 ∩B(a * ,2) c p(t/2, x, z)p(t/2, z, y)m p (dz) ≤ sup z∈D 0 ∩B(a * ,2) c p(t/2, y, z) 1 t . (5.29) On the other hand, by Proposition 5.13, for t ≥ 8, R + p(t/2, x, z)p(t/2, z, y)m p (dz) = E x p(t/2, X t/2 , y); X t/2 ∈ R + = E x p(t/2, X t/2 , y); σ a * < t/2 and X t/2 ∈ R + = E x E a * [p(t/2, X t/2−s , y)]\| s=σ a * ; σ a * < t/2 and X t/2 ∈ R + = E x p(t − σ a * , a * , y); σ a * < t/2 and X t/2 ∈ R + ≍ 1 t P x σ a * ≤ t/2 and X t/2 ∈ R + ≤ 1 t . (5.30) The estimates (5.28)-(5.30) imply that 2) p(t/2, a * , y)p(t/2, y, x)m p (dy) + 2) p(t/2, a * , y)p(t/2, y, x)m p (dy) p(t, a * , x) = D 0 ∩Bρ(a * ,D 0 ∩B c ρ (a * ,+ R + p(t/2, a * , y)p(t/2, y, x)m p (dy) 1 t . On the other hand, there is a constant c 7 > 0 so that P x (σ a * ≤ 1) ≥ c 7 for all x ∈ D 0 with \|x\| ρ ≤ 2. Hence we have by Proposition 5.13 that for t ≥ 2 and x ∈ D 0 with \|x\| ρ ≤ 1, p(t, x, y) ≥ 1 0 p(t − s, a * , y)P x (σ a * ∈ ds) 1 t P x (σ a * ≤ 1) 1 t . Therefore we have p(t, x, y) ≍ 1 t for t ≥ 8 and x, y ∈ D 0 with \|x\| ρ + \|y\| ρ ≤ 2. This completes the proof of the theorem. Theorem 5.15. There exist constants C i > 0, 19 ≤ i ≤ 26, such that the following estimates hold for (t, x, y) ∈ [4, ∞) × R + × D 0 : when \|y\| ρ < 1, C 19 t 1 + \|x\| log t √ t e −C 20 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 21 t 1 + \|x\| log t √ t e −C 22 ρ(x,y) 2 /t ; (5.31) while for \|y\| ρ ≥ 1, C 23 t 1 + \|x\| √ t log 1 + √ t \|y\| ρ e −C 24 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 25 t 1 + \|x\| √ t log 1 + √ t \|y\| ρ e −C 26 ρ(x,y) 2 /t . (5.32) Proof. First, note that by Proposition 5.13 and Theorem 4.7, 1 t e −c 1 \|y\| 2 ρ /t p(t, x, y) 1 t e −c 2 \|y\| 2 ρ /t for t ≥ 1 and y ∈ D 0 (5.33) and 1 √ t e −c 3 \|y\| 2 ρ /t p(t, a * , y) 1 √ t e −c 4 \|y\| 2 ρ /t for t ≤ 1 and y ∈ D 0 with \|y\| ρ ≤ 1. (5.34) By (5.8), p(t, x, y) = t 0 p(t − s, a * , y)P x (σ a * ∈ ds) = t 0 p(t − s, a * , y) \|x\| √ 2πs 3 e −x 2 /(2s) ds. (5.35) It follows from (5.33) and Lemma 5.11 that for every y ∈ D 0 and t ≥ 4, t/2 0 p(t − s, a * , y) \|x\| √ 2πs 3 e −x 2 /(2s) ds − 1 t e −c 2 \|y\| 2 ρ /t t/2 s=0 e − \|x\| 2 2s d \|x\| √ s 1 t e −c 2 \|y\| 2 ρ /t 1 1 + \|x\|/ √ t e −\|x\| 2 /t ≤ 1 t e −c 3 ρ(x,y) 2 /t . (5.36) Similarly, we have t/2 0 p(t − s, a * , y) x √ 2πs 3 e −x 2 /(2s) ds 1 t e −(\|x\| 2 +c 4 \|y\| 2 ρ )/t 1 t e −c 5 ρ(x,y) 2 /t . (5.37) We now consider two cases depending on the range of the values of \|y\| ρ . Case 1. y ∈ D 0 with \|y\| ρ < 1. In this case, we have by (5.33) t−1 t/2 p(t − s, a * , y) x √ 2πs 3 e −x 2 /(2s) ds t−1 t/2 1 t − s e −c 2 \|y\| 2 ρ /(t−s) \|x\| √ s 3 e −x 2 /(2s) ds \|x\| t 3/2 e −\|x\| 2 /2t t−1 t/2 1 t − s ds \|x\| log t t 3/2 e −\|x\| 2 /(2t) . Similarly, we have t−1 t/2 p(t − s, a * , y) x √ 2πs 3 e −x 2 /(2s) ds \|x\| log t t 3/2 e −\|x\| 2 /t . On the other hand, by (5.34), t t−1 p(t − s, a * , y) x √ 2πs 3 e −x 2 /(2s) ds \|x\| √ t 3 e −x 2 /(2t) t t−1 1 √ t − s e −c 4 \|y\| 2 ρ /(t−s) ds \|x\| √ t 3 e −x 2 /(2t) t t−1 1 √ t − s ds \|x\| √ t 3 e −x 2 /(2t) , and similarly t t−1 p(t − s, a * , y) x √ 2πs 3 e −x 2 /(2s) ds \|x\| √ t 3 e −x 2 /t . These estimates together with (5.35)-(5.37) establishes (5.31). Case 2. y ∈ D 0 with \|y\| ρ > 1. Note that by (5.8), p(t, x, y) = t 0 p(t − s, a * , y)P x (σ a * ∈ ds) (5.38) = t/2 0 p(t − s, a * , y) \|x\| √ 2πs 3 e −x 2 /2s ds + t t/2 p(t − s, a * , y) \|x\| √ 2πs 3 e −x 2 /2s ds. By Theorem 4.7, (5.33) and a change of variable r = \|y\| ρ / √ t − s, we have t t/2 p(t − s, a * , y) \|x\| √ 2πs 3 e −x 2 /2s ds 1 t − s e −c 6 \|y\| 2 ρ /(t−s) \|x\| √ s 3 e −x 2 /2s ds \|x\| t 3/2 e −x 2 /2t t t/2 1 t − s e −c 6 \|y\| 2 ρ /(t−s) ds = \|x\| t 3/2 e −x 2 /2t ∞ \|y\|ρ/ √ 2/t 2 r e −c 6 r 2 dr. Note that for each fixed a > 0, ∞ λ r −1 e −ar 2 dr = 1 λ r −1 e −ar 2 dr + ∞ 1 r −1 e −c 2 r 2 dr is comparable to log(1/λ) when 0 < λ ≤ 1/2. For λ ≥ 1/2, by Lemma 5.11 ∞ λ r −1 e −ar 2 dr ≤ 2 ∞ λ e −ar 2 dr = 2 √ 2a ∞ √ 2aλ e −s 2 /2 ds 1 1 + √ aλ e −aλ 2 ≤ e −aλ 2 and ∞ λ r −1 e −ar 2 dr ∞ λ e −2ar 2 dr = 1 2 √ a ∞ 2 √ aλ e −s 2 /2 ds 1 1 + √ aλ e −2aλ 2 e −3aλ 2 Hence we have log(1 + λ −1 )e −3aλ 2 ∞ λ r −1 e −ar 2 dr ≤ log(1 + λ −1 )e −p(t − s, a * , y) \|x\| √ 2πs 3 e −x 2 /2s ds \|x\| t 3/2 e −x 2 /2t log 1 + √ t \|y\| ρ e −2c 6 \|y\| 2 ρ /t ≤ \|x\| t 3/2 log 1 + √ t \|y\| ρ e −c 7 ρ(x,y) 2 /t . Similarly, we have t t/2 p(t − s, a * , y) \|x\| √ 2πs 3 e −x 2 /2s ds \|x\| t 3/2 log 1 + √ t \|y\| ρ e −c 8 ρ(x,y) 2 /t . These together with (5.36)-(5.37) establishes (5.32). We will need the following elementary lemma. Lemma 5.16. For every c > 0, there exists C 27 ≥ 1 such that for every t ≥ 3 and 0 < y ≤ √ t, C −1 27 log t ≤ t 2 1 s 1 + y log s √ s e −c\|y\| 2 /s ds ≤ C 27 log t. Proof. By a change of variable r = y/ √ s, while since 0 < y ≤ √ t, t 2 1 s e −c\|y\| 2 /s ds ≍ t 2 1 s ds ≍ log t. This proves the lemma. Theorem 5.17. There exist constants C i > 0, 28 ≤ i ≤ 34, such that the following estimate holds for all (t, x, y) ∈ [8, ∞) × R + × R + : C 28 √ t 1 ∧ \|x\| √ t 1 ∧ \|y\| √ t e −C 29 \|x−y\| 2 /t + C 28 t 1 + (\|x\| + \|y\|) log t √ t e −C 30 (x 2 +y 2 )/t ≤ p(t, x, y) ≤ C 31 √ t 1 ∧ \|x\| √ t 1 ∧ \|y\| √ t e −C 32 \|x−y\| 2 /t + C 31 t 1 + (\|x\| + \|y\|) log t √ t e −C 33 (x 2 +y 2 )/t . (5.40) Proof. When either x = a * or y = a * , this has been established in Proposition 5.12 so we assume \|x\| ∧ \|y\| > 0. For simplicity, denote R + \ {a * } by (0, ∞). Since p (0,∞) (t, x, y) = (2πt) −1/2 e −\|x−y\| 2 /2t − e −\|x+y\| 2 /2t = (2πt) −1/2 e −\|x−y\| 2 /2t 1 − e −2xy/t , there are constants c 1 > c 2 > 0 so that 1 √ t 1 ∧ \|x\| √ t 1 ∧ \|y\| √ t e −c 1 \|x−y\| 2 /t p (0,∞) (t, x, y) 1 √ t 1 ∧ \|x\| √ t 1 ∧ \|y\| √ t e −c 2 \|x−y\| 2 /t . (5.41) for all t > 0 and x, y ∈ R + . Note also p(t, x, y) = p (0,∞) (t, x, y) + t 0 p(t − s, a * , y)P x (σ a * ∈ ds). (5.42) We prove this theorem by considering two cases. Case 1. \|x\| ∧ \|y\| ≥ √ t. In this case , p(t, x, y) ≥ p (0,∞) (t, x, y) t −1/2 e −c 3 \|x−y\| 2 /t . Thus we have by Proposition 3.3, 1 √ t e −c 3 \|x−y\| 2 /t p(t, x, y) 1 √ t e −c 4 \|x−y\| 2 /t . Case 2. 0 < \|x\| ∧ \|y\| < √ t. Without loss of generality, we may and do assume \|y\| < √ t. By (5.8), t 0 p(t − s, a * , y)P x (σ a * ∈ ds) = t 0 p(t − s, a * , y) \|x\| √ 2πs 3 e −x 2 /2s ds. (5.43) By Proposition 5.12 and Lemma 5.11 t/2 0 p(t − s, a * , y) \|x\| √ 2πs 3 e −x 2 /2s ds t/2 0 1 t − s 1 + \|y\| log(t − s) √ t − s e −y 2 /2(t−s) \|x\| √ 2πs 3 e −x 2 /2s ds 1 t 1 + \|y\| log t √ t t/2 0 \|x\| s 3/2 e −x 2 /2s ds 1 t 1 + \|y\| log t √ t 1 1 + \|x\|/ √ t e −x 2 /t 1 t 1 + \|y\| log t √ t e −(x 2 +y 2 )/t , while by Lemma 5.16, t−2 t/2 p(t − s, a * , y) \|x\| √ 2πs 3 e −x 2 /2s ds t−2 t/2 1 t − s 1 + \|y\| log(t − s) √ t − s e −y 2 /2(t−s) \|x\| √ 2πs 3 e −x 2 /2s ds \|x\| t 3/2 e −x 2 /2t t−2 t/2 1 t − s 1 + \|y\| log(t − s) √ t − s e −y 2 /2(t−s) ds r=t−s = \|x\| t 3/2 e −x 2 /2t t/2 2 1 r 1 + \|y\| log r √ r e −y 2 /2r dr \|x\| t 3/2 e −x 2 /2t log t ≍ \|x\| log t t 3/2 e −(x 2 +y 2 )/2t . A similar calculation shows t/2 0 p(t − s, a * , y) \|x\| √ 2πs 3 e −x 2 /2s ds 1 t 1 + \|y\| log t √ t e −2(x 2 +y 2 )/t and t−2 t/2 p(t − s, a * , y) \|x\| √ 2πs 3 e −x 2 /2s ds \|x\| t 3/2 e −x 2 /2t log t ≍ \|x\| log t t 3/2 e −2(x 2 +y 2 )/t . By Theorem 4.5, t t−2 p(t − s, a * , y) \|x\| √ 2πs 3 e −x 2 /2s ds t t−2 1 √ t − s e −c 5 y 2 /(t−s) \|x\| √ 2πs 3 e −x 2 /2s ds \|x\| √ t 3 e −x 2 /t t t−2 1 √ t − s e −c 5 y 2 /(t−s) ds \|x\| t 3/2 e −x 2 /2t \|x\| t 3/2 e −(x 2 +y 2 )/2t . These estimates together with (5.41)-(5.43) establish the theorem. Theorem 5.17 together with Theorems 5.14 and 5.15 gives Theorem 1.4. Hölder regularity of Parabolic Functions As we noted in Remark 4.8(iii), parabolic Harnack principle fails for the BMVD X. However we show in this section that Hölder regularity holds for the parabolic functions of X. In the elliptic case (that is, for harmonic functions instead of parabolic functions), this kind of phenomenon has been observed for solutions of SDEs driven by multidimensional Lévy processes with independent coordinate processes; see [6]. To show the Hölder-continuity of parabolic functions of X, we begin with the following two lemmas. Lemma 6.1. There exist C 1 > 0 and 0 < C 2 ≤ 1/2 such that for every x 0 ∈ E and R > 0, p B(x 0 ,R) (t, x, y) ≥ 1 2 p(t, x, y) for t ∈ (0, C 1 /(R ∨ 1) 2 ] and x, y ∈ B(x 0 , C 2 R). Proof. By Theorem 1.3, there exist constants c i > 0, 1 ≤ i ≤ 4, such that for all t ≤ 1 and x, y ∈ E, c 3 √ t e −c 4 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ c 1 t e −c 2 ρ(x,y) 2 /t . (6.1) We choose 0 < c 5 < 1/2 sufficiently small such that (1 − c 5 ) 2 (2c 5 ) 2 ≥ c 2 c 4 . (6.2) As t → t −1 e −c 0 /t is increasing in t ∈ (0, 1/c 0 ], we have for 0 < t ≤ 1/(c 2 (1 − c 5 ) 2 R 2 ) and x, y ∈ B(x 0 , c 5 R), p B(x 0 ,c 5 R) (t, x, y) := E x [p(t − τ B(x 0 ,c 5 R) , X B(x 0 ,c 5 R) , y); τ B(x 0 ,c 5 R) < t] E x [(t − τ B(x 0 ,c 5 R) ) −1 e −c 2 ((1−c 5 )R) 2 /(t−τ B(x 0 ,c 5 R) ) ; τ B(x 0 ,c 5 R) < t] ≤ t −1 e −c 2 ((1−c 5 )R) 2 /t e −c 2 (1−c 5 ) 2 R 2 /2t , while p(t, x, y) ≥ c 3 √ t e −c 4 (2c 5 R) 2 /t (6.2) ≥ c 3 √ t e −c 2 (1−c 5 ) 2 R 2 /2t . Hence there is c 6 ≤ 1/(c 2 (1 − c 5 ) 2 ) so that p(t, x, y) ≥ 1 2 p B(x 0 ,c 5 R) (t, x, y) for every R > 0, x 0 ∈ E, 0 < t ≤ c 6 /(R∨1) 2 and x, y ∈ B(x 0 , c 5 R). This proves the lemma as p B(x 0 ,c 5 R) (t, x, y) = p(t, x, y) − p B(x 0 ,c 5 R) (t, x, y). Let Z s = (V s , X s ) be the space-time process of X where V s = V 0 + s. In the rest of this section, Q(t, x, R) := (t, t + R 2 ) × B ρ (x, R). For any Borel measurable set A ⊂ Q(t, x, R), we use \|A\| to denote its measure under the product measure dt × m p (dx). Lemma 6.2. Fix R 0 ≥ 1. There exist constants 0 < C 3 ≤ 1/2 and C 4 > 0 such that for all 0 < R ≤ R 0 , x 0 ∈ E, x ∈ B ρ (x 0 , C 3 R) and any A ⊂ Q(0, x 0 , C 3 R) with \|A\| \|Q(0,x 0 ,C 3 R)\| ≥ 1 3 , P (0,x) (σ A < τ R ) ≥ C 4 ,(6.3) where τ R = τ Q(0,x 0 ,R) = inf{t ≥ 0 : X t / ∈ B ρ (x 0 , R)} ∧ R 2 and σ A := inf{t ≥ 0 : (V t , X t ) ∈ A}. Proof. Let C 1 and C 2 be the constants in Lemma 6.1. Define C 3 = (C 1 /R 3 0 ) 1/2 ∧ C 2 . For x 0 ∈ E and R ∈ (0, R 0 ], denote by X Bρ(x 0 ,R) the subprocess of X killed upon exiting the ball B ρ (x 0 , R) and p Bρ(x 0 ,R) its transition density with respect to the measure m p . As \|(0, C 2 3 R 2 /6)× B ρ (x 0 , C 3 R)\| = \|Q(0, x 0 , C 3 R)\|/6 and \|A\| ≥ \|Q(0, x 0 , C 3 R)\|/3, we have \|{(t, x) ∈ A : t ∈ [C 2 3 R 2 /6, C 2 3 R 2 ]}\| ≥ \|Q(0, x 0 , C 3 R)\|/6. For s > 0, let A s := {x ∈ E : (s, x) ∈ A}. Note that E x τ R 0 1 A (s, X s )ds = E x τ R ∧(C 3 R) 2 0 1 A (s, X Bρ(x 0 ,R) )ds = C 2 3 R 2 0 P x X Bρ(x 0 ,R) s ∈ A s ds = C 2 3 R 2 0 As p Bρ(x 0 ,R) (s, x, y)m p (dy)ds ≥ C 2 3 R 2 C 2 3 R 2 /6 p Bρ(x 0 ,R) (s, x, y)m p (dy)ds. (6.4) We now consider two cases. Case 1. m p (B ρ (x 0 , R)) > pR/6. In this case, we have by (6.4), Lemma 6.1 and (6.1) that E x τ R 0 1 A (s, X s )ds C 2 3 R 2 C 2 3 R 2 /6 As 1 √ t e −c 2 ρ(x,y) 2 /t m p (dy)ds 1 R \|{(t, x) ∈ A : t ∈ [C 2 3 R 2 /6, C 2 3 R 2 ]}\| \|Q(0, x 0 , C 3 R)\|/R R 2 . Case 2. m p (B ρ (x 0 , R)) ≤ pR/6. In this case, x 0 must be in D 0 with ρ(x 0 , a * ) ≥ 5R 6 and so m p (B ρ (x 0 , R)) ≥ (5R/6) 2 . Thus we have by (6.4) and Theorem 1.3(iii), E x τ R 0 1 A (s, X s )ds C 2 3 R 2 C 2 3 R 2 /6 As 1 t e −c 2 ρ(x,y) 2 /t m p (dy)ds 1 R 2 \|{(t, x) ∈ A : t ∈ [C 2 3 R 2 /6, C 2 3 R 2 ]}\| \|Q(0, x 0 , C 3 R)\|/R 2 R 2 . Thus in both cases, there is a constant c 0 > 0 independent of x 0 and R ∈ (0, R 0 ] so that E x τ R 0 1 A (s, X s )ds ≥ c 0 R 2 . (6.5) On the other hand, E x τ R 0 1 A (s, X s )ds = ∞ 0 P x τ R 0 1 A (s, X s )ds > u du = R 2 0 P x τ R 0 1 A (s, X s )ds > u du ≤ R 2 P x (σ A < τ R ). The desired estimate now follows from this and (6.5). Theorem 6.3. For every R 0 > 0, there are constants C = C(R 0 ) > 0 and β ∈ (0, 1) such that for every R ∈ (0, R 0 ], x 0 ∈ E, and every bounded parabolic function q in Q(0, x 0 , 2R), it holds that \|q(s, x) − q(t, y)\| ≤ C q ∞,R R −β \|t − s\| 1/2 + ρ(x, y) β (6.6) for every (s, x), (t, y) ∈ Q(0, x 0 , R/4), where q ∞,R := sup (t,y)∈(0,4R 2 ]×Bρ(x 0 ,2R) \|q(t, y)\|. Proof. With loss of generality, assume 0 ≤ q(s) ≤ q ∞,R = 1. We first assume x 0 = a * and show that (6.6) holds for all (s, x), (t, y) ∈ Q(0, x 0 , R) (instead of Q(0, x 0 , R/4)). Let C 3 ∈ (0, 1/2] and C 4 ∈ (0, 1) be the constants in Lemma 6.2. Let η = 1 − C 4 /4 ≥ 3/4 and γ = C 3 /2 ≤ 1/4. Note that for every (s, x) ∈ Q(0, a * , R), q is parabolic in Q(s, x, R) ⊂ Q(0, a * , 2R). We will show by induction that sup Q(s,x,γ k R) \|q\| − inf Q(s,x,γ k R) \|q\| ≤ η k for all integer k. For notation convenience, we denote Q(s, x, γ k R) by Q k . Define a i = inf Q i q, b i = sup Q i q. Clearly, b i − a i ≤ 1 ≤ η i for all i ≤ 0. Now suppose b i − a i ≤ η i for all i ≤ k and we wil show that b k+1 − a k+1 ≤ η k+1 . Observe that Q k+1 ⊂ Q k and so a k ≤ q ≤ b k on Q k+1 . Define A ′ := z ∈ s + (γ k+1 R) 2 , s + (C 3 γ k R) 2 × B ρ (x, C 3 γ k R) : q(z) ≤ (a k + b k )/2 , which is a subset of Q k . Note that s + (γ k+1 R) 2 , s + (C 3 γ k R) 2 × B ρ (x, γ k R) = 3 4 (C 3 γ k R) 2 m p (B ρ (x, C 3 γ k R)). We may suppose \|A ′ \| ≥ 1 2 (C 3 γ k R) 2 m p (B(x, C 3 γ k R)); otherwise we consider 1 − q instead of q. Let A be a compact subset of A ′ such that \|A\| ≥ 1 2 (C 3 γ k R) 2 m p (B(x, C 3 γ k R)). For any given ε > 0, pick z 1 = (t 1 , x 1 ), z 2 ∈ Q k+1 so that q(z 1 ) ≥ b k+1 − ε and q(z 1 ) ≤ a k+1 + ε. Note that Z τ k ∈ ∂Q k as BMVD X t has continuous sample paths. So by Lemma 6.2, b k+1 − a k+1 − 2ε ≤ q(z 1 ) − q(z 2 ) = E z 1 [q(Z σ A ∧τ k ) − q(z 2 )] = E z 1 [q(Z σ A ) − q(z 2 ); σ A < τ k ] + E z 1 [q(Z τ k ) − q(z 2 ); τ k < σ A ] ≤ a k + b k 2 − a k P z 1 (σ A < τ k )) + (b k − a k )P z 1 (σ A > τ k ) = (b k − a k ) (1 − P z 1 (σ A < τ k )/2) ≤ η k (1 − C 4 /2) ≤ η k+1 . Since ε is arbitrary, we get b k+1 − a k+1 ≤ η k+1 . This proves that b k − a k ≤ η k for all integer k. For z = (s, x) and w = (t, y) in Q(0, a * , R) with s ≤ t, let k be the smallest integer such that \|z − w\| := \|t − s\| 1/2 + ρ(x, y) ≤ γ k R. Then \|q(z) − q(w)\| ≤ η k = γ k log η/ log γ ≤ \|z − w\| γR log η/ log γ . This establishes (6.6) for x 0 = a * and for every (s, x), (t, y) ∈ Q(0, a * , R) with β = log η/ log γ. Note that β ∈ (0, 1) since 0 < γ < η < 1. For general x 0 ∈ E, we consider two cases based on the distance ρ(x, a * ): Case 1. \|x\| ρ < R/2. In this case, Q(0, x 0 , R/4) ⊂ Q(0, a * , 3R/4) ⊂ Q(0, a * , 3R/2) ⊂ Q(0, x 0 , 2R). By what we have established above, \|q(s, x) − q(t, y)\| ≤ C(R 0 ) q ∞,R R −β \|t − s\| 1/2 − ρ(x, y) β for (s, x), (t, y) ∈ Q(0, x 0 , R/4). Case 2. \|x\| ρ ≥ R/2. Since a * / ∈ Q(0, x 0 , R/2), it follows from the classical results for Brownian motion in R d with d = 1 and d = 2 that for every (s, x), (t, y) ∈ Q(0, x 0 , R/4) \|q(s, x) − q(t, y)\| ≤ C(R 0 ) q ∞,R/2 R −β \|t − s\| 1/2 + ρ(x, y) β ≤ C(R 0 ) q ∞,R R −β \|t − s\| 1/2 + ρ(x, y) β . This completes the proof of the theorem. Green Function Estimates In this section, we establish two-sided bounds for the Green function of BMVD X killed upon exiting a bounded connected C 1,1 open set D ⊂ E. Recall that the Green function G D (x, y) is defined as follows: G D (x, y) = ∞ 0 p D (t, x, y)dt, where p D (t, x, y) is the transition density function of the subprocess X D with respect to m p . We assume a * ∈ D throughout this section, as otherwise, due to the connectedness of D, either D ⊂ R + or D ⊂ D 0 . Therefore G D (x, y) is just the standard Green function of a bounded C 1,1 domain for Brownian motion in one-dimensional or two-dimensional spaces, whose two-sided estimates are known, see [18]. It is easy to see from p D (t, x, y) = p(t, x, y) − E x [p(t − τ D , X τ D , y); τ D < t], that p D (t, x, y) is jointly continuous in (t, x, y). Recall that for any bounded open set U ⊂ E, δ U (·) := ρ(·, ∂U ) denotes the ρ-distance to the boundary ∂U . For notational convenience, we set D 1 := D ∩ (R + \ {a * }) and D 2 := D ∩ D 0 . Note that a * ∈ ∂D 1 ∩ ∂D 2 . The following theorem gives two-sided Green function estimates for X in bounded C 1,1 domains. Theorem 7.1. Let G D (x, y) be the Green function of X killed upon exiting D, where D is a connected bounded C 1,1 domain of E containing a * . We have for x = y in D, G D (x, y) ≍              δ D (x) ∧ δ D (y), x ∈ D 1 ∪ {a * }, y ∈ D 1 ∪ {a * }; δ D (x)δ D (y) + ln 1 + δ D 2 (x)δ D 2 (y) \|x−y\| 2 , x ∈ D 2 , y ∈ D 2 ; δ D (x)δ D (y), x ∈ D 1 ∪ {a * }, y ∈ D 2 . This combined with the Green function estimates of G D 2 (x, y) (see [18]) yields G D (x, y) ≍ ln 1 + δ D 2 (x)δ D 2 (y) \|x − y\| 2 + δ D (x)δ D (y). (iii) We now consider the last case that x ∈ D 1 ∪ {a * } and y ∈ D 2 . When x = a * , the desired estimates follows from (7.4) and so it remains to consider x ∈ D 1 and y ∈ D 2 . By the strong Markov property of X, (7.3) and (7.4), G D (x, y) = E x [G D (X σ a * , y); σ a * < τ D ] = G D (a * , y)P x (σ a * < τ D ) ≍ δ D (x)δ D (y). This completes the proof of the theorem. Some other BMVD In this section, we present two more examples of spaces with varying dimension that are variations of R 2 ∪ R considered in previous sections. The existence and uniqueness of BMVD on these spaces can be established in a similar way as Theorem 2.2 in Section 2. We will concentrate on short time two-sided estimates on the transition density function of these two BMVD. One can further study their large time heat kernel estimates. Due to the space limitation, we will not pursue it in this paper. A square with several flag poles In this subsection, we study the BMVD on a large square with multiple flag poles of infinite length. The state space E of the process embedded in R 3 is defined as follows. Let {z j ; 1 ≤ j ≤ k} be k points in R 2 that have distance at least 4 between each other. Fix a finite sequence {ε j ; 1 ≤ j ≤ k} ⊂ (0, 1/2) and a sequence of positive constants p := {p j ; 1 ≤ j ≤ k}. For 1 ≤ j ≤ k, denote by B j the closed disk on R 2 centered at z j with radius ε j . Clearly, the distance between two distinct balls is at least 3. Let D 0 = R 2 \ (∪ 1≤i≤k B i ). For 1 ≤ j ≤ k, denote by L j the half-line {(z j , w) ∈ R 3 : w > 0}. By identifying each closed ball B j with a singleton denoted by a * j , we equip the space E := D 0 {a * 1 , · · · , a * k } (∪ k i=1 L i ) with induced topology from R 2 and the half-lines L j , 1 ≤ j ≤ k, with the endpoint of the half-line L i identified with a * i and a neighborhood of a * i defined as {a * i } ∪ (U 1 ∩ L i ) ∪ (U 2 ∩ D 0 ) for some neighborhood U 1 of 0 in R and U 2 of B i in R 2 . Let m p be the measure on E whose restriction on D 0 is the two-dimensional Lebesgue measure, and whose restriction on L j is the one-dimensional Lebesgue measure multiplied by p j for 1 ≤ j ≤ k. So in particular, m p ({a * i }) = 0 for all 1 ≤ i ≤ k. We denote the geodesic distance on E by ρ. Similar to Definition 1.1, BMVD on the plane with multiple half lines is defined as follows. Definition 8.1. Given a finite sequence ε := {ε j ; 1 ≤ j ≤ k} ⊂ (0, 1/2) and a sequence of positive constants p := {p j ; 1 ≤ j ≤ k}. A Brownian motion with varying dimension with parameters ( ε, p) on E is an m p -symmetric diffusion X on E such that (i) its subprocess in L i , 1 ≤ i ≤ k, or D 0 has the same law as that of standard Brownian motion in R + or D 0 ; (ii) it admits no killings on any a * i , 1 ≤ i ≤ k. Recall that the endpoint of the half-line L i is identified with a * i , 1 ≤ i ≤ k. Similar to Theorem 2.2, we have the following theorem stating the existence and uniqueness of the planary BMVD X with multiple half lines. Theorem 8.2. For each k ≥ 2, every ε := {ε j ; 1 ≤ j ≤ k} ⊂ (0, 1/2) and p := {p j ; 1 ≤ j ≤ k} ⊂ (0, ∞), BMVD X on E with parameter ( ε, p) exists and is unique. Its associated Dirichlet form (E, F) on L 2 (E; m p ) is given by F = f : f \| R 2 ∈ W 1,2 (R 2 ); f \| L i ∈ W 1,2 (R), f \| B i = f \| L i (a * i ) for 1 ≤ i ≤ k , E(f, g) = 1 2 D 0 ∇f (x) · ∇g(x)dx + k i=1 p i 2 L i f ′ (x)g ′ (x)dx. It is not difficult to see that BMVD X has a continuous transition density p(t, x, y) with respect to the measure m p . Proposition 8.3. There exist constants C 1 , C 2 > 0 such that p(t, x, y) ≤ C 1 1 t + 1 t 1/2 e −C 2 ρ(x,y) 2 /t for all x, y ∈ E and t > 0. Proof. By an exactly the same argument as that for Proposition 3.1, we can establish Nash-type inequality for X. From it, the off-diagonal upper bound can be derived using Davies' method as in Propositions 3.2 and 3.3. The following theorem gives two-sided bounds for the transition density function p(t, x, y) when t ∈ (0, T ] for each fixed T > 0. Theorem 8.4. Let T ≥ 2 be fixed. There exist positive constants C i , 3 ≤ i ≤ 16 so that the transition density p(t, x, y) of BMVD X on E satisfies the following estimates when t ∈ (0, T ]. Case 1. For x, y ∈ D 0 ∩ B ρ (a * i , 1) for some 1 ≤ i ≤ k, C 3 √ t e −C 4 ρ(x,y) 2 /t + C 3 t 1 ∧ ρ(x, a * i ) √ t 1 ∧ ρ(y, a * i ) √ t e −C 5 \|x−y\| 2 /t ≤ p(t, x, y) ≤ C 6 √ t e −C 7 ρ(x,y) 2 /t + C 6 t 1 ∧ ρ(x, a * i ) √ t 1 ∧ ρ(y, a * i ) √ t e −C 8 \|x−y\| 2 /t ; Case 2. For some 1 ≤ i ≤ k, x ∈ L i , y ∈ L i ∪ B ρ (a * i , 1), C 9 √ t e −C 10 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 11 √ t e −C 12 ρ(x,y) 2 /t ; Case 3. For all other cases, C 13 t e −C 14 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 15 t e −C 16 ρ(x,y) 2 /t . (8.1) Proof. The idea of the proof is to reduce it to the heat kernel for BMVD on plane with one vertical half-line. For an open subset D of E, we use X D to denote the subprocess of X killed upon leaving D and p D (t, x, y) the transition density function of X D with respect to m p . Let C 1 > 0 and C 2 ∈ (0, 1/2) be the constants in Lemma 6.1. (i) We first show that the desired estimates hold for any x, y ∈ E with ρ(x, y) < 2C 2 and for every t ∈ (0, T ]. In this case, let z 0 ∈ E so that {x, y} ∈ B ρ (z 0 , C 2 ). Since ρ(a * i , a * j ) > 3 for i = j, without loss of generality we may and do assume that a * 1 is the base that is closest to z and so min 2≤j≤k ρ(z 0 , a * j ) > 3/2. We have by Lemma 6.1 that p Bρ(z 0 ,1) (t, w, z) ≥ 1 2 p 1 (t, w, z) for t ∈ (0, C 1 ] and w, z ∈ B ρ (z 0 , C 2 ), (8.2) where p 1 (t, x, y) stands for the transition density function of BMVD on the plane with one vertical halfline L 1 at base a * 1 and vertical half line L 1 . This together with Theorem 1.3 in particular implies that there is a constant c 1 > 0 so that p Bρ(z 0 ,1) (t, w, z) ≥ c 1 for every t ∈ [C 1 /2, C 1 ] and w, z ∈ B ρ (z 0 , C 2 ). It thus follows from the Chapman-Kolmogorov's equation that there is a constant c 2 > 0 so that p Bρ(z 0 ,1) (t, w, z) ≥ c 2 for every t ∈ [C 1 , T ] and w, z ∈ B ρ (z 0 , C 2 ). (8.3) This together with (8.2) implies that there is a constant c 3 > 0 so that p(t, w, z) ≥ p Bρ(z 0 ,1) (t, w, z) ≥ c 3 p 1 (t, w, z) for t ∈ (0, T ] and w, z ∈ B ρ (z 0 , C 2 ). (8.4) On the other hand, we have by Proposition 3.3 and the fact that s → s −1 e −a 2 /s is increasing in (0, a 2 ) and decreasing in (a 2 , ∞) that for t ∈ (0, T ] and ρ(x, y) < 2C 2 < 1, p Bρ(z 0 ,1) (t, x, y) := E x [p(t − τ Bρ(z 0 ,1) , X τ Bρ (z 0 ,1) , y); τ Bρ(z 0 ,1) < t] t −1 e −c 4 /t e −c 5 /t e −c 6 ρ(x,y) 2 /t . Consequently, p(t, x, y) = p Bρ(z 0 ,1) (t, x, y) +p Bρ(z 0 ,1) (t, x, y) p 1 (t, x, y) + e −c 6 ρ(x,y) 2 /t . (8.5) This together with (8.4) and Theorem 1.3 establishes the desired estimate of the theorem for any x, y ∈ E with ρ(x, y) < 2C 2 and t ∈ (0, T ]. (ii) We now consider the case when x ∈ L i and y ∈ L i ∪ B ρ (a * i , C 2 ) for some 1 ≤ i ≤ k with ρ(x, y) ≥ 2C 2 . Without loss of generality, we may and do assume i = 1 and ρ(x, a * 1 ) < ρ(y, a * 1 ) if y ∈ L 1 . Let z 0 ∈ L 1 with ρ(z 0 , a * 1 ) = ρ(y, a * 1 ) + C 2 . By the strong Markov property of X, (8.4)-(8.5) and Theorem 1.3, we have for t ∈ (0, T ], p(t, x, y) = E x [p(t − σ z 0 , z 0 , y); σ z 0 < t] ≥ c 3 E x [p 1 (t − σ z 0 , z 0 , y); σ z 0 < t] = c 3 p 1 (t, x, y) t −1/2 e −c 7 ρ(x,y) 2 /y , and p(t, x, y) = E x [p(t − σ z 0 , z 0 , y); σ z 0 < t] (8.5) E x [p 1 (t − σ z 0 , z 0 , y); σ z 0 < t] + E x [e −c 6 C 2 2 /(t−σz 0 ) ; σ z 0 < t] ≤ p 1 (t, x, y) + e −c 6 C 2 2 /t P x (σ z 0 < t) t −1/2 e −c 8 ρ(x,y) 2 /y + e −c 6 C 2 2 /t e −\|x−z 0 \| 2 /t t −1/2 e −c 9 ρ(x,y) 2 /y . In the second to last inequality, we used crossing estimate for one-dimensional Brownian motion Now for case (a) when x ∈ L i ∪ B ρ (a * i , C 2 ) and y ∈ L j ∪ B ρ (a * j , C 2 ) for i = j, let z 0 ∈ D 0 so that both ρ(z, a * i ) and ρ(z, a * j ) take values within (ρ(a * i , a * j )/3, 2ρ(a * i , a * j )/3). We then have by (8.7) that for all t ∈ (0, T ], p(t, x, y) ≥ D 0 ∩Bρ(z 0 ,C 2 ) p(t/2, x, z)p(t/2, z, y)m p (dz) t −1 e −c 16 ρ(x,z 0 ) 2 /t t −1 e −c 16 ρ(y,z 0 ) 2 /t m p (D 0 ∩ B ρ (z 0 , C 2 )) t −1 e −c 17 ρ(x,y) 2 /t , where in the last inequality, we used the fact that ρ(x, y) ≥ 3. This completes the proof that the lower bound in (8.1) holds for all three cases (a)-(c). The theorem is now proved. A large square with an arch In this subsection, we study Brownian motion on a large square with an arch. The state space E of the process is defined as follows. Let z 1 , z 2 ∈ R 2 with \|z 1 − z 2 \| ≥ 6. Fix constants 0 < ε 1 , ε 2 < 1/2 and p > 0. For i = 1, 2, denote by B i the closed disk on R 2 centered at z i with radius ε i . Let D 0 = R 2 \ (B 1 ∪ B 2 ). We short B i into a singleton denoted by a * i . Denote by L a one dimensional arch with two endpoints a * 1 and a * 2 . Without loss of generality, we assume L is isometric to an closed interval [−b, b] for some b ≥ 4. We equip the space E := D 0 ∪ {a * 1 , a * 2 } ∪ L with the Riemannian distance ρ induced from D 0 and L, analogous to the last example of a large square with multiple flag poles. Let m p be the measure on E whose restriction on L and D 0 is the arch length measure and the Lebesgue measure multiplied by p and 1, respectively. In particular, we have m p ({a * 1 , a * 2 }) = 0. As before, BMVD on E is defined as follows. Definition 8.5. Given 0 < ε 1 , ε 2 < 1/2 and p > 0, BMVD on E with parameters (ε 1 , ε 2 , p) on E is an m p -symmetric diffusion X on E such that (i) its subprocess process in L or D 0 has the same distribution as the one-dimensional or two-dimensional Brownian motion in L or D 0 , respectively. (ii) it admits no killings at {a * 1 , a * 2 }. Similar to Theorem 2.2, we have the following. Theorem 8.6. For every 0 < ε 1 , ε 2 < 1/2 and p > 0, BMVD X on E with parameter (ε 1 , ε 2 , p) exists and is unique. Its associated Dirichlet form (E, F) on L 2 (E; m p ) is given by F = f : f \| R 2 ∈ W 1,2 (R 2 ), f \| L ∈ W 1,2 (L), f \| B i = f \| L (a * i ) , i = 1, 2 , E(f, g) = 1 2 D 0 ∇f (x) · ∇g(x)dx + p 2 L f ′ (x)g ′ (x)dx. It is easy to see that BMVD X has a continuous transition density function p(t, x, y) with respect to the measure m p . Similar to that for Proposition 3.1, Propositions 3.2 and 3.3, using the classical Nash's inequality for one-and two-dimensional Brownian motion and Davies method, one can easily establish the following. Proposition 8.7. Let T ≥ 2. There exist C 1 , C 2 > 0 such that p(t, x, y) ≤ C 1 1 t + 1 t 1/2 e −C 2 ρ(x,y) 2 /t for all x, y ∈ E, t ∈ (0, T ]. The next theorem gives short time sharp two-sided estimates on p(t, x, y). Theorem 8.8. Let T ≥ 2 be fixed. There exist positive constants C i , 3 ≤ i ≤ 16, so that the transition density p(t, x, y) of BMVD X on E satisfies the following estimates when t ∈ (0, T ]: (i) For x ∈ L and y ∈ E, C 3 √ t e −C 4 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 5 √ t e −C 6 ρ(x,y) 2 /t . (8.8) (ii) For x, y ∈ D 0 ∪ {a * 1 , a * 2 }, when ρ(x, a * i ) + ρ(y, a * i ) < 1 for some i = 1, 2, C 7 √ t e −C 8 ρ(x,y) 2 /t + C 7 t 1 ∧ ρ(x, a * i ) √ t 1 ∧ ρ(y, a * i ) √ t e −C 9 \|x−y\| 2 /t (8.9) ≤ p(t, x, y) ≤ C 10 √ t e −C 11 ρ(x,y) 2 /t + C 10 t 1 ∧ ρ(x, a * i ) √ t 1 ∧ ρ(y, a * i ) √ t e −C 12 \|x−y\| 2 /t ; otherwise, C 13 t e −C 14 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ C 15 t e −C 16 ρ(x,y) 2 /t . (8.10) (iii) x ∈ L ∪ B ρ (a * 1 , C 2 /2) ∪ B ρ (a * 2 , C 2 /2) and y ∈ D 0 \ (B ρ (a * 1 , 2C 2 ) ∪ B ρ (a * 2 , 2C 2 )). Without loss of generality, we assume x is closer to a * 1 than to a * 2 . Let D 3 := {z ∈ D 0 : C 2 /2 ≤ ρ(z, a * 1 ) ≤ C 2 }. Note that ρ(x, y) ≥ (ρ(x, z) + ρ(y, z))/5 for z ∈ D 3 . By Markov property, p(t, x, y) ≥ D 3 p(t/2, x, z)p(t/2, z, y)m p (dz). The desired lower bound for p(t, x, y) follows from the results obtained in (i) and (ii). Proposition 2 . 1 . 21For any p > 0, volume doubling property fails for measure m p . Theorems 4.5 and 4.6 establish Theorem 1.3(i). We next consider part (ii) of Theorem 1.3 when both x and y are in D 0 . s, a * , a * )ds. a change of variable s = x 2 /r gives t ≥ 0. On the other hand, there are unique solutions to e −cr 2 dr log t, {Yt>0} dt + sgn(Y t )(2πε − p)dL 0 t (X) + dL 0 t (Y ) = sgn(Y t )dB t + 1 2(Y t + ε) 1 {Yt>0} dt + (2πε − p)sgn(Y t )dL 0 t (X) + dL 0 t (Y ), t t/2 Proof. We first show that G D (x, a * ) is a bounded positive continuous function on D. By Theorem 1.3, there is a constant c 1 > 0 so that for every x ∈ D,Thus P x (τ D ≥ 1) ≤ 1 − c 1 for every x ∈ D. By the strong Markov property of X, there are constants c 2 , c 3 > 0 so that P x (τ D ≥ t) ≤ c 2 e −c 3 t for every x ∈ D and t > 0. For t ≥ 2 and x, y ∈ D, we thus have by Theorem 1.3,By Theorem 1.3 again, we conclude thatconverges and is a bounded positive continuous function in x ∈ D. In particular, G D (a * , a * ) < ∞.We further note that x → G D (x, a * ) is a harmonic function in D 1 and so it is a linear function. As it vanishes at b := ∂D ∩ R + , we have(i) Assume x, y ∈ D 1 ∪ {a * } and x = y. If x = a * or y = a * , the desired estimate holds in view of (7.2). Thus we assume neither x nor y is a * . By the strong Markov property of X,We know from the one-dimensional Green function estimates,On the other hand, x → P x (σ a * < τ D ) is a harmonic function in D 1 that vanishes at b. Thus by the same reasoning as that for (7.2), we haveIt follows then(ii) Assume that x, y ∈ D 2 . By the strong Markov property of X,Since both y → G D (a * , y) and x → P x (σ a * < τ D ) are bounded positive harmonic functions on D ∩ D 0 that vanishes on D 0 ∩ ∂D, it follows from the boundary Harnack inequality for Brownian motion in R 2 that G D (a * , y) ≍ δ D (y) and P x (σ a * < τ D ) ≍ δ D (x) (7.4) and Lemma 5.11. The above two estimates give the desired estimates.There are three remaining cases:We claim that (8.1) holds for all these three cases. The upper bound in (8.1) holds due to Proposition 3.3 so it remains to establish the lower bound.It follows from the Dirichlet heat kernel estimate for Brownian motion in C 1,1 -domain[40,16]that in case (c), for any t ∈ (0, T ],For case (b), without loss of generality, we assume i = 1. Define u 1 (w) = −ρ(w, a * 1 ) for w ∈ L 1 and u 1 (w) = ρ(w, a * 1 ) analogous to (4.1). Let Y = u(X) and τ 1 ::= ∂ for t ≥ τ 1 has the same distribution as the killed radial process radial process Y in Section 4 for BMVD on plane with one vertical half-line. By Proposition 4.3 and the arguments similar to Proposition 4.4 of this paper and that of[16], one can show that Y (−∞,2C 2 ) t has a transition density function p 0 (t, w, z) with respect to the Lebesgue measure on R and it has the following two-sided estimates:for t ∈ (0, T ] and x, y ∈ (−∞, 2C 2 ). Thus we have for t ∈ (0, T ], x ∈ L 1 ∪ B ρ (a * 1 , C 2 ) and y ∈ D 1 with ρ(x, y) ≥ 2C 2 ,where in the second inequality it was used that ρ(z, y) ≤ 5ρ(a * 1 , y)/2, and in the last two inequalities we used the fact that \|6C 2 /4 − u 1 (x)\| ≥ C 2 /2 and ρ(x, y) ≥ 2C 2 .Proof. This theorem can be established by a similar consideration as that for Theorem 8.4. Here we only give a brief sketch. Let C 1 > 0 and C 2 ∈ (0, 1/2) be the constants in Lemma 6.1. Case 1. ρ(x, y) < 2C 2 . The desired estimates can be obtained in a similar way as that for Case 1 in the proof of Theorem 8.4, by using Lemma 6.1. Case 2. ρ(x, y) ≥ 2C 2 . Due to the upper bound estimate in Theorem 8.7, it suffices to show the following lower bound estimate hold: there exists some c 1 , c 2 > 0, such that p(t, x, y) ≥ c 1 e −c 2 ρ(x,y) 2 /t for all t ∈ (0, T ].(8.11)We divided its proof into three cases.(i) Both x and y are in L ∪ B ρ (a * 1 , C 2 ) ∪ B ρ (a * 2 , C 2 ). Without loss of generality, we assume x ∈ L∪B ρ (a * 1 , C 2 ) and x is closer to a * 1 if both x and y are on the arch L. Denote by ℓ(w, z) the arch length in L between two points w, z ∈ L, and D 1 := L ∪ B ρ (a * 1 , 3C 2 )∪ B ρ (a * 2 , 3C 2 ). 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[ "Stacked Grenander and rearrangement estimators of a discrete distribution", "Stacked Grenander and rearrangement estimators of a discrete distribution" ]	[ "Vladimir Pastukhov [email protected] \nDepartment of Computer Science and Engineering\nChalmers University of Technology\n\n" ]	[ "Department of Computer Science and Engineering\nChalmers University of Technology\n" ]	[]	In this paper we consider the stacking of isotonic regression and the method of rearrangement with the empirical estimator to estimate a discrete distribution with an infinite support. The estimators are proved to be strongly consistent with √ n-rate of convergence. We obtain the asymptotic distributions of the estimators and construct the asymptotically correct conservative global confidence bands. We show that stacked Grenander estimator outperforms the stacked rearrangement estimator. The new estimators behave well even for small sized data sets and provide a trade-off between goodness-of-fit and shape constraints. MSC2020 subject classifications: 62E20, 62G07, 62G20.	null	[ "https://export.arxiv.org/pdf/2106.00560v3.pdf" ]	235,368,383	2106.00560	5e9e2b4d6a006d782ce6d27d04344edf8b944010	Stacked Grenander and rearrangement estimators of a discrete distribution Aug 2022 Vladimir Pastukhov [email protected] Department of Computer Science and Engineering Chalmers University of Technology Stacked Grenander and rearrangement estimators of a discrete distribution Aug 2022and phrases: Constrained inferencecross-validationdiscrete distributionGrenander estimatorisotonic regressionmodel stackingre- arrangementsmoothing In this paper we consider the stacking of isotonic regression and the method of rearrangement with the empirical estimator to estimate a discrete distribution with an infinite support. The estimators are proved to be strongly consistent with √ n-rate of convergence. We obtain the asymptotic distributions of the estimators and construct the asymptotically correct conservative global confidence bands. We show that stacked Grenander estimator outperforms the stacked rearrangement estimator. The new estimators behave well even for small sized data sets and provide a trade-off between goodness-of-fit and shape constraints. MSC2020 subject classifications: 62E20, 62G07, 62G20. Introduction This work is largely inspired by recent papers in the estimation of discrete distributions with shape constraints. The first paper in this area is [19], where the authors studied the method of rearrangement and maximum likelihood estimator (MLE) of probability mass function (p.m.f.) under monotonicity constraint. The MLE under monotonicity constraint is also known as Grenander estimator. Next, in the paper [13] the authors introduced the least squares estimator of a discrete distribution under the constraint of convexity and, further, its limiting distribution was obtained in [1]. Furthermore, the MLE of log-concave p.m.f. was studied in detail in [4], and in [20] the problem was generalised to the case of multidimensional discrete support. Next, in paper [3] the authors introduced the MLE of unimodal p.m.f. with unknown support, proved the consistency and obtained the asymptotic distribution. The problem of least squares estimation of a completely monotone p.m.f. was considered in papers [2,5]. In most of the papers listed above the authors considered both the well-and the mis-specified cases and studied the asymptotic properties of the estimators in both cases. In this work we do not have the mis-specified case in a sense that we assume that the true p.m.f. can be non-monotone and our estimators are strongly consistent even if the true p.m.f. is not decreasing. The estimators introduced and studied in this paper are in some sense similar to nearly-isotonic regression approach, cf. [33] and [23] for multidimensional case. Nearly-isotonic regression is a convex optimisation problem, which provides intermediate less restrictive solution and the isotonic regression is included in the path of the solutions. At the same time, our approach is in some sense opposite to liso (lassoisotone), cf. [14], and to bounded isotonic regression, cf. [22]. The liso is a combination of isotonic regression and lasso penalties, and bounded isotonic regression imposes additional penalisation to the range of the fitted model. In this paper we combine Grenander estimator and the method of rearrangement with cross-validation-based model-mix concept, cf. [31]. The estimator is constructed as a convex combination of the empirical estimator and Grenander estimator or the empirical estimator and rearrangement estimator. Following the terminology for regression and classification problems in [9,21,35], we call the resulting estimators as stacked Grenander estimator and stacked rearrangement estimator, respectively. Therefore, we do not impose the strict monotonic restriction and let the data decide. There are several papers where the authors studied a convex combination of the empirical estimator with a prescribed probability vector, cf. [15,16,31,34]. In particular, in [31] the authors proposed the combination of the empirical estimator and a constant p.m.f. with a mixture parameter selected by crossvalidation. Also, the minimax estimator of a p.m.f. with respect to ℓ 2 -loss with a fixed known finite support and sample size n is given by a convex combination of the empirical estimator and the uniform distribution with a mixture parameter equal to √ n n+ √ n , cf. [34]. Furthermore, in [16] the authors provide a geometrical explanation on the gain from stacking the empirical estimator with a fixed probability vector and show that the improvement of the estimation increases as the size of the support becomes larger. In the case of continuous support the first paper on the density estimation via stacking is [30], where it is shown that the method of stacking performs better than selecting the best model by cross-validation. Next, in [27] the authors studied the approach of linear and convex aggregation of density estimators and, in particular, proved that the aggregation of two estimators allows to combine the advantages of both. To the authors' knowledge the constrained stacked estimators have not been investigated for the case of continuous density. To the authors' knowledge, the problem of staking the shape constrained estimators has not been studied much even in a regression setup, except for the paper [36]. In the paper [36] the author used a convex combination of linear regression with isotonic regression to obtain a strictly monotonic solution. Also, it is worth to mention the paper [18], where it was shown that in terms of prediction accuracy the simplified relaxed lasso (which is stacking of least squares estimator and lasso) performs almost equally to the lasso in low signalto-noise ratio regimes, and nearly as well as the best subset selection in high signal-to-noise ratio scenarios. The paper is organised as follows. In Section 2 we state the problem and introduce notation. The derivation of cross-validation based mixture parameter is given in Section 3. Section 4 is dedicated to the theoretical properties of the estimators such as consistency, rate of convergence and asymptotic distribution. Also, in Section 4 we construct asymptotic confidence bands. In Section 5 we do simulation study to compare the performance of the estimators with empirical, minimax, rearrangement and Grenander estimators. The article closes with a conclusion and a discussion of possible generalisations in Section 6. The ancillary results and the proofs of some statements are given in Appendix. The R code for the simulations is available upon request. Statement of the problem and notation First, let us introduce notation and several definitions. Assume that z 1 , z 2 , . . . , z n is a sample of n i.i.d. random variables with values in N and generated by a p.m.f. p. For a given data sample let us create the frequency data x = (x 0 , . . . , x tn ), where x j = n i=1 1{z i = j} and t n = sup{j : x j > 0} denotes the largest order statistic for the sample. The empirical estimator of p is given bŷ p n,j = x j n , j ∈ N, and it is strongly consistent, unbiased and asymptotically normal in ℓ 2 -space. The rearrangement estimator studied in [19] is defined aŝ r n = rear(p n ),(2.1) where rear(w) denotes the reversed-ordered vector. Also, equivalently, the rearrangement estimator can be written asr n,j = sup{u : Q n (u) ≤ j}, where Q n (u) = #{k :p n,k ≥ u}. The MLE of decreasing p.m.f., or Grenander estimator, which we denote byĝ n , is equivalent to the isotonic regression of the empirical estimator, cf. [6,19,28], i.e.ĝ n = Π(p n \|F decr ) := argmin f ∈F decr j [p n,j − f j ] 2 , (2.2) where F decr is the monotonic cone in ℓ 2 , i.e. F decr = f ∈ ℓ 2 : f 0 ≥ f 1 ≥ . . . , p n is the empirical estimator and Π(p n \|F decr ) denotes the ℓ 2 -projection ofp n onto F decr 1 . In our work we construct the estimator in the following way: φ n = βĥ n + (1 − β)p n ,(2.3) whereĥ n = r n , for the stacked rearrangement estimator, g n , for the stacked Grenander estimator, with the data-driven selection of β: β n = argmin β∈[0,1] CV (β), where CV (β) is a cross-validation criterion, which we introduce and study below. We associate each component x j of the frequency vector x with multinomial indicator δ [j] ∈ R tn+1 , given by δ [j] = (0, . . . , 0, 1, 0, . . . , 0) (2.4) for j = 0, . . . , t n , cf. [31]. All elements of δ [j] are zeros, except for the one with index j. Next, letp \[j] n for j = 0, . . . , t n denote the leave-one-out version of the empirical estimatorφ n for the frequency data x = (x 0 , . . . , x tn ), i.e. for j such that x j > 0 letp \[j] n = x − δ [j] n − 1 . Next, for the rearrangement estimator, the leave-one-out version is given bŷ r \[j] n = rear(p \[j] n ), and for Grenander estimator: g \[j] n = Π p \[j] n \|F decr . Therefore, for j such that x j > 0 the leave-one-out versions of stacked rearrangement and stacked Grenander estimators for a fixed misture parameter β are given byφ For an arbitrary vector f ∈ ℓ k we define ℓ k -norm \[j] n = βĥ \[j] n + (1 − β)p \[j] n ,(2.\|\|f \|\| k =    ∞ j=0 \|f j \| k 1/k , if k ∈ N\{0}, sup j∈N \|f j \|, if k = ∞, and for v ∈ ℓ 2 and w ∈ ℓ 2 let v, w = ∞ j=0 v j w j denote the inner product on ℓ 2 . For a random sequence b n ∈ R we will use the notation b n = O p (n q ) if for any ε > 0 there exists a finite M > 0 and a finite N > 0 such that P[n −q \|b n \| > M ] < ε, n = tn j=0φ n,j p j and L (3) n = tn j=0 p 2 j . We aim to minimise L n . Obviously, p is unknown, and we will use the approach introduced in [24] to estimate L n . First, note that L (3) n is a constant and can be omitted. Next, note that for a given n we have for L 2 we have L 2 = tn j=0φ n,j p j = E[φ n ], and following [24] we estimate L n,j , withφ \[j] n defined in (2.5). Therefore, we select the mixture parameter β to minimise CV (β) = L (1) n − 2L (2) n ,(3.2) i.e.β n = argmin β∈[0,1] CV (β). This cross-validation approach for estimation of discrete distributions was first introduced in [24] for smoothing kernel estimator and was also used in, for example, [11,12,25]. The mixture parameterβ n is given in the following theorem. Theorem 1. The leave-one-out least-squares cross-validation mixture parameterβ n is given byβ n =      bn an , if a n = 0 and 0 ≤ b n ≤ a n , 1, if 0 < a n ≤ b n , 0, otherwise, where a n = tn j=0 (ĥ n,j −p n,j ) 2 , and b n = tn j=0p n,j (ĥ \[j] n,j −p \[j] n,j ) − tn j=0p n,j (ĥ n,j −p n,j ), withĥ \[j] n =r \[j] n for the case of stacked rearrangement estimator, andĥ \[j] n = g \[j] n for the case of stacked Grenander estimator, respectively. In the sequel of the paper we always assume that bothφ n andφ \[j] n are constructed with the leave-one-out least-squares cross-validation mixture parameter β n . Theoretical properties of the estimator In this section we study theoretical properties of stacked rearrangement and stacked Grenander estimators. First, let us assume that p ∈ F decr , i.e. the underlying p.m.f. is decreasing. Note that from the subadditivity of the norms for \|\|φ n − p\|\| k , with 1 ≤ k ≤ ∞, we have \|\|φ n − p\|\| k = \|\|β nĥn + (1 −β n )p n − p\|\| k ≤ β n \|\|ĥ n − p\|\| k + (1 −β n )\|\|p n − p\|\| k . From the error reduction property of the rearrangement and Grenander estimators, i.e. \|\|ĥ n − p\|\| k ≤ \|\|p n − p\|\| k , with 1 ≤ k ≤ ∞, cf. Theorem 2.1 in [19], we have \|\|φ n − p\|\| k ≤ \|\|p n − p\|\| k (4.1) for all 1 ≤ k ≤ ∞. Therefore, in the case of a decreasing true p.m.f. both the stacked rearrangement and stacked Grenander estimators also provide the error reduction. Assume that the true p.m.f. is not decreasing. Let r = rear(p) and g = Π p\|F decr . Note that r = p nor g = p, if p ∈ F decr , i.e. the vector r is reversed ordered vector p and g is decreasing vector in ℓ 2 which is closest in ℓ 2 -norm to the true p.m.f. p. Then, since the isotonic regression and the rearrangement, viewed as a mapping from ℓ 2 into ℓ 2 , are continuous in the case of a finite support, and the empirical estimator is strongly consistent, then r n a.s. → r, andĝ n a.s. → g, pointwise. Note that from the statements (i), (ii) and (iv) of Lemma 3 in Appendix it follows thatĝ n always exists, and it is a probability vector for all n. Clearly, the same result holds for the rearrangement estimatorr n for all n. The almost sure convergence in ℓ k -norm, for 1 ≤ k ≤ ∞, ofr n andĝ n to r and g, respectively, now follows from Lemma C.2 in the supporting material of [4]. Consistency First, let us study the leave-one-out versions of the empirical, rearrangement and Grenander estimators. Recall that p \[j] n = x − δ [j] n − 1 ,r \[j] n = rear(p \[j] n ) andĝ \[j] n = Π p \[j] n \|F decr , for j such that x j > 0. Let us define vectorsπ n ∈ ℓ 1 ,ρ n ∈ ℓ 1 , andγ n ∈ ℓ 1 aŝ π n,j = p \[j] n,j , if x j > 0, 0, otherwise, ρ n,j = r \[j] n,j , if x j > 0, 0, otherwise, γ n,j = ĝ \[j] n,j , if x j > 0, 0, otherwise. (4.2) Lemma 1. The sequences of vectorsπ n ,ρ n andγ n converge pointwise a.s. to p, r, and g, respectively. Proof. The proof is given in Appendix. ✷ Next, we prove the following important lemma. Lemma 2. For the vectorsπ n we havê π n,j ≤p n,j for all j, and forρ n andγ n we havê ρ n,j ≤ n n − 1r n,j andγ n,j ≤ n n − 1ĝ n,j for all j. Proof. The proof is given in Appendix. ✷ In Lemma C.2 in the supporting material of [4] it was proved that for probability mass functions the pointwise convergence and the convergence in ℓ k for 1 ≤ k ≤ ∞ are all equivalent. Note, in our case the sequencesπ n ,ρ n andγ n are not probability vectors. Nevertheless, as we prove below, allπ n ,ρ n andγ n converge a.s. to p, r and g, respectively, in ℓ k -norm for 1 ≤ k ≤ ∞. Theorem 2. For the vectorsπ n ,ρ n andγ n we havê π n a.s. → p, ρ n a.s. → r, andγ n a.s. → g in ℓ k -norm for 1 ≤ k ≤ ∞. Proof. The proof starts in a similar way as the one for Lemma C.2 in [4]. Let us, first, study the case ofπ n . Fix some ε > 0. Then, we can choose K such that j≤K p j ≥ 1 − ε 4 . Since both π n and the empirical estimator p n converge to p pointwise, then there exists random n 0 such that for all n ≥ n 0 sup j≤K \|p n,j − p j \| ≤ ε 4(K + 1) , sup j≤K \|π n,j − p j \| ≤ ε 4(K + 1) , almost surely. This implies that for all n ≥ n 0 we have j≤Kp n,j ≥ 1− ε 2 and j≤K \|π n,j − p j \| ≤ ε 4 , almost surely. Next, for any n ∞ j=0 \|π n,j −p j \| = j≤K \|π n,j −p j \|+ j>K \|π n,j −p j \| ≤ j≤K \|π n,j −p j \|+ j>Kπ n,j + j>K p j . Furthermore, j>Kπ n,j ≤ j>Kp n,j since 0 <π n,j ≤p n,j . Then, for all n > n 0 we have proved that ∞ j=0 \|π n,j − p j \| ≤ ε 4 + ε 2 + ε 4 = ε, almost surely. This means that for any ε > 0 there exists random n 0 , such that for all n > n 0 \|\|π n − p\|\| 1 ≤ ε, almost surely. Furthermore, since ℓ 1 ⊂ ℓ k , for all k > 1, then a.s. convergence holds in ℓ k , for all 1 ≤ k ≤ ∞. Let us prove the convergence forγ n . First, from Lemma 2 it follows that n − 1 nγ n,j ≤ĝ n,j . Then, since both n−1 nγ n andĝ n converge to g a.s., we can use the same approach as forπ above, and prove that n − 1 nγ n a.s. → g, in ℓ k , for 1 ≤ k ≤ ∞, which means that γ n a.s. → g, in ℓ k , for 1 ≤ k ≤ ∞. Now, using the result of Lemma 2, we can prove the result forρ n in the same way as we did forγ n . ✷ Now we can summarize the above results in the following theorem. Theorem 3. For any underlying distribution p, both the stacked rearrangement and stacked Grenander estimators are strongly consistent: φ n a.s. → p in ℓ k -norm for 1 ≤ k ≤ ∞. Proof. Firs, let us assume that p is decreasing. Then the result of the theorem follows from the strong consistency ofĝ n ,r n andp n . Next, assume that p is not decreasing. From Theorem 2 it follows that for the case of stacked rearrangement estimator we have a n a.s. → \|\|r − p\|\| 2 2 , and b n a.s. → p, (r − p) − p, (r − p) = 0, and for the case of stacked Grenander estimator we have a n a.s. → \|\|g − p\|\| 2 2 , and b n a.s. → p, (g − p) − p, (g − p) = 0. Therefore,β n a.s. → 0. Next, since \|\|φ n − p\|\| k ≤β n \|\|ĥ n − p\|\| k + (1 −β n )\|\|p n − p\|\| k for all 1 ≤ k ≤ ∞, it followsφ n a.s. → p in ℓ k -norm for 1 ≤ k ≤ ∞. ✷ Rate of convergence In this section we study the rate of convergence of stacked estimator. In the case of bounded support the √ n-rate of convergence follows from pointwise convergence of the vectorsπ n ,ρ n andγ n . In this work we assume that the support can be infinite. Theorem 4. Stacked rearrangement and Grenander estimators have √ n-rate of convergence for any underlying p.m.f. p: √ n\|\|φ n − p\|\| k = O p (1) for 1 < k ≤ ∞. Next, if ∞ j=0 √ p j < ∞, then √ n\|\|φ n − p\|\| 1 = O p (1). Proof. Assume that p is decreasing. Then the result follows from (4.1) and Corollaries 4.1 and 4.2 in [19]. Next, assume that p is not decreasing. Let us, first, prove the case of stacked Grenander estimator. Recall that β n =      bn an , if a n = 0 and 0 ≤ b n ≤ a n , 1, if 0 < a n ≤ b n , 0, otherwise, where a n = tn j=0 (ĝ n,j −p n,j ) 2 , and in the notation introduced in 4.2, we can write b n as b n = tn j=0p n,j (γ n,j −π n,j ) − tn j=0p n,j (ĝ n,j −p n,j ). First, as we proved in Theorem 3 a n a.s. → \|\|g − p\|\| 2 2 > 0. (4.3) Second, note that from Lemma 2 it follows that for all n we have b n = tn j=0p n,j (γ n,j −ĝ n,j ) + tn j=0p n,j (p n,j −π n,j ) ≤ n n − 1 tn j=0p n,jĝn,j − tn j=0p n,jĝn,j + tn j=0p n,j (p n,j −π n,j ). Recall thatπ n,j = xj −1 n−1 = n n−1p n,j − 1 n−1 , if x j = 0, 0, otherwise, which leads to tn j=0p n,j (p n,j −π n,j ) = tn j=0p n,j (p n,j −π n,j ) = 1 − tn j=0p 2 n,j n − 1 . Therefore, the upper bound for b n is given by b n ≤ tn j=0p n,jĝn,j n − 1 + 1 − tn j=0p 2 n,j n − 1 , and, consequently, √ nb n a.s. → 0,(4.4) since both sequences tn j=0p n,jĝn,j andp 2 n,j are bounded. Next, since β n ≥ 0, from (4.3) and (4.4) it follows that √ nβ n a.s. → 0. (4.5) Then, from (4.5) for any p and all 1 ≤ k ≤ ∞ the following holdŝ β n √ n\|\|ĝ n − p\|\| k a.s. → 0, for all 1 ≤ k ≤ ∞. Further, as it follows from Corollary 4.2 in [19], if ∞ j=0 √ p j < ∞, then √ n\|\|p n − p\|\| 1 = O p (1). Therefore, for all 2 ≤ k ≤ ∞ and all p we have (1 −β n ) √ n\|\|p n − p\|\| k = O p (1), and, if ∞ j=0 √ p j < ∞, then we have (1 −β n ) √ n\|\|p n − p\|\| 1 = O p (1). Finally, recall that √ n\|\|φ n − p\|\| k ≤β n √ n\|\|ĝ n − p\|\| k + (1 −β n ) √ n\|\|p n − p\|\| k , which finishes the prove of theorem for the case of Grenander estimator. Similarly, using the results of Lemma 2, for the case of stacked rearrangement estimator we can show that b n ≤ tn j=0p n,jrn,j n − 1 + 1 − tn j=0p 2 n,j n − 1 , for all n. Then, the rest of the proof is the same as for Grenander estimator withĝ n and g suitably changed tor n and r, respectively. ✷ Asymptotic distribution and global confidence band In this section we study the asymptotic distribution of stacked rearrangement and Grenander estimators and discuss calculation of global confidence band for p. The limit distribution of rearrangement and Grenanader estimators were obtained in [19]. The asymptotic distribution of stacked Grenander estimator for the case when true p.m.f. p is either not decreasing with a countable support or strictly decreasing with a finite support is given in the next theorem. Theorem 5. Assume that p is either not decreasing with a countable support or strictly decreasing with a finite support. Then stacked rearrangement and Grenander estimators are asymptotically normal √ n(φ n − p) d → Y 0,C , in ℓ 2 , where Y 0,C is a Gaussian process in ℓ 2 with mean zero and the covariance operator C such that Ce i , e i ′ = p i δ i,i ′ − p i p i ′ , with e i ∈ ℓ 2 the orthonormal basis in ℓ 2 such that in a vector e i all elements are equal to zero but the one with the index i is equal to 1, and δ i,j = 1, if i = j and 0 otherwise, cf. [19]. Proof. The proof is given in Appendix. ✷ For the case of a general decreasing underlying p.m.f. with some constant regions the limit distribution of the stacked estimator remains an open problem. Figure 1 illustrates the difference of the asymptotic distributions of the empirical estimator, monotonically constrained estimators and the stacked estimators. Let U (s) denote the uniform distribution over {0, . . . , s} and T d (s) be strictly decreasing triangular function with the support {0, . . . , s} (for the definition of triangular function see e.g. [13]). Figure 1 shows standard normal QQ-plots of 1000 samples of √ n(p n,1 − p 1 ), √ n(ĝ n,1 − p 1 ), √ n(r n,1 − p 1 ) and √ n(φ n,1 − p 1 ) for bothĥ n =ĝ n andĥ n =r n , with n = 1000 for the following distributions: (a) (left) p = U (11), (b) (middle) p = 0.15U (3) + 0.1U (7) + 0.75U (11), (c) (right) p = T d (11). From Figure 1 we can conclude that, first, in the case of a decreasing p.m.f. the distributions of stacked estimators asymptotically are not equivalent to the distribution of the empirical estimator, and, second, stacked estimators and constrained estimators have different asymptotic distribution if the underlying p.m.f. has constant regions. For the process Y 0,C defined in Theorem 5 let q α denote the α-quantile of its ℓ ∞ -norm, i.e. P[\|\|Y 0,C \|\| ∞ > q α ] = α. Then, if p is not decreasing or strictly decreasing, from Theorem 5 for stacked estimator we have lim n P[ √ n\|\|φ n − p\|\| ∞ ≤ q α ] = 1 − α. Next, note that in the case of a decreasing p.m.f. p from (4.1) it follows P[ √ n\|\|φ n − p\|\| ∞ ≤ q α ] ≥ P[ √ n\|\|p n − p\|\| ∞ ≤ q α ] for all n. Therefore, in the case of a decreasing p we have lim inf n P[ √ n\|\|φ n − p\|\| ∞ ≤ q α ] ≥ 1 − α. In the same way as in [3], to estimate q α we can use the stacked estimator φ n in place of p in Y 0,C , and then each quantile can be estimated using Monte-Carlo method. In Proposition B.7 in the supplementary material of [3] it was proved thatq α a.s. → q α . Therefore, the following confidence band max (φ n,j −q α √ n ), 0 ,φ n,j +q α √ n , for j ∈ N is asymptotically correct global confidence band if p is either not decreasing or strictly decreasing, and it is asymptotically correct conservative global confidence band if p is decreasing with some constant regions. n(r n,1 − p 1 ) and √ n(φ n,1 − p 1 ) for bothĥ n =ĝ n andĥ n =r n , with n = 1000 for uniform distribution (left), decreasing distribution (middle) and strictly decreasing distribution (right). Simulation study of performance of the stacked estimators In this section we do simulation study to compare the performance of stacked estimators with the empirical, Grenander, rearrangement and the minimax estimators. For the p.m.f. with finite support {0, . . . , s} and for a given sample size n the minimax estimator of p with respect to ℓ 2 -loss is given bŷ p mm n = α mm n λ + (1 − α mm n )p n ,(5.1) with λ = ( 1 s+1 , . . . , 1 s+1 ) and α mm n = √ n n+ √ n , cf. [34]. To the authors' knowledge, the minimax estimation with respect to ℓ 2 -loss for infinitely supported p.m.f. is an open problem. With some abuse of notation, in this and next sections for infinitely supported distributions we refer the estimator defined in (5.1) with s = t n as "minimax". Performance of the estimators We study the cases of decreasing and not decreasing true p.m.f. p separately. True p.m.f. is decreasing Let us consider the following uniform and decreasing p.m.f.: where Geom(θ) is Geometric distribution, i.e. p j = (1 − θ)θ j for j ∈ N with 0 < θ < 1. The models M 2, M 3 and M 4 were used in [19] to assess the performance of Grenander estimator and compare its performance with empirical and rearrangement estimators. First, we compare the performance of the estimators in ℓ 1 (Figure 2) and ℓ 2 ( Figure 3) distances for small n = 20 and moderate n = 300 sample sizes with 1000 Monte Carlo simulations. From the boxplots at Figure 2 and Figure 3 we can conclude that for both small and moderate sized data sets stacked Grenander estimator outperforms in ℓ 1 and ℓ 2 norms both the empirical estimator and minimax estimator ("minimax" for the case of Geometric distribution). Further, stacked Grenander estimator outperforms stacked rearrangement estimator when the underlying distribution has constant regions and it performs almost the same in the case of strictly decreasing p.m.f. The superiority of Grenander estimator over the rearrangement estimator was proved in [19]. Next, in order to summarise the results and demonstrate the superiority of stacked Grenander estimator we plot the estimates of scaled risk nE[\|\|ξ n − Figure 4. We can conclude that in the case of a decreasing underlying distribution stacked Grenander estimator performs almost as good as Grenander estimator and it performs significantly better than the empirical and the minimax estimators. True p.m.f. is not decreasing Now let us consider the case when the underlying distributions are not decreasing: where T i (s) stands for strictly increasing triangular function; N Bin(r, θ) is the negative binomial distribution with r the number of failures until the experiment is stopped and θ the success probability; P ois(λ) is Poisson distribution with rate λ. Therefore, we consider very non-monotonic distributions. Indeed, model M 5 is a strictly increasing p.m.f., M 6 is a unimodal distribution, and M 7 is bimodal. From Figure 5 and Figure 6 we can conclude that stacked Grenander estimator outperforms in ℓ 1 and ℓ 2 norms the empirical, rearrangement and minimax estimators ("minimax" for the cases of Negative Binomial and Poisson mixture). Next, it is interesting to note that even if the underlying distribution is not monotone, Grenander estimator can still outperform the empirical estimator in both ℓ 1 and ℓ 2 norms for small sample size. This happens because the isotonisation decreases the variance of the estimator though bias becomes larger. Let us summarise the results at Figure 7 by plotting the estimates of the scaled risk nE[\|\|ξ n − p\|\| 2 2 ] (withξ n one of the following estimators: empirical, minimax or stacked Grenander estimator). Note that in the case of nondecreasing true p.m.f. we do not plot the risk for Grenander estimator, because, obviously, in the miss-specified case the scaled risks of the constrained estimators are worse than the risk of consistent estimators. Based on the simulations we can conclude that stacked Grenander estimator performs better than empirical and minimax estimators even when the underlying distribution is not decreasing. The result might look surprising at the first sight. Nevertheless, the explanation of the effect of ℓ 2 -risk reduction by stacking empirical estimator with some fixed probability vector was explained in [16]. Further, let us consider the case of model M5, i.e. very non-decreasing case when the underlying distribution is strictly increasing. Then, since the empirical estimator is strongly consistent there exist a random n 1 such that for all n > n 1 the vector isp n is strictly increasing almost surely. Next, note that from Lemma 4 it follows that for all n > n 1 we haveĝ j = 1/12, for all j = 0, . . . 11, almost surely. Therefore, for n > n 1 stacked Grenander estimator becomes the stacking of the empirical estimator with a uniform distribution U (11) almost surely, which is similar to what, for example, minimax estimator in (5.1) does. One can also see from Figure 7 that in the case of model M5 stacked Grenander estimator performs very similarly to the minimax estimator in a sense of ℓ 2 -risk. Coverage probabilities for the confidence bands The Table 1 presents the proportion of times that max (φ n,j −q α √ n ), 0 ≤ p j ≤φ n,j +q α √ n , for all j ∈ N among 1000 runs for the models M1-M7. The quantilesq α are estimated based on 100000 Monte-Carlo simulations. First, one can see that the proposed global confidence band performs well. Second, note that for the decreasing p.m.f (models M1-M4) the coverage probabilities mostly larger than 0.95, while for non-decreasing p.m.f (models M5-M7) the coverage probabilities are closer to 0.95 when n becomes large, because in the former case the confidence band is asymptotically conservative, while in the later case it is asymptotically correct. Table 1 Empirical coverage probabilities for the confidence bands for α = 0.05 of the empirical estimator (e), stacked rearrangement estimator (sr) and stacked Grenander estimator (sG). Estimator Computational times First, note, that in general the complexity of the solution for the mixture parameterβ n depends on the largest order statistic t n . In Table 2 we provide the "worst case" computational times, i.e. we computeβ n for the estimator based on the following strictly increasing frequency data vector x ′ = (x ′ 0 , . . . , x ′ s ), with x ′ j = j + 1 for the different values of s, averaged over 10 runs for every s. Table 2 The "worst case" averaged over 10 runs computational times of the mixture parameterβn for stacked rearrangement (SR) and stacked Grenander (SG) estimators for different sizes s of the frequency data vector x. Estimator s=500 s=1000 s=3000 s=5000 Second, recall that in order to compute the confidence band for a given estimated distributionθ n for estimation of the coverage probability in Table 1 we performed 100000 Monte-Carlo simulations of the multivariate normal distribution N (0, Σ(θ n )), with Σ i,j (θ n ) =θ n,j δ i,j −θ n,iθn,j (i, j = 0, . . . , t n ) to estimate the quantileq α . The Table 3 shows the averaged over 10 runs computational times of the estimation ofq α of N (0, Σ(θ)) for a fixed non-random p.m.f. vector θ = T d (s) (recall that T d (s) is a strictly decreasing triangular function), for different values of s based on 100000 Monte-Carlo simulations. Table 3 The averaged over 10 runs computational times of the quantileqα for different values of the support size s. All the computations were performed on MacBook Air (Apple M1 chip), 16 GB RAM. We can conclude that both stacked rearrangement and stacked Grenander estimators are computationally feasible. Conclusion and discussion In this paper we introduced and studied estimation of a discrete infinitely supported distribution by stacking the empirical estimator with Grenander estimator and the empirical estimator with rearrangement estimator. The main results of the paper: the stacked Grenander estimator is computationally feasible, it outperforms the empirical estimator, and it is almost as good as Grenander estimator for the case of decreasing true p.m.f. Also, stacked Grenander estimator outperforms the stacked rearrangement estimator, except for the case of a strictly decreasing p.m.f. The same effect was shown in [19] for rearrangement and Grenander estimators in the case when underlying p.m.f. is decreasing. We proved that even when the true distribution is not decreasing, the estimator remains strongly consistent with √ n-rate of convergence. Therefore, the stacked Grenander estimator provides a trade-off between goodness of fit and monotonicity. The first natural generalisation of stacked Grenander estimator could be stacking with isotonic regression for a general isotonic constraint (cf. Appendix for the definition). Throughout the paper, in almost all the proofs we used properties of a general isotonic regression, cf. Lemma 3. However, the proof of Lemma 2 is based on the maximum upper sets algorithm, which is given in Lemma 4 in Appendix, and this algorithm is valid only for one dimensional monotonic case. Therefore, the generalisation of stacked Grenander estimator to the general isotonic case for finite support is straightforward, though the case of an infinite support remains an open problem. Second, it is also important to consider other shape constraints, such as unimodal, convex and log-concave cases. Stacking these estimators is, in effect, similar to the generalisation of nearly-isotonic regression to the nearly-convex regression in [33]. Third, in this work we studied the case of discrete distribution with infinite support. The empirical estimator is closely related to estimation of probability density functions via histograms. Therefore, another direction is stacking the histogram estimators with isotonised histogram. Forth, as mentioned in the introduction, the constrained stacked estimators have not been investigated for the case of continuous density. The interesting property of Grenander estimator in a continuous case is that the distributional pointwise rate of convergence depends on the local behaviour of the underlying distribution: if the true distribution is flat, the Grenander estimator has n 1/2 -rate of convergence cf. [10], and n 1/3 -rate otherwise, cf. [26]. Therefore, in the case of a continuous support it would be interesting to study stacking, for example, Grenander estimator and kernel density estimator. Another interesting direction of research concerns the stacking with a crossvalidation based on other loss functions. For the overview and theoretical properties of different loss functions for evaluation of discrete distributions we refer to the paper [17]. Finally, as we mentioned in the introduction, the problem of stacking shaped constrained regression estimators has not been studied much. Therefore, since stacked Grenander estimator performs quite well, it would be interesting to explore, for example, the prediction performance of stacked isotonic regression. Appendix We start with the definition of a general isotonic regression. Let J = {j 1 , . . . , j s }, with s ≤ ∞, be some index set. Next, let us define the following binary relations on J : A binary relation on J is a simple order if (i) it is reflexive, i.e. j j for j ∈ J ; (ii) it is transitive, i.e. j 1 , j 2 , j 3 ∈ J , j 1 j 2 and j 2 j 3 imply j 1 j 3 ; (iii) it is antisymmetric, i.e. j 1 , j 2 ∈ J , j 1 j 2 and j 2 j 1 imply j 1 = j 2 ; (iv) every two elements of J are comparable, i.e. j 1 , j 2 ∈ X implies that either j 1 j 2 or j 2 j 1 . A binary relation on J is a partial order if it is reflexive, transitive and antisymmetric, but there may be noncomparable elements. A pre-order is reflexive and transitive but not necessary antisymmetric and the set J can have noncomparable elements. Note, that in some literature the pre-order is called as a quasi-order. Next, a vector v with the elements indexed by J is isotonic if j 1 j 2 implies v j1 ≤ v j2 . We denote the set of all isotonic square summable vectors by F is , which is also called isotonic cone. Furthermore, a vector v * ∈ R s , with s ≤ ∞, is the isotonic regression of an arbitrary vector v ∈ R s (or v ∈ ℓ 2 , if s = ∞) over the pre-ordered index set J if v * = argmin f ∈F is j∈J (f j − v j ) 2 . In Lemma 3 we provide properties of a general isotonic regression which are referred to in the paper. Lemma 3. [Properties of a general isotonic regression] Let v * n ∈ ℓ 2 be the isotonic regressions of some set of vectors v n ∈ ℓ 2 , for n = 1, 2 . . . . Then, the following holds. (i) v * n exists and it is unique. (ii) j v n,j = j v * n,j , for all n = 1, 2, . . . . (iii) v * n , viewed as a mapping from ℓ 2 into ℓ 2 , is continuous. (iv) v * n satisfies the same bounds as the basic estimator, i.e. a ≤ v * n,j ≤ b, for all n = 1, 2, . . . and j = 1, 2, . . . . (v) Π(av n \|F is ) = aΠ(v n \|F is ) for all a ∈ R + . Proof. Statements (i), (ii) and (iii) follow from Theorem 8.2.1, Corollary B of Theorem 8.2.7 and Theorem 8.2.5, respectively, in [28], statements (iv), (v) and (vi) follow from Corollary B of Theorem 7.9, Theorems 7.5, respectively, in [6]. ✷ In the next lemma we describe the maximum upper sets algorithm for the solution to the isotonic regression in the monotone case. We continue this process and get −1 = m(−1) < m(1) < · · · < m(l) = t n . The solution x * (i.e. the isotonic regression of x) is given by x * j = m(r) k=m(r−1)+1 x k m(r) − m(r − 1) for j ∈ [m(r − 1) + 1, m(r)] and r ∈ [0, l]. Proof. The proof is given on p. 77 in [6] and p. 26 in [28], and, also, for simpler explanation of the algorithm we refer to [37]. ✷ Proof of Lemma 1. In order to prove the statement of the lemma, we show that the pointwise convergence almost surely ofp \[j] n ,r \[j] n andĝ \[j] n for a fixed j holds. First, note that for j such that p j = 0 the statement holds, since in this case we haveπ n,j =ρ n,j =γ n,j = 0 for all n almost surely. Second, let us fix some 0 ≤ j ≤ t n , such that p j = 0. Next, clearly, p \[j] n a.s. → p (7.1) in ℓ k -norm for 1 ≤ k ≤ ∞. Next, from (7.1) for the sequenceĝ \[j] n we havê g \[j] n = Π p \[j] n \|F decr a.s. → g in ℓ 2 -norm, since the isotonic regression is a continuous map (cf. statement (iii) in Lemma 3). Therefore, we have proved the statement of the lemma for the sequencesπ n andγ n . Next, we prove the statement forρ n . Let us fix some s > j such that p k < p j for all k > s. Next, let n,j ). p (k) = Then, after simplification we get CV (β) = a n β 2 − 2b n β + c n , where the term c n does not depend on β, and a n = n,j ) − tn j=0p n,j (ĥ n,j −p n,j ). Assume, that a n = 0. Then, CV (β) is minimised by β n =      bn an , if 0 ≤ b n ≤ a n , 1, if a n ≤ b n , 0, if b n ≤ 0. Next, note that ifp n =ĥ n , thenφ n =p n =ĥ n for any 0 ≤ β n ≤ 1, and, therefore, for consistency of notation we defineβ n = 0 when a n = 0. ✷ Proof of Lemma 2. First, we prove the statement forπ n . Assume that for some j we havep \[j] n,j = 0 and recall that π n,j =p \[j] n,j = x j − 1 n − 1 . Next, note that x j − 1 n − 1 − x j n = −n + x j n(n − 1) < 0. Let us study the case ofγ n . To prove the statement of the lemma we will use maximum upper sets algorithm, which is given in Lemma 4 in the Appendix. Let x = (x 0 , . . . , x tn ) be frequency data from p. Next, let x * = (x * 0 , . . . , x * tn ) be the isotonic regression of x and assume that x * has (l + 1) constant regions. Let m(0) < · · · < m(l) = t n be the indices of the last elements in the constant regions of x * and m(−1) = −1. Therefore, we have x * j = m(r) k=m(r−1)+1 x k m(r) − m(r − 1) for j ∈ [m(r − 1) + 1, m(r)] and r ∈ [0, l]. Let us consider the first constant region of x * and for some integer q ∈ [0, m(0)] define vector y ∈ R t+1 + y j = x j − 1, if j = q, x j , otherwise, and let y * be isotonic regression of y. Recall, m(0) is the largest non-negative integer which maximizes the following mean Further, let m ′ (0) be the largest non-negative integer which maximizes the following mean for the vector y S 2 = m ′ (0) k=0 y k m ′ (0) + 1 . Let us prove that S 2 ≤ S 2 . First, assume that m ′ (0) = m(0), then, clearly, S 2 ≤ S 1 since y j ≤ x j . Second, let us assume that m ′ (0) = m(0). Then, from the definitions of m(0) and m ′ (0) it follows S 2 = m ′ (0) k=0 y k m ′ (0) + 1 ≤ m ′ (0) k=0 x k m ′ (0) + 1 ≤ m(0) k=0 x k m(0) + 1 = S 1 . Next, assume that q is not in the first constant region. Then in this case from maximum upper sets algorithm it follows that the constant regions in the isotonic regressions x * and y * are the same up to the region which contains element with index m. Then, we can use the same approach as for the first region. Therefore, we have proved that y * q ≤ x * q . Next, from statement (v) of Lemma 3 forĝ n andγ n we havê g n,j = x * j n , andγ n,j = y * j n − 1 , therefore, we proved thatγ n,j ≤ n n − 1ĝ n,j . Finally, we prove the inequality forρ n . Analogously to the case ofγ n , let us consider the vectors x and y, discussed above. Note that y j ≤ x j for all j, therefore, the same componentwise inequality holds for the sorted vectors rear(x) and rear(y). Next, using the definition ofr n andρ n we prove that ρ n,j ≤ n n − 1r n,j . ✷ Proof of Theorem 5. Assume that the p.m.f. p is not decreasing. Note that \|\| √ n(φ n − p) − √ n(p n − p))\|\| 2 = √ n\|\|φ n −p n \|\| 2 ≤ β n √ n\|\|ĥ n −p n \|\| 2 + (1 −β n ) √ n\|\|p n −p n \|\| 2 =β n √ n\|\|ĥ n −p n \|\| 2 . → 0. The statement of the theorem now follows from Theorem 3.1 in [8]. Assume that p is a strictly decreasing p.m.f. over {0, . . . , s}, with s < ∞. Next, let ε = inf{\|p j − p j+1 \| : j = 0, . . . , s − 1} and note that {sup j {\|p j − p j \| < ε/2} ⊆ {ĥ n,j =p n,j } for bothĥ n =ĝ n andĥ n =r n . Therefore, this implies that for any j = {0, . . . , s we have P[φ n,j =p n,j ] ≥ P[sup j {\|p n,j − p j \| < ε/2] → 1, since the empirical estimator is strongly consistent. The statement of the theorem follows from Theorem 3.1 in [8]. ✷ Acknowledgments This work was partially supported by the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundataion. for any n > N . 3 . 3Data-driven selection of the mixture parameter β Let us consider squared ℓ 2 -distance between the true p.m.f. p and the stacked estimatorφ n : L n = \|\|φ n − p\|\| 2 2 :≡ L (1) n − 2L (2) n + L Fig 1 : 1Standard normal QQ-plots of 1000 samples of √ n(p n,1 −p 1 ), √ n(ĝ n,1 −p 1 ), √ M 1 1: p = U (11), M 2 : p = 0.15U (3) + 0.1U (7) + 0.75U (11), M 3 : p = 0.25U (1) + 0.2U (3) + 0.15U (5) + 0.4U (7), M 4 : p = Geom(0.25), Fig 2 : 2The boxplots for ℓ 1 -distances of the estimators: the empirical estimator (e), minimax estimator (mm), rearrangement estimator (r), Grenander estimator (G), the stacked rearrangement estimator (sr) and the stacked Grenander estimator (sG) for the models M1, M2, M3 and M4. Fig 3 :Fig 4 : 34The boxplots for ℓ 2 -distances of the estimators: the empirical estimator (e), minimax estimator (mm), rearrangement estimator (r), Grenander estimator (G), the stacked rearrangement estimator (sr) and the stacked Grenander estimator (sG) for the models M1, M2, M3 and M4. The estimates of the scaled risk for the models M1, M2, M3 and M4. (withξ n one of the following estimators: empirical, minimax Grenander or stacked Grenander estimator) versus the sample size n, based on 1000 Monte Carlo simulations, cf. M 5 5: p = T i (11), M 6 : p = N Bin(7, Fig 5 : 5The boxplots for ℓ 1 -distances of the estimators: the empirical estimator (e), minimax estimator (mm), rearrangement estimator (r), Grenander estimator (G), the stacked rearrangement estimator (sr) and the stacked Grenander estimator (sG) for the models M5, M6 and M7. Fig 6 : 6The boxplots for ℓ 2 -distances of the estimators:the empirical estimator (e), minimax estimator (mm), rearrangement estimator (r), Grenander estimator (G), the stacked rearrangement estimator (sr) and the stacked Grenander estimator (sG) for the models M5, M6 and M7. Fig 7 : 7The estimates of the scaled risk for the models M5, M6 and M7. Lemma 4 . 4[Maximum upper sets algorithm] For a given x ∈ R t+1 + the solution x * of a simple order isotonic regression, i.e. j − f j ] 2 is given by the following algorithm. First, let us define m(−1) = −1. Second, we choose m(0) > m(−1) to be the largest integer which maximizes the following mean Next, let us choose m(1) > m(0) to be the largest integer which maximizes for the case of stacked Grenander estimator, respectively.5) withĥ \[j] n =r \[j] n for the case of stacked rearrangement estimator, andĥ \[j] n = g \[j] n the kth largest of {p \[j] n,0 , . . . ,p \[j] n,tn }.Further, from (7.1) it follows that there exists n 1 such that for all n > n 1[r \[j] n ] (0,j) = {p (1) , . . . ,p (j) } ⊂ {p \[j] n,0 , . . . ,p \[j] n,s },almost surely, where [·] (0,j) denotes the first (j + 1) elements of the vector. Finally, since the rearrangement operator is continuous map in a finite dimensional case (Lemma 6.1 in[19]), the result of the lemma follows from continuous mapping theorem. ✷ Proof of Theorem 1. Recall that the least-squares cross-validation criterion is given by CV (β) = (βĥ n,j + (1 − β)p n,j ) 2 − 2tn j=0φ 2 n,j − 2 tn j=0p n,jφ \[j] n,j = tn j=0 tn j=0p n,j (βĥ \[j] n,j + (1 − β)p \[j] Then, since \|\|r n −p n \|\| 2 a.s. → \|\|r − p\|\| 2 < ∞, \|\|ĝ n −p n \|\| 2 a.s. → \|\|g − p\|\| 2 < ∞,and using (4.5) we haveβ n √ n\|\|r n −p n \|\| 2 a.s. → 0, β n √ n\|\|ĝ n −p n \|\| 2 a.s. → 0, which leads to √ n\|\|φ n −p n \|\| 2 a.s. The notion of "isotonic regression" in (2.2) might be confusing. Though, for historical reasons, it is a standard notion in the subject of constrained inference, cf. the monographs[28,29] and also papers[7,32], dedicated to the computational aspects, where the notation "isotonic regression" is used for the isotonic projection of a general vector. On asymptotics of the discrete convex LSE of a p.m.f. F Balabdaoui, C Durot, F Koladjo, Bernoulli. 23Balabdaoui, F., Durot, C., Koladjo, F. (2017). On asymptotics of the discrete convex LSE of a p.m.f. 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[ "Stochastic geodesics", "Stochastic geodesics" ]	[ "Ana Bela Cruzeiro ", "Jean-Claude Zambrini " ]	[]	[]	We describe, in an intrinsic way and using the global chart provided by Itô's parallel transport, a generalisation of the notion of geodesic (as critical path of an energy functional) to diffusion processes on Riemannian manifolds. These stochastic processes are no longer smooth paths but they are still critical points of a regularised stochastic energy functional. We consider stochastic geodesics on compact Riemannian manifolds and also on (possibly infinite dimensional) Lie groups. Finally the question of existence of such stochastic geodesics is discussed: we show how it can be approached via forward-backward stochastic differential equations.	10.1007/978-3-030-87432-2_4	[ "https://arxiv.org/pdf/2007.05291v1.pdf" ]	220,487,155	2007.05291	37bc430d75e73d5d80dced273c94d8d9d4993f28	Stochastic geodesics arXiv:2007.05291v1 [math.PR] 10 Jul 2020 Ana Bela Cruzeiro Jean-Claude Zambrini Stochastic geodesics arXiv:2007.05291v1 [math.PR] 10 Jul 2020 We describe, in an intrinsic way and using the global chart provided by Itô's parallel transport, a generalisation of the notion of geodesic (as critical path of an energy functional) to diffusion processes on Riemannian manifolds. These stochastic processes are no longer smooth paths but they are still critical points of a regularised stochastic energy functional. We consider stochastic geodesics on compact Riemannian manifolds and also on (possibly infinite dimensional) Lie groups. Finally the question of existence of such stochastic geodesics is discussed: we show how it can be approached via forward-backward stochastic differential equations. Introduction The notion of geodesic in Riemannian manifolds appeared first in a lecture of Riemann, in 1854. Originally, it was referring to the shortest path between two points on Earth's surface. Nowadays, given an affine connection like the one of Levi-Civita, it can also be defined as a curve whose tangent vectors remain parallel when transported along the curve. In Theoretical Physics it is in General Relativity that this notion played a key rôle. In a stochastic framework, a generalisation of geodesic curve is described. It corresponds to a critical path for some generalised action functional. The concept is reminiscent of Feynman path integral approach to Quantum Mechanics ( [1]) but for well defined probability measures on path spaces. It involves, in particular, regularisation of the second order in time classical dynamical equations, which is not traditional in Stochastic Analysis. The derived equations of motion are of Burgers type. When considering flows which keep the volume measure invariant one obtains Navier-Stokes equations. This point of view was developed in in [9], [3] and [2] in particular. It is currently being investigated (c.f [8] as well as [10] for a review on this subject). After a short survey of classical geodesics on Riemannian manifolds, Cartan's frame bundle approach and its relation with the horizontal and Laplace-Beltrami operators are recalled. Stochastic Analysis of diffusions on manifolds along the line of Itô-Ikeda-Watanabe is given, together with Itô's associated notion of parallel transport. Then one comes back to one of the historic definitions of geodesics, namely as critical points of an Action functional. The regularisations associated with the critical diffusion provide the appropriate generalised energy functional. The same strategy applies to geodesics on Lie groups. It is also shown how, if needed, stochastic geodesics can be characterised via stochastic forward-backward SDEs. It is a special pleasure to dedicate this paper to Sergio Albeverio as a modest sign of recognition for his faithful friendship along the years. Geodesics on Riemannian manifolds We shall denote by M a d-dimensional compact Riemannian manifold and g its metric tensor. Given m ∈ M, if u, v are vectors in the tangent space T m (M) the Riemannian inner product is given in local chart by g m (u, v) = (g i, j u i v j )(m) Here and in the rest of the paper we adopt Einstein summation convention. The Levi-Civita covariant derivative of a vector field z has the expression [∇ k z] j = ∂ ∂ m k z j + Γ j k,l z l , where Γ denotes the corresponding Christoffel symbols in the local chart; explicitly, Γ j k,l = 1 2 ∂ ∂ m k g i,l + ∂ ∂ m l g k,i − ∂ ∂ m i g k,l g j,i(1) Given a smooth curve t → ϕ(t) ∈ M, the parallel transport of a vector field z along this curve is defined by the condition of zero covariant derivative of z in theφ direction, ∇φ (t) z(t) = 0 orż j = −Γ j k,lφ k z l .(2) Its solution, z(t) = t ϕ t←0 (z(0)), the parallel transport of z along the curve, is an Euclidean isomorphism between tangent spaces: t ϕ t←0 : T ϕ(0) (M) → T ϕ(t) (M). Consider the curves minimising the length J (γ) = l(γ,γ)dt, l(γ,γ) = g i, jγ iγ j and therefore satisfy the Euler variational equation d dt ( g i, jγ j l ) = 1 2l ∂ i (g j,k )γ jγ k . Replacing dt by ds (where s is the arc lenght) we obtain d dt (g i, jγ j 1 ds ) − 1 2 1 ds ∂ i (g j,k )γ jγ k = 0 and also g i, j d 2 γ j ds 2 + ∂ k (g i, j ) dγ k ds dγ j ds − 1 2 ∂ i (g j,k ) dγ k ds dγ j ds = 0 Multiplying both members by g α,i we obtain the following classical form of the geodesic equation: d 2 γ α ds 2 + Γ α j,k dγ j ds dγ k ds = 0(3) or ∇γγ = 0. A curve satisfying the last equation is called a geodesic for the corresponding Riemannian metric. It is also well known that geodesics (defined in a time interval [0, T ]) are characterised as being critical paths of the (kinetic) energy functional E (γ) = T 0 \|\|γ(t)\|\| 2 dt = T 0 g i, j (γ(t))γ i (t)γ j (t)dt(4) By critical it is meant that, for every family of smooth curves (variations of γ) γ ε starting (at time 0) and ending (at time T ) at γ(0) and γ(T ) resp., we have d dε \| ε=0 E (γ ε ) = 0. The frame bundle and the Laplacians The bundle of orthonormal frames over M is defined by O(M) = {(m, r) : m ∈ M, r : R d → T m (M) is an Euclidean isometry} The map π : O(M) → M, π(m, r) = m is the canonical projection. Let e i , i = 1, ...d denote the vectors of the canonical basis of R d and γ i denote the (unique) geodesic on M such that γ i (0) = m, d dt t=0 γ i (t) = r(e i ). Let (γ i (t), r i (t)) represent the parallel transport of r along γ i , ∇γ i r i = 0, r i (0) = Id. Then A i (m, r) = d dt t=0 r i (t) are called the horizontal vector fields on M. Denote by Θ the one-form defined on O(M) with values in R d × so(d) such that < Θ , A i >= (e i , 0); Θ = (θ , ω), with ω(m, r) = r −1 dr the Maurer-Cartan form on the orthogonal group O(d). Its structure equations are given by dθ = ω ∧ θ dω = ω ∧ ω + Ω (θ ∧ θ ), where Ω denotes the curvature tensor: Ω (X,Y, Z) = (∇ X ∇ Y − ∇ Y ∇ X − ∇ [X,Y ] )Z, where [X,Y ] denotes the bracket of two vector fields. Recall also that the Ricci tensor (Ricci kl ) is the trace of the curvature, taken in the second and third entries. In particular θ (A k ) = e k and ω(A k ) = 0. The horizontal Laplacian on O(M) is the second order differential operator ∆ O(M) = d ∑ k=1 L 2 A k (5) where L A k denotes the Lie derivative along the vector field A k . For every smooth function f defined on M we have ∆ O(M) ( f • π) = (∆ M f ) • π where ∆ M is the Laplace-Beltrami operator on M. This operator is expressed in local coordinates by ∆ M f = g i, j [ ∂ 2 f ∂ m i ∂ m j − Γ k i, j ∂ f ∂ m k ].(6) Stochastic Analysis on manifolds We are going to consider stochastic diffusions associated to elliptic operators on M of the form L u f := 1 2 ∆ M f + ∂ u f(7) in the sense of Itô stochastic calculus. Here u is a possibly time-dependent, smooth (at least C 2 ) vector field on M. In local coordinates the diffusion with generator L u can be written as dm j (t) = σ j k dx k (t) − 1 2 g m,n Γ j m,n − u j dt(8) where σ = √ g and x k are independent real-valued Brownian motions. We consider the horizontal lift of these M-valued diffusion processes. Denote by u k the functions defined on O(M) by U k (r) = r(e k ).u π(r) . Thenũ = ∑ k U k A k satisfies π ′ (ũ) = u (π ′ being the derivative of the canonical pro- jection π). Denoting by x a sample path of the standard Brownian motion on R d , x(t),t ∈ [0, T ], x(0) = 0,dm i (t) = e i α • (dx α (t) + u α dt) de i α (t) = −Γ i j,k (m(t))e k α (t) • dm j (t), If a ∈ M, we denote the path space of the manifold-valued paths starting from a by P a (M) = {p : [0, T ] → M, p(0) = a, p continuous}. The diffusion m(t) has for generator the operator L u . We refer to [16] for a detailed exposition of diffusions on Riemannian manifolds constructed on the frame bundle. For each vector field u the operator L u and the operator L U = 1 2 ∆ O(M) + ∂ U induce on the path spaces P m 0 (M) and P m 0 (O(M)), respectively, two probability measures, namely the laws of the corresponding diffusion processes. The projection map π realizes an isomorphism between these two probability spaces. Let the path space P 0 (R d ) be endowed with the law of the process dy(t) = (•dx(t) + U)(y(t)), t ∈ [0, T ] and P m 0 (M) with the law of the diffusion p with generator L u ). Consider the Itô map I : P 0 (R d ) → P m 0 (M) defined by I(x)(t) = π(r x (t)) This map is a.s. bijective and provides an isomorphism between the corresponding probability measures ( [18]). Even though p is not differentiable in time, Itô has shown that one can still define a parallel transport along p, which is the isomorphism from T p(s) (M) → T p(t) (M) given by t p t←s := r x (t)r x (s) −1 . The differentiability of r x (t) with respect to variations of the Brownian motion x was studied in [17] and [12] within the framework of Malliavin Calculus [19], [5] (c.f. also [13] for the case of the Brownian motion with drift). Denote D β α = L A α u β . The following result holds: Proposition 1. Given a process of bounded variation in time h : P 0 (R d ) × [0, T ] → R d , we have, using the notations of section 3, < θ , d dε \| ε=0 r x+εh >= ζ , < ω, d dε \| ε=0 r x+εh >= ρ(10) where ζ and ρ are determined by Itô (and Stratonovich) stochastic differential equations dζ (t) =ḣ(t)dt − [ 1 2 Ricci + D](h(t))dt − ρ(t)dx(t) (11) dρ = Ω (•dx + udt, h) with initial conditons ζ (0) = 0, ρ(0) = 0. The result above is still valid for pinned Brownian motion, namely when p(T ) is fixed. Then the variations are equal to zero not only at the initial but also at this final time. The sigma-algebra and filtration on the corresponding path space are the usual ones, generated by the coordinate maps and generated by the coordinate maps up to time t, respectively. We refer to [15] for more details. Stochastic geodesics We shall consider stochastic geodesics as processes which are critical points of some energy functional generalising the classical deterministic one. Since the stochastic processes, diffusions on the manifold, are no longer differentiable in time, some notion of generalised velocity has to replace the usual time derivative. If ξ (·) is a semimartingale with respect to an increasing filtration P t , t ∈ [0, T ] and with values in a manifold M, we consider the process η defined by the Stratonovich integral η(t) := t 0 t ξ 0←s odξ (s) This is a semimartingale taking values in T ξ (0) (M). We consider its (generalised) right-hand time derivative (or drift) by taking conditional expectations: D t η(t) = lim ε→0 E P t [ η(t + ε) − η(t) ε ] Notice that if ξ is a differentiable deterministic path, this notion of derivative reduces to the usual one. Then we define the generalised (forward) derivative D ∇ t ξ (t) := t t←0 D t η(t)(12) We use the symbol ∇ to stress that the derivative depends on the choice of covariant derivative used to define the parallel transport, although in this work we are only consider the Levi-Civita covariant derivative. For a (possibly time dependent) vector field Z computed along a semimartingale ξ , the generalised derivative is defined as D ∇ t Z(t) = lim ε→0 E P t [t t←t+ε Z(t + ε, ξ (t + ε)) − Z(t, ξ (t)] Let us consider our base manifold M and, for a M-valued semimartingale ξ , define the corresponding kinetic energy by E (ξ ) = E T 0 \|\|D ∇ ξ (t)\|\| 2 dt(13) Next Theorem characterises the critical paths of E . Allowed variations are processes of bounded variation h satisfying h(0) = h(T ) = 0. We have the following result: Theorem 1. A diffusion process m(·) with generator L u , u ∈ C 2 (M), is a critical path for the energy functional E if and only if D ∇ t u(t, m(t)) = 0 almost everywhere or, equivalently, ∂ ∂t u + (∇ u u) + 1 2 [(∆ u) + Ricci(u)] = 0(14) Notice that, in particular, we obtain the expression derived in [20] using local coordinates. It is shown in [4] (c.f, more generally, [20]) that the symmetries of the critical diffusion coincide with the regularisation of its classical counterpart. In other words, if the diffusion coefficient in (8), regarded now as variable, tends to zero, D ∇ ξ in (12) reduces to an ordinary (strong) derivative, Eq. (14) and the symmetries of the critical diffusion reduce to those of the classical functional (4). Proof. We first write the energy functional via the lift of the process to the frame bundle, as explained in the last paragraph: E = E T 0 \|\|D t π(r x (t))\|\| 2 dt where D t refers to the generalised derivative for processes defined on the flat space (of the Brownian motion x). Then we perform variations of the Brownian motion x along directions h(·), processes of bounded variation with h(0) = h(T ) = 0. Using Proposition 1, these variations will give rise to variations on the path space of the manifold M along semimartingales ζ (·), where ζ is given by (11). More precisely we have, d dε \| ε=0 E T 0 \|\|D t π(r x+εh (t))\|\| 2 dt = 2E T 0 < D t π(r x (t)), D t π ′ ( d dε \| ε=0 r x+εh (t)) > dt = 2E T 0 < D t π(r x (t)), D t (ζ )(t)) > dt = 2E T 0 < D t π(r x (t)),ḣ − 1 2 Ricci(h) − D(h)(t) > dt Using integration by parts in time, the assumption h(0) = h(t) = 0 and the fact that there is no Itô's extra term in the integration since h is of bounded variation, the first term is equal to −2E T 0 < D t D t π(r x ), h(t) >. We arrive to the conclusion that a process r x of the form (9) is critical for the action functional E if and only if D ∇ t u = 0 almost everywhere, which proves the Theorem. Stochastic geodesics on Lie groups Let G denote a Lie group endowed with a left invariant metric < > and a left invariant connection ∇, that we assume here to be the Levi-Civita connection. The corresponding Lie algebra G can be identified with the tangent space T e G, where e is the identity element of the group. Taking a sequence of vectors H k ∈ G , consider the following Stratonovich stochastic differential equation on the group: dg(t) = T e L g(t) ∑ k H k • dx k (t) − 1 2 ∇ H k H k dt + u(t)dt(15) with g(0 = e, where T a L g(t) : T a G → T g(t)a G is the differential of the left translation L g(t) (x) = g (t)x, x ∈ G and where x k (t) are independent real valued Brownian motions. The vector u(·) is assumed to be non random, u(·) ∈ C 2 ([0, T ]; G ). The stochastic energy functional for a general G-vaued semimartingale ξ (t),t ∈ [0, T ], reads: E (ξ ) = E T 0 \|\|T ξ (t) L ξ (t) −1 D ∇ t ξ (t)\|\| 2 dt(16) Assume furthermore that ∇ H k H k = 0 for all k (in particular the Stratanovich integral in (15) coincides with the Itô one). Then the following result holds: (15) is critical for the energy functional (16) if and only if the vector field u(·) satisfies the equation Theorem 2 ([2]). A G-valued semimartingale of the form d dt u(t) = ad u(t) u(t) − 1 2 ∑ k ∇ H k ∇ H k u(t) + Ricci(u(t)) When H k = 0 for all k the equation reduces to the well known Euler-Poincaré equation for (deterministic) geodesics in Lie groups d dt u(t) = ad u(t) u(t). Up to some sign changes, the right invariant case is analogous. The theorem also holds for infinite-dimensional Lie groups and allows, as a particular case, to derive the Navier-Stokes equation, when the problem is formulated on the diffeomorphisms group (c.f. [2]). Relation with stochastic forward-backward differential equations Deterministic geodesics solve second order differential equations and as such can be obtained using standard methods for such equations, with given initial position and velocity as well as with initial and final given positions. The meaning of "second order" stochastic differential equations is not so clear. A possible method is its characterisation via stochastic forward-backward differential equations. In local coordinates (c.f. notations defined in (8) Given m j (0) and y(T ) = u(T, m(T )) these kind of systems may provide solutions of the form (m(t), y(t)) with y(t) = u(t, m(t)) corresponding to our stochastic geodesics (c.f., for example, [14]). The term Z is an a priori unknown of the equation, but is in fact determined a posteriori by the solution (m, y). In the case of stochastic geodesics on Lie groups, the characterisation via forward-backward equations was described in [6]. An extension to infinite dimensional Lie groups and, in particular, to the Navier-Stokes equation framework, is also possible ( [11], [7]). we consider the following Stratonovich stochastic differential equation on O(M):dr x (t) = d ∑ k=1 A k (•dx k (t) + U k dt), r x (0) = 0(9)with π(r 0 ) = m 0 . In local coordinates (m i , e i α ) on O(M) and if r(t) = (m(t), e(t)) we have, , a stochastic geodesic in the time interval [0, T ] reads m j (t) = m j AcknowledgementsThe authors acknowledge the support of the FCT Portuguese grant PTDC/MAT-STA/28812/2017. Mathematical Theory of Feynman Path Integrals: An Introduction. S Albeverio, R Hoegh-Krohn, S Mazzucchi, Lecture Notes in Math. 523Springer2nd EditionAlbeverio, S., Hoegh-Krohn, R., Mazzucchi, S., Mathematical Theory of Feynman Path Inte- grals: An Introduction, 2nd Edition, Lecture Notes in Math. 523, Springer (2008). Stochastic Euler Poincaré reduction. M Arnaudon, X Chen, A B Cruzeiro, J. Math. Physics. 5581507Arnaudon, M., Chen, X., Cruzeiro, A.B., Stochastic Euler Poincaré reduction, J. Math. Physics 55 (2014), 081507. 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[ "Absolute Maximal Entanglement and Quantum Secret Sharing", "Absolute Maximal Entanglement and Quantum Secret Sharing" ]	[ "Wolfram Helwig \nCenter for Quantum Information and Quantum Control (CQIQC)\nDepartment of Physics\nDepartment of Electrical & Computer Engineering\nUniversity of Toronto\nM5S 3G4TorontoOntarioCanada\n", "Wei Cui \nCenter for Quantum Information and Quantum Control (CQIQC)\nDepartment of Physics\nDepartment of Electrical & Computer Engineering\nUniversity of Toronto\nM5S 3G4TorontoOntarioCanada\n", "José Ignacio Latorre \nDept. d'Estructura i Constituents de la Matèria\nUniversitat de Barcelona\n647 Diagonal08028BarcelonaSpain\n", "Arnau Riera \nMax Planck Institute for Gravitational Physics\nAlbert Einstein Institute\nAm Mühlenberg 1D-14476GolmGermany\n\nDahlem Center for Complex Quantum Systems\nFreie Universität Berlin\n14195BerlinGermany\n", "Hoi-Kwong Lo \nCenter for Quantum Information and Quantum Control (CQIQC)\nDepartment of Physics\nDepartment of Electrical & Computer Engineering\nUniversity of Toronto\nM5S 3G4TorontoOntarioCanada\n" ]	[ "Center for Quantum Information and Quantum Control (CQIQC)\nDepartment of Physics\nDepartment of Electrical & Computer Engineering\nUniversity of Toronto\nM5S 3G4TorontoOntarioCanada", "Center for Quantum Information and Quantum Control (CQIQC)\nDepartment of Physics\nDepartment of Electrical & Computer Engineering\nUniversity of Toronto\nM5S 3G4TorontoOntarioCanada", "Dept. d'Estructura i Constituents de la Matèria\nUniversitat de Barcelona\n647 Diagonal08028BarcelonaSpain", "Max Planck Institute for Gravitational Physics\nAlbert Einstein Institute\nAm Mühlenberg 1D-14476GolmGermany", "Dahlem Center for Complex Quantum Systems\nFreie Universität Berlin\n14195BerlinGermany", "Center for Quantum Information and Quantum Control (CQIQC)\nDepartment of Physics\nDepartment of Electrical & Computer Engineering\nUniversity of Toronto\nM5S 3G4TorontoOntarioCanada" ]	[]	We study the existence of absolutely maximally entangled (AME) states in quantum mechanics and its applications to quantum information. AME states are characterized by being maximally entangled for all bipartitions of the system and exhibit genuine multipartite entanglement. With such states, we present a novel parallel teleportation protocol which teleports multiple quantum states between groups of senders and receivers. The notable features of this protocol are that (i) the partition into senders and receivers can be chosen after the state has been distributed, and (ii) one group has to perform joint quantum operations while the parties of the other group only have to act locally on their system. We also prove the equivalence between pure state quantum secret sharing schemes and AME states with an even number of parties. This equivalence implies the existence of AME states for an arbitrary number of parties based on known results about the existence of quantum secret sharing schemes.PACS numbers:	10.1103/physreva.86.052335	[ "https://arxiv.org/pdf/1204.2289v1.pdf" ]	119,229,329	1204.2289	6b5ac5f0545a5f2c524e5eaf1cc0b36481d6066c	Absolute Maximal Entanglement and Quantum Secret Sharing (Dated: May 10, 2014) Wolfram Helwig Center for Quantum Information and Quantum Control (CQIQC) Department of Physics Department of Electrical & Computer Engineering University of Toronto M5S 3G4TorontoOntarioCanada Wei Cui Center for Quantum Information and Quantum Control (CQIQC) Department of Physics Department of Electrical & Computer Engineering University of Toronto M5S 3G4TorontoOntarioCanada José Ignacio Latorre Dept. d'Estructura i Constituents de la Matèria Universitat de Barcelona 647 Diagonal08028BarcelonaSpain Arnau Riera Max Planck Institute for Gravitational Physics Albert Einstein Institute Am Mühlenberg 1D-14476GolmGermany Dahlem Center for Complex Quantum Systems Freie Universität Berlin 14195BerlinGermany Hoi-Kwong Lo Center for Quantum Information and Quantum Control (CQIQC) Department of Physics Department of Electrical & Computer Engineering University of Toronto M5S 3G4TorontoOntarioCanada Absolute Maximal Entanglement and Quantum Secret Sharing (Dated: May 10, 2014) We study the existence of absolutely maximally entangled (AME) states in quantum mechanics and its applications to quantum information. AME states are characterized by being maximally entangled for all bipartitions of the system and exhibit genuine multipartite entanglement. With such states, we present a novel parallel teleportation protocol which teleports multiple quantum states between groups of senders and receivers. The notable features of this protocol are that (i) the partition into senders and receivers can be chosen after the state has been distributed, and (ii) one group has to perform joint quantum operations while the parties of the other group only have to act locally on their system. We also prove the equivalence between pure state quantum secret sharing schemes and AME states with an even number of parties. This equivalence implies the existence of AME states for an arbitrary number of parties based on known results about the existence of quantum secret sharing schemes.PACS numbers: Introduction. Entanglement is at the core of the power of quantum information processing and has been extensively studied for few qubits. The classification of entanglement classes for three and four qubits is well understood [1][2][3][4][5][6][7] and canonical forms of pure states under local unitary transformations of each local Hilbert space have also been analyzed [6,8,9]. As the number of local quantum degrees of freedom increases, our understanding of entanglement gets poorer. The number of independent invariants that classify entanglement grows exponentially and it is unclear which purpose each category of entanglement serves [10,11]. In recent years, there has been an important progress in the classification of the maximally multipartite entangled states composed of qubits [7,[12][13][14][15]. Nevertheless, a complete understanding of the structure, classification and usefulness of quantum states with the largest possible entanglement for arbitrary dimension is still missing. Another motivation for studying multipartite entanglement is its connection to other apparently unrelated areas of physics, like string theory and black-holes [16,17]. Quantum teleportation is one of the most intriguing utilizations of entanglement. It allows distant parties, who share a resource of entanglement, to transport a quantum state from one party to the other by only exchanging classical information and using up said entanglement. The first proposal of such a protocol used the resource of bipartite entanglement between two parties [18]. Later teleportation protocols using genuine multipartite entanglement between more than two parties were proposed theoretically for four qubit entanglement [19], and experimentally in the form of open-destination teleportation for five qubits [20]. This manuscript is devoted to initiate the study of a class of states with genuine multipartite entanglement. These states, which we call absolutely maximally entangled (AME) states, are defined as having the strict maximal entanglement in all bipartitions of the system. Up until now, AME states have been thought to be a rather limited concept, because only few AME states exist for qubits [21], specifically no AME states exist for four, or eight and more qubits [15,22]. In this work, we consider the qudit problem, and show that AME states exist for any number of parties by choosing an appropriate qudit dimension. The fact that AME states contain maximal entanglement makes them the natural candidates to implement novel multipartite communication protocols. Indeed, we shall here show how they can be used to implement novel parallel teleportation scenarios that postpone the choice of senders and receivers until after the state has been distributed. These protocols require that either the senders or receivers perform joint quantum operations, while the respective other parties only have to act locally on their systems. We further establish a one-to-one correspondence between pure state quantum secret sharing (QSS) schemes [23,24] and even-party AME states, which also proves the existence of AME states for any number of parties given an appropriate choice of the system dimensions. This follows from the existence of pure state QSS schemes for any odd number of parties [23]. It should be mentioned that, while our parallel teleportation protocol is different from the aforementioned open-destination teleportation, it is also possible to implement open-destination teleportation with AME states [25]. Definition of AME states. An AME(n, d) state (absolutely maximally entangled state) of n qudits of dimension d, \|ψ ∈ C ⊗n d , is a pure state for which every bipartition of the system into the sets B and A, with m = \|B\| ≤ \|A\| = n − m, is strictly maximally entangled such that S(ρ B ) = m log 2 d .(1) Consequently, every partition of m local degrees of freedom arXiv:1204.2289v1 [quant-ph] 10 Apr 2012 is represented by a reduced density matrix proportional to the identity ρ B = T r A \|ψ ψ\| = 1 d m I d m , 1 ≤ m ≤ n 2 .(2) In practice, to detect an AME state it is sufficient to check that all the n n/2 possible bipartitions of n/2 qudits are maximally entangled, since all subsequent traces of the identity matrix are again identity matrices. A state is an AME state iff it can be written as \|AME = 1 √ d m k∈Z m d \|k 1 B1 · · · \|k m Bm \|φ(k) A ,(3) with φ(k)\|φ(k ) = δ kk , for every partition into m = \|B\| ≤ \|A\| = n − m disjoint sets B and A. Two obvious examples of AME states are the Einstein-Rosen-Podolsky (EPR) and the Greenberger-Horne-Zeilinger (GHZ) states for two and three qubits, respectively. In both cases, the entanglement entropy is maximal for all their partitions. It has been proven that there are no absolutely maximally entangled states for four qubits [15]. AME states exist for five and six qubits [26], and a possible form for them will be given later in Example 1. No AME states exist for eight or more qubits [15,22]. The existence of an AME(7, 2) state is still an open question, but it has been conjectured in Ref [26] that no such state exists. By increasing the party dimension, AME states can be found for these cases in which no qubit AME states exist. We remark, however, that, although we will show that for each n, AME(n, d) states exist for some appropriate choice of d, finding the conditions for the existence of AME(n,d) states, depending on n and d, is generally a nontrivial problem. In a future publication [25], we will show that, interestingly, a special class of AME states can be constructed from certain classical error correcting codes, namely those that satisfy the singleton bound [27]. Parallel Teleportation. The maximal entanglement property of an AME(n, d) state for any bipartition into the sets A and B can be used to teleport quantum states between those two sets. In contrast to the teleportation scenario where A and B share a maximally entangled state that is not an AME state, in the AME scenario the sets A and B do not have to be specified when the state is created, but instead can be chosen after the AME state has been distributed. There are essentially three different ways in which the teleportation can be performed, depending on which parties can perform joint quantum operations, and which are separated and only able to perform local operations on their own quantum systems. In the first case, the parties within each set, A and B, are able to perform joint quantum operations. A standard teleportation of an arbitrary d m -dimensional state, where m = min(\|A\|, \|B\|), can be performed in either direction. In the second case, the sending parties A can perform a joint quantum operation, but the parties in B are only able to perform local quantum operations. Additionally we require In the third and probably the most interesting case, the sending parties can only perform local operations, but the receiving parties can perform joint quantum operations. In this case, a teleportation is possible if the number of receiving parties is larger or equal n/2. Hence, sticking to our convention m = \|B\| ≤ \|A\|, we now consider a teleportation from B to A. See the right hand side of Figure 1 for an illustration. The first scenario is just a straightforward teleportation between maximally entangled parties. The second and third scenarios are presented in the following theorem. Theorem 1. Given an AME(n, d) state, and a bipartition of the n parties into the sets A and B such that m = \|B\| ≤ \|A\| = n − m, then the following two parallel teleportation scenarios are possible (i) A can teleport one qudit to each party in B by performing a joint quantum operation and communicating two classical "dits" to each party in B. Each party in B can then locally recover their respective qudit with a local operation. (ii) Each party in B can locally teleport one qudit to A. After receiving the measurement outcomes, consisting of two "dits" of classical information from each party in B, the parties in A are able to recover all m qudits by performing a joint quantum operation. Proof. In both scenarios the parties in set A perform a joint quantum operation to transform the AME state into m ddimensional EPR pairs. Then these pairs are used to teleport m qudits from the sending to the receiving parties. This is done by performing the joint unitary operation U A \|φ(k) A = \|k 1 A1 · · · \|k m Am \|0 A .(4) on the initial AME(n, d) state \|Φ = 1 √ d m k∈Z m d \|k 1 B1 · · · \|k m Bm \|φ(k) A ,(5) with φ(k)\|φ(k ) = δ kk . This results in the state U A \|Φ = \|Ψ B1A1 · · · \|Ψ BmAm \|0 A ,(6) where \|Ψ = \|i \|i are d-dimensional EPR pairs. These EPR pairs can now be used to teleport a qudit from A i to B i in case (i) (B i to A i in case (ii)). This requires A i (B i ) to perform a generalized Bell measurement on her qudit and the qudit she wants to teleport, and communicate the measurement result to B i (A i ). This amounts to sending the classical information of two "dits" for each EPR pair. Upon reception of the measurement result, B i (A i ) can recover the teleported qudit by performing an appropriate unitary on his qudit. Quantum Secret Sharing. The last teleportation scenario suggests a close relationship between AME states and quantum secret sharing (QSS) schemes [23]. In a QSS protocol [23,24], a dealer encodes a secret S in a quantum state that is shared among n players in such a way that only special subsets of players are able to recover the secret. The set of all subsets that are able to recover the secret form the access structure and the set of all subsets that can gain no information about the secret form the adversary structure. If the encoded state is a pure state, we call it a pure state QSS scheme. We are only interested in pure state QSS schemes here. Additionally, we restrict our attention to threshold QSS schemes [23], which means that the access structure is formed by all sets that contain k or more number of parties, while any set with less than k parties cannot obtain any information about the secret. Thus k is the threshold number of parties required to recover the secret. Such a QSS scheme is denoted as a ((k, n)) threshold QSS scheme. For pure state threshold QSS schemes, n and k are always related by n = 2k − 1. To see the relation between AME states and threshold QSS schemes, we consider an AME(2m, d) state with an even number of parties and divide the parties into two sets A = {A 1 , . . . , A m } and B = {D, B 1 , . . . , B m−1 } of equal size m. In set B we have singled out one party D, which will act as the dealer of the QSS scheme. Now we perform the protocol of Theorem 1 (ii), but only D ∈ B performs the final teleportation operation. Also note that the unitary operation in Equation (4) and the Bell measurement by the dealer commute. Thus, D can first perform her Bell measurement, effectively encoding the teleported qudit onto the residual AME state, from which it can be recovered by the players in A. Furthermore, instead of the bipartition into the sets A and B, we could have equally well chosen any other bipartition into two sets A and B of cardinality m with D ∈ B . Then, without changing the operations that D has to perform, the parties in A are able to recover the teleported qudit (see Figure 2 for an illustration). Thus, any set of at least m of the residual 2m − 1 parties without D can recover the teleported state, given that the measurement outcome is broadcasted to all parties. Furthermore, the no-cloning theorem guarantees that any set of less than m players cannot gain any information about the state [24]. Hence we accomplished to construct a ((m, 2m − 1)) threshold QSS scheme from an AME(2m, d) state. Before stating the theorem that formulates this observation concisely, we shortly review how a QSS protocol works. A secret of dimension d, \|S = a i \|i , is encoded into the state a i \|Φ i which is shared by the players such that each authorized set can deterministically recover \|S from its reduced state, while the reduced state of unauthorized sets is independent of the encoded secret. We call \|Φ i the basis states of the QSS scheme, and we show in [25] that they are AME states for pure state threshold QSS schemes with equal share and dimension size. Theorem 2. There is a one to one correspondence between an AME(2m, d) state and a pure state ((m, 2m − 1)) threshold QSS scheme, whose share and secret dimensions are d. Proof. AME to QSS: For any bipartition into parties A = {A 1 , . . . , A m } and B = {D, B 1 , . . . , B m−1 }, the AME(2m, d) states has the form \|Φ = 1 √ d m (i,k)∈Z m d \|i D \|k 1 B1 · · · \|k m−1 Bm−1 \|φ(i, k) A , with φ(k, i)\|φ(k , j) = δ kk δ ij . We define the QSS basis states \|Φ i = √ d D i\|Φ = 1 √ d m−1 k∈Z m−1 d \|k 1 · · · k m−1 B \|φ(k, i) A .(7) A secret encoded as \|a = a i \|i → a i \|Φ i ,(8) satisfies the requirement of a threshold QSS scheme, because the parties B have a completely mixed states, independent of the encoded secret. Additionally, the set A, which can be chosen to be any set of n players, can restore the secret \|a by performing the joint unitary operation U A \|φ(k, i) A = \|k 1 A1 · · · \|k m−1 Am−1 \|i Am .(9) QSS to AME: For any bipartition into m authorized parties A = {A 1 , . . . , A m } and m − 1 unauthorized parties B = {B 1 , . . . , B m−1 }, the AME basis states of the QSS scheme can be written in the form \|Φ i = 1 √ d m−1 k∈Z m−1 d \|k 1 B1 · · · \|k m−1 Bm−1 \|φ(k, i) A , where φ(k, i)\|φ(k , i) = δ kk , because the states are AME states, and φ(k, i)\|φ(k, j) = δ ij , because the authorized parties can recover the secret deterministically. Thus, φ(k, i)\|φ(k , j) = δ kk δ ij .(10) From these basis states, define the state \|Φ = 1 √ d i∈Z d \|i \|Φ i = 1 √ d m (i,k)∈Z m d \|i D \|k 1 B1 · · · \|k m−1 Bm−1 \|φ(k, i) . Because of Equation (10), \|Φ is a maximally entangled state with respect to the bipartition B ∪ {D} vs. A. Since the original bipartition into A and B was arbitrary, \|Φ is maximally entangled with respect to any bipartition into two cardinality m sets and thus is an AME(2m, d) state. Since it is known that ((m, 2m−1)) threshold QSS scheme exist for any number of m and an appropriate choice of d [23], Theorem 2 proves the existence of AME states for any number of parties. Example 1. In this example, we show how the five qubit code can be used to construct AME(5, 2) and AME(6, 2) states. From the five qubit code a ((3, 5)) threshold QSS scheme can be constructed [23]. The corresponding basis states are \|0 L = 1 4 ( \|00000 + \|10010 + \|01001 + \|10100 + \|01010 − \|11011 − \|00110 − \|11000 − \|11101 − \|00011 − \|11110 − \|01111 − \|10001 − \|01100 − \|10111 + \|00101 ),(11)\|1 L = 1 4 ( \|11111 + \|01101 + \|10110 + \|01011 + \|10101 − \|00100 − \|11001 − \|00111 − \|00010 − \|11100 − \|00001 − \|10000 − \|01110 − \|10011 − \|01000 + \|11010 ).(12) These states are AME(5, 2) states as required. Following the receipe of Theorem 2, we obtain the AME(6, 2) state \|Φ = 1 √ 2 [\|0 \|0 L + \|1 \|1 L ] = 1 4 [\|000 (\|+ − + + \|− + − ) + \|001 (− \|+ − − + \|− + + ) + \|010 (\|+ + − − \|− − + ) + \|011 (− \|+ + + − \|− − − ) + \|100 (− \|+ + + + \|− − − ) + \|101 (− \|+ + − − \|− − + ) + \|110 (− \|+ − − − \|− + + ) + \|111 (− \|+ − + + \|− + − )].(13) Conclusion. In this manuscript, we have introduced AME states, a class of highly entangled states, for n qudits shared among n locally separated parties. Previous investigations of maximal entanglement showed that AME states do not exist when the number of qubits is eight or larger. Here we proved the existence of AME states for any number of parties with the appropriate qudit dimension. Moreover, we have shown how they can be utilized in different parallel teleportation scenarios, which require some parties to perform joint quantum operations, while others' capabilities may be restricted to local operations. In those scenarios the advantage of AME states over less entangled states like a collection of EPR pairs lies in the fact that the partition into senders and receivers may be chosen after the state has been distributed. Furthermore, we have investigated the relationship of AME states with QSS schemes and established a one-to-one correspondence between even party AME states and pure state threshold QSS schemes. This correspondence allows us to prove the existence of AME states for any number of parties with the appropriate dimension. In future work we further explore this very intuitive approach to develop new communication protocols from AME states as well as extending the range of QSS schemes that can be derived from AME states. For instance, instead of assigning the role of the dealer to only one of the parties in the AME state, we can imagine multiple dealers who encode independent secrets onto the residual AME states, resulting in QSS schemes with more involved access structures. The established connection to QSS schemes also confirms a relation between AME states and quantum error correction codes that was already suggested in Ref. [28]. A better understanding of this relation will allow us to construct new quantum error correction codes from AME states as well as deriving AME states from already known quantum codes. This might also shed light upon the open question of existence of AME states for a given number of parties and system dimension. FIG. 1 : 1Parallel Teleportation scenarios of Theorem 1. Scenario (i) is on the left, and (ii) on the right. Parties in A perform joint quantum operations, parties in B only local quantum operations. m = \|B\| ≤ \|A\| = n − m. Then one qudit can be teleported from A to each of the parties in B, and thus in total m qudits are teleported from A to B. This is illustrated in the left hand side of Figure 1. online) After D (blue) performs her teleportation operation, any set of m parties (red), A, A , A etc., can recover the teleported state. Any set of parties with m − 1 or less parties (any set consisting only of green parties) cannot gain any information about the teleported state. Acknowledgments. W.H., W.C., and H.K.L. acknowledge financial support from funding agencies including NSERC, Quantum-Works, the CRC program and CIFAR. 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[ "Subgroups of 3-factor direct products", "Subgroups of 3-factor direct products" ]	[ "Daniel Neuen \nRWTH Aachen University\n\n", "Pascal Schweitzer \nRWTH Aachen University\n\n" ]	[ "RWTH Aachen University\n", "RWTH Aachen University\n" ]	[]	Extending Goursat's Lemma we investigate the structure of subdirect products of 3-factor direct products. We give several example constructions and then provide a structure theorem showing that every such group is essentially obtained by a combination of the constructions. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. In the spirit of Goursat's Lemma, for two special cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphism between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups.	10.2478/tmmp-2019-0003	[ "https://arxiv.org/pdf/1607.03444v1.pdf" ]	119,644,964	1607.03444	4e71cabd92ffd0d517cd7984cbcc4a733e43ac08	Subgroups of 3-factor direct products Jul 2016 Daniel Neuen RWTH Aachen University Pascal Schweitzer RWTH Aachen University Subgroups of 3-factor direct products Jul 2016 Extending Goursat's Lemma we investigate the structure of subdirect products of 3-factor direct products. We give several example constructions and then provide a structure theorem showing that every such group is essentially obtained by a combination of the constructions. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. In the spirit of Goursat's Lemma, for two special cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphism between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups. Introduction The lemma of Goursat [7] is a classic result of group theory that characterizes the subgroups of a direct product of two groups G 1 × G 2 . A version of the lemma also provides means to describe the subgroups of G 1 × G 2 by inspecting the subgroup lattices of G 1 and G 2 and considering isomorphisms between arising factor groups. In an expository article, Anderson and Camillo [1] demonstrate for example the applicability of Goursat's lemma to determine normal subgroups of G 1 × G 2 , to count the number of subgroups of S 3 × S 3 , and to prove the Zassenhaus Lemma. They also describe how Goursat's Lemma can be stated in the context of rings, ideals, subrings and in modules. The Lemma itself can also be found in various introductory algebra and group theory texts (e.g., [8,10]). While Goursat's Lemma applies to subgroups of the direct product of two groups, in this work we are concerned with subgroups of the direct product of three groups. It seems that there is no straightforward generalization to three factors. Indeed, Bauer, Sen, and Zvengrowski [2] develop a generalization to an arbitrary finite number of factors by devising a non-symmetric version of Goursat's lemma for two factors that can then be applied recursively. A more category theory focused approach is taken by Gekas. However no simple correspondence theorem between the subdirect products of 3-factor direct products and data depending on the sublattice of the subgroups of the factors and isomorphisms between them is at hand. In fact, in [2] the authors exhibit two Abelian examples that stand in the way of such a correspondence theorem by sharing the various characteristic subgroups and isomorphisms between them and yet being distinct. Both these papers are able to recover several identities provided by Remak [13] who is explicitly concerned with 3-factor subdirect products. In this paper we analyze the structure of subdirect products of 3-factor direct products. To this end we give several example constructions of such groups and then provide a structure theorem showing that every such group is essentially obtained by a combination of the constructions. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. We call a subdirect product of G 1 × G 2 × G 3 2-factor injective if each of the three projections onto two factors is injective. By taking suitable quotients, it is possible to restrict our investigations to 2-factor injective subdirect products, for which we obtain the following theorem. Theorem 1.1 (Characterization of 2-factor injective subdirect products of 3-factor products). Let ∆ ≤ G 1 ×G 2 ×G 3 be a 2-factor injective subdirect product. Then there is a normal subgroup of H ∆ with [π i (∆) : π i (H)] = [∆ : H] for i ∈ {1, 2, 3} and H is isomorphic to a group of the following form: there are three finite groups H 1 , H 2 , H 3 that all have an Abelian subgroup M contained in their the center such that H is isomorphic to the factor group of {((h 2 , h ′ 3 −1 ), (h 3 , h ′ 1 −1 ), (h 1 , h ′ 2 −1 )) \| h i , h ′ i ∈ H i , h i h ′ i −1 ∈ M, h 1 h ′ 1 −1 h 2 h ′ 2 −1 h 3 h ′ 3 −1 = 1}, by the normal subgroup {((m 1 , m 1 ), (m 2 , m 2 ), (m 3 , m 3 )) \| m i ∈ M }. In the spirit of Goursat's Lemma we then investigate correspondence theorems between data obtained from the subgroup lattices of the G i (as well as isomorphism between arising factor groups) and the subdirect products of G 1 × G 2 × G 3 . For two special cases, namely the cases H = ∆, and M = {1}, we obtain such a correspondence theorem for three factors. Here the second case is a particular special case hinted at in [2], which is indeed describable by a symmetric version of a generalized Goursat's Lemma. In a third special case, where one of the G i is the semi-direct product of the projection of H onto the i-th component and some other group, we also obtain a partial correspondence theorem. As demonstrated by Petrillo [12], Goursat Lemma can readily be applied to count subgroups of the product of two Abelian groups. For a direct product of an arbitrary number of Abelian groups the number of subgroups has been extensively studied. We refer to the monograph of Butler [5]. In fact there are also explicit formulas for the counts of subgroups of Z p × Z q × Z r (see for example [9]). In line with the papers and as an application of our characterization, we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups S n1 × S n2 × S n3 . It is also possible for example to count the normal subgroups of such direct products. In fact, the normal subgroups can be also characterized for arbitrary finite products of symmetric groups [11]. Let us finally also point to some literature concerning finiteness properties of groups [3,4] which also contains some structural results on 3-factor direct products (in particular on the case we call 3-factor surjective). Goursat's Lemma Let G = G 1 × G 2 × · · · × G t be a direct product of groups. We define for i ∈ {1, . . . , t} the map π i as the projection to the i-th coordinate and we define the homomorphism ψ i : ∆ → G 1 × · · · × G i−1 × G i+1 × · · · × G t : (g 1 , g 2 , . . . , g t ) → (g 1 , . . . , g i−1 , g i+1 , . . . , g t ). A subgroup ∆ ≤ G of the direct product is said to be a subdirect product if π i (∆) = G i for all 1 ≤ i ≤ t. Goursat's Lemma is a classic statement concerned with the structure of subdirect products of direct products of two factors. Theorem 2.1 (Goursat's Lemma). Let ∆ ≤ G 1 × G 2 = G be a subdirect product and de- fine N 1 = {g 1 ∈ G 1 \| (g 1 , 1) ∈ ∆} as well as N 2 = {g 2 ∈ G 1 \| (1, g 2 ) ∈ ∆}. Then G 1 /N 1 is isomorphic to G/N 2 via an isomorphism ϕ for which (g 1 , g 2 ) ∈ ∆ if and only if ϕ(g 1 ) = g 2 . This gives a natural homomorphism ∆ → G 1 /N 1 × G 2 /N 2 defined as (g 1 , g 2 ) → (g 1 N 1 , g 2 N 2 ) with image {(g 1 , g 2 ) \| ϕ(g 1 ) = g 2 }. Thus we can view ∆ as a fiber product (or pull back) of G 1 and G 2 over G i /N i . A typical application of the lemma is a proof of the fact that subdirect products of non-Abelian finite simple groups are direct products of diagonal subgroups. Furthermore, the lemma can be applied to count subgroups of direct products. For this, the following correspondence version of the lemma is more convenient. Theorem 2.2. There is a natural one-to-one correspondence between the subgroups of G 1 × G 2 and the tuples (P 1 , P 2 , N 1 , N 2 , ϕ) for which for i ∈ {1, 2} we have 1. N i P i ≤ G i and 2. P 1 /N 1 ϕ ∼ = P 2 /N 2 . Here, we write G 1 ϕ ∼ = G 2 to denote that G 1 and G 2 are isomorphic via an isomorphism ϕ. The subdirect products correspond to those tuples for which P 1 = G 1 and P 2 = G 2 . Diagonal subgroups are those subdirect products that also satisfy N 1 = N 2 = 1. Subproducts are those for which N 1 = P 1 and N 2 = P 2 . Three factors We now focus on 3-factor subdirect products. Before we investigate the general case, we consider four examples of subdirect products. In our first examples, we consider groups that are 2-factor surjective. We say ∆ ≤ G 1 ×G 2 ×G 3 is 2-factor surjective if ψ i is surjective for all 1 ≤ i ≤ 3. Note that the analogous definition of 1factor surjectivity (i.e., all π i are surjective) means then the same as being subdirect. Similarly, we say ∆ is 2-factor injective if ψ i is injective for all 1 ≤ i ≤ 3. Note that this assumption is equivalent to saying that two components of an element of ∆ determine the third. Analogously 1-factor injective then means that one component determines the other two. Examples of 3-factor direct products Example 3.1. The subgroup of (G 1 ) 3 comprised of the set {(g, g, g) \| g ∈ G 1 )} is called the diagonal subgroup. It is not difficult to see that the only 1-factor injective subdirect products are diagonal subgroups. As a second example, let G 1 be an Abelian group. Then the group Example 3.2. ∆ := {(a, b, c) ∈ (G 1 ) 3 \| abc = 1}. (3.1) is a subdirect product of (G 1 ) 3 that is 2-factor surjective and 2-factor injective. It turns out that this is the only type of group with these properties. Lemma 3.3. Let G = G 1 × G 2 × G 3 be a group and ∆ a subdirect product of G that is 2-factor surjective and 2-factor injective. Then G 1 , G 2 and G 3 are isomorphic Abelian groups and ∆ is isomorphic to the subgroup of G 3 1 given by {(a, b, c) ∈ (G 1 ) 3 \| abc = 1}, which in turn is isomorphic to (G 1 ) 2 as an abstract group. Proof. Let g 1 and g ′ 1 be elements of G 1 . Then by 2-factor surjectivity there are elements g 2 and g 3 such that (g 1 , g 2 , 1) ∈ ∆ and (g ′ 1 , 1, g 3 ) ∈ ∆. We thus have that (g 1 , g 2 , 1) (g ′ 1 ,1,g3) = (g g ′ 1 1 , g 2 , 1). By 2-factor injectivity it follows that g g ′ 1 1 = g 1 and thus G 1 is Abelian. For every g 1 ∈ G 1 there is exactly one element of the form (g 1 , g 2 , 1) in ∆. The map ϕ that sends every g 1 to the corresponding g 2 provides us with a map from G 1 to G 2 . By 2-factor surjectivity and 2-factor injectivity, this map is an isomorphism from G 1 to G 2 . Similarly, G 1 and G 3 are isomorphic. Finally, note that the map that sends (g 1 , g 2 , g 3 ) to (g 1 , ϕ −1 (g 2 ) −1 , g −1 1 ϕ −1 (g 2 )) is an isomor- phism from ∆ to {(a, b, c) ∈ (G 1 ) 3 \| abc = 1}. We now drop the requirement for the group to be 2-factor surjective. Our next examples of 2-factor injective subdirect products will be non-Abelian. Example 3.4. Let G 1 = H ⋊ K be a semidirect product with an Abelian normal subgroup H. Then ∆ = {(ak, bk, ck) ∈ (G 1 ) 3 \| a, b, c ∈ H, k ∈ K, abc = 1}. (3.2) is a 2-factor injective subdirect product of (G 1 ) 3 . To see this we verify that ∆ is closed under multiplication. Let d = (ak, bk, ck), d ′ = (a ′ k ′ , b ′ k ′ , c ′ k ′ ) ∈ ∆. Then dd ′ = (ak, bk, ck)(a ′ k ′ , b ′ k ′ , c ′ k ′ ) = (aka ′ k ′ , bkb ′ k ′ , ckc ′ k ′ ) = (a(ka ′ k −1 )kk ′ , b(kb ′ k −1 )kk ′ , c(kc ′ k −1 )kk ′ ) and a(ka ′ k −1 )b(kb ′ k −1 )c(kc ′ k −1 ) = (abc)k(a ′ b ′ c ′ )k −1 = 1 implying dd ′ ∈ ∆. So ∆ ≤ (G 1 ) 3 . The fact that ∆ is a subdirect product and 2-factor injective follows directly from the definition. Example 3.5. As a next example suppose G 1 = H 2 × H 3 , G 2 = H 1 × H 3 and G 3 = H 1 × H 2 with arbitrary finite groups H i . Then the group consisting of the set {((h 2 , h 3 ), (h 1 , h 3 ), (h 1 , h 2 )) \| h i ∈ H i } is a 2-factor injective subdirect product of G 1 × G 2 × G 3 . Finally, it is not difficult to construct subdirect products that are not 2-factor injective by considering extensions of the factors. Example 3.6. Let ∆ ≤ G 1 × G 2 × G 3 be a subdirect product and let G 1 be a surjective homomorphism κ : G 1 → G 1 . Then {(g 1 , g 2 , g 3 ) ∈ G 1 × G 2 × G 3 \| (κ(g 1 ), g 2 , g 3 ) ∈ ∆} is a subdirect product of G 1 × G 2 × G 3 that is not 2-factor injective if κ is not injective. The structure of subgroups of 3-factor direct products We now analyze the general case, showing that it must essentially be a combination of the examples presented above. We first argue that we can focus our attention on 2-factor injective subdirect products. Lemma 3.7. Let ∆ ≤ G 1 ×G 2 ×G 3 be a subdirect product. Let N i = π i (ker(ψ i )) for i ∈ {1, 2, 3}. Then ∆ ′ = ∆/(N 1 × N 2 × N 3 ) is a 2-factor injective subdirect product and ∆ = {(g 1 , g 2 , g 3 ) \| (g 1 N 1 , g 2 N 2 , g 3 N 3 ) ∈ ∆ ′ }. Thus in the following suppose ∆ is a 2-factor injective subdirect product of G 1 × G 2 × G 3 . Let H i = ker(π i ) ∩ ∆ = {(g 1 , g 2 , g 3 ) ∈ ∆ \| g i = 1}. Then H = H 1 , H 2 , H 3 is a normal subgroup of ∆. Lemma 3.8. For i, j ∈ {1, 2, 3} with i = j we have [H i , H j ] = 1 that is, all elements in H i commute with all elements in H j . Proof. Without loss of generality assume i = 1 and j = 2. For (1, g 2 , g 3 ) ∈ H 1 and (h 1 , 1, h 3 ) in H 2 we get that (1, g 2 , g 3 ) (h1,1,h3) = (1, g 2 , g h3 3 ) . By 2-factor injectivity we conclude that g h3 3 = g 3 and thus the two elements commute. Define M i := π i (H k ) ∩ π i (H j ), where j and k are chosen so that {i, j, k} = {1, 2, 3}. Lemma 3.9. Let i, j, k be integers such that {i, j, k} = {1, 2, 3}. Then there is a canonical isomorphism ϕ := ϕ i j,k from π j (H i ) to π k (H i ) that maps M j to M k . Proof. Assume without loss of generality that i = 1, j = 2 and k = 3. Define a map ϕ : π 2 (H 1 ) → π 3 (H 1 ) such that (1, g 2 , ϕ(g 2 ) −1 ) ∈ ∆ for all g 2 ∈ π 2 (H 1 ) . Such a map exists and is well defined since ∆ is a 2-factor injective subdirect product. Suppose g 2 ∈ M 2 then (1, g 2 , ϕ(g 2 ) −1 ) ∈ ∆ and there is a g 1 such that (g 1 , g 2 , 1) ∈ ∆. Then (1, g 2 , ϕ(g 2 ) −1 )(g 1 , g 2 , 1) −1 = (g −1 1 , 1, ϕ(g 2 ) −1 ) so ϕ(g 2 ) ∈ M 3 . It follows by symmetry that all M i are isomorphic and that ϕ\| M2 is an isomorphism from M 2 to M 3 . Note that the canonical isomorphisms behave well with respect to composition. In particular we have ϕ i j,k = (ϕ i k,j ) −1 and ϕ i j,k \| Mj • ϕ j k,i \| M k = ϕ k j,i \| Mj . This implies for example that the composition of the canonical isomorphism from M 1 to M 2 and the canonical isomorphism from M 2 to M 3 is exactly the canonical isomorphism from M 1 to M 3 . We can thus canonically identify the subgroups M 1 , M 2 and M 3 with a fixed subgroup M . Moreover, we can canonically associate the elements in H 1 with the elements in π 2 (H 1 ) and with the elements in π 3 (H 1 ) by associating (1, g 2 , ϕ(g 2 ) −1 ) ∈ H 1 with the element g 2 ∈ G 2 and ϕ(g 2 ) in G 3 . Similarly we can associate elements (ϕ(g 3 ) −1 , 1, g 3 ) ∈ H 2 with g 3 ∈ G 3 and ϕ(g 3 ) ∈ G 1 and also associate (g 1 , ϕ(g 1 ) −1 , 1) ∈ H 3 with g 1 ∈ G 1 and ϕ(g 1 ) ∈ G 2 . Theorem 1.1 (restated). Let ∆ ≤ G 1 × G 2 × G 3 be{((h 2 , h ′ 3 −1 ), (h 3 , h ′ 1 −1 ), (h 1 , h ′ 2 −1 )) \| h i , h ′ i ∈ H i , h i h ′ i −1 ∈ M, h 1 h ′ 1 −1 h 2 h ′ 2 −1 h 3 h ′ 3 −1 = 1}, by the normal subgroup {((m 1 , m 1 ), (m 2 , m 2 ), (m 3 , m 3 )) \| m i ∈ M }. Proof. As before define H i = ker(π i ) = {(g 1 , g 2 , g 3 ) ∈ ∆ \| g i = 1} and H = H 1 , H 2 , H 3 . By Lemma 3.9 and the comment about the compatibility of the isomorphism between the groups we can canonically associate the elements of M i with those of M j . Moreover we can assume that there is an Abelian group M that is isomorphic to the intersection of every pair of {H 1 , H 2 , H 3 }. All elements of H commute with all elements in such an intersection. If (g 1 , g 2 , g 3 ) is an element of H then g i can be written as c i · b −1 i with c i ∈ C i := π i (H i+1 ) and b i ∈ B i := π i (H i+2 ) , where indices will always be taken modulo 3. We can set (g 1 , g 2 , g 3 ) = (c 1 b −1 1 , c 2 b −1 2 , c 3 b −1 3 ). Since c 1 ∈ π 1 (H 2 ) we conclude that (c −1 1 , 1, ϕ 2 1,3 (c 1 )) ∈ H. We also have (b 1 , ϕ 3 1,2 (b 1 ) −1 , 1) ∈ H. This implies that (g 1 , g 2 , g 3 )(c −1 1 , 1, ϕ 2 1,3 (c 1 ))(b 1 , ϕ 3 1,2 (b 1 −1 ), 1) = (1, c 2 b 2 −1 ϕ 3 1,2 (b 1 −1 ), c 3 b 3 −1 ϕ 2 1,3 (c 1 )) ∈ H. We thus see by looking at the third component that c 3 b 3 −1 ϕ 2 1,3 (c 1 ) ∈ π 3 (H 1 ) which implies that b 3 −1 ϕ 2 1,3 (c 1 ) ∈ π 3 (H 1 ) and thus b 3 −1 ϕ 2 1,3 (c 1 ) ∈ M 3 . By symmetry we conclude that b i+1 −1 ϕ i i−1,i+1 (c i−1 ) ∈ M i+1 for i ∈ {1, 2, 3}. (3.3) Looking at the second component, we also see that c 2 b 2 −1 ϕ 3 1,2 (b 1 −1 ) ∈ π 2 (H 1 ) and conclude that ϕ 1 2,3 (c 2 b 2 −1 ϕ 3 1,2 (b 1 −1 )) −1 = c 3 b 3 −1 ϕ 2 1,3 (c 1 ). Recalling that H i and H j commute for i = j, we thus conclude that c 3 ϕ 1 2,3 (b 2 ) −1 · ϕ 2 1,3 (c 1 )b 3 −1 · ϕ 1 2,3 (c 2 ϕ 3 1,2 (b 1 −1 )) = 1. (3.4) Thus, since all involved isomorphisms are compatible we can reinterpret this equation in M and we see that c 3 (b 2 ) −1 · c 1 (b 3 ) −1 · c 2 (b 1 ) −1 = 1. Suppose that (g 1 , g 2 , g 3 ) = ( c 1 b 1 −1 , c 2 b 2 −1 , c 3 b 3 −1 ) is a second representation of the element (g 1 , g 2 , g 3 ) of H. Then we have for the first component that c 1 b 1 −1 = c 1 b 1 −1 and thus c −1 1 c 1 = b 1 −1 b 1 ∈ M 1 or, in other words, c 1 m 1 = c 1 and b 1 −1 m 1 = b −1 1 for some element m 1 ∈ M . Similarly there are elements m 2 and m 3 for the other components. We conclude that the map sending (g 1 , g 2 , g 3 ) to ((c 1 , b 1 −1 ), (c 2 , b 2 −1 ), (c 3 , b 3 −1 ) ) is a homomorphism from H to the group described in the theorem. It remains to show injectivity of this homomorphism. Suppose (g 1 , g 2 , g 3 ) is mapped to the trivial element. This implies for the first component of the image ((c 1 , b 1 −1 ), (c 2 , b 2 −1 ), (c 3 , b 3 −1 )) that (c 1 , b 1 −1 ) = (m 1 , m 1 ) for some m 1 implying that g 1 = m 1 m 1 −1 = 1. Repeating the same argument for the other components we see that the homomorphism is an isomorphism. Correspondence theorems We now investigate the possibility of having a correspondence theorem in the style of Theorem 2.2 for 3 factors. As before, we can readily reduce to the case of 2-factor injective subdirect products. Lemma 3.10. There is a natural one-to-one correspondence between subdirect products of G 1 × G 2 × G 3 and the tuples (N 1 , N 2 , N 3 , ∆ ′ ), where N i G i for every i ∈ {1, 2, 3} and ∆ ′ is a 2-factor injective subdirect product of G 1 /N 1 × G 2 /N 2 × G 3 /N 3 . Proof. Let ∆ ≤ G 1 × G 2 × G 3 be a subdirect product. Choose N i = π i (ker(ψ i )) for i ∈ {1, 2, 3} and let ∆ ′ = ∆/(N 1 × N 2 × N 3 ). Then N i G i , because ∆ is a subdirect product, and ∆ ′ is 2-factor injective by Lemma 3.7. Conversely, let N i G i and let ∆ ′ be a 2-factor injective subdirect product of G 1 /N 1 ×G 2 /N 2 × G 3 /N 3 . Then ∆ = {(g 1 , g 2 , g 3 ) \| (g 1 N 1 , g 2 N 2 , g 3 N 3 ) ∈ ∆ ′ } is subdirect product of G 1 × G 2 × G 3 . Suppose ∆ is a 2-factor injective subdirect product. Then we can define for i ∈ {1, 2, 3} the groups H i = {(g 1 , g 2 , g 3 ) ∈ ∆ \| g i = 1} and with them the groups B i = π i (H i+2 ) and C i = π i (H i+1 ). As we will see, the canonical isomorphism ϕ i := ϕ i+2 i,i+1 that exists by Lemma 3.9 can be extended to an isomorphism from G i /C i to G i+1 /B i+1 . We would like to have a correspondence theorem in the style of Theorem 2.2 for 3 factors. For this, in principle, we would like to relate the 2-factor injective subdirect products of G 1 , G 2 , G 3 to the tuples (B 1 , B 2 , B 3 , C 1 , C 2 , C 3 , ϕ 1 , ϕ 2 , ϕ 3 ) that satisfy certain consistency properties. However, in general, neither is it clear which consistency properties to choose so that every tuple corresponds to a subdirect product, nor do distinct subdirect products always correspond to distinct tuples. Indeed, in [2] the authors describe two distinct Abelian subdirect products of the same group G 1 × G 2 × G 3 for which the corresponding tuples agree. In the light of that we content ourselves with study two special cases, namely those where ∆ = H and those where M i = B i ∩ C i = 1. Theorem 3.11. There is a natural one-to-one correspondence between subdirect products of ∆ ≤ G 1 × G 2 × G 3 which are 2-factor injective satisfying ∆ = H, and tuples of the form (B 1 , B 2 , B 3 , C 1 , C 2 , C 3 , ϕ 1 , ϕ 2 , ϕ 3 ) for which for all i ∈ {1, 2, 3} (indices taken modulo 3) we have 1. B i , C i G i , 2. B i C i = G i , 3. B i ϕi ∼ = C i+1 , 4. [B i , C i ] = 1, 5. ϕ i (B i ∩ C i ) = B i+1 ∩ C i+1 , 6. ϕ 3 \| B3∩C3 • ϕ 2 \| B2∩C2 • ϕ 1 \| B1∩C1 = id. Proof. Let ∆ ≤ G 1 × G 2 × G 3 be a subdirect product. Define H i := {(g 1 , g 2 , g 3 ) ∈ ∆ \| g i = 1} for i ∈ {1, 2, 3} and H = H 1 , H 2 , H 3 . Suppose that H = ∆. For i ∈ {1, 2, 3}, define B i = π i (H i+2 ) and C i = π i (H i+1 ). Clearly B i , C i G i . By Lemma 3.8 we get that [B i , C i ] = 1 and B i C i = π i (H). The assumption ∆ = H implies B i C i = G i . By Lemma 3.9 the groups B i and C i+1 are isomorphic via an isomorphism ϕ i = ϕ i+2 i,i+1 , which maps M i = B i ∩ C i to M i+1 = B i+1 ∩ C i+1 . Finally Property 6 follows directly from the comment below Lemma 3.9. This gives us the tuple (B 1 , B 2 , B 3 , C 1 , C 2 , C 3 , ϕ 1 , ϕ 2 , ϕ 3 ) with the desired properties. On the other hand suppose we are given a tuple (B 1 , B 2 , B 3 , C 1 , C 2 , C 3 , ϕ 1 , ϕ 2 , ϕ 3 ) with the desired properties. Let M i = B i ∩ C i . Define ∆ = (g 1 , g 2 , g 3 ) ∈ G 1 × G 2 × G 3 g i = c i b −1 i for b i ∈ B i , c i ∈ C i , c i+1 M i+1 = ϕ i (b i M i ) and c 3 ϕ 2 (b 2 ) −1 · ϕ −1 3 (c 1 )b 3 −1 · ϕ 2 (c 2 ϕ 1 (b 1 −1 )) = 1 . For i ∈ {1, 2, 3} suppose g i = c i b −1 i = c ′ i b ′−1 i . Then there is some m i ∈ M i with b i = b ′ i m i and c i = c ′ i m i . Hence, b i M i = b ′ i M i and c i M i = c ′ i M i . Furthermore, we have c ′ 3 ϕ 2 (b ′ 2 ) −1 · ϕ −1 3 (c ′ 1 )b ′ 3 −1 · ϕ 2 (c ′ 2 ϕ 1 (b ′ 1 −1 )) = c 3 m −1 3 ϕ 2 (b 2 m −1 2 ) −1 · ϕ −1 3 (c 1 m −1 1 )(b 3 m −1 3 ) −1 · ϕ 2 (c 2 m −1 2 ϕ 1 ((b 1 m −1 1 ) −1 )) = c 3 ϕ 2 (b 2 ) −1 · ϕ −1 3 (c 1 )b 3 −1 · ϕ 2 (c 2 ϕ 1 (b 1 −1 )), so the membership in ∆ is independent from the representation of g i ∈ G i . Also, it is easy to check that ∆ is closed under multiplication because [B i , C i ] = 1. The group ∆ is a subdirect product, since for g 1 = c 1 b −1 1 we have (c 1 b −1 1 , ϕ 1 (b −1 1 ) −1 b −1 2 , ϕ −1 3 (c 1 )ϕ 2 (b 2 ) ) ∈ ∆, and it can be checked that the group is 2-factor injective. Define H i = {(g 1 , g 2 , g 3 ) ∈ ∆ \| g i = 1} for i ∈ {1, 2, 3} and H = H 1 , H 2 , H 3 . Then B i = π i (H i+2 ) and C i = π i (H i+1 ), which means that H = ∆ by Property 2 and Theorem 1.1. Finally, it can be checked that ϕ i = ϕ i+2 i,i+1 . It remains to show that for each 2-factor injective subdirect product ∆ ≤ G 1 × G 2 × G 3 with ∆ = H, the group ∆ is of the form described above. But this follows from Theorem 1.1, Equation (3.3) and (3.4). The previous theorem shows that for the subdirect products with ∆ = H we can devise a correspondence theorem. As a second case, on the other end of the spectrum, we can also devise a correspondence theorem if the Abelian part that interlinks the three components is trivial. In fact this case corresponds to the case discussed by Bauer, Sen, and Zvengrowski [2, 5.1 Remark], who already suspect that a theorem like the previous can be obtained. We remark that Example 3.5 from the previous section is of this form. In fact, we can already conclude from Theorem 3.11 that for every group ∆, where the Abelian part interlinking the components is trivial, the group H essentially has the form of Example 3.5. Definition 3.12. Let ∆ be a subdirect product of G 1 × G 2 × G 3 . We say ∆ is degenerate if π i (ker(π i+1 )) ∩ π i (ker(π i+2 )) = π i (ker(ψ i )) (i.e. M i = 1) for some, and thus every, i ∈ {1, 2, 3}. Lemma 3.13. For i ∈ {1, 2, 3} let B i , C i G i , such that B i ∩ C i = 1 and [B i , C i ] = 1. Further- more let G i /C i ϕi ∼ = G i+1 /B i+1 and suppose ϕ i (B i C i ) = C i+1 B i+1 and ϕ 3 (ϕ 2 (ϕ 1 (g 1 B 1 C 1 ))) = g 1 B 1 C 1 for all g 1 ∈ G 1 . Define ∆ = {(g 1 , g 2 , g 3 ) ∈ G 1 × G 2 × G 3 \| ϕ i (g i C i ) = g i+1 B i+1 }. (3.5) Then π i (∆) = G i and ∆ is a degenerate 2-factor injective subdirect product. Proof. We first show, that ∆ is closed under multiplication. Let (g 1 , g 2 , g 3 ), (g ′ 1 , g ′ 2 , g ′ 3 ) ∈ ∆. Then ϕ i (g i g ′ i C i ) = ϕ i (g i C i )ϕ i (g ′ i C i ) = g i+1 g ′ i+1 B i+1 for all i ∈ {1, 2, 3}, so (g 1 g ′ 1 , g 2 g ′ 2 , g 3 g ′ 3 ) ∈ ∆. Let E 1 = B 1 , C 1 and pick e 1 ∈ E 1 . The element e 1 can uniquely be written as e 1 = b 1 c 1 with b 1 ∈ B 1 , c 1 ∈ C 1 . For each i ∈ {1, 2, 3} define ϕ * i : B i → C i+1 with ϕ * i (b i ) = c i+1 for the unique c i+1 ∈ C i+1 with ϕ i (b i C i ) = c i+1 B i+1 . Then (b 1 c 1 , b 2 ϕ * 1 (b 1 ), (ϕ * 3 ) −1 (c 1 )ϕ * 2 (b 2 )) ∈ ∆. So E 1 ≤ π 1 (∆). The argument for the other components is analogous. Now let n 1 1 , . . . , n 1 t be a transversal of E 1 in G 1 . Let n 2 i B 2 = ϕ 1 (n 1 i C 1 ) and n 3 i C 3 = ϕ −1 3 (n 1 i B 1 ) for i ∈ {1, . . . , t}. Then ϕ 2 (n 2 i C 2 ) ⊆ n 3 i E 3 and hence there is some b 2 i ∈ B 2 with ϕ 2 (n 2 i b 2 i C 2 ) = n 3 1 B 3 . So (n 1 i , n 2 i b 2 i , n 3 i ) ∈ ∆ and G 1 ≤ π 1 (∆). It remains to prove that ∆ is 2-factor injective. Let (g 1 , g 2 , g 3 ) ∈ ∆ with g 2 = g 3 = 1. Then g 1 ∈ B 1 , because ϕ 3 (C 3 ) = B 1 = g 1 B 1 , and g 1 ∈ C 1 , because ϕ 1 (g 1 C 1 ) = g 2 B 2 = B 2 . So g 1 = 1. Again, the argument for the other components is analogous. Lemma 3.14. Let ∆ be a 2-factor injective subdirect product of G 1 × G 2 × G 3 . Furthermore, let H i = {(g 1 , g 2 , g 3 ) ∈ ∆ \| g i = 1} for i ∈ {1, 2, 3} and H = H 1 , H 2 , H 3 . Define B i = π i (H i+2 ) and C i = π i (H i+1 ). Suppose B i ∩ C i = 1 for all i ∈ {1, 2, 3}. Then there are canonical isomorphisms ϕ 1 , ϕ 2 , ϕ 3 with G i /C i ϕi ∼ = G i+1 /B i+1 and ϕ i (B i C i ) = C i+1 B i+1 such that ϕ 3 (ϕ 2 (ϕ 1 (g 1 B 1 C 1 ))) = g 1 B 1 C 1 for all g 1 ∈ G 1 . Furthermore ∆ is given by Equation (3.5). Proof. For i ∈ {1, 2, 3} define a homomorphism ϕ i : G i /C i → G i+1 /B i+1 by setting ϕ i (g i C i ) = g i+1 B i+1 if (g 1 , g 2 , g 3 ) ∈ ∆ for some g i ∈ G i . We first have to show that ϕ i is well-defined. Without loss of generality consider i = 1 and let (g 1 , g 2 , g 3 ), (g ′ 1 , g ′ 2 , g ′ 3 ) ∈ ∆ with g 1 C 1 = g ′ 1 C 1 . Then there is a (c, 1, h 2 ) ∈ ∆ with g ′ 1 c = g 1 . We obtain (g ′ 1 , g ′ 2 , g ′ 3 )(c, 1, h 2 )(g 1 , g 2 , g 3 ) −1 = (1, g ′ 2 g −1 2 , g ′′ 3 ) for some g ′′ 3 ∈ G 3 and hence, g 2 B 2 = g ′ 2 B 2 . So ϕ i is well-defined. Since ∆ is a subdirect product, ϕ i is a surjective homomorphism. Suppose ϕ 1 (g 1 C 1 ) = B 2 . Then (g 1 c 1 , b 2 , g 3 ) ∈ ∆ for some c 1 ∈ C 1 , b 2 ∈ B 2 and g 3 ∈ G 3 . Also there is h 3 ∈ G 3 with (1, b 2 , h 3 ) ∈ ∆ and hence, (g 1 c 1 , 1, g 3 h −1 3 ) ∈ ∆ implying that g 1 ∈ C 1 . So G i /C i ϕi ∼ = G i+1 /B i+1 . For every b 1 ∈ B 1 there is a c 2 ∈ C 2 with (b 1 , c 2 , 1) ∈ ∆ and ϕ 1 (b 1 C 1 ) = c 2 B 2 ∈ C 2 B 2 . By symmetry it follows that ϕ i (B i C i ) = C i+1 B i+1 for all i ∈ {1, 2, 3}. Now let ∆ ′ be the group defined in Equation (3.5). Clearly ∆ ≤ ∆ ′ by the definition of ϕ i for i ∈ {1, 2, 3}. So let (g ′ 1 , g ′ 2 , g ′ 3 ) ∈ ∆ ′ . Since ∆ is subdirect there is a (g ′ 1 , g 2 , g 3 ) ∈ ∆ with g 2 B 2 = g ′ 2 B 2 . So we can assume that g 2 = g ′ 2 . But then, by 2-factor injectivity of ∆ ′ , we get that g 3 = g ′ 3 . Finally for (g 1 , g 2 , g 3 ) ∈ ∆ we have that ϕ i (g i B i C i ) = ϕ i (g i C i )ϕ i (B i C i ) = g i+1 C i+1 B i+1 = g i+1 B i+1 C i+1 . So ϕ 2 (ϕ 1 (g 1 B 1 C 1 )) = ϕ −1 3 (g 1 B 1 C 1 ) for all g 1 ∈ G 1 . Theorem 3.15. There is a natural one-to-one correspondence between degenerate 2-factor injective subdirect products of G 1 × G 2 × G 3 and the tuples (B 1 , B 2 , B 3 , C 1 , C 2 , C 3 , ϕ 1 , ϕ 2 , ϕ 3 ) for which for all i ∈ {1, 2, 3} (indices taken modulo 3) we have Theorem 3.17. Suppose G 1 = E 1 ⋊ K is a semidirect product. There is an injective mapping from the set of 2-factor injective subdirect products ∆ of G 1 × G 2 × G 3 with π 1 (H) = E 1 and B 1 = C 1 to the tuples (κ, ι) where G 2 κ ∼ = G 3 and ι is an automorphism of G 1 that fixes K as a set. Moreover if (κ, ι) is in the image of this mapping then (κ, ι ′ ) is also in the image for every automorphism ι ′ of G 1 that fixes K as a set. Proof. Let ∆ be a 2-factor injective subdirect product of G 1 × G 2 × G 3 satisfying π 1 (H) = E 1 and B 1 = C 1 . For every element g 2 ∈ G 2 there is exactly one element (k 1 , g 2 , g 3 ) ∈ ∆ with k 1 ∈ K. We obtain a well defined isomorphism κ from G 2 to G 3 . Suppose now that ∆ ′ is a second 2-factor injective subdirect product G 1 × G 2 × G 3 satisfying π 1 (H) = E 1 and B 1 = C 1 for which we obtain the same isomorphism κ. Then we can construct an automorphism ι of G 1 as follows. For every g 2 ∈ G 2 there is exactly one (k, g 2 , κ(g 2 )) ∈ ∆. There is also an element of the form (k ′ , g 2 , κ(g 2 )) ∈ ∆ ′ . We define ι K : K → K so that it maps k to k ′ , this gives us an automorphism ι K of K. For every e ∈ E 1 there is an element (e, g 2 , 1) ∈ ∆. There is also an element (e ′ , g 2 , 1) ∈ ∆ ′ and define the map ι E : E 1 → E 1 by mapping e to e ′ . Then the map ι E is an automorphism of E. We claim that the map that sends e · k to ι E (e) · ι K (k) is an automorphism of G 1 . To see this suppose a = (e 1 , h, 1)(k, g, κ(g)) and a = (e 1 , h, 1)(k, g, κ(g)) are two elements in ∆. Then ι(a) = (ι E (e 1 ), h, 1)(ι K (k), g, κ(g)) and ι(a) = (ι E (e 1 ), h, 1)(ι K (k), g, κ(g)) are elements of ∆ ′ . For the products we obtain that aa = (e 1 e 1 k , hh g , 1)(kk, gg, κ(g)κ(g)) and ι(a)ι(a) = (ι E (e 1 )ι E (e 1 ) ιK (k) , hh g , 1)(ι K (k)ι K (k), gg, κ(g)κ(g)). To conclude that ι is an isomorphism we now only need to argue that ι E (e 1 ) ιK (k) is equal to ι E (e 1 k ). However, this is the case since (e 1 k , h g , 1) ∈ ∆ and (ι E (e 1 ) ιK (k) , h g , 1) ∈ ∆ ′ . Now suppose that ∆ is a subdirect product with π 1 (H) = B 1 = C 1 and let κ : G 2 → G 3 be defined as above. Let ι be an automorphism of G 1 that fixes K then ∆ ′ = {(ι(g 1 ), g 2 , g 3 ) \| (g 1 , g 2 , g 3 ) ∈ ∆} is a subdirect product of G 1 × G 2 × G 3 . If we apply the above construction for the automorphism of G 1 we reobtain ι. This shows that the construction of ι from ∆ ′ and the construction of ∆ ′ from ι are inverses to one another. Note that in the theorem, the isomorphism κ associated with a subdirect product is canonical (it only depends on the choice of K) but the choice of ι is not. Subdirect products of two or three symmetric groups In this section we apply the correspondence theorems to count the subdirect products of the direct product of three symmetric groups. We first reduce this problem to counting the number of 2factor injective subdirect products. For finite groups G 1 , . . . , G k let ℓ(G 1 , . . . , G k ) be the number of subdirect products of G 1 × · · · × G k . Furthermore, for k = 3, we denote by ℓ 2-inj (G 1 , . . . , G 3 ) the number of 2-factor injective subdirect products. Lemma 4.1. Let G 1 , G 2 , G 3 be finite non-trivial groups. Then ℓ(G 1 , G 2 , G 3 ) = Ni Gi ℓ 2-inj (G 1 /N 1 , G 2 /N 2 , G 3 /N 3 ) = ℓ(G 1 , G 2 ) + ℓ(G 2 , G 3 ) + ℓ(G 1 , G 3 ) − 2 + Ni⊳Gi ℓ 2-inj (G 1 /N 1 , G 2 /N 2 , G 3 /N 3 ). Proof. The first equality follows from the correspondence described in Lemma 3.10. The second equality follows by noting that the direct product is counted three times, so 2 has to be subtracted. We are now interested in the number ℓ(n 1 , n 2 , n 3 ) := ℓ(S n1 , S n2 , S n3 ), where S ni is the symmetric group of a set with n i elements. Recall, that every factor group of a symmetric group is isomorphic to a symmetric group over another set. Thus, by the previous lemma, it suffices to compute the numbers ℓ(n 1 , n 2 ) and ℓ 2-inj (n 1 , n 2 , n 3 ) := ℓ 2-inj (S n1 , S n2 , S n3 ). We start by analyzing the situation for two factors. if n 1 = n 2 = 2, n 1 ! + 8 if n 1 = n 2 = 4, 2n 1 ! + 2 if n 1 = n 2 = 6. Proof. We assume for our considerations that n 1 ≥ n 2 . Let (S n1 , S n2 , N 1 , N 2 , ϕ) be a tuple corresponding to a subdirect product via the correspondence of Theorem 2.2. • If N 1 = 1 then N 2 = 1 and n 1 = n 2 . The number of isomorphisms from S n1 to S n1 is i(n 1 ) =      1 if n 1 = 2 2n! if n 1 = 6 n! otherwise . This corresponds to the number of possible choices for ϕ. (These are the diagonal subgroups.) • If N 1 = S n1 then N 2 = S n2 . There is only one subgroup of this type. (This is the direct product). • If N 1 = A n1 (n 1 ≥ 3) then N 2 = A n2 , since the only index 2 subgroup that a symmetric group can have is the alternating group. There is only one subgroup of this type. If n 1 = 4 then N 1 ∈ {1, A n1 , S n1 }, and we already considered all these cases. Suppose now that n 1 = 4 and n 2 ≤ 4. Then N 1 ∈ {1, V, A n1 , S n1 }, where V is the Klein-four-group. Three of cases are considered above. • If N 1 = V then N 2 = V and n 2 = 4 or N 2 = 1 and n 2 = 3. In either case there are 6 options for ϕ. In the following we use "{ {" and "} }" to denote multisets. Lemma 4.3. Let n 1 , n 2 , n 3 ≥ 2. For the number ℓ 2-inj (n 1 , n 2 , n 3 ) of 2-factor injective subdirect products of S n1 × S n2 × S n3 we have ℓ 2-inj (n 1 , n 2 , n 3 ) =                                  (n 1 !) 2 if n 1 = n 2 = n 3 / ∈ {2, 3, 4, 6}, 2 if n 1 = n 2 = n 3 = 2, (n 1 !) 2 + 2n 1 ! if n 1 = n 2 = n 3 = 3, (n 1 !) 2 + 6n 1 ! if n 1 = n 2 = n 3 = 4, (2n 1 !) 2 if n 1 = n 2 = n 3 = 6, 1440 if { {n 1 , n 2 , n 3 } } = { {2, 6, 6} } n! if { {n 1 , n 2 , n 3 } } = { {2, n, n} } for n / ∈ {2, 6}, 144 if { {n 1 , n 2 , n 3 } } = { {3, 4, 4} } 0 otherwise. Proof. Suppose without loss of generality that n 1 ≥ n 2 ≥ n 3 . Let ∆ ≤ S n1 × S n2 × S n3 be a 2-factor injective subdirect product. , B 3 , C 1 , C 2 , C 3 , ϕ 1 , ϕ 2 , ϕ 3 ) with subgroups B i , C i E i , such that B i ∩ C i is Abelian and B i C i = E i . We obtain the following options. • If E 1 = 1 then n 1 = n 2 = n 3 and E 2 = E 3 = 1. We have B i = C i = 1 for i ∈ {1, 2, 3} In this case H = 1 and ∆ is degenerate. By Theorem 3.15 there are i(n 1 ) 2 groups of this type, where i(n 1 ) is the number of isomorphisms from S n1 to S n1 . This corresponds to the choices for ϕ 1 and ϕ 2 . For ϕ 3 we get ϕ −1 3 = ϕ 1 • ϕ 2 . • If E 1 = V then n 1 = 4 and S n1 /E 1 ∼ = S 3 . In this case {n 1 , n 2 , n 3 } ⊆ {3, 4}. Let us first consider the case that n 3 = 3. Then B 3 = C 3 = E 3 = 1 and we can thus apply Theorem 3.15. By Lemma 3.9 we conclude that C 1 = B 2 = 1 and E 1 = B 1 = C 2 = V . Using the correspondence given in Theorem 3.15 the number of such groups equals the number of pairs (ϕ 1 , ϕ 2 ), where ϕ 1 is an isomorphism from S 4 to S 4 and ϕ 2 is an isomorphism from S 3 to S 3 . There are 144 such pairs. Next let us consider the case n 1 = n 2 = n 3 = 4. This implies that B i = C i = E i = V for all i ∈ {1, 2, 3}. Since S 4 is the semidirect product V ⋊ S 3 , we can apply Theorem 3.17. For every isomorphism κ from G 2 to G 3 we can find a subdirect product realizing κ by setting ∆ = {k, g 1 , κ(g 1 ) \| g 1 ∈ G 1 , k ∈ S 3 ∩ g 1 V } ∪ {(a, a −1 , 1) \| a ∈ V } . Thus, by Theorem 3.17 the number of such subdirect products is equal to the number pairs (κ, ι) where κ is an isomorphism from S 4 to S 4 and ι is an automorphism of S 4 that fixes V as a set. There are 6n 1 ! = 144 such pairs. • If E 1 = A n1 (and n 1 ≥ 3) then E 2 = A n2 and E 3 = A n3 . By applying Theorem 3.11 to H it follows that either n 1 = n 2 = n 3 = 3 or n 1 = n 2 > n 3 = 2. In the former case B i = C i = A 3 = Z 3 for i ∈ {1, 2, 3} and in total there are 2n 1 ! = 12 groups of this type by Theorem 3.17 (by the same arguments as in the previous case). In the latter case B 1 = C 2 = A n1 and C 1 = B 2 = B 3 = C 3 = 1. So ∆ is degenerate and by Theorem 3.15 there are in total i(n 1 ) · i(2) = i(n 1 ) groups of this type, where again i(n 1 ) is the number of isomorphisms from S n1 to S n1 . n 1 = 4 4 3 2 1 4 1386 282 66 32 3 282 90 18 8 2 66 18 6 2 1 32 8 2 1 n 1 = 3 3 2 1 3 90 18 8 2 18 6 2 1 8 2 1 n 1 = 2 2 1 2 6 2 1 2 1 Table 1: The numbers ℓ(n 1 , n 2 , n 3 ) for n 1 , n 2 , n 3 ∈ {1, 2, 3, 4} • If E 1 = S n1 then E 2 = S n2 and E 3 = S n3 . In this case H = ∆. Using the correspondence described in Theorem 3.11 we get that n 1 = n 2 = n 3 = 2 and B i = C i = S 2 for i ∈ {1, 2, 3}. Since there is only one isomorphism from S 2 to S 2 there is exactly one option in this case, namely the group given in Example 3.2 for G 1 = S 2 . The finitely many cases not covered by the previous corollary are listed in Table 1. These numbers were calculated using the Lemmas 4.1, 4.2 and 4.3. However, these numbers were also double-checked with the computer algebra system gap [6]. a 2-factor injective subdirect product. Then there is a normal subgroup of H ∆ with [π i (∆) : π i (H)] = [∆ : H] for i ∈ {1, 2, 3} and H is isomorphic to a group of the following form: there are three finite groups H 1 , H 2 , H 3 that all have an Abelian subgroup M contained in their the center such that H is isomorphic to the factor group of Lemma 4. 2 . 2Let n 1 , n 2 ≥ 2. For the number ℓ(n 1 , n 2 ) of subdirect products of S n1 × S n2 we have n 1 = n 2 and {n 1 , n 2 } = {3, 4}, 8 if {n 1 , n 2 } = {3, 4}, n 1 ! + 2 if n 1 = n 2 / ∈ {2, 4, 6}, 2 Corollary 4. 4 . 4Let n 1 ≥ n 2 ≥ n 3 ≥ 2, n 1 ≥ 5. For the number ℓ(n 1 , n 2 , n 3 ) of subdirect products of S n1 × S n2 × S n3 we haveℓ(n 1 , n 2 , n 1 !) 2 + 6n 1 ! + 6 if n 1 = n 2 = n 3 / ∈ {6}, 2082246 if n 1 = n 2 = n 3 = 6, 66 if n 2 = n 3 = 4, 18 if n 2 ∈ {3, 4}, n 3 = 3, 2886 if { {n 1 , n 2 , n 3 } } = { {6, 6, m 2 } }, m 2 = 6, 2m 1 ! + 6 if { {n 1 , n 2 , n 3 } } = { {m 1 , m 1 , m 2 } }, m 1 = m 2 , 6= m 1 inj (n 1 , n 1 , n 1 ) + 3ℓ 2-inj (2, n 1 , n 1 ) if n 1 = n 2 = n 3 , ℓ 2-inj (2, 4, 4) + ℓ 2-inj(2,3,3) if n 2 = n 3 = 4, n 2 ∈ {3, 4}, n 3 = 3, ℓ 2-inj (2, m 1 , m 1 ) if { {n 1 , n 2 , n 3 } } = { {m 1 , m 1 , m 2 } } for m 1 = m 2 , m 1 Define H 1 , H 2 , H 3 , H as in Section 3.2. Let E i = π i (H). Then H ∆ and E i S ni for i ∈ {1, 2, 3}. By Theorem 1.1, it holds that S n1 /E 1 ∼ = S n2 /E 2 ∼ = S n3 /E 3 . Also, by Theorem 3.11, for H we get a canonical tuple (B 1 , B 2 Acknowledgments. Funded by the Excellence Initiative of the German federal and state governments. By combining Theorem 3.15 with Lemma 3.10 we obtain the following correspondence result for degenerate subdirect products. By combining Theorem 3.15 with Lemma 3.10 we obtain the following correspondence result for degenerate subdirect products. ) for which for all i ∈ {1, 2, 3} (indices taken modulo 3) we have 1. 3N 1 , N 2 , N 3 , B 1 , B 2 , B 3 , C 1 , C 2 , C; N i G i , 2. B i , C i G i /N i , 3. B i ∩ C i = 1, 4Corollary 3.16. There is a natural one-to-one correspondence between degenerate subdirect products of G 1 × G 2 × G 3 and the tuples. B i , C i ] = 1, 5. (G i /N i )/C i ϕi ∼ = (G i+1 /N i+1 )/B i+1Corollary 3.16. There is a natural one-to-one correspondence between degenerate subdirect products of G 1 × G 2 × G 3 and the tuples (N 1 , N 2 , N 3 , B 1 , B 2 , B 3 , C 1 , C 2 , C 3 , ϕ 1 , ϕ 2 , ϕ 3 ) for which for all i ∈ {1, 2, 3} (indices taken modulo 3) we have 1. N i G i , 2. B i , C i G i /N i , 3. B i ∩ C i = 1, 4. [B i , C i ] = 1, 5. (G i /N i )/C i ϕi ∼ = (G i+1 /N i+1 )/B i+1 , Example 3.4 described in the previous section is of this form. For this case we only obtain an injection to tuples. For Some I ∈ {1, 23rather than a one-to-one correspondence. We will exploit havingfor some i ∈ {1, 2, 3}. Example 3.4 described in the previous section is of this form. For this case we only obtain an injection to tuples, rather than a one-to-one correspondence. We will exploit having Subgroups of direct products of groups, ideals and subrings of direct products of rings, and Goursat's lemma. D D Anderson, V Camillo, Rings, modules and representations. Providence, RIAmer. Math. Soc480D. D. Anderson and V. Camillo. Subgroups of direct products of groups, ideals and subrings of direct products of rings, and Goursat's lemma. In Rings, modules and representations, volume 480 of Contemp. Math., pages 1-12. Amer. Math. Soc., Providence, RI, 2009. A generalized Goursat lemma. K Bauer, D Sen, P Zvengrowski, Tatra Mt. Math. Publ. 64K. Bauer, D. Sen, and P. Zvengrowski. A generalized Goursat lemma. Tatra Mt. Math. Publ., 64, 2015. On the finite presentation of subdirect products and the nature of residually free groups. M R Bridson, J Howie, C F Miller, H Iii, Short, Amer. J. Math. 1354M. R. Bridson, J. Howie, C. F. Miller, III, and H. Short. On the finite presentation of subdirect products and the nature of residually free groups. Amer. J. Math., 135(4):891- 933, 2013. Structure and finiteness properties of subdirect products of groups. M R Bridson, C F Miller, Iii , Proc. Lond. Math. Soc. 983M. R. Bridson and C. F. Miller, III. Structure and finiteness properties of subdirect products of groups. Proc. Lond. Math. Soc. (3), 98(3):631-651, 2009. Subgroup lattices and symmetric functions. L M Butler, Mem. Amer. Math. Soc. 112539160L. M. Butler. Subgroup lattices and symmetric functions. Mem. Amer. Math. Soc., 112(539):vi+160, 1994. Gap The, Group, GAP -Groups, Algorithms, and Programming. Version 4.8.3The GAP Group. GAP -Groups, Algorithms, and Programming, Version 4.8.3, 2016. Sur les substitutions orthogonales et les divisions régulières de l'espace. E Goursat, Ann. Sci. École Norm. Sup. 63E. Goursat. Sur les substitutions orthogonales et les divisions régulières de l'espace. Ann. Sci. École Norm. Sup. (3), 6:9-102, 1889. The theory of groups. M HallJr, Chelsea Publishing CoNew YorkReprinting of the 1968 editionM. Hall, Jr. The theory of groups. Chelsea Publishing Co., New York, 1976. Reprinting of the 1968 edition. On the subgroups of finite abelian groups of rank three. M Hampejs, L Tóth, Ann. Univ. Sci. Budapest. Sect. Comput. 39M. Hampejs and L. Tóth. On the subgroups of finite abelian groups of rank three. Ann. Univ. Sci. Budapest. Sect. Comput., 39:111-124, 2013. . S Lang, Algebra, Graduate Texts in Mathematics. 211Springer-Verlagthird editionS. Lang. Algebra, volume 211 of Graduate Texts in Mathematics. Springer-Verlag, New York, third edition, 2002. On the lattice of normal subgroups of a direct product. M D Miller, Pacific J. Math. 602M. D. Miller. On the lattice of normal subgroups of a direct product. Pacific J. Math., 60(2):153-158, 1975. Counting subgroups in a direct product of finite cyclic groups. J Petrillo, College Math. J. 423J. Petrillo. Counting subgroups in a direct product of finite cyclic groups. College Math. J., 42(3):215-222, 2011. Über Unterguppen direkter Produkte von drei Faktoren. R Remak, J. Reine Angew. Math. 166R. Remak. Über Unterguppen direkter Produkte von drei Faktoren. J. Reine Angew. Math., 166:65-100, 1932.	[]
[ "State-Based Confidence Bounds for Data-Driven Stochastic Reachability Using Hilbert Space Embeddings ⋆", "State-Based Confidence Bounds for Data-Driven Stochastic Reachability Using Hilbert Space Embeddings ⋆", "State-Based Confidence Bounds for Data-Driven Stochastic Reachability Using Hilbert Space Embeddings ⋆", "State-Based Confidence Bounds for Data-Driven Stochastic Reachability Using Hilbert Space Embeddings ⋆" ]	[ "Adam J Thorpe \nElectrical and Computer Engineering\nUniversity of New Mexico\nAlbuquerqueNMUSA\n", "Kendric R Ortiz \nElectrical and Computer Engineering\nUniversity of New Mexico\nAlbuquerqueNMUSA\n", "Meeko M K Oishi \nElectrical and Computer Engineering\nUniversity of New Mexico\nAlbuquerqueNMUSA\n", "Adam J Thorpe \nElectrical and Computer Engineering\nUniversity of New Mexico\nAlbuquerqueNMUSA\n", "Kendric R Ortiz \nElectrical and Computer Engineering\nUniversity of New Mexico\nAlbuquerqueNMUSA\n", "Meeko M K Oishi \nElectrical and Computer Engineering\nUniversity of New Mexico\nAlbuquerqueNMUSA\n" ]	[ "Electrical and Computer Engineering\nUniversity of New Mexico\nAlbuquerqueNMUSA", "Electrical and Computer Engineering\nUniversity of New Mexico\nAlbuquerqueNMUSA", "Electrical and Computer Engineering\nUniversity of New Mexico\nAlbuquerqueNMUSA", "Electrical and Computer Engineering\nUniversity of New Mexico\nAlbuquerqueNMUSA", "Electrical and Computer Engineering\nUniversity of New Mexico\nAlbuquerqueNMUSA", "Electrical and Computer Engineering\nUniversity of New Mexico\nAlbuquerqueNMUSA" ]	[]	In this paper, we compute finite sample bounds for data-driven approximations of the solution to stochastic reachability problems. Our approach uses a nonparametric technique known as kernel distribution embeddings, and provides probabilistic assurances of safety for stochastic systems in a model-free manner. By implicitly embedding the stochastic kernel of a Markov control process in a reproducing kernel Hilbert space, we can approximate the safety probabilities for stochastic systems with arbitrary stochastic disturbances as simple matrix operations and inner products. We present finite sample bounds for pointbased approximations of the safety probabilities through construction of probabilistic confidence bounds that are state-and input-dependent. One advantage of this approach is that the bounds are responsive to non-uniformly sampled data, meaning that tighter bounds are feasible in regions of the state-and input-space with more observations. We numerically evaluate the approach, and demonstrate its efficacy on a neural network-controlled pendulum system.	10.1016/j.automatica.2021.110146	[ "https://arxiv.org/pdf/2010.08036v2.pdf" ]	223,953,393	2010.08036	560af177c4dde4246f12ee4aa93fca3ea41f55b7	State-Based Confidence Bounds for Data-Driven Stochastic Reachability Using Hilbert Space Embeddings ⋆ 7 Dec 2021 Adam J Thorpe Electrical and Computer Engineering University of New Mexico AlbuquerqueNMUSA Kendric R Ortiz Electrical and Computer Engineering University of New Mexico AlbuquerqueNMUSA Meeko M K Oishi Electrical and Computer Engineering University of New Mexico AlbuquerqueNMUSA State-Based Confidence Bounds for Data-Driven Stochastic Reachability Using Hilbert Space Embeddings ⋆ 7 Dec 2021arXiv:2010.08036v2 [math.OC]Machine LearningStochastic SystemsReachabilitySafetyOptimal Control In this paper, we compute finite sample bounds for data-driven approximations of the solution to stochastic reachability problems. Our approach uses a nonparametric technique known as kernel distribution embeddings, and provides probabilistic assurances of safety for stochastic systems in a model-free manner. By implicitly embedding the stochastic kernel of a Markov control process in a reproducing kernel Hilbert space, we can approximate the safety probabilities for stochastic systems with arbitrary stochastic disturbances as simple matrix operations and inner products. We present finite sample bounds for pointbased approximations of the safety probabilities through construction of probabilistic confidence bounds that are state-and input-dependent. One advantage of this approach is that the bounds are responsive to non-uniformly sampled data, meaning that tighter bounds are feasible in regions of the state-and input-space with more observations. We numerically evaluate the approach, and demonstrate its efficacy on a neural network-controlled pendulum system. Introduction In expensive, high risk, or safety-critical systems, tools for verification are important for ensuring correctness before testing, implementation, or deployment. As autonomy grows in prevalence, and systems continue to grow in size and complexity, there is a need to extend ⋆ Corresponding author Meeko Oishi ([email protected]). This material is based upon work supported by the National Science Foundation under NSF Grant Number CNS-1836900. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. This research was supported in part by the Laboratory Directed Research and Development program at Sandia National Laboratories, a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA-0003525. The views expressed in this article do not necessarily represent the views of the U.S. Department of Energy or the United States Government. This work was also supported in part by the University of New Mexico's Vice President for Research Office. Email addresses: [email protected] (Adam J. Thorpe), [email protected] (Kendric R. Ortiz), [email protected] (Meeko M. K. Oishi). such tools to accommodate learning-enabled components in autonomous systems that resist traditional modeling. Stochastic reachability is an established tool for model-based verification and probabilistic safety [4,52] that has been applied to safety-critical problems in a variety of domains, including spacecraft rendezvous and docking [27], robotics and path planning [31], and vehicle control [23]. Safety refers to the ability of trajectories of the system to respect known constraints on the state space, with at least a desired likelihood, despite bounded control authority. In this paper, we characterize confidence bounds on a data-driven approach for stochastic reachability, enabling rigorous assurances of safety in a model-free manner. Model-based stochastic reachability has received considerable attention, with methods and tools specific to Markov decision processes [12,49], polynomial dynamical systems [37,38], and linear dynamical systems with log-concave disturbances [57][58][59], as well as the development of various benchmarks for comparisons [1][2][3]. Additional work has been done in the model checking community to develop tools for probabilistic, bounded-time reachability problems on parametric stochastic systems which provide interval-based assurances on the safety probabilities [43,44]. However, far less work has been done on data-driven stochastic reachability. Methods for statistical verification [40,61,62], forward reachable set estimation [29,54], and data-driven interval-based methods based in adaptive sampling and Gaussian processes [16] have been explored, with application to powertrains, bipedal robots, and learning-enabled cyber-physical systems. We focus in particular on a framework for stochastic reachability that is based in dynamic programming [4,52]. In contrast to model checking approaches, methods based on this framework are amenable to simultaneous controller synthesis, and the underlying theory accommodates a wide range of dynamical systems that can be captured as a Markov decision process with both discrete-and continuous-valued variables. This approach is general, in that it encompasses formulations for the terminal-hitting time problem [4,52], as well as for the first-hitting time problem, which is far more computationally complex [52]. We employ a point-based formulation, in which we seek to find the likelihood of maintaining desirable time-varying state constraints over a finite time horizon, from a known initial condition. In contrast to set-based methods [16,18,19], we do not seek to compute the stochastic reachable set, the largest set of states for which the desired likelihood of safety can be maintained. We draw in particular on tools based in reproducing kernel Hilbert spaces (RKHS), a family of machine learning techniques which enable an implicit approximation of the underlying stochastic kernel. In our previous work [53], we integrated the RKHS framework with dynamic programming-based approaches for stochastic reachability, to facilitate a data-driven approach to stochastic reachability [53,55]. Key to the utility of this approach are bounds on the approximation error associated with the RKHS methods. In [53], we described the existence of an upper bound on the asymptotic convergence rate, which can be used to infer the existence of a confidence bound on the stochastic reachability probability. However, these bounds are difficult to compute, and hence are not practicable for ascertaining the quality of the result. In this paper, we present computable confidence bounds that are state-and input-dependent. The benefit of this approach is that it enables higher accuracy bounds in regions of the state-and input-space for which more training data exists. Kernel methods have long been used in probability and statistics, [7,36], and more recently applied to Markov models [21], partially observable systems [35,46], and policy synthesis [28] in the context of optimal control. Finite sample bounds for conditional distribution embeddings in these contexts are presented in [20][21][22][46][47][48], which show that the estimated expectation converges in probability to the true expected value with a known, asymptotic convergence rate. The main drawback of kernel-based approaches is the tradeoff that can occur between computability and accuracy. Higher volumes of data are required to ensure an accurate result, which inevitably increases the computational burden associated with inversion of a matrix whose size is proportional to the sample size [34,39]. Fortunately, several methods have been developed to significantly reduce the computational overhead [26,39,55]. In this paper, we construct finite sample bounds for kernel embedding-based computation of the stochastic reachability probability, that are specific to the stochastic reachability problem. Our main contribution is the construction of computable state-and input-based upper and lower bounds, obtained via concentration inequalities [32] and tools from statistical learning theory [56]. Our proposed approach accommodates the fact that the gathered data may not be available uniformly through the state and input space. That is, the bound is dependent upon the sample set from which the kernel embedding is inferred. The paper is organized as follows. In Section 2, we formulate the problem and provide relevant background information. In Section 3, we derive the finite sample bounds. We discuss implications of the proposed bounds in Section 4, and the problem of parameter selection. In Section 5, we numerically validate the bounds on a stochastic chain of integrators, and demonstrate its utility on a nonlinear pendulum system and a nonlinear cart-pole system with black-box neural network controllers. Concluding remarks are provided in Section 6. Preliminaries We employ the following notational conventions. Let E be an arbitrary nonempty space. The indicator function 1 A : E → {0, 1} of A ⊆ E is defined such that 1 A (x) = 1 if x ∈ A, and 1 A (x) = 0 if x / ∈ A. Let E denote the σ-algebra on E. If E is a topological space [14], the σ-algebra generated by the set of all open subsets of E is called the Borel σ-algebra, denoted by B(E). Let (Ω, F , P) denote a probability space, where F is the σalgebra on Ω and P : F → [0, 1] is a probability measure on the measurable space (Ω, F ). A measurable function X : Ω → E is called a random variable taking values in (E, E). The image of P under X, P(X −1 A), A ∈ E is called the distribution of X. Let T be an arbitrary set, and for each t ∈ T , let X t be a random variable. The collection of random variables {X t : t ∈ T } on (Ω, F ) is a stochastic process. System Model & Stochastic Reachability Problem Consider a discrete-time stochastic dynamical system described by a Markov control process as defined in [52]. Definition 1 (Markov Control Process). A Markov control process H is defined as a 3-tuple, H = (X , U, Q), consisting of: a Borel space X ⊆ R n representing the state space, a compact Borel space U ⊂ R m representing the control space, and Q : B(X ) × X × U → [0, 1], a Borel-measurable stochastic kernel which assigns a probability measure Q(· \| x, u) to every x ∈ X and u ∈ U on the Borel space (X , B(X )). The system evolves from an initial condition x 0 ∈ X over a finite time horizon t = 0, 1, . . . , N , N < ∞. The control inputs are chosen according to a Markov control policy π = {π 0 , π 1 , . . . , π N −1 }, which is a sequence of universally-measurable [9] maps π i : X → U, i = 0, 1, . . . , N − 1. For simplicity, we assume that the policy π is stationary, meaning π(x) = π 0 (x) = π 1 (x) = · · · = π N −1 (x) for all x ∈ X . This is a simplifying assumption for the purpose of analysis, and the extension to non-stationary policies is trivial. We consider a probabilistic reach/avoid problem known as the terminal-hitting time problem where the objective is to determine the likelihood that the system starting at an initial condition x 0 ∈ X will remain in a pre-defined safe set and reach a target set at the final time instant. Let K, T ∈ B(X ), denote the safe set and target set, respectively. From [52], the terminal-hitting time safety probability r π x0 (K, T ) is defined as the probability that a system H following a Markov policy π from the initial condition x 0 will reach the target set T at time N while remaining within the safe set K for all t = 0, 1, . . . , N −1, r π x0 (K, T ) = E π x0 N −1 i=0 1 K (x i ) 1 T (x N ) .(1) For a fixed Markov policy π, we define the value functions V π t : X → [0, 1], t = 0, . . . , N , via backward recursion: V π N (x) = 1 T (x),(2)V π t (x) = 1 K (x) X V π t+1 (y)Q(dy \| x, π(x)).(3) Then according to [52], we have that V π 0 (x 0 ) = r π x0 (K, T ) for every x 0 ∈ X . We consider the case where the stochastic kernel Q is unknown, which means the integral in (3) becomes intractable. Assumption 2. The stochastic kernel Q is unknown. Instead, we presume that a finite sample S of M ∈ N observations is available, taken independently and identically distributed (i.i.d.) from the stochastic kernel Q, S = {(y i , x i , u i )} M i=1 ,(4) where x i are taken i.i.d. from a probability measure with support X , u i = π(x i ), and y i ∼ Q(· \| x i , u i ). Conditional Distribution Embeddings Because Q is assumed to be unknown, we cannot compute the safety probabilities r π x0 (K, T ) directly. Thus, we seek to approximate the safety probabilities by numerically approximating the integral with respect to Q(· \| x, π(x)) in (3) using the sample S. In order to approximate the integral of V π t+1 in (3), we choose to represent the integral operator with respect to Q(· \| x, π(x)) as an element in a high-dimensional space of functions known as a reproducing kernel Hilbert space (RKHS). Define a positive-definite [13,Definition 4.15] function k : X × X → R known as a kernel function and let H be a Hilbert space of functions of the form X → R equipped with an inner product ·, · H and the induced norm · H . Definition 3 (RKHS). Let E be an arbitrary space. A Hilbert space H is a reproducing kernel Hilbert space (RKHS) if there exists a positive-definite kernel function k : E × E → R that satisfies the following properties [5]: (1) k(x, ·) ∈ H for all x ∈ E, and (2) f (x) = f, k(x, ·) H for all f ∈ H and x ∈ E. where the second property is known as the reproducing property. Remark 4. Conversely, by the Moore-Aronszajn theorem [5], for any positive-definite kernel k, there exists a unique RKHS with k as its reproducing kernel, where H is defined as the closure of the linear span of kernel functions, i.e. H = span{k(x, ·) \| x ∈ X }. In short, this means that by defining a reproducing kernel, we obtain a corresponding RKHS. See, e.g., [7,13,42] for more information on reproducing kernel Hilbert spaces. The reproducing property is central to our approach, since it allows us to evaluate any function in the RKHS as a Hilbert space inner product. By embedding the integral operator with respect to Q(· \| x, π(x)) as an element in the RKHS, we can use the reproducing property to evaluate the integral in (3). For the measurable space X , define the kernel function k : X ×X → R with the associated RKHS H . We impose a mild simplifying assumption for the purpose of analysis and assume that the kernel k is bounded above by a real number ρ < ∞, where sup x∈X (k(x, x)) 1/2 ≤ ρ < ∞. Such a kernel is called a bounded kernel [13, §4.3]. This assumption (along with the measurability of k) ensures that we can represent the integral operator as an element in the RKHS. Let P denote the set of probability measures on X which are densities of Y ∈ X conditioned on (X, U ) ∈ X ×U (of which the probability measures defined by the stochastic kernel Q are a part). For any probability measure Q(· \| x, u) ∈ P, if the following necessary and suffi- [51] (which is satisfied if k is bounded and measurable on X ), there exists an element m(x, u) ∈ H called the conditional distribution embedding [48], defined as: cient condition E Y ∼Q(·\|x,u) [k(Y, Y )] < ∞ is satisfiedm : P → H , Q(· \| x, u) → m(x, u) := X k(y, ·)Q(dy \| x, u).(5) Using this representation, we can embed the integral with respect to Q(· \| x, u) as an element in the RKHS H . Furthermore, if the kernel is characteristic [50], then the embedding is unique, meaning the embedding captures all statistical information of the underlying distribution, and no information is lost by this representation. As shown in [53], we can use conditional distribution embeddings to solve the stochastic reachability problem. By the reproducing property, we can evaluate the integral of any function f ∈ H with respect to Q(· \| x, π(x)) as a Hilbert space inner product with the embedding m(x, π(x)). Thus, assuming the value functions V π t , t = 1, . . . , N , are in H , we can evaluate the integral in (3) as: V π t+1 , m(x, π(x)) H = V π t+1 , X k(y, ·)Q(dy \| x, π(x)) H (6) = X V π t+1 , k(y, ·) H Q(· \| x, π(x)) (7) = X V π t+1 (y)Q(· \| x, π(x)).(8) Using the embedding m(x, π(x)), we can substitute the integral expression in the stochastic reachability backward recursion (3) with a Hilbert space inner product in order to compute the safety probabilities r π x0 (K, T ). This reduces the evaluation of potentially expensive integrals to a simple linear operation in Hilbert space. Empirically Approximating the Value Functions In practice, we do not have access to the true embedding m(x, π(x)) since the stochastic kernel Q is unknown, meaning we cannot compute (3) directly. Thus, we compute an empirical approximation of the embedding m(x, π(x)) using a finite sample S as in (4) of size M ∈ N collected i.i.d. from Q. Following [33], using a sample S, we can compute an empirical estimatem(x, u) of an embedding m(x, u) as the solution to the following regularized least-squares problem [20,33], min m 1 M M i=1 k(y i , ·) −m(x i , u i ) 2 H + λ m 2 Γ ,(9) where Γ is a vector-valued RKHS [33] and λ > 0 is the regularization parameter. By the representer theorem [33], the solution to (9) is unique and has the following form:m = M i=1 α i k(x i , ·)l(u i , ·),(10) where α ∈ R M is a vector of real-valued coefficients and l : U × U → R is a reproducing kernel function over U. For simplicity, we assume that the kernel l is bounded and the bound ρ of the kernel k is also a bound for l. By substituting (10) into (9) and taking the derivative with respect to α, we obtain the following closed-form solution,m (x, u) = Φ ⊤ W Ψk(x, ·)l(u, ·),(11) where Φ and Ψ are known as feature vectors, with ele- ments Φ i = k(y i , ·) and Ψ i = k(x i , ·)l(u i , ·), respectively, and W = (ΨΨ ⊤ + λM I) −1 . For simplicity, let β(x, u) := W Ψk(x, ·)l(u, ·)(12) be a vector of real-valued coefficients that depends on the value of the conditioning variables x and u, such that m(x, u) = Φ ⊤ β(x, u). As shown in [53], using an estimatem(x, π(x)) of m(x, π(x)), we can approximate the integral of the value functions with respect to Q(· \| x, π(x)) in (3) as an inner product, V π t+1 ,m(x, π(x)) H = V π t+1 ⊤ β(x, π(x)) ≈ X V π t+1 (y)Q(dy \| x, π(x)),(13)where V π t+1 = [V π t+1 (y 1 ), . . . , V π t+1 (y M )] ⊤ . Thus, using the estimatem(x, π(x)), we can recursively approximate and substitute the value functions in the stochastic reachability backward recursion in order to approximate the safety probabilities. We summarize this procedure as Lemma 5. Lemma 5 (Approximate Backward Recursion [53]). Let π be a fixed Markov policy. Define the approximate value functionsV π t : X → [0, 1], t = 0, . . . , N by the backward recursion: V π N (x) = V π N (x),(14)V π t (x) = 1 K (x) V π t+1 ,m Y \|x,π(x) H .(15) Then r π x0 (K, T ) ≈V π 0 (x 0 ). Note that since the estimatem(x, π(x)) is conditioned on a particular value of x ∈ X , by approximating the safety probability r π x0 (K, T ) using Lemma 5, we obtain a point-based approximation of the safety probability at a particular value of x 0 ∈ X . Lemma 5 provides a model-free approach to approximate the stochastic reachability probability, and can easily be extended to solve related problems, including the first-hitting time problem [52] and the multiplicative and maximal cost stochastic reachability problems in [4]. Finite Sample Bounds The difficulty in finding bounds on the stochastic reachability probability stems from the underlying structure of the conditional distribution embedding estimate. Unlike the embedding for a marginal distribution [45], which has uniform coefficients 1/M , the conditional distribution embedding has non-uniform coefficients β(x, u) (12) which depend upon the value of the conditioning variables. This complicates the application of existing mathematical techniques from statistical learning theory. It is worth noting that in our case, we directly bound the variation of the estimator at a particular value of the conditioning variables x and u. This means the bounds we derive provide a localized result, which can be used to assess the quality of the approximation at a particular point. This is not a significant limitation in the context of Lemma 5, since in the case of the stochastic reachability backward recursion, we seek to evaluate the safety probability at a single point. In order to determine a bound on the quality of the approximation obtained using Lemma 5, we seek a bound on the difference between the expectation of the value function and its empirical counterpart. Worst-Case Difference Between the True and the Empirical Expectation Assume that V π t ∈ H , t = 1, . . . , N , and assume that for all f ∈ H , f ∈ [0, 1] and f H ≤ 1. We begin by upper bounding the deviation of the empirical expectation computed using an estimatem(x, u) from the true expectation for any value function V π t . In other words, we seek a state-based bound B(x, u) ∈ R such that for any value function V π t , t = 1, . . . , N , \|E Y ∼Q(·\|x,u) [V π t (Y )] − V π t ⊤ β(x, u)\| ≤ B(x, u). (16) For simplicity of notation, let Ef : = E Y ∼Q(·\|x,u) [f (Y )]. We can uniformly bound the difference between the value function expectation and the empirical expectation computed using an estimatem(x, u). Note that for any value function V π t ∈ H , not necessarily attaining the supremum, \|EV π t − V π t ⊤ β(x, u)\| ≤ sup h H ≤1 \|Eh − h ⊤ β(x, u)\|, (17) where h is some function in H , h = [h(y 1 ), . . . , h(y M )] ⊤ , and β(x, u) is defined as in (12). This means that in the worst case, the deviation of the estimated value function expectation from the true value function expectation is less than or equal to the deviation of a function h * ∈ H which satisfies the supremum. We then bound the right-hand side of (17) using Mc-Diarmid's inequality. In simple terms, McDiarmid's inequality states that if the empirical estimate h ⊤ β(x, u) computed using S has bounded variation when a single observation in the sample S is changed, then the deviation of the estimate from the true expectation is bounded by some quantity that depends on the variation bound. Definition 6 (Bounded Differences Condition). Given coefficients c i ≥ 0, i = 1, . . . , M , a function f : E M → R satisfies the bounded differences condition if sup x1,...,xM x ′ i ∈E \|f (x 1 , . . . , x M ) − f (x 1 , . . . , x ′ i , . . . , x M )\| ≤ c i (18) for every i = 1, . . . , M . Lemma 7 (McDiarmid's inequality [32]). Let X = {X 1 , . . . , X M } be independent random variables taking values in a set E, and assume that the function f : E M → R satisfies the bounded differences condition (Definition 6). Then for every ε > 0, Pr(\|f (X) − E[f (X)]\| ≥ ε) ≤ exp − 2ε 2 M i=1 c 2 i .(19) Alternatively, Lemma 7 implies that, given a small probability δ/2 ∈ (0, 1), then with probability 1 − δ/2, the deviation of the function f from the expectation E[f (X)] is bounded by: f (X) − E[f (X)] ≤ M C 2 log(2/δ) 2 ,(20) where C ≥ c i for all i = 1, . . . , M . In order to determine the bound on (17), we seek some constant C that satisfies the bounded differences condition. However, the effect of changing an individual observation in the empirical expectation term in (17) is non-trivial, since changing a single observation affects all elements of the coefficient vector β(x, u). Therefore, in order to determine C, we make use of a well-known result in linear algebra known as the pushthrough identity [ Let A ∈ R n×m and B ∈ R m×n , and assume that AB + I is non-singular. Then BA + I is non-singular and (AB + I) −1 A = A(BA + I) −1 .(21) We now prove the following theorem: Theorem 9. The variation of h ⊤ β(x, u) by changing a single observation is at most ρ/(λM ). PROOF. For some h ∈ H , let h ⊤ β(x, u) be defined as in (17), such that β(x, u) = W Ψk(x, ·)l(u, ·), where Ψ is a feature vector with elements Ψ i = k(x i , ·)l(u i , ·), W = (ΨΨ ⊤ + λM I) −1 , and the kernels k and l are bounded by some constant ρ < ∞. Using Lemma 8, we can write W Ψ as: W Ψ = (ΨΨ ⊤ + λM I) −1 Ψ = Ψ(Ψ ⊤ Ψ + λM ) −1 ,(22) where we note that (Ψ ⊤ Ψ + λM ) −1 is scalar. Using the identity in (22), we can rewrite the estimate h ⊤ β(x, u) as: h ⊤ W Ψk(x, ·)l(u, ·) = 1 Ψ ⊤ Ψ + λM h ⊤ Ψk(x, ·)l(u, ·). (23) Since h ∈ [0, 1] by assumption, we have from (23) that the variation for changing a single observation is at most ρ/(λM ), which proves the result. Remark 10. Note that the result provided by Theorem 9 is strong, and is verified by the result presented in [10,Theorem 22] (with a scaling factor of 1 instead of 1/2, which comes from the worst-case bound). From Theorem 9, we have that C = ρ/(λM ) satisfies the bounded differences condition (Definition 6). Continuing from (17), using McDiarmid's inequality (19), we have that given δ/2 ∈ (0, 1), with probability 1 − δ/2, sup h H ≤1 \|Eh − h ⊤ β(x, u)\| ≤ E S sup h H ≤1 \|Eh − h ⊤ β(x, u)\| + ρ 2 log(2/δ) 2λ 2 M .(24) We now have an expression for the worst-case difference between the true and empirical stochastic reachability probabilities. However, because the expectation on the right-hand side of (24) relies upon the true expectation of h, it is not directly computable. Removing Reliance Upon the True Expectation In order to enable computability of the bound in (24), we bound the first term on the right-hand side of (24) via symmetrization and then utilize the properties of the RKHS to bound the expectation of the worst-case empirical estimate. In particular, we make use of the reproducing property and the definition of the dual norm for Hilbert spaces, which we present here adapted from [41,Theorem 4.3]. Definition 11 (Dual Norm). Let H be a Hilbert space. For any f, g ∈ H , f H = sup g H ≤1 \| f, g H \|. In addition, we rely upon a special type of random variable known as a Rademacher variable (cf. [6]), which is a uniform random variable taking values in {−1, 1}. Definition 12 (Rademacher Variable [6]). A random variable σ is called a Rademacher variable if it is independent uniform, such that Pr(σ = 1) = Pr(σ = −1) = 1/2. We now prove the following lemma: Lemma 13. Let h ∈ H and h ⊤ β(x, u) be defined as in (17). Given δ/2 ∈ (0, 1), then with probability 1 − δ/2, E S sup h H ≤1 \|Eh − h ⊤ β(x, u)\| ≤ 2 tr(β ⊤ ΦΦ ⊤ β) + 2 ρ 2 log(2/δ) 2λ 2 M .(25) PROOF. We begin by bounding the first term on the right-hand side of (24) via symmetrization [56]. LetS be a ghost sample, that is, an independent copy of S that is drawn from the same sampling distribution as S [56]. We replace the expectation in the first term on the right-hand side of (24) with a second empirical estimate computed usingS, to obtain: E S sup h H ≤1 \|Eh − h ⊤ β(x, u)\| ≤ E S sup h H ≤1 \|ES[h ⊤β (x, u) − h ⊤ β(x, u)]\| (26) ≤ E SS sup h H ≤1 \|h ⊤β (x, u) − h ⊤ β(x, u)\| ,(27) whereβ(x, u) is computed usingS as in (12) We next exploit the symmetry of the empirical distributions to upper bound (27). Let σ be Rademacher variables with σ i ∈ {−1, 1}, and let β := diag(β(x, u)). Since the distribution of the difference in empirical expectationsh ⊤β (x, u) − h ⊤ β(x, u) is symmetric around 0, which follows since f ⊤ β(x, u) ∈ [0, 1] for all f ∈ H , we see thath ⊤β σ − h ⊤ βσ has the same distribution. In effect, the Rademacher variables randomly exchange observations in S andS with probability 1/2. When we take the expectation over σ, the expectations of the empirical estimates computed using S andS are the same. Using this fact, we obtain the following: E SS sup h H ≤1 \|h ⊤β (x, u) − h ⊤ β(x, u)\| = E SSσ sup h H ≤1 \|h ⊤β σ − h ⊤ βσ\| (28) ≤ 2E Sσ sup h H ≤1 \|h ⊤ βσ\| .(29) Then by applying the reproducing property (Definition 3) and the definition of the dual norm for Hilbert spaces (Definition 11), we remove the dependence on h: 2E Sσ sup h H ≤1 \|h ⊤ βσ\| = 2E Sσ sup h H ≤1 \| h, Φ ⊤ βσ H \| (30) = 2E Sσ [ Φ ⊤ βσ H ].(31) We then utilize the definition of the Hilbert space norm and the concavity of the square root and note that the expectation of the Rademacher variables E σ [σ i σ j ] vanishes except when i = j, 2E Sσ [ Φ ⊤ βσ H ] ≤ 2E S E σ σ ⊤ β ⊤ ΦΦ ⊤ βσ 1/2 (32) = 2E S tr β ⊤ ΦΦ ⊤ β 1/2 .(33) By bounding the expectation in (33) by McDiarmid's inequality again, we obtain: 2E S tr β ⊤ ΦΦ ⊤ β 1/2 ≤ 2 tr(β ⊤ ΦΦ ⊤ β) + 2 ρ 2 log(2/δ) 2λ 2 M ,(34) which proves the result. Continuing from (24), using McDiarmid's inequality with C = ρ/(λM ) via Theorem 9 and Lemma 13, we have that given δ/2 ∈ (0, 1), with probability 1 − δ/2, sup h H ≤1 \|Eh − h ⊤ β(x, u)\| ≤ 2 tr(β ⊤ ΦΦ ⊤ β) + 3 ρ 2 log(2/δ) 2λ 2 M .(35) Thus, we have a computable bound for the worst-case difference between the true and empirical expectation of a function h ∈ H . Remark 14. We note that even in the worst case, the first term on the right-hand side of (35) is bounded, since Φ ⊤ β(x, u) H ≤ 1 by assumption. However, in practice, with appropriate kernel selection and choice of regularization parameter λ, this term will often be significantly less than 1. Finite Sample Bound on the Safety Probability We now can state the main result, which we present as Theorem 15. Theorem 15. For any value function V π t , given δ/2 ∈ (0, 1), with probability 1 − δ/2, the difference between the true and empirical expectation of the value functions is bounded by: \|EV π t − V π t ⊤ β(x, u)\| ≤ 2 tr(β ⊤ ΦΦ ⊤ β) + 3 ρ 2 log(2/δ) 2λ 2 M .(36) PROOF. The proof follows from the arguments presented for (35). In (17), we uniformly bound the difference between the true and empirical expectation of the value functions by the worst-case function h ∈ H . We then use McDiarmid's inequality (Lemma 7) with the bound C = ρ/(λM ) satisfying the bounded differences condition (Theorem 9) to obtain (24). Using a ghost sample, the symmetrization argument, and the definition of the dual norm for Hilbert spaces, we then bound the first term on the right-hand side of (24) in order to remove the dependence on h, E S sup h H ≤1 \|Eh − h ⊤ β(x, u)\| ≤ 2 tr(β ⊤ ΦΦ ⊤ β) + 2 ρ 2 log(2/δ) 2λ 2 M .(37) We then substitute the bound in (37) into (24) and ob-tain the result, \|EV π t − V π t ⊤ β(x, u)\| ≤ 2 tr(β ⊤ ΦΦ ⊤ β) + 3 ρ 2 log(2/δ) 2λ 2 M ,(38) which proves (36). Let B(x, u) be the bound on the difference between the expected value of the value functions and its empirical counterpart in (36), given by B(x, u) = 2 tr(β ⊤ ΦΦ ⊤ β) + 3 ρ 2 log(2/δ) 2λ 2 M .(39) This means that for any value function V π t , given δ/2 ∈ (0, 1), with probability 1 − δ/2, that the absolute difference between the actual expectation and the empirical expectation computed usingm(x, π(x)) is bounded by − B(x, u) ≤ EV π t − V π t ⊤ β(x, u) ≤ B(x, u).(40) Thus, by applying this bound to the expectations in the backward recursion, we obtain an overall bound on the approximation of the safety probabilities obtained using Lemma 5. Furthermore, the bound in (39) depends on the value of the conditioning variables x and u, which means the bound can serve as an indication of the quality of the approximation at a particular point. Note that the bound in (39) applies to the error of a value function at a single time step. This means that the total error on the approximation of the safety probabilities increases linearly with the number of time steps used in the backward recursion in Lemma 5. Following [53,Corollary 3], if the error of each value function is bounded by B(x, u) in (39), we obtain an overall bound on the safety probabilities V π 0 of N B(x, u), where N is the time horizon. We summarize this result in the following lemma: Given a bound B(x, u), the error in the safety probabilities obtained via Lemma 5 is given by Lemma 16.\|EV π 0 − V π 0 ⊤ β(x, u)\| ≤ N B(x, u),(41) where N is the time horizon. PROOF. The proof is by induction. If the error of a single value function is B(x, u) as in (39), and the safety probabilities are computed via recursive substitution as in Lemma 5, then at time t = 0, the maximum error of V π 0 is N B(x, u). Kernel and Parameter Selection The quality of the approximation of the stochastic reachability probability is governed by a number of factors, including the choice of kernel function, parameters associated with the kernel function, the regularization parameter of the least-squares problem (9), and the sample, S. In a realistic setting, we typically do not have explicit control over the sample S or the number of observations in the sample. Thus, the choice of kernel function and the model parameters plays an important role in the quality of the kernel-based approximation. Kernel Selection The performance of kernel-based learning algorithms is closely tied to the choice of kernel function and the structure of the RKHS. In essence, the Hilbert space needs to be rich enough to model the set of probability measures underlying the observed data without overfitting. Thus, we frame the problem of kernel selection as a problem of limiting the complexity of the RKHS [6,56], where complexity in statistical learning literature refers to the ability of a function class to fit random noise. Intuitively, by choosing a function class that lowers the complexity term, we reduce the possibility that our function class H will overfit the observed data. We propose a complexity term that is closely related to the Rademacher complexity [6] from statistical learning theory, but instead based on the bound presented in Theorem 15 that accommodates non-uniform coefficients β(x, u). Define the random variablê C (H ) = E σ sup h H ≤1 \|h ⊤ βσ\| .(42) Then the conditional complexity of H is defined as C (H ) = E S [Ĉ (H )] . Using the bound in (33), we have C (H ) ≤ E S tr β ⊤ ΦΦ ⊤ β 1/2 .(43) This choice of complexity term is equivalent to the first term on the right-hand side of (39). Since the conditional complexity appears in the bounds presented in Theorem 15, minimizing the complexity term also minimizes the finite sample bounds on the difference in expectations. Thus, we can choose the kernel functions which minimize the complexity term for all (x, u), effectively minimizing the finite sample bound on the difference in expectations of the value functions. The kernel should also be chosen to satisfy universality [50,51] (resp. characteristic) and boundedness properties. One common choice of kernel that satisfies these properties is the Gaussian RBF kernel k(x, x ′ ) = exp(− x − x ′ 2 2 /2σ 2 ), where σ > 0. Universal [45,51] kernels are so-named because they satisfy a universal approximation property and are able to learn any real-valued function arbitrarily well. Because the mapping from the set of all probability measures P into the RKHS (5) is injective for universal (resp. characteristic, see [50]) kernels, this means there is a unique element in the RKHS H for any P, Q ∈ P, such that m P −m Q H = 0 if and only if P = Q. This ensures that the conditional distribution embedding admits a unique solution [51], and that we can distinguish between distributions in Hilbert space. By choosing a bounded kernel function, we ensure that ρ < ∞, and we can achieve tighter bounds by selecting a kernel function with small ρ. The Gaussian kernel function, for example, has ρ = 1. Typically, a parameterized kernel which is known to satisfy these properties is chosen, and then kernel parameters are tuned via cross-validation techniques. However, nascent work has posed kernel synthesis for a given sample S as an optimization problem. This approach has been demonstrated via convex optimization [24] and semi-definite programming [25] for marginal distributions with scalar-valued regression. While this idea is promising, the connection between conditional distribution embeddings and the underlying regression problem has only recently begun to be explored [20]. Further, the extension from scalar-valued regression to vector-valued regression [33] that is required by the objective function in (9) is not straightforward. Hence additional work will be needed to evaluate the feasibility of kernel synthesis methods for this problem. Parameter Selection Tighter bounds may be possible by identifying a strict upper bound on the elements of β(x, u), which in turn are influenced by the parameters ρ, the upper bound on the kernel function, and λ, the least-squares regularization coefficient in (9). The value of ρ is determined primarily by the choice of kernel and its parameters, and can be tuned by minimizing the complexity term C (H ). The value of λ affects the convergence rate associated with the uniform finite sample bound. The convergence guarantees in [48] and [21] typically depend upon λ going to zero as the number of observations increases. See [11,15] for a discussion of optimal values of λ. Scalability One significant advantage of using the kernel-based approach in Lemma 5 is that approximating the value function expectation in (3) using conditional distribution embeddings does not scale exponentially as the system dimensionality increases (also known as the curse of dimensionality). This is primarily due to the fact that the system dimensionality (as well as the input dimension) only has an effect on the evaluation of the kernel functions k and l. Instead, the complexity of computing the conditional distribution embedding estimate is generally O(M 3 ), which is primarily driven by the matrix inverse W in (12), and scales solely with the sample size M used to construct the embedding estimate. This is demonstrated empirically in [53], which shows that the computational complexity increases roughly linearly as the system dimensionality is increased. However, in order to adequately characterize the stochastic kernel of a high-dimensional state space, a large sample size may be required. In effect, this means that the quality of the approximation is in large part governed by the sample used to characterize the region of interest-which further motivates our approach to state-based finite sample bounds. As the time horizon N increases, the backward recursion in Lemma 5 shows that if the policy π is time-invariant, meaning π 0 = π 1 = · · · = π N −1 , we do not need to recompute the embedding estimatem(x, π(x)) at every time step. This means we only need to compute the estimate once for a given sample S, and indicates that the complexity of Lemma 5 increases linearly as the time horizon increases. In the case of time-varying policies, this of course means that we must recompute the embedding at every time step since the closed-loop dynamics of the system vary with time in accordance with the policy. Examples We implemented Lemma 5 on a stochastic chain of integrators for the purpose of validation, and on a nonlinear pendulum system [17] and a closed-loop nonlinear cart-pole benchmark system [30] with black-box neural network controllers to demonstrate the capabilities of the proposed approach. For each problem, we generated a sample S of observations via simulation, and then assumed no knowledge of the system dynamics or the structure of the disturbance for the purposes of computing the stochastic reachability probability. We then computed finite sample bounds via Theorem 15. For all problems, we chose a Gaussian kernel k(x, x ′ ) = exp(− x − x ′ 2 2 /2σ 2 ) with σ = 0.1, and chose λ using the optimal rate computed in [11]. All computations were done in Matlab, and code to reproduce the analysis and figures is available at: github.com/unm-hscl/ajthor-Automatica2020a. Stochastic Chain of Integrators We first consider a 2-D stochastic chain of integrators [60], in which the input appears at the second derivative and each element of the state vector is the discretized integral of the element that follows it. The dynamics with sampling time T = 0.25 are given by: x k+1 = 1 T 0 1 x k + T 2 2 T u k + w k(44) where w k is an i.i.d. disturbance defined on the probability space (W, B(W), Pr w ). We presume a Gaussian disturbance w k ∼ N (0, Σ) with Σ = 0.01I, a control policy π(x) = 0, and target and safe sets T = [−1, 1] 2 and K = [−1, 1] 2 . We consider a sample S of M = 2500 observations drawn i.i.d. from Q, a representation of (44) as a Markov control process (Definition 1). The initial conditions x ∈ X in the sample were chosen uniformly in the interval [−1.1, 1.1] × [−1.1, 1.1] in order to ensure that a subset of the initial conditions violates the safety constraints, K and T . We do this to ensure the "learned" model does not map all initial conditions to a safe set. Otherwise, in the regression, the value function estimate maps all values to 1. The resulting state y ∈ X is drawn from Q(· \| x, π(x)) using the dynamics in (44). We then presumed no knowledge of the system dynamics or Q and computed the estimatem(x, π(x)) according to (11), with β(x, π(x)) computed as in (12). Usingm(x, π(x)), we then computed the stochastic reachability probability using Lemma 5 for a time horizon of N = 5. We compare the stochastic reachability probability computed according to Lemma 5 against the solution via dynamic programming, that presumes the stochastic kernel Q is known. The absolute error \|V π 0 (x) −V π 0 (x)\| between the results obtained from Lemma 5 and the dynamic programming solution is shown in Figure 1(c). As expected, the stochastic reachability probabilities computed using Lemma 5, show low absolute error as compared with the dynamic programming solution, with a maximum absolute error of 0.1158 and a mean absolute error of 0.0122. We then evaluated the finite sample bounds of the approximation using Theorem 15. We computed the bound B(x, π(x)) as in (39) with δ = 0.1 to obtain bounds on the safety probabilities. The bounds obtained from Theorem 15 are probabilistic upper and lower bounds, meaning that given any δ/2 ∈ [0, 1], with probability 1 − δ/2, the approximation from Lemma 5 is bounded by Theorem 15, which validates the result. Figure 1(d) shows the upper bound on the safety probabilities, while Figure 1(e) shows the lower bound. We can see that the difference between the upper and lower bounds is small, which indicates that the quality of the approximation obtained via Lemma 5 is close to the true solution with high probability. Further, the absolute difference is larger than the error values in Figure 1(c), as expected, meaning that the computed bound is a reasonable probabilistic upper bound. Figure 2 shows the mean of B(x, π(x)) (39), the finite sample bounds computed using Theorem 15 for the five samples of length M ∈ [100, 2500] as a function of δ. As expected, we can see that as the size of the samples increases, we obtain tighter probabilistic bounds B(x, π(x)) in the region [−1, 1] × [−1, 1] via Theorem 15. Effectively, this means that as the number of observations from the stochastic kernel increases, we obtain a better estimate of the conditional distribution embedding m(x, π(x)), and thus a better estimate of the safety probabilities via Lemma 5. Also as expected, as the violation threshold decreases (i.e., δ increases), the mean of the probabilistic bound decreases. Figure 2 also shows that for low values of delta, we obtain higher values of the probabilistic bound B(x, π(x)). This corresponds to a high desired confidence. Further, we can see in Figure 2 that the finite sample bounds do not improve appreciably as the sample size increases beyond a certain point. Linearized Cart-Pole System We then considered a benchmark cart-pole system [30] with a black-box neural network feedback controller. The dynamics for the linearized cart-pole system [30] are given by:ẍ = 0.0043θ − 2.75θ + 1.94u − 10.95ẋ θ = 28.58θ − 0.044θ − 4.44u + 24.92ẋ (45) with state x = [x,ẋ, θ,θ] ⊤ ∈ R 4 and control input u ∈ R. The dynamics are then discretized in time with sampling time T s = 0.2 s. We add an additional Gaussian disturbance w k ∼ N (0, Σ) with Σ = 0.01I to the dynamical state equations, which can simulate dynamical uncertainty or minor system perturbations. The control input is computed via a neural network controller [30], which takes the current state and outputs a real number u ∈ R, which can be interpreted as the input torque. The benchmark is defined [30] such that the neural network controller must keep the lateral position of the cart x within [−0.7, 0.7], maintain a low cart velocityẋ ∈ [−1, 1], and keep the pendulum angle θ within [−π/6, π/6] while the angular velocityθ is unconstrained. We define the safe set K according to the above constraints, and define the target set T such that the pendulum angle θ must be within [−0.05, 0.05]. K = {x ∈ R 4 \| \|x 1 \| ≤ 0.7, \|x 2 \| ≤ 1, \|x 3 \| ≤ π/6} (46) T = {x ∈ R 4 \| \|x 3 \| ≤ 0.05}(47) We simulated 10 trajectories from initial conditions taken uniformly from the ranges specified above, and extracted a sample S of M = 12,234 observations taken i.i.d. from the stochastic kernel Q, a representation of the dynamics (45) as a Markov control process (Definition 1). We then computed the safety probabilities for the system over a time horizon N = 10 using Lemma 5 to demonstrate the feasibility of the approach. The safety probabilities computed using Lemma 5 at k = 0 for N = 10 are shown in Figure 3(a). We can see that the closed-loop system has a high probability of stabilizing the pendulum from an initial condition within the range θ ∈ [−π/6, π/6] and the results also show an underlying symmetry around θ = 0, as expected. We then computed the finite sample bounds on the approximation using Theorem 15 with δ = 0.1 to obtain probabilistic upper and lower bounds on the safety probabilities. This means that using the proposed data-driven approach, we can utilize stochastic reachability to analyze the safety properties of a dynamical system with a blackbox neural network controller. Similarly, we can expose the underlying structure of the closed-loop system to reveal useful knowledge of the system properties, such as symmetry. Nonlinear Cart-Pole System We then analyzed a nonlinear cart-pole system with a neural network controller [30], with dynamics given by: x = u + mlω 2 sin(θ) where g = 9.8 is the gravitational constant, the pole mass is m = 0.1, half the pole's length is l = 0.5, and m t = 1.1 is the total mass. The dynamics are then discretized in time with sampling time T s = 0.015 s. The control input, u ∈ {−10, 10}, which affects the lateral position of the cart, is chosen by the neural network controller [30]. This means the controller is less "smooth" than the neural network controller for the linearized cartpole system, because when the pendulum is near vertical, the controller rapidly switches between a high positive and negative control input value. Thus, the velocity components of the system state,ẋ andθ, will not stabilize to zero, as shown in Figure 4. As before, we added a Gaussian disturbance w k ∼ N (0, Σ) with Σ = 0.01I to the dynamical equations and represent the system as a Markov control process. The benchmark is defined such that the pole angle θ will remain within [−π/6, π/6], while the other variables are unconstrained. As such, we define the target set T such that θ ∈ [−0.05, 0.05] as with the linearized cart-pole system, but define the safe set K to be the entire state space, meaning there is no unsafe region. T = {x ∈ R 4 \| \|x 3 \| ≤ 0.05}(49) This allows us to analyze the behavior of the controller to reach a pre-specified objective without enforcing constraints on the system before the terminal time. We then simulated 10 trajectories from initial conditions sampled uniformly from the ranges specified above, and collected a sample S of M = 10,000 observations. Then, we computed the safety probabilities using Lemma 5 with N = 600. The results are shown in Figure 5(a). We then computed the finite sample bounds using Theorem 15 for δ = 0.1, and plotted the lower bound on the safety probabilities in Figure 5(b). As expected, we see that the safety probabilities show that the nonlinear pendulum system is able to stabilize a pendulum starting within a small range of θ. Interestingly, it is revealed in Figure 5(a) that the controller is not completely symmetric, meaning it has a higher chance of stabilizing the pendulum for positive values of θ than negative values of θ. Fig. 5(b) shows the probabilistic lower bound on safety probabilities computed using Theorem 15 with δ = 0.1. Conclusion We provided state-and input-based finite sample bounds for the stochastic reachability probability constructed via conditional distribution embeddings. Our approach is based on an application of statistical learning theory, that relates the observed data to the quality of the approximation of the stochastic reachability probability at a given state and input. This approach enables rigorous bounds on model-free stochastic reachability. We validated our approach on a nonlinear dynamical system with a neural net controller, and numerically characterized our approach on the stochastic double integrator. andh = [h(ỹ 1 ), . . . , h(ỹ M )] ⊤ . Equation (26) follows almost surely from the properties of conditional expectations and (27) follows by the convexity of the supremum. Fig. 1 . 1(a) Safety probabilities at k = 0, N = 5, for a double integrator computed using dynamic programming. (b) Safety probabilities at k = 0 for a double integrator computed using Lemma 5 for N = 5. (c) Absolute error \|V π 0 (x) −V π 0 (x)\| between the dynamic programming solution and Lemma 5. (d, e) Upper and lower finite sample bounds, respectively, of the safety probabilities of a double integrator system computed using Theorem 15 with δ = 0.1, where (d) isV π 0 (x) + B(x, π(x)) and (e) isV π 0 (x) − B(x, π(x)), where B(x, π(x)) is computed as in(39). Fig. 2 . 2Figure showing the mean of the finite sample bounds B(x, π(x)) in the region [−1, 1] × [−1, 1] for a double integrator system as a function of δ. Fig. 3 . 3(a) Safety probabilities for a linearized cart-pole system computed using Lemma 5 for N = 10. (b) Upper bound on the safety probabilities computed using Theorem 15. (c) Lower bound on the safety probabilities computed using Theorem 15. We then consider the effect of varying the parameters M and δ in (39) on the finite sample bound computed using Theorem 15. In order to demonstrate the effect of the parameter M , we drew 5 new samples from the stochastic kernel Q, where the number of observations in each sample was chosen to be of size M ∈ [100, 2500]. An estimate was then computed for each sample and the finite sample bounds were computed for each estimate. For each sample of length M ∈ [100, 2500], we computed the mean of the finite sample bounds in the region [−1, 1] × [−1, 1] for different values of δ ∈ [0.1, 1.9]. Fig. 4 . 4Sample realization of the nonlinear cart-pole system (48) over 600 time steps. Fig. 5 . 5(a) Safety probabilities of the nonlinear cart-pole system computed using Lemma 5 for a time horizon of N = 600. 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[ "Quench Dynamics of Edge States in 2-D Topological Insulator Ribbons", "Quench Dynamics of Edge States in 2-D Topological Insulator Ribbons" ]	[ "Aavishkar A Patel \nDepartment of Physics\nIndian Institute of Technology Kanpur\n208016KanpurIndia\n", "Shraddha Sharma \nDepartment of Physics\nIndian Institute of Technology Kanpur\n208016KanpurIndia\n", "Amit Dutta \nDepartment of Physics\nIndian Institute of Technology Kanpur\n208016KanpurIndia\n" ]	[ "Department of Physics\nIndian Institute of Technology Kanpur\n208016KanpurIndia", "Department of Physics\nIndian Institute of Technology Kanpur\n208016KanpurIndia", "Department of Physics\nIndian Institute of Technology Kanpur\n208016KanpurIndia" ]	[]	We study the dynamics of edge states of the two dimensional BHZ Hamiltonian in a ribbon geometry following a sudden quench to the quantum critical point separating the topological insulator phase from the trivial insulator phase. The effective edge state Hamiltonian is a collection of decoupled qubit-like two-level systems which get coupled to bulk states following the quench. We notice a pronounced collapse and revival of the Loschmidt echo for low-energy edge states illustrating the oscillation of the state between the two edges. We also observe a similar collapse and revival in the spin Hall current carried by these edge states, leading to a persistence of its time-averaged value. PACS numbers: 64.70.Tg, 03.65.Pm, 74.40.Kb Topological insulators (TIs) are novel materials with an insulating bulk and conducting edges which are of extensive contemporary interest [1-4]. The low-energy electrons in two dimensional (2-D) Hg-Te/Cd-Te quantum well TIs, which display conducting helical edge state solutions that exist within the bulk bandgap in the TI phase, are described by the 2-D BHZ Hamiltonian [2, 4]. The bulk states undergo a quantum phase transition (QPT) [5-7] with the low-energy modes satisfying a 2-D Dirac Hamiltonian (DH) with a linear dispersion at the quantum critical point (QCP) (which is a 2-D Dirac point). Additionally, the chiral edge states with linear dispersion in the TI phase are described by an effective 1-D DH [2].At the same time, there is a recent upsurge in studies of quenching dynamics of quantum many body systems across QCPs [8-10], essentially because of the possibility of experimental realization of the same in optical lattices[11]. The scaling of the defect density generated in the final state following a slow[12,13]or a sudden quench[14,15], or generation of quantum correlations which are otherwise absent in the defect free final state[16]or the possibility thermalization with an effective temperature[17]are some of the topics which are being explored thoroughly.In this communication, we focus on the dynamics of edge states of the BHZ Hamiltonian when the system is suddenly quenched from the TI phase to either the QCP or the trivial insulator (TrI) phase. The question here is whether there is a surviving edge current following the quench. (It is to be noted that when a one-dimensional chain of hard core Bosons is quenched from the superfluid to the Mott Insulator state, there is a surviving supercurrent in the insulator phase which oscillates in time[18]). In our problem, the quench couples the twolevel subspace of the edge states to a multi-level environment of bulk states. We study the decoherence of these edge states using the Loschmidt echo (LE) [19] when the 2-D Hamiltonian is quenched from the TI phase to the QCP (or to the TrI phase). We observe a strong oscillation of the low-energy edge states between the two edges of the system and the time-averaged persistence of the spin Hall current (SHC) when the system is quenched to the QCP, which we attribute directly to the linear low-energy dispersion at this point (which is also found in other models describing 2-D TIs). The experimental prospect of real-time tuning of parameters controlling QPTs of TIs in optical [20] and photonic lattices [21] as well as by exploiting Floquet dynamics[22], has made the study of quenching dynamics of TI Hamiltonians relevant and important. It is to be noted that slow quenching results in a violation of the Kibble-Zurek scaling of defects in systems with edge states[23].The 4 × 4 BHZ Hamiltonian comprising of two 2×2 blocks (for opposite electron spins) is given bywhereHere, A, B, C, D and m are determined by the thickness of the quantum well and the material parameters; the parameter m controls the phase of the system and changes sign relative to B when the system crosses from the TI phase (where edge states are present) to the TrI phase (with no edge states) via a DP at m/B = 0. Although the results presented here are valid in generic situations, we shall set D = 0 for simplicity, which also ensures an electron-hole symmetric spectrum. We consider a ribbon geometry extending from −L/2 to L/2 in the y direction (with the wavefunction vanishing at the edges) and apply periodic boundary conditions in the x-direction[25][26][27].To obtain the spectrum of Hamiltonian (1), we consider the 2 × 2 block H(k), fix k x , and use k y → −i∂ y , with the trial solution ψ = (A 1 , A 2 )e λy [25]. The condition that the wavefunctions must vanish at the edges of the ribbon quantizes the energies E of the eigenstates at a given k x , which are given by the solutions of the following transcendental equation: tanh(λ + L/2) tanh(λ − L/2) + tanh(λ − L/2) tanh(λ + L/2) = λ 2 + + λ 2 − − (B/A) 2 (λ 2arXiv:1304.2248v1 [cond-mat.mes-hall]	10.1140/epjb/e2013-40657-2	[ "https://arxiv.org/pdf/1304.2248v1.pdf" ]	119,178,913	1304.2248	70babe5ba5aac2c5586d4f081f51e664508cfdf5	Quench Dynamics of Edge States in 2-D Topological Insulator Ribbons 8 Apr 2013 Aavishkar A Patel Department of Physics Indian Institute of Technology Kanpur 208016KanpurIndia Shraddha Sharma Department of Physics Indian Institute of Technology Kanpur 208016KanpurIndia Amit Dutta Department of Physics Indian Institute of Technology Kanpur 208016KanpurIndia Quench Dynamics of Edge States in 2-D Topological Insulator Ribbons 8 Apr 2013 We study the dynamics of edge states of the two dimensional BHZ Hamiltonian in a ribbon geometry following a sudden quench to the quantum critical point separating the topological insulator phase from the trivial insulator phase. The effective edge state Hamiltonian is a collection of decoupled qubit-like two-level systems which get coupled to bulk states following the quench. We notice a pronounced collapse and revival of the Loschmidt echo for low-energy edge states illustrating the oscillation of the state between the two edges. We also observe a similar collapse and revival in the spin Hall current carried by these edge states, leading to a persistence of its time-averaged value. PACS numbers: 64.70.Tg, 03.65.Pm, 74.40.Kb Topological insulators (TIs) are novel materials with an insulating bulk and conducting edges which are of extensive contemporary interest [1-4]. The low-energy electrons in two dimensional (2-D) Hg-Te/Cd-Te quantum well TIs, which display conducting helical edge state solutions that exist within the bulk bandgap in the TI phase, are described by the 2-D BHZ Hamiltonian [2, 4]. The bulk states undergo a quantum phase transition (QPT) [5-7] with the low-energy modes satisfying a 2-D Dirac Hamiltonian (DH) with a linear dispersion at the quantum critical point (QCP) (which is a 2-D Dirac point). Additionally, the chiral edge states with linear dispersion in the TI phase are described by an effective 1-D DH [2].At the same time, there is a recent upsurge in studies of quenching dynamics of quantum many body systems across QCPs [8-10], essentially because of the possibility of experimental realization of the same in optical lattices[11]. The scaling of the defect density generated in the final state following a slow[12,13]or a sudden quench[14,15], or generation of quantum correlations which are otherwise absent in the defect free final state[16]or the possibility thermalization with an effective temperature[17]are some of the topics which are being explored thoroughly.In this communication, we focus on the dynamics of edge states of the BHZ Hamiltonian when the system is suddenly quenched from the TI phase to either the QCP or the trivial insulator (TrI) phase. The question here is whether there is a surviving edge current following the quench. (It is to be noted that when a one-dimensional chain of hard core Bosons is quenched from the superfluid to the Mott Insulator state, there is a surviving supercurrent in the insulator phase which oscillates in time[18]). In our problem, the quench couples the twolevel subspace of the edge states to a multi-level environment of bulk states. We study the decoherence of these edge states using the Loschmidt echo (LE) [19] when the 2-D Hamiltonian is quenched from the TI phase to the QCP (or to the TrI phase). We observe a strong oscillation of the low-energy edge states between the two edges of the system and the time-averaged persistence of the spin Hall current (SHC) when the system is quenched to the QCP, which we attribute directly to the linear low-energy dispersion at this point (which is also found in other models describing 2-D TIs). The experimental prospect of real-time tuning of parameters controlling QPTs of TIs in optical [20] and photonic lattices [21] as well as by exploiting Floquet dynamics[22], has made the study of quenching dynamics of TI Hamiltonians relevant and important. It is to be noted that slow quenching results in a violation of the Kibble-Zurek scaling of defects in systems with edge states[23].The 4 × 4 BHZ Hamiltonian comprising of two 2×2 blocks (for opposite electron spins) is given bywhereHere, A, B, C, D and m are determined by the thickness of the quantum well and the material parameters; the parameter m controls the phase of the system and changes sign relative to B when the system crosses from the TI phase (where edge states are present) to the TrI phase (with no edge states) via a DP at m/B = 0. Although the results presented here are valid in generic situations, we shall set D = 0 for simplicity, which also ensures an electron-hole symmetric spectrum. We consider a ribbon geometry extending from −L/2 to L/2 in the y direction (with the wavefunction vanishing at the edges) and apply periodic boundary conditions in the x-direction[25][26][27].To obtain the spectrum of Hamiltonian (1), we consider the 2 × 2 block H(k), fix k x , and use k y → −i∂ y , with the trial solution ψ = (A 1 , A 2 )e λy [25]. The condition that the wavefunctions must vanish at the edges of the ribbon quantizes the energies E of the eigenstates at a given k x , which are given by the solutions of the following transcendental equation: tanh(λ + L/2) tanh(λ − L/2) + tanh(λ − L/2) tanh(λ + L/2) = λ 2 + + λ 2 − − (B/A) 2 (λ 2arXiv:1304.2248v1 [cond-mat.mes-hall] We study the dynamics of edge states of the two dimensional BHZ Hamiltonian in a ribbon geometry following a sudden quench to the quantum critical point separating the topological insulator phase from the trivial insulator phase. The effective edge state Hamiltonian is a collection of decoupled qubit-like two-level systems which get coupled to bulk states following the quench. We notice a pronounced collapse and revival of the Loschmidt echo for low-energy edge states illustrating the oscillation of the state between the two edges. We also observe a similar collapse and revival in the spin Hall current carried by these edge states, leading to a persistence of its time-averaged value. Topological insulators (TIs) are novel materials with an insulating bulk and conducting edges which are of extensive contemporary interest [1][2][3][4]. The low-energy electrons in two dimensional (2-D) Hg-Te/Cd-Te quantum well TIs, which display conducting helical edge state solutions that exist within the bulk bandgap in the TI phase, are described by the 2-D BHZ Hamiltonian [2,4]. The bulk states undergo a quantum phase transition (QPT) [5][6][7] with the low-energy modes satisfying a 2-D Dirac Hamiltonian (DH) with a linear dispersion at the quantum critical point (QCP) (which is a 2-D Dirac point). Additionally, the chiral edge states with linear dispersion in the TI phase are described by an effective 1-D DH [2]. At the same time, there is a recent upsurge in studies of quenching dynamics of quantum many body systems across QCPs [8][9][10], essentially because of the possibility of experimental realization of the same in optical lattices [11]. The scaling of the defect density generated in the final state following a slow [12,13] or a sudden quench [14,15], or generation of quantum correlations which are otherwise absent in the defect free final state [16] or the possibility thermalization with an effective temperature [17] are some of the topics which are being explored thoroughly. In this communication, we focus on the dynamics of edge states of the BHZ Hamiltonian when the system is suddenly quenched from the TI phase to either the QCP or the trivial insulator (TrI) phase. The question here is whether there is a surviving edge current following the quench. (It is to be noted that when a one-dimensional chain of hard core Bosons is quenched from the superfluid to the Mott Insulator state, there is a surviving supercurrent in the insulator phase which oscillates in time [18]). In our problem, the quench couples the twolevel subspace of the edge states to a multi-level environment of bulk states. We study the decoherence of these edge states using the Loschmidt echo (LE) [19] when the 2-D Hamiltonian is quenched from the TI phase to the QCP (or to the TrI phase). We observe a strong oscillation of the low-energy edge states between the two edges of the system and the time-averaged persistence of the spin Hall current (SHC) when the system is quenched to the QCP, which we attribute directly to the linear low-energy dispersion at this point (which is also found in other models describing 2-D TIs). The experimental prospect of real-time tuning of parameters controlling QPTs of TIs in optical [20] and photonic lattices [21] as well as by exploiting Floquet dynamics [22], has made the study of quenching dynamics of TI Hamiltonians relevant and important. It is to be noted that slow quenching results in a violation of the Kibble-Zurek scaling of defects in systems with edge states [23]. The 4 × 4 BHZ Hamiltonian comprising of two 2×2 blocks (for opposite electron spins) is given by H BHZ = H( k) 0 0 H * (− k) ,(1) where H(k) = [C − D(k 2 x + k 2 y )]I 2×2 + A[k x σ x + k y σ y ] + [m − B(k 2 x + k 2 y )]σ z . Here, A, B, C, D and m are determined by the thickness of the quantum well and the material parameters; the parameter m controls the phase of the system and changes sign relative to B when the system crosses from the TI phase (where edge states are present) to the TrI phase (with no edge states) via a DP at m/B = 0. Although the results presented here are valid in generic situations, we shall set D = 0 for simplicity, which also ensures an electron-hole symmetric spectrum. We consider a ribbon geometry extending from −L/2 to L/2 in the y direction (with the wavefunction vanishing at the edges) and apply periodic boundary conditions in the x-direction [25][26][27]. To obtain the spectrum of Hamiltonian (1), we consider the 2 × 2 block H(k), fix k x , and use k y → −i∂ y , with the trial solution ψ = (A 1 , A 2 )e λy [25]. The condition that the wavefunctions must vanish at the edges of the ribbon quantizes the energies E of the eigenstates at a given k x , which are given by the solutions of the following transcendental equation: where tanh(λ + L/2) tanh(λ − L/2) + tanh(λ − L/2) tanh(λ + L/2) = λ 2 + + λ 2 − − (B/A) 2 (λ 2 + − λ 2 − ) 2 λ + λ − ,(2)k x (nm -1 ) k x (nm -1 ) k x (nm -1 ) E (meV) E (meV) E (meV) (a) TI Phase (b) QCP (c) TrI Phaseλ ± = k 2 x + F ± F 2 − (M 2 − E 2 )/B 2 , with F = (A 2 − 2M B)/(2B 2 ) . The corresponding wavefunctions are given by ψ(x, y) = e ikxx (c + f + (y) + c − f − (y)) ,(3) where c ± are two-component spinors whose entries are determined by the boundary conditions, with f + (y) = cosh(yλ + )/ cosh(Lλ + /2)− cosh(yλ − )/ cosh(Lλ − /2), and f − (y) = sinh(yλ + )/ sinh(Lλ + /2)− sinh(yλ − )/ sinh(Lλ − /2). One can show that in the TI phase (m/B > 0), there are two types of eigenstates of the Hamiltonian, i.e., edge states (localized towards the edges and decaying exponentially over a length 1/λ − (k x , m)) and bulk states (spreading across the whole ribbon). In order to have true edge states, one needs L 1/λ − . The spectrum, which is symmetric in ±k x and ±E, is displayed in Fig. 1. The edge states in the TI phase exist for \|k x \| < k 0 , k 0 depends on m and L [26]. The solutions of the two 2 × 2 blocks are time reversed conjugates of each other, with the same set of energies but opposite momentum and spin. We now perform a sudden quench of the parameter m going from m/B > 0 to m/B ≤ 0 and look at the subsequent evolution of an edge state and its spin current. The edge states, which originally formed a qubit-like two level system with an effective Hamiltonian H edge ≈ Ak x σ z at each k x [4] now get coupled to several bulk modes and subsequently decohere. Following a sudden quench, the evolution of an edge state is given by \|ψ edge (k x , t) = ∞ n=−∞ ψ n (k x )\|ψ edge (k x ) e −iEnt \|ψ n (k x ) ,(4) where \|ψ edge (k x ) is an edge eigenstate of the Hamiltonian at the initial value of m (= m 1 ) and \|ψ n (k x ) are the eigenstates of the Hamiltonian at the final value m 2 . The index n runs from −∞ to ∞ excluding n = 0 and denotes the -ve and +ve energy bulk modes, respectively. Since all the modes are plane waves along the x direction, different k x modes do not couple to each other. To study the dynamics of a single edge state and quantify its decay, we calculate the LE L(t) = \| ψ edge \|e iH(m1)t e −iH(m2)t \|ψ edge \| 2 , which using Eq.(4) can be put in the form L edge (k x , t) = ∞ n=−∞ \| ψ n (k x )\|ψ edge (k x ) \| 2 e −iEnt 2 .(5) In general, the LE defined above initially drops rapidly with time and turns into a rapidly oscillating noisy function of small amplitude (Fig. 2(a)), indicating that the edge state decoheres significantly. However, there is a striking difference when one looks at the evolution of a low-energy (k x << k 0 ) edge state following a quench to the QCP at m = 0; the LE of the edge state shows a pronounced collapse and nearly complete revival for several oscillation cycles (Fig. 2(b)). This is a consequence of the nearly equal spacing of the first few energy levels at low k x near m = 0 (arising due to confinement of the linearly dispersing particles (ν = 1 at the QCP) in a ribbon geometry) where the overlap with the edge state is the most significant (see Fig. 1). We then have E n ≈ sign(n)[E g /2+(\|n\|−1)∆E] for all significant terms in Eq.(4), where E g ∼ 1/L is the bulk bandgap. The summations over n > 0 and n < 0, then represent Fourier series of a peroidic function with period τ = 2π ∆E ≈ 2h L A ,(6) making the LE a periodic function with this period. Since L(t = 0) = 1, the LE shows a near-complete revival at t = nτ . Eventually, after several oscillations, the slight non-uniformity in spacing becomes significant and the revival of the LE weakens. Since ∆E ∼ 1/L, the period of this revival scales as L. For small quench amplitudes (\|m\| << A/L), the edge state does not decay significantly. Interestingly, we find that the edge state travels from one edge to the other and back, existing on opposite edges at the points of maxima and minima of the LE (Fig. 3(a)). This effect is due to the finite width of the ribbon and will not be seen in an infinite system. Since, for a significantly large system, the edge states at opposite edges do not overlap, the LE drops to zero when the edge state reaches the opposite edge, and revives again when it comes back. For very low values of momentum k x → 0, the edge state exists with peaks on both edges of a finite ribbon [25]. Hence, when a peak on a given The probabilty current density of the state in the x direction (Jx) at t = 0 (solid) and t = τ /2 (dashed). The same state carries currents of opposite direction on opposite edges. The system parameters are the same as Fig. (2). edge travels to the opposite edge, the peak on the opposite edge also travels simultaneously to the given edge, resulting in a maximum of the LE at (2n + 1)τ /2 instead of a minimum, and hence a doubling of the frequency of oscillation of the LE. The LE is now minimum at times when the state is concenterated near the middle of the ribbon (Fig. 2(c)). The probability current carried near the edge in the x direction by the edge states over their decay length 1/λ − is proportional to the net SHC carried by the state and its time-reversed conjugate in the opposite spin sector. It can be calculated using the continuity equation for the probability current density J in conjunction with the Schrodinger time evolution equation ∂ ∂t ψ † (x, y, t)ψ(x, y, t) + ∇ · J = 0, i ∂ ∂t ψ(x, y, t) = Hψ(x, y, t),(7) where ψ(x, y, t) is the time dependent twocomponent wavefunction of the form ψ(x, y, t) = (φ 1 (y, t), φ 2 (y, t))e ikxx . Using k → −i ∇, we obtain J x (y, t) = 2A (φ 1 (y, t)φ * 2 (y, t) + φ * 1 (y, t)φ 2 (y, t)) + 2Bk x \|φ 2 (y, t)\| 2 − \|φ 1 (y, t)\| 2 ,(8) with A and B as defined in (1). For k x → 0, the first Dirac-like term dominates the second Schrodinger-like term, implying that all low-energy edge states carry virtually the same current. The profile of J x is shown at the instants of time when the edge state exists on opposite edges in Fig. 3(b). The sign of the current reverses when the state moves to the opposite edge. The evolution of the SHC carried by an edge state near a given edge (say, −L/2) can be obtained by integrating J x from −L/2 to −L/2 + 1/λ − ; i.e., I edge (t) = −L/2+1/λ −L/2 J x (y, t)dy. Since there are two oppositely propagating edge states at a given energy which exist on opposite edges of the system, we must add the currents carried by both of them. The time evolution of such a current following a sudden quench to the QCP is shown in Fig. 4. Due to the oscillation of the edge states between the two edges, this current also displays a pronounced collapse and revival for several cycles. The time-averaged value of the current is non-zero and is a significant fraction of the original value of the current. Thus, there is a persistence of the SHC carried by lowenergy edge states following a sudden quench to the QCP. After several oscillations, the edge state disperses and does not regain it's original character again. This dispersion is introduced by the non-linear Bk 2 term in the Hamiltonian, and must be sufficiently small if sustained oscillations are to be observed. Using the condition that the spread of the edge state over one time period must be significantly smaller than it's initial decay length 1/λ − , we obtain the condition (BL)/A < ∼ 1/λ 2 − . Since we already have L 1/λ − , this condition implies that these oscillations will be seen over an intermediate range of ribbon widths roughly determined by the values of the parameters A and B. It has already been mentioned that recently a realization of a Floquet TI has been experimentally achieved in photonic lattices [21], where the quenching dynamical study discussed above can possibly be verified. The dynamics of a Floquet quantum system driven with a time period T are determined by its Floquet evolution operator U (T ), which may be associated with an effective Hamiltonian H ef f [28] as U (T ) = T e −(i/h) T 0 H(t)dt = e −(i/h)H ef f T , where T denotes the time ordering symbol. Thus, in a Floquet system, the evolution of an edge eigenstate of H ef f (m 1 ) following a sudden quench to m 2 is like a normal Schrodinger evolution under H ef f (m 2 ) at instants of time t = nT . For these oscillations to be visible in experimental studies of such a system, one needs T τ (see, Eq. (6)). To summarize, we observe interesting decoherence dynamics of the low-energy edge states of a TI Hamiltonian when quenched to the QCP displayed in the temporal evolution of their LE. Given the recent prospects of tuning of control parameters of the related Hamiltonians, we believe that this study is of experimental relevance. Furthermore, one can show that the predicted oscillations of edge modes should occur in any 1-D or effectively 1-D system with a linearly dispersing (ν = 1) QCP (such as the 1-D p-wave superconducting chain with Majorana boundary modes), due to the fact that the energy levels at the QCP in a finite system will be more or less equally spaced [29]. We acknowledge Apoorva Patel and Diptiman Sen for helpful discussions. AAP acknowledges the KVPY fellowship and AD and SS acknowledge CSIR, New Delhi, for financial support. PACS numbers: 64.70.Tg, 03.65.Pm, 74.40.Kb FIG. 1 : 1Spectrum of the BHZ Hamiltonian in a ribbon geometry with ribbon width L = 200 nm. The parameters used are A = 364.5 meV /nm, B = −686 meV /nm 2 and C = D = 0. The values of m used are −10 meV (a), 0 (b) and +10 meV (c). There is a small gap of O(A/L) in the spectrum near E = 0 at m = 0 because of the finite width of the ribbon.There also exists an exponentially small gap between the edge state bands in the TI phase. Note the (almost) equal spacing of energy levels for kx = 0 at m = 0, characterized by the solutions of a Dirac particle geometrically confined in a 1-D box. FIG. 2 : 2Loschmidt Echo for various edge states and quenches. The system parameters are A = 364.5 meV /nm, B = −686 meV /nm 2 , C = D = 0 and L = 400 nm. LE for an edge state with: (a) kx = 0.01 meV /nm and m = −10 meV after quenching to m = +10 meV . There is no significant revival of the edge state. (b) kx = 0.001 meV /nm and m = −10 meV after quenching to the QCP at m = 0. There is a pronounced collapse to 0 and a nearly full recovery of the LE for several cycles. (c) kx = 0 and m = −10 meV after quenching to the QCP. There is a doubling of the frequency of oscillation as compared to the previous case, as this edge state exists on both edges. probability density ρ = ψ † ψ of an edge state (kx = 0.001 meV /nm) following a sudden quench from m = −10 meV to m = 0 shown at t = 0 (solid) and t = τ /2 (dashed). The edge state travels between the two edges. (b) FIG. 4 : 4The net probabilty current carried by the pair of edge states with E = 3.65 meV (near the −L/2 edge over a length 1/λ− ≈ 37 nm) following a quench from m = −10 meV to m = 0. The current collapses and revives like the LE, and is always greater than zero, leading to a persistence of its time-averaged value. The system parameters are A = 364.5 meV /nm, B = −686 meV /nm 2 , C = D = 0 and L = 200 nm. . 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[ "Pseudonatural Inflation", "Pseudonatural Inflation" ]	[ "Nima Arkani-Hamed \nJefferson Physical Laboratory\nHarvard University\n02138CambridgeMAUSA\n", "Hsin-Chia Cheng \nJefferson Physical Laboratory\nHarvard University\n02138CambridgeMAUSA\n", "Paolo Creminelli \nJefferson Physical Laboratory\nHarvard University\n02138CambridgeMAUSA\n", "Lisa Randall \nJefferson Physical Laboratory\nHarvard University\n02138CambridgeMAUSA\n" ]	[ "Jefferson Physical Laboratory\nHarvard University\n02138CambridgeMAUSA", "Jefferson Physical Laboratory\nHarvard University\n02138CambridgeMAUSA", "Jefferson Physical Laboratory\nHarvard University\n02138CambridgeMAUSA", "Jefferson Physical Laboratory\nHarvard University\n02138CambridgeMAUSA" ]	[]	We study how to obtain a sufficiently flat inflaton potential that is natural from the particle physics point of view. Supersymmetry, which is broken during inflation, cannot protect the potential against non-renormalizable operators violating slow-roll. We are therefore led to consider models based on non-linearly realized symmetries. The basic scenario with a single four-dimensional pseudo Nambu Goldstone boson requires the spontaneous breaking scale to be above the Planck scale, which is beyond the range of validity of the field theory description, so that quantum gravity corrections are not under control. A nice way to obtain consistent models with large field values is to consider simple extensions in extra-dimensional setups. We also consider the minimal structures necessary to obtain purely four-dimensional models with spontaneous breaking scale below M P ; we show that they require an approximate symmetry that is supplemented by either the little-Higgs mechanism or supersymmetry to give trustworthy scenarios.	10.1088/1475-7516/2003/07/003	[ "https://arxiv.org/pdf/hep-th/0302034v2.pdf" ]	118,904,004	hep-th/0302034	211ac7f0cb84af9df290092ccabc067986718d42	Pseudonatural Inflation May 2003 Nima Arkani-Hamed Jefferson Physical Laboratory Harvard University 02138CambridgeMAUSA Hsin-Chia Cheng Jefferson Physical Laboratory Harvard University 02138CambridgeMAUSA Paolo Creminelli Jefferson Physical Laboratory Harvard University 02138CambridgeMAUSA Lisa Randall Jefferson Physical Laboratory Harvard University 02138CambridgeMAUSA Pseudonatural Inflation May 2003arXiv:hep-th/0302034v2 13numbers: 9880Cq1130Pb1110Kk We study how to obtain a sufficiently flat inflaton potential that is natural from the particle physics point of view. Supersymmetry, which is broken during inflation, cannot protect the potential against non-renormalizable operators violating slow-roll. We are therefore led to consider models based on non-linearly realized symmetries. The basic scenario with a single four-dimensional pseudo Nambu Goldstone boson requires the spontaneous breaking scale to be above the Planck scale, which is beyond the range of validity of the field theory description, so that quantum gravity corrections are not under control. A nice way to obtain consistent models with large field values is to consider simple extensions in extra-dimensional setups. We also consider the minimal structures necessary to obtain purely four-dimensional models with spontaneous breaking scale below M P ; we show that they require an approximate symmetry that is supplemented by either the little-Higgs mechanism or supersymmetry to give trustworthy scenarios. I. INTRODUCTION Inflation is surely the most compelling paradigm for solving many problems of the standard big bang cosmology [1,2,3]. Besides its theoretical appeal, its basic predictions of a flat Universe with a nearly scale-invariant spectrum of adiabatic perturbations are now experimentally well tested by the Cosmic Microwave Background Radiation (CMBR) anisotropies and the Large Scale Structures (LSS) galaxy surveys. The basic framework can be realized in models as simple as a single scalar field with a monomial potential. Although such simple toy-models can be attractive, they are tremendously unnatural from the particle physics point of view. In scenarios where the inflaton takes values above the Planck mass (M P = (8πG) −1/2 ) [4], the use of a simple potential requires the fine-tuning of an infinite number of non-renormalizable operators, suppressed by powers of M P . The inflaton potential must be sufficiently flat to allow a slow-rolling phase, but at the same time it must couple to other fields to provide an efficient reheating and, in hybrid models, to trigger the final phase transition. There are only two known candidates for keeping a scalar potential nearly flat and stable under radiative corrections: supersymmetry (SUSY) and non-linearly realized symmetries. The latter mechanism applies both to a pseudo Nambu-Goldstone boson (PNGB) and to the extra components of gauge fields propagating in extra dimensions; both are protected at lowest order by a shift symmetry. So far most of the attention in inflation model-building has been devoted to supersymmetry, but this symmetry alone cannot naturally provide potentials that are flat enough for inflation, once supergravity effects are included. In the next section (section II), we will describe this problem and review the proposed solutions. None of them is completely compelling. In section III we turn our attention to the other, much less studied, candidate: the shift symmetry. Even if the Goldstone theorem ready provides flat directions for inflation, it is not trivial to build inflationary models based on PNGBs, essentially because both the potential and its slope vanish in the limit in which the explicit breaking is turned off. The simplest scenario with a single PNGB does not work unless the symmetry breaking scale is higher than the Planck scale, which is presumably outside the range of validity of an effective field theory description. Moreover it is expected that quantum gravity effects will explicitly break the global symmetry, giving a typical scale for the potential of order M P , far too big to satisfy the COBE constraint. We show that these problems are not present in theories with extra dimensions. In particular, the extra components of gauge fields living in extra dimensions provide natural candidates for the inflaton [5]. In the rest of the paper we concentrate on the requirements for building purely 4d models with PNGBs, with a symmetry breaking scale below M P . This requires more complicated structures such as hybrid inflation models [6]. In section IV we discuss the necessary ingredients for building natural 4d models. The PNGB potential of the inflaton needs protection from the interactions which are required to end inflation and to reheat the Universe. We present a SUSY model as well as non-SUSY models based on the same recent ideas which were used to build new models of electroweak symmetry breaking. Some of the details are left to the Appendix. We draw our conclusions in section V. II. HIGHER DIMENSION OPERATORS AND THE SUGRA η PROBLEM As noted in the introduction, non-renormalizable operators are clearly very crucial in models of inflation in which the inflaton variation is bigger than the Planck scale, because they are naively more important than lower dimension operators. This makes it very hard to justify any 4d model with a big variation of field values. We will discuss in the next section how this problem can be solved in models in which the 4d effective field theory is the dimensional reduction of an higher dimensional theory. Non-renormalizable operators are important also in theories where the inflaton variation is much smaller than M P . This is clear if one considers operators of dimension 6, which can give a mass term V M 2 P φ 2 ∼ H 2 φ 2(1) to the inflaton, spoiling slow-roll. One would think that supersymmetry can provide flat directions for inflation in a rather natural way; however, it is known that this is not quite true once supergravity corrections are included [7]. Non minimal terms in the Kähler potential can obviously give contributions like (1), but the same kind of corrections are present also with a minimal Kähler potential. The supergravity potential, neglecting for the moment the D-term contribution, can be expressed as a function of the Kähler potential K and the holomorphic superpotential W as V = e K/M 2 P (K −1 ) i j L i L j − 3 \|W \| 2 M 2 P ,(2) where L i ≡ W i + K i · W M 2 P . During inflation supersymmetry is broken because the vacuum energy is positive. Taking, at the lowest order, a canonically normalized Kähler potential K = φ * φ, the exponential factor in front of V gives a mass to any flat direction of order V /M 2 P ∼ H 2 . This point is quite clear in the superconformal formalism, where the kinetic term for φ can be expressed using a superconformal compensator Φ as d 2 θd 2θ ΦΦ † φφ † .(3) As ΦΦ † contains the Ricci scalar, we obtain a non-minimal coupling of φ to gravity which gives the mass correction during inflation. This effect gives a tilt to the inflaton potential and it is simple to check that its contribution to the slow-roll η parameter (η ≡ M 2 P V ′′ /V ) is exactly 1, while a slow-roll phase requires η ≪ 1. There can be additional contributions in the potential (2) which are of the same order of magnitude and a cancellation is possible. Nevertheless this required cancellation introduces a fine-tuning problem, which is often referred to as the η-problem. The on-going experiments on the CMBR and on LSS are making the problem increasingly acute. A conservative limit on the spectral index is now \|n − 1\| < 0.1, which turns into a limit for η: η < 0.05. Unless a better reason for the cancellation is found, a fine-tuning of at least 1/20 is required. The situation somewhat resembles the Higgs hierarchy problem: the top Yukawa and the gauge and quartic couplings would drive the Higgs mass towards the scale Λ where new physics comes in, but a certain separation is required to account for electroweak precision tests. Here gravity itself drives the inflaton mass towards H, but again the two scales must be separated to allow a sufficient amount of inflation. Several ways to overcome the problem have been proposed. Before reviewing them, we want to stress that none of them is entirely convincing. Most of them rely on assumptions about the fundamental theory, which cannot be justified from the effective low energy point of view. This is not better than assuming a certain cancellation among the various terms in (2). Below we discuss some proposed solutions to the η-problem. For additional references see [8]. Superpotential linear in the inflaton [7]. It is easy to verify that in this case the contribution to η coming from the exponential factor in (2) is canceled. However, one is left to assume a small quartic term (φφ * ) 2 in K. Even if the situation might be considered better than in the general case, the fine-tuning problem is still there, as there is no symmetry which can protect the smallness of this term. Particular form of the Kähler potential. If T is a modulus (e.g., a compactification radius) and φ is the inflaton and the Kähler potential depends only on the combination ρ ≡ (T + T * − φ * φ) (e.g., K = −3 log ρ), a flat direction is preserved [9]. This could be ensured by a so-called Heisenberg symmetry and it seems to be quite generic in orbifold compactifications of superstrings. The problem is that, during inflation, one also gets a runaway potential in the ρ direction. It is hard to justify why a stabilization mechanism should depend on the ρ variable and not on T itself, as the Heisenberg symmetry is not a symmetry of the full theory. On the other hand, if T is stabilized, corrections to the inflaton potential are reintroduced, giving η ∼ 1. All this kind of solutions relies on particular features of the Kähler potential, which cannot be justified in term of symmetries of the low energy theory, but must be taken from the UV stringy completion. D-term inflation. In addition to the F-term potential (2), D-term contributions are also present: V D = g 2 2 Ref −1 AB D A D B , D A = K i (T A ) j i φ j + ξ A ,(4) where f is the gauge kinetic function, T A 's are the gauge group generators and ξ is a Fayet-Iliopoulos (FI) term, which is admissible only for U(1) groups. If the vacuum energy during inflation is dominated by a D-term, the η-problem is simply not there [10]. One can easily build a hybrid inflation model taking the inflaton to be a neutral superfield S, coupled to two charged multiplets φ + and φ − through a superpotential W = λSφ + φ − . For large values of the scalar component of S, φ + and φ − are stuck at the origin, so that the gauge symmetry is unbroken and the vacuum energy is dominated by the FI term. For smaller values, the negatively charged scalar becomes tachyonic and we go to a vacuum where the U(1) is broken and the vacuum energy is zero. The S direction is classically flat, but it is lifted by quantum corrections as supersymmetry is broken. The potential is generically flat enough to allow slow-roll and no η-problem seems to be present. However, after a closer look, it seems difficult to get a viable scenario of inflation both with an anomalous U(1) and with a non-anomalous one. 1. Anomalous U(1) with Green-Schwarz mechanism of anomaly cancellation [11]. As in this case the non-linear transformation of the dilaton cancels the anomaly, its behavior is clearly crucial: during inflation the dilaton gets a runaway potential and it must be stabilized. The stabilization mechanism generically gives F-term contributions bigger than D-terms [12], thus reintroducing the η-problem. 2. An explicit FI term for a non-anomalous U(1) is introduced. Supergravity requires this U(1) to be an R-symmetry [13]. [13]. This implies that the U(1) symmetry must in fact be a gauged version of the U(1) R symmetry: the gravitino must be charged. The framework is now quite constrained: taking the normalization in which the charge of the gravitino is −1, the vacuum energy during inflation is given by V 0 = g 2 2 ξ 2 = g 2 2 (2M 2 P ) 2 ;(5) where the FI term is fixed by the gravitino charge. The R-symmetry requires that the superpotential has charge +2: as the scalar component of S must be neutral, φ + and φ − have charge 2 + Q and −Q respectively. The classically flat direction S is lifted by quantum corrections and it gets a potential 2 : V 1−loop = g 2 ξ 2 2 1 + 2 + 2Q + Q 2 8π 2 g 2 log λ 2 \|S\| 2 µ 2 ,(6) 1 We thank M. Luty for stressing this point to us. 2 Note that the effective potential for S is different from the case [10] of a U(1) which is not an R-symmetry, because here the charges of φ + and φ − do not add up to zero and an additional term ∝ \|S\| 2 log(\|S\| 2 /µ 2 ) is induced. Anyway its contribution is at most comparable to the log \|S\| 2 piece, so that neglecting it does not alter the conclusions. where µ is the renormalization scale. It is easy to obtain the constraint from the COBE normalization (see e.g. [8]), which is independent of the coupling constant g and requires a huge charge Q: ξ Q ≃ 10 16 GeV ⇒ Q ∼ 10 6 .(7) This is technically natural but quite unreasonable, especially because anomaly cancellation gives strong contraints on the spectrum [14,15]. Even if we allow additional contributions to lift the inflaton potential, the vacuum energy (5) is clearly too big to satisfy the COBE normalization unless g is very small. In summary, due to the large vacuum energy during inflation, supersymmetry is badly broken in such a way that it lost the power to preserve the flat direction required by inflation. We need some other mechanism to obtain a natural flat potential for inflation. III. SHIFT SYMMETRIES I: GENERAL DISCUSSION If we want to explain the lightness of the inflaton from the low energy point of view, we must rely on symmetry arguments. Supersymmetry alone is insufficient, as we explained above, so that one is naturally led to consider approximate bosonic shift symmetries: i.e., the inflaton as a pseudo Nambu-Goldstone Boson. This is certainly not a new idea. Models of inflation based on PNGBs were discussed in [16,17] and in many subsequent works 3 . In this paper we want to emphasize that this seems to be the only natural way to keep the potential flat for a slow-roll inflation. However, it is not straightforward to obtain satisfactory models. The original "natural inflation" model is based on a single PNGB field parametrized by an angular variable θ ∼ θ + 2π. In the limit of exact symmetry θ is a flat direction. With the addition of an explicit breaking term the Lagrangian is of the form L = 1 2 f 2 (∂θ) 2 − V 0 (1 − cos θ) ,(8) where f is the spontaneous breaking scale. The canonically normalized field is φ = f θ, so the potential is naturally a function of φ/f , which can be flat for large f . This scenario is however problematic, because the requirements ǫ ≪ 1 and η ≪ 1 on the slow-roll parameters ǫ ≡ M 2 P 2 V ′ V 2 ∼ M 2 P f 2 , η ≡ M 2 P V ′′ V ∼ M 2 P f 2(9) gives f ≫ M P . If we interpret f as some symmetry breaking vacuum expectation value (VEV), then this would require that the field theory is valid above M P , which is hard to justify. But the real problem is that we expect that quantum gravity effects, such as the virtual appearance of black holes, will explicitly break the approximate symmetry 4 . These effects, usually suppressed by powers of f /M P , are here unsuppressed 5 , so that it is hard to justify why V 0 should be smaller than M P , as required by the COBE bound on the overall normalization of density perturbations: δρ/ρ ∼ 10 −5 . It is the same problem of higher dimension operators we discussed in the previous section: here the inflaton variation is bigger than M P so that non-renormalizable operators are important. Quantum gravity effects will induce higher-dimension operators which badly break the symmetry, changing the potential in (8). Therefore a single PNGB in a 4d field theory with the simple potential in (8) cannot provide a satisfactory model of inflation. The situation is changed when we consider theories with extra dimensions [5]. If the 4d effective field theory description comes from the dimensional reduction of a higher dimensional theory, it is possible to build models with variation of the inflaton field bigger than similarly to what happens to a four-dimensional PNGB. A non-local potential as a function 4 Quantum gravity effects on a PNGB potential are known to be dangerous in the case of the axion. See e.g. [21]. 5 Naive dimensional analysis suggests, in the limit of strong coupling gravity, that higher dimension operators arising from quantum gravity effects are suppressed by M P = (8πG) −1/2 and not by the alternative definition of the Planck mass G −1/2 . of the gauge invariant Wilson loop e iθ = e i A 5 dx 5(10) will however be generated in presence of charged fields in the bulk. At energies below 1/R, θ is a 4d field with a Lagrangian of the form L = 1 2 g 2 4 (2πR) 2 (∂θ) 2 − V (θ) + · · ·(11) where g 2 4 = g 2 5 /(2πR) is the 4D gauge coupling, and the potential V (θ) is given at one-loop by [22,23,24,25,26] V (θ) = − 1 R 4 I (−1) F I 3 64π 6 ∞ n=1 cos(nqθ) n 5 ,(12) where F I = 0(1) for massless bosonic (fermionic) fields of charge q coupled to A 5 . Note that the potential is of the same form as in natural inflation (with small corrections from additional terms in the sum), with the effective decay constant given by f eff = 1 2πg 4d R .(13) It is easily seen that f eff can be bigger than M P for sufficiently small g 4d ; the slow-roll condition f eff ≫ M P requires only that 2πg 4d M P R ≪ 1 .(14) The canonically normalized field is φ = θf eff . Due to the higher dimensional nature of the model, the potential (12) can be trusted even when the 4d field φ takes values above M P ; no dangerous higher-dimension operator can be generated in a local higher-dimensional theory. This conclusion is quite important as it is commonly believed that any inflation model with field values above M P cannot be justified from a particle physics point of view; we see that It is worthwhile stressing that a variation of the inflaton field during inflation bigger than M P is required to have a significant and measurable production of gravitational waves [27]. It seems that the only way to get a realistic scenario of this kind is in an extra-dimensional setup. Another example using extra dimensions is the idea of "brane inflation" [28]. Also this model can be considered based on a PNGB. In fact, the inflaton is the field which describes the distance between two branes. It is massless in the limit in which we neglect the interactions between the two branes, because it is the Goldstone boson of the broken translational invariance. The non-trivial potential generated by the interactions between the two branes has to be very flat when two branes are far apart, again by the locality in extra dimensions. From the 4d point of view the inflaton takes values above the Planck scale, but the extra dimensional completion allows to control higher-dimension operators. Moreover quantum gravity effects are again suppressed by locality, which is really the key ingredient of this type of models. One could ask whether it is possible to derive a purely 4d theory from the simple 5d model based on the Wilson line by applying the recent idea of deconstructing dimensions, where the Wilson line in the extra dimension corresponds to a 4d PNGB [29,30,31,32]. In this case one replaces the 5d gauge theory by a chain of 4d gauge groups, with the adjacent gauge groups connected by the link fields, which get nonzero VEVs and break the gauge groups down to the diagonal one. There is one linear combination of the Nambu-Goldstone bosons not eaten by the massive gauge fields. It remains light and corresponds to the nonlocal Wilson line field in the 5d case. However the symmetry breaking scale, f link = √ Nf eff , where f link is the VEV of the link fields and N is the number of the sites, is still required to be bigger than M P . In the rest of the paper we concentrate on 4d models, by which we mean that there is no (gravitational) extra dimension with size larger than the Planck length and the theory is 4-dimensional all the way up to the 4d M P . In this case we can not use the locality in extra dimensions to protect the flat inflaton potential and it is only sensible to consider f ≪ M P . As explained, this is not consistent with slow-roll in a scenario with the simplest PNGB potential, so that one is naturally led to consider models which involve more than one field. With f ≪ M P the corrections to the inflaton potential due to quantum gravity effects can be sufficiently suppressed if the explicit symmetry breaking operators arising from quantum gravity are prohibited up to a high dimension. Operators of dimension six contributing directly to the inflaton mass are still dangerous because their effect on the mass can be of order V /M 2 P ∼ H 2 . We would like to emphasize that quantum gravity corrections crucially depend on the UV completion of each model below the Planck scale. There are many ways to suppress quantum gravity effects. Besides locality in the extra space discussed above, additional (discrete or continuous) gauge symmetries in the UV theory can forbid dangerous operators. For example, in a dimensionally deconstructed gauge theory with many sites, the PNGB is the product of many link variables, so that the only gauge invariant operators are of very high dimensions and the Planck scale effects are suppressed by many powers of f /M P [33]. IV. SHIFT SYMMETRIES II: FIELD VALUES SMALLER THAN M P To obtain trustworthy 4d inflation models we must require the symmetry breaking scale f and field values smaller than M P so that the simple potential (8) does not work. A more complicated PNGB potential is needed. In particular, the variation of the potential during inflation and the total height of the potential should be controlled by different terms with different scales. A sharp drop of the potential is therefore needed at some point to end inflation. However, such sharp drop in the potential explicitly breaks the shift symmetry of the inflaton field and may spoil the flatness of the inflaton potential through radiative corrections. We need to examine this point in more detail. Let us first consider the case of single field inflation. Being a PNGB, its potential is periodic. To have a separation of scales inflation must occur near the maximum so that the potential is sufficiently flat, otherwise we come back to the requirement of natural inflation f ≫ M P . Near the maximum (chosen to be at ψ = 0) we can expand the potential V (ψ) = V 0 − m 2 2 ψ 2 − λ 4 ψ 4 + · · · .(15)If V 0 ∼ m 2 f 2 , where f is the symmetry breaking scale and is also the maximal variation of ψ, as in the case of (8), then again we are stuck with the troublesome relation f ≫ M P . Therefore, we must demand that the total potential near the maximum is dominated by the higher order terms, e.g., m 2 f 2 ≪ \|λ\|f 4 ;(16) the potential is flattened near the maximum with respect to a conventional PNGB. On the other hand, a quadratically divergent contribution to the mass term will be generated by the quartic coupling, ∆m 2 = − 3λ Λ 2 16π 2 .(17) Comparing it with (16), we see that the cutoff Λ of (17) must be much smaller than the naively expected value 4πf . In other words, there must be other fields with masses much below 4πf which cut off the quadratic divergence. This is possible if their interactions soften the symmetry breaking due to the quartic term. This requires some special structure of the theory which will be discussed in subsection IV A. Another possibility is that the total vacuum energy during inflation is carried by another field as in hybrid inflation models [6]. The slow-roll field acts as a trigger of the phase transition of the other field. In this case there is a similar worry that the coupling between these fields, which is an explicit breaking contribution, can destroy the flatness of PNGB potential through radiative corrections. We will discuss this case in subsection IV B. A. A model of single field inflation based on little Higgs theories From the discussion above we see that a single field inflation with the field value smaller than M P requires that the quadratic term is smaller than that naively induced by higher order terms. This is quite similar to what happens for the Higgs potential, so that we can use the same ideas of little Higgs theories [32,34,35,36,37,38], recently proposed as a new solution to the hierarchy problem in electroweak symmetry breaking. c 1 g 2 1 f 2 φ + i h 2 f 2 + c 2 g 2 2 f 2 φ − i h 2 f 2 ,(18) where f is the symmetry breaking scale, g 1 and g 2 represent the two sets of couplings, c 1 , c 2 are order 1 constants, h is the little Higgs, and φ is a "fat" Higgs which receives large contributions to its mass both from g 1 and g 2 alone. The first term preserves a shift symmetry h → h + ǫ, φ → φ − 2 i ǫ h f ,(19) while the second one preserves a different symmetry, h → h + ǫ, φ → φ + 2 i ǫ h f ;(20) each one forbids a mass term for h. For c 1 g 2 1 + c 2 g 2 2 > 0, we can integrate out the heavy φ field and obtain a quartic coupling for h, 4c 1 c 2 g 2 1 g 2 2 c 1 g 2 1 + c 2 g 2 2 \|h\| 4 .(21) In addition, the radiative contribution to h mass squared is c 3 g 2 1 g 2 2 16π 2 f 2 ,(22) with c 3 = O(1). The coefficients c 1 , c 2 , c 3 can be either positive or negative depending on the model and the types of interactions. In little Higgs theories of electroweak symmetry breaking, one requires c 1 , c 2 > 0 and c 3 < 0. To obtain a model for inflation we make a different choice, c 1 · c 2 < 0, c 3 < 0, so that both the squared mass term and the quartic coupling are negative. In terms of the inflaton ψ, which is assumed to be the real part of h, ψ = √ 2ℜ(h), the potential is V (ψ) = V 0 − m 2 2 ψ 2 − λ 4 ψ 4 ,(23) where m 2 = \|c 3 \|g 2 1 g 2 2 16π 2 f 2 , λ = 4c 1 c 2 g 2 1 g 2 2 c 1 g 2 1 + c 2 g 2 2 , V 0 ≈ λf 4 .(24) Generically one expects a cutoff of the order of 4πf , therefore the quadratic term is suppressed by a loop factor with respect to that naively induced by the quartic coupling. The form of the inflaton potential is similar to the one proposed in Ref. [39,40] in a gauge mediated SUSY breaking model (though the η-problem was not addressed there). Note that the real potential is not unbounded from below because ψ is a PNGB; the true minimum occurs at ψ ∼ f . The Universe is assumed to start near ψ = 0. In the beginning, when the m 2 ψ 2 dominates the tilt of the potential, the Universe undergoes slow-roll inflation. Inflation ends after ψ grows and the λψ 4 term becomes dominant. To be specific, we will assume that c 1 > 0, c 2 < 0, and \|c 2 \|g 2 2 ≪ c 1 g 2 1 , then λ ≈ 4\|c 2 \|g 2 2 . During inflation the slow-roll parameter η is given by η = M 2 P V ′′ V ≈ − g 2 1 64π 2 c 3 c 2 M 2 P f 2 .(25) The observational constraint \|η\| < 1/20 requires f M P > g 1 4π 5c 3 c 2 .(26) It can be satisfied with f < M P if g 1 O(1). One can easily check that with this choice of parameters also the other slow-roll parameter ǫ is small. The number of e-foldings the Universe expands after the λψ 4 term dominates is roughly given by \|η\| −1 , so if we assume that the COBE scale occurs when the m 2 ψ 2 term is still more important, \|η\| can not be too small and should be close to the current limit. In the opposite limit only the quartic term is relevant during observable inflation. The tilt of the spectral index n − 1 ≈ 2η in this model is predicted to be negative. From the COBE normalization for curvature perturbation, we have 5.3 × 10 −4 = V 3/2 M 3 P V ′ ≈ \|η\| −3/2 m ψ ≈ \|η\| −3/2 √ λ ,(27) where the last relation is obtained because the COBE scale should be near the point where the m 2 ψ 2 term and the λψ 4 term are comparable. This requires g 2 ≈ 2.7 × 10 −3 \|η\| 3/2 \|c 2 \| −1/2 .(28) The very small coupling g 2 can be seen as a weak point of this model, though it is natural in the 't Hooft's sense [41], because a larger symmetry is recovered in the limit g 2 → 0. Another concern is whether the flat potential is preserved in the presence of quantum gravity effects. One expects that, in addition to the couplings g 1 and g 2 , higher dimensional operators generated by quantum gravity effects may also break the global symmetry explicitly and give rise to a potential for the PNGB. These effects are suppressed by powers of f /M P . Which higher dimensional operators can be generated depends on the specific little Higgs theory and its UV completion as discussed in the previous section. B. Hybrid inflation models In this subsection we consider hybrid inflation models with the inflaton being a PNGB 6 In these models the slow-rolling field is protected by an approximate symmetry, while the vacuum energy is dominated by another (waterfall) field. The first field acts like a trigger on the other one: when a critical value is reached we are quickly driven to the true vacuum and inflation ends. At first sight, however, this introduces another problem: how can the approximate symmetry protect the flatness of the potential without suppressing the coupling between the slow-rolling field and the other one? Another way of phrasing this problem is that since the coupling between the two fields breaks the global symmetry, one may worry that it generates a large potential for the inflaton and spoil slow-roll inflation. Assuming that the slow-rolling inflaton ψ and the waterfall field φ couple through the interaction λψ 2 φ 2 ,(29) this will generate a correction to the mass of the inflaton field, ∆m 2 ψ ∼ λ 16π 2 Λ 2 ,(30) where Λ is the cutoff of the integral. In order for ψ to act as a switch on the waterfall field, we need λψ 2 0 > \|m 2 φ \| ,(31) where ψ 0 is the initial value of ψ. This implies m 2 ψ > 1 16π 2 Λ 2 \|m 2 φ \| ψ 2 0 .(32) We can see that the cutoff Λ can not be too high. The hybrid inflation requires m 2 ψ ≪ \|m 2 φ \|. It would not work with a naive cutoff Λ ∼ 4πf expected in strong dynamics, which would yield m 2 ψ > \|m 2 φ \|. Therefore, we need a much lower cutoff for the corrections to m 2 ψ . The only known ways to have such a low cutoff are supersymmetry and little Higgs theories. In these cases, one may (at best) cut off the integral at Λ 2 ∼ \|m 2 φ \|, then we have m 2 ψ > 1 16π 2 m 4 φ ψ 2 0 .(33) On the other hand, the current constraint on the slow-roll parameter \|η\| < 0.05 implies m 2 ψ < 0.05 × V M 2 P .(34) Comparing the above two equations, the requirement ψ 0 ≪ M P implies m 4 φ ≪ V ; the waterfall field φ has to be light compared to the scale of the total vacuum energy it controls. To get a natural small mass for φ again requires some symmetry reason. In contrast with the case of the slow-roll field, SUSY can protect the lightness of the waterfall field because we only need \|m φ \| < V 1/4 , not \|m φ \| ≪ H. Another possibility is that also the waterfall field is a PNGB, protected by a shift symmetry. From these general arguments, we see that we are led to very specific structures for any natural hybrid 4d models of inflation, if all field values are required to be smaller than M P . Either we need both SUSY and PNGBs, or we need a little Higgs structure with all relevant fields being PNGBs. We present two examples to demonstrate it explicitly. A SUSY model The idea of SUSY hybrid inflation with the inflaton as a PNGB was discussed in Ref. [43,44], in the context of non-Abelian discrete symmetries. To illustrate our point and to clarify the requirements, let us study a very simplified model. Consider the superpotential W = λ 0 S(ψ 2 1 + ψ 2 2 − f 2 ) + λ 1 2 ψ 1 φ 2 + λ 2 X(φ 2 − v 2 ) ,(35) with λ 2 1 f 2 > 2λ 2 2 v 2 .(36) The first term preserves a U(1) symmetry which is spontaneously broken. We can parametrize the flat directions as follows, ψ 1 + iψ 2 ≡ √ 2Q = (f + σ) e iχ/f , ψ 1 − iψ 2 ≡ √ 2Q = (f − σ) e −iχ/f ,(37) where χ is the Nambu-Goldstone boson of the broken U(1) symmetry, and σ is the other flat modulus due to SUSY. When SUSY is broken, σ receives a potential and we assume that it is stabilized at σ = 0. We will only consider the field χ, which plays the role of the inflaton. For convenience, in the following we will also use ψ 1 and ψ 2 to simply represent their values along this direction, ψ 1 = f cos χ f , ψ 2 = f sin χ f .(38) The U(1) symmetry is also explicitly broken by the coupling λ 1 . For the moment we assume that this is the only explicit breaking effect and that the Kähler potential preserves the U(1) symmetry up to corrections of order λ 2 1 /(16π 2 ). A potential is generated for χ due to the λ 1 coupling. We assume the initial condition of the early Universe to be χ ≈ 0; this forces φ = 0 because this field receives a large mass from ψ 1 . SUSY is broken by F X and the vacuum energy density is V 0 ≈ \|F X \| 2 = λ 2 2 v 4 .(39) There are two kinds of contributions which lift the potential of χ. First, supergravity induces a soft SUSY breaking mass of order H for every scalar (ψ 1 , ψ 2 , φ). However, because χ is a PNGB, it only receives a potential due to the presence of the explicit breaking λ 1 . The corresponding contribution is loop-suppressed, m 2 χ (SUGRA) ∼ λ 2 1 16π 2 3H 2 .(40) One can see that there is no SUGRA η-problem if λ 1 1 In addition to the corrections due to supergravity, there is a direct Yukawa mediated contribution through a φ loop, arising from the splitting of the spectrum of the φ supermultiplet due to F X . The potential receives χ dependence at one loop, V (χ) ≈ V 0 1 + λ 2 2 4π 2 ln λ 1 ψ 1 µ = V 0 1 + λ 2 2 4π 2 ln λ 1 cos(χ/f ) µ/f .(41) The derivatives are easy to calculate, V ′ (χ) = −V 0 λ 2 2 4π 2 sin(χ/f ) f cos(χ/f ) = −V 0 λ 2 2 4π 2 ψ 2 f ψ 1 , (42) V ′′ (χ) = −V 0 λ 2 2 4π 2 1 f 2 cos 2 (χ/f ) = −V 0 λ 2 2 4π 2 1 ψ 2 1 .(43) We see that χ is rolling in the right direction (0 → πf /2), and eq. (43) agrees with eqs. (32), (33) with the cutoff Λ 2 ∼ \|m 2 φ \| = λ 2 2 v 2 . The slow roll parameter η is now η = M 2 P V ′′ V 0 = − λ 2 2 4π 2 M 2 P ψ 2 1 .(44) We have λ 2 = 2π \|η\| ψ 1 M P .(45) Given the current constraint \|η\| < 0.05, ψ 1 ≪ M P requires λ 2 ≪ 1, which is equivalent to say that φ has to be light. In the limit λ 2 → 0, the potential becomes flat because SUSY breaking vanishes. However, in this model there is no enhanced symmetry in the Lagrangian in the λ 2 → 0 limit. This is technically natural in SUSY theories though because of the non-renormalization theorem. For a non-SUSY theory the smallness of λ 2 would be unstable against radiative corrections from the other interactions. Let us examine the other constraints. We will assume that the Yukawa mediated contribution, eq. (41), dominates over the supergravity contributions, eq. (40). The number of e-folds of slow-roll inflation after a given epoch is N(χ) ≃ χ χ end M −2 P V V ′ dχ = ψ 1 (χ) ψ 1end M −2 P dψ 1 λ 2 2 4π 2 1 ψ 1 = 4π 2 λ 2 2 M 2 P ψ 2 1 2 − ψ 2 1end 2 = 1 2\|η(χ)\| − 1 2\|η end \| ,(46) where the subscript "end" represents the end of inflation. Generally it is dominated by the first term. We obtain a prediction for the deviation of the spectral index from 1 n − 1 ≃ 2 η COBE ≈ − 1 N COBE ,(47) where the subscript "COBE" denotes the values corresponding to the scale of COBE measurement, and N COBE is typically 40 − 60. From the constraint on curvature perturbation measured by COBE, 5.3 × 10 −4 = V 3/2 M 3 P \|V ′ \| ≈ 1 M 3 P λ 3 2 v 6 λ 2 2 v 4 λ 2 2 4π 2 ψ 2 f ψ 1 = 2πv 2 f M 2 P ψ 2 \|η\| −1/2 = 2πv 2 M 2 P sin(χ/f ) \|η\| −1/2 . (48) we obtain v 2 M 2 P ≈ 8 × 10 −6 50 N COBE sin χ COBE f ⇒ v M P ≈ 3 × 10 −3 50 N COBE 1 4 sin 1 2 χ COBE f .(49) In this simple model we did not address why there is an approximate U(1) global symmetry, broken only by λ 1 . We left out many other possible terms which do not respect the U(1) symmetry. While it is technically natural in SUSY theories, it is not very well motivated. In addition, explicit breaking terms can also arise from quantum gravity effects. Our main point here is to demonstrate how an approximate shift-symmetry can protect a [43,44], and suppresses any dangerous symmetry breaking terms. Another natural way to obtain an accidental global symmetry is to exploit the locality in (deconstructed) extra dimensions, which we will study next. In particular, one can find natural hybrid models even without SUSY. A 6d hybrid model and its 4d deconstruction In this subsection we present a non-SUSY hybrid inflation model. As we argued earlier, to get such a model in 4d without SUSY, we need the little Higgs structure with both the inflaton field and the waterfall field being PNGBs. Little Higgs theories were first motivated from deconstructing extra-dimensional theories, where the PNGBs correspond to the extra components of gauge fields in extra dimensions [32,34]. We will first discuss a 6d model where the extra components of the gauge field, A 5 , A 6 , play the roles of the inflaton and waterfall fields. It provides simple physics intuition and a clear picture of inflation dynamics. Later we will show that it can safely be deconstructed to purely 4d models. We consider an SU(2) gauge theory in 4 ordinary infinite dimensions and 2 extra compact dimensions. We assume that the x 5 direction is compactified on a circle with radius R 5 , and the x 6 direction is compactified on an S 1 /Z 2 orbifold with radius R 6 (Fig. 1). Furthermore we assume that the orbifold projection breaks the SU(2) gauge symmetry down to U(1) at the orbifold fixed points x 6 = 0, πR 6 . The assumed parities of the various gauge components under the Z 2 projection are shown in Table I. We see that in the 4d picture, the T 3 1 2 g 2 (A 3 5 ) 2 (A 1 6 ) 2 + (A 2 6 ) 2 ,(50) where g is the 4d gauge coupling. For simplicity, we used A 5 , A 6 to represent the zero modes and we will omit the generator indices in the rest of the discussion. Note that the full theory is periodic under the transformations A 5 → A 5 + 1/(gR 5 ), A 6 → A 6 + 1/(gR 6 ), where the Kaluza-Klein modes simply shift by one unit. At tree level, there are flat directions along A 5 (with A 6 = 0), and A 6 (with A 5 = 0). As discussed in section III, A 5,6 can not have local mass terms by gauge invariance. They can only get non-local contributions from Wilson lines and these contributions are indeed generated by radiative corrections, lifting the flat directions. One can imagine that at the origin (A 5 = A 6 = 0) the radiative corrections generate a positive squared mass for one direction, say A 6 , and a (larger) negative mass squared for the other (A 5 ). This can be achieved if there are charged fermions living at the orbifold fixed lines, x 6 = 0 or πR 6 . In this case A 6 can play the role of the slow-roll field and A 5 can be the waterfall field. The Universe with the initial condition A 6 = 0, A 5 = 0 will slowly roll to the origin until the squared mass of A 5 turns negative and A 5 jumps down to the true vacuum. To satisfy the slow-roll condition during inflation and to have a sufficiently fast waterfall process it is required that m 2 A 6 ≪ H 2 ≪ m 2 A 5 ,(51) which implies R 6 ≫ R 5 . Similarly to the case of one extra dimension in sec. III, the potential can be computed for a given particle content, and is well known in the literature. the waterfall field) while still allowing significant couplings which trigger the waterfall phase transition and reheat the Universe after inflation. From the constraints on the parameters discussed in the Appendix, one can check that the effective decay constants or symmetry breaking scales 1/(2πgR 5 ), 1/(2πgR 6 ) can be smaller than M P in this model. Therefore, a valid 4d model can easily be obtained by deconstruction. In fact, there is more freedom in the 4d deconstructed theories, since the various couplings are not required to be related as in the 6d theory by the higher dimensional gauge symmetry. The hierarchy between the scales of the inflaton field and the waterfall field can either come from the symmetry breaking scales or the couplings. In the 4d picture, the inflaton and the waterfall fields are PNGBs, whose masses are protected by many approximate symmetries. Because in the limit in which one of them is restored the PNGB is exactly massless, its mass can be quite small even in the presence of a large coupling to additional fields. Let us start from two sites with an SU(2) symmetry on each site, with four link fields X i , i = 1, 2, 3, 4 (see fig. 2), which transform as fundamentals under both SU(2)'s: the VEVs of the X i break SU(2) × SU(2) to the diagonal subgroup. To reproduced the orbifold projection we gauge the full SU(2) group on the first site but only the U(1) subgroup corresponding to the T 3 generator on the second site. We add to the potential two plaquette operators: V = −κ 1 f 4 Tr(X 1 X † 2 X 3 X † 4 ) − κ 2 f 4 Tr(X 2 X † 3 X 4 X † 1 ) .(52) Gauge fixing X 1 = 1 1, it is easy to find that the classically flat directions can be parameterized by X 2 and X 4 , (X 3 = X 2 X 4 ), with the additional constraint X 2 X 4 = X 4 X 2 , coming from the second plaquette [35,37]. In this way we have reproduced the commutator potential between A 5 and A 6 . As only T 3 is gauged in the second site, we can add plaquette operators which include the projection Ω ≡ diag(1, −1) on this site to get rid of useless light states. The operator −κ 3 f 4 Tr(X 1 ΩX † 2 X 3 ΩX † 4 ) ,(53) forces X 4 to commute with Ω so that only the neutral component survives, and When a critical point is reached, the potential of X 4 becomes unstable and the waterfall process starts, ending inflation. −κ 4 f 4 Tr(X 1 ΩX † 3 X 4 ΩX † 2 ) ,(54) The model is quite similar to the 6d one, except that there are differences due to couplings and volume factors, and to the fact that now gravity is 4-dimensional. As gravity is now 4-dimensional, one needs to worry about the corrections of the potential due to explicit symmetry breaking operators generated by quantum gravity effects (which were exponentially suppressed in "real" extra dimensions). For the simplest 2-site model, there are gauge invariant operators involving only two links. If one imagines that the links come from bilinear fermion condensates, they correspond to dimension-6 operators which are still dangerous as discussed before. These operators can be eliminated by additional (continuous or discrete) gauge symmetries 7 . A deconstruction with more sites (gauge groups) so that the gauge invariant operators require more links is also sufficient to suppress quantum gravity effects [33]. As there is more freedom in the 4d deconstructed model, changing the model parameters we can have a very fast waterfall process along X 4 as in the original hybrid inflation [6] or a quite slow one as in the SUSY inspired "supernatural models" [45]. In the above discussion we assumed that A 6 (X 2 ) is the slow-roll field and A 5 (X 4 ) is the waterfall field. One can also consider the opposite case where the roles of the two fields are reversed. In that case one requires m 2 A 5 > 0, m 2 A 6 < 0, and R 5 ≫ R 6 . Cosmic strings will be generated at the end of inflation because U(1) is broken; this process can have interesting but complicated consequences [46]. V. CONCLUSIONS From the point of view of cosmology, inflation is definitely the most attractive scenario describing the very early Universe. On the other hand, from the particle physics point of view, the required inflaton potential is extremely unnatural and seems to need a lot of fine tuning. In this paper we examined in detail the physics ideas which may be used to naturally obtain a viable inflaton potential. We emphasize that SUSY, although being a popular paradigm for inflation models, can not adequately preserve the flat potential for inflaton by itself. The only natural way to obtain a flat potential for inflation is to incorporate some With extra dimensions, locality in the extra space allows to get a trustworthy potential even if the variation of the inflaton field is bigger than M P , with exponentially suppressed quantum gravity corrections [5]. On the other hand, purely 4d models require more sophisticated structures. The inflaton, besides being a PNGB, must have a potential with further protection from the potentially dangerous explicit symmetry breaking interactions which are required to end inflation. We conclude that supersymmetry or little Higgs structure is necessary for such protection. Discrete or continuous gauge symmetries are required in the UV completion of the 4d models below the Planck scale to control quantum gravity effects. A generic prediction of the 4d models is that the contribution to the density perturbations from gravitational waves is unobservably small, because the field values are smaller than M P . This has to be contrasted with extra-dimensional setups, where a significant production of gravitational waves is possible. In hybrid models density perturbations produced during the final phase transition can give interesting phenomenological signatures [46]. The prediction of the spectral index depend on the individual models. However, as it is quite difficult to preserve a flat direction during inflation, it seems quite generic that the spectral index n should deviate from unity considerably. In our models the slow-roll parameters are small because of loop-factor suppressions, so that we do not expect them to be utterly small. Anyway, the same conclusion holds if the η-problem is solved just by a certain amount of fine-tuning. Therefore, a small deviation from scale-invariance seems to be a smoking gun for the inflationary paradigm itself. In some sense the inflaton mass cannot be too separated from the Hubble scale during inflation for the same reason we do not expect the Higgs mass to be very far from the scale of new physics (whatever it is), in which the SM is embedded. Note added: As this work was completed ref. [47] appeared, where hybrid models based on PNGBs are discussed. For the 6d model, the potential for the extra components of the gauge field consists of a sum of cosine functions which are periodic in A 5 → A 5 + 1/(gR 5 ) and A 6 → A 6 + 1/(gR 6 ). Here we will just expand it around the origin to obtain the mass terms for A 5 and A 6 at the origin. To satisfy eq. (51) we have to assume that R 5 ≪ R 6 . The gauge loops give positive contributions to the squared mass for both A 5 and A 6 . For R 5 ≪ R 6 they are given by m 2 A 5 (gauge) ≈ 2g 2 ζ(4) π 5 R 6 R 3 5 ,(A1)m 2 A 6 (gauge) ≈ 3g 2 ζ(3) 4π 4 1 R 2 6 .(A2) To make m 2 A 5 negative, we can introduce fermions extended along the x 5 direction but localized in the x 6 direction, so that they only contribute to the mass of A 5 . A natural choice is to have charged fermions living at the orbifold fixed lines, x 6 = 0 or πR 6 , which preserves only the U(1) gauge symmetry (corresponding to A 3 ). Their contribution to m 2 A 5 is 8 m 2 A 5 (fermion) ≈ − 3g 2 ζ(3) 4π 4 1 R 2 5 i 2Q 2 i ,(A3) where Q i is the U(1) charge of the fermion i. From the constraint (which will be discussed later) on the large density perturbations generated at the beginning of waterfall, R 6 /R 5 is required to be 20. We can see that for somewhat large Q i or many fermions, m 2 A 5 can become negative. The exact values of m 2 A 5 , m 2 A 6 depends on the field content. Nevertheless, they have to be cut off by 1/R 2 5 and 1/R 2 6 as they should vanish in the R 5,6 → ∞ limit. Below we will simply parametrize them by m 2 A 5 = − c 5 g 2 π 4 1 R 2 5 ,(A4)m 2 A 6 = c 6 g 2 π 4 1 R 2 6 ,(A5) where c 5 and c 6 are constants of order 1 or larger. Note that we require that the positive squared mass for A 5 from the tree-level potential eq. (50) to be larger than the negative radiative contribution in the beginning of inflation, i.e., \|m 2 A 5 \| = c 5 g 2 π 4 R 2 5 < g 2 A 2 6 (initial) 1 4R 2 6 ⇒ R 5 > 2 √ c 5 g π 2 R 6 . (A6) 8 In fact, this contribution is localized at the orbifold fixed lines, which can mix KK modes in the x 6 direction. However, as we see later, this term is required to be smaller than the tree-level KK masses n 2 /R 2 6 , so we can treat it as a small perturbation without re-diagonalizing the mass eigenstates. We can see that eqs.(A5), (A6) are consistent with eqs.(32), (33) as Λ 2 ∼ 1/R 2 6 > \|m 2 A 5 \|. During inflation, the vacuum energy is dominated by the A 5 potential, V ∼ \|m 2 A 5 \| g 2 (2πR 5 ) 2 ∼ c 5 4π 6 1 R 4 5 ,(A7) while the slope is determined by the A 6 potential, V ′ ∼ m 2 A 6 g(2πR 6 ) ∼ c 6 g 2π 5 1 R 3 6 , V ′′ ∼ m 2 A 6 .(A8) There are several constraints for the parameters g, R 5 , R 6 . For the slow roll condition, we require η = M 2 P V ′′ V ≈ g 2 (2πR 5 ) 2 M 2 P c 6 c 5 R 2 5 R 2 6 ≪ 1 .(A9) From the COBE measurement of curvature perturbations, it is required that 5.3 × 10 −4 = V 3/2 M 3 P V ′ ≈ η −3/2 c 1/2 6 2 π g 2 .(A10) Finally, as discussed in Ref. [45], in hybrid inflation models, large density perturbations will be generated during the period when both fields are light. A rough condition for a fast enough end of inflation, so that the large density perturbations do not fall inside the observable window is m A 5 H m A 6 H ≈ η c 5 c 6 R 6 R 5 1 .(A11) For some sample numbers, if we assume c 5 , c 6 ∼ 1, all constraints can be satisfied with R 6 R 5 ∼ 100, g ∼ 2 × 10 −3 , 2πR 5 M P ∼ 10 4 , η ∼ 0.03 . One can check that the effective decay constants or symmetry breaking scales 1/(2πgR 5 ), 1/(2πgR 6 ) can be smaller than M P in this model. Therefore, in this case it is possible to obtain valid 4d models by dimensionally deconstructing this model. For 4d models there is more freedom in the choice of parameters as couplings are not related by higher dimensional gauge symmetry. For simplicity of discussion, we assume κ ∼ g 2 for the couplings of the plaquette operators (52), where g is the SU(2) gauge coupling: in this way the commutator potential is ∼ g 2 as happens in the 6d model. We also assume that the VEVs for X i 's are equal (∼ f ), and the required hierarchy between the scales of the slow-roll field and the waterfall field is generated by the couplings to fermions. The charged scalars from X 2 have a positive mass squared from gauge loops: m 2 2 ∼ g 4 /(16π 2 )f 2 ( 9 ). The neutral scalar of X 4 , on the other hand, receives a negative contribution to its mass squared from couplings to fermions, m 2 4 ∼ −λ 4 /(16π 2 )f 2 , where λ is the analogue of the top Yukawa coupling 10 . As emphasized, the fermion couplings should be introduced in a "delocalized way," i.e., preserving enough global symmetries, to avoid the quadratic divergences. The required hierarchy can be obtained if λ ≫ g. Inflation evolves as in the 6d model: we start away from the origin along the X 2 direction and we roll towards the origin. For sufficiently big values of X 2 , X 4 is stuck at the origin because of the commutator potential. For this to happen, the positive contribution from the commutator potential must be able to overcome the negative one from the fermion couplings: g 2 f 2 > λ 4 16π 2 f 2 .(A13) The vacuum energy during the slow-roll is approximately constant and given by V 0 ∼ λ 4 16π 2 f 4 . Before we get to the origin the mass along X 4 becomes negative and we end up to the true minimum with restored gauge symmetry. We can easily estimate the slow-roll parameters: ǫ ≡ M 2 P 2 V ′ V 0 2 ≃ M 2 P f 2 g 4 λ 4 2 , (A14) η ≡ M 2 P V ′′ V 0 ≃ M 2 P f 2 g 4 λ 4 .(A15) To have slow-roll we must assume that f ≫ M P g 2 λ 2 .(A16) Note that, assuming a certain hierarchy between the coupling constant g ≪ λ, we have slow roll, even if f is smaller than the Planck scale, in constrast to what happens in natural inflation. This is possibile in a hybrid model because the vacuum energy depends on λ, 9 With the assumption κ ∼ g 2 , the combined contribution of the plaquette operators is comparable to the gauge term. 10 These 1-loop contributions are slightly enhanced with respect to the explicit UV operator, which can be estimated from the two loop quadratic divergence through naive dimensional analysis, by the logarithmic factor log(Λ 2 /(gf ) 2 ) ≃ 2 log(4π/g). We will see in the following that a viable model of inflation requires a small g so that this enhancement is quite consistent, while it is rather small in models of EWSB, where g ∼ 1. while the slow-roll potential is lifted by g. Let us now look at the COBE normalization for the large scale perturbations CMBR V 0 ǫ 1/4 ≃ 0.027 · M P ; (A17) it gives the constraint λ 3 √ 4π · g 2 f M P 3/2 = g · η −3/4 √ 4π ≃ 0.027 .(A18) Using the experimental limit η < 1/20, we get a rather strong constraint on g: g 0.01. M P , while keeping the effects of higher-dimension operators under control. Locality in extra dimensions can in fact prevent large corrections to the inflaton potential from quantum gravity effects. Consider a 5d model with the extra dimension compactified on a circle of radius R. The extra component A 5 of an abelian gauge field propagating in the bulk cannot have a local potential, due to the higher dimensional gauge invariance; a shift symmetry protects it this conclusion is valid only if we restrict to purely 4d models. Quantum gravity corrections to the potential(12) are negligible if the extra dimension is bigger than the Planck length, different from what is expected in a 4d PNGB model. Again locality in the extra space is the key feature; virtual black holes cannot spoil the gauge invariance and do not introduce a local potential for A 5 , while non-local effects are exponentially suppressed by ∼ e −2πM 5 R , because the typical length scale of quantum gravity effects (the 5d Planck length M −1 5 ) is much smaller than the size of the extra dimension. from SUGRA corrections during inflation and the subtleties involved when one needs to couple the PNGB to other fields for the end of inflation. One can extend the model in such a way that a non-Abelian discrete symmetry gives rise to the approximate global symmetry as in FIG. 2 : 2A more detailed discussion of the parameters required for slow-roll inflation and all the other observational constraints can be found in the Appendix. The most important feature of this model is again that the locality in extra dimensions protects the flat potential for the inflaton (and also the The moose diagram for the deconstructed model of hybrid inflation. keeps only the charged components of X 2 . It is easy to verify that there are no one-loop quadratically divergent contributions which lift the flat directions[35,37]: each of the plaquette interactions and gauge couplings respect a subgroup of the SU(2) 8 global symmetry of the link fields, leaving the PNGBs exactly massless. Only the combination of two sources of explicit breaking lift the flat directions, so that we have only logarithmic divergences.As in the 6d model the charged scalars from X 2 take a positive mass squared, from gauge loops: m 2 2 ∼ g 4 /(16π 2 )f 2 . Instead we want that the neutral component of X 4 receives a tachyonic contribution to the potential: this can be done by introducing fermions in the theory, coupled to X 4 . To avoid the presence of quadratic divergences, fermions must be introduced in a "delocalized way", similarly to what one does to control the top loop corrections to the Higgs mass in little Higgs models. We get a contribution: m 2 4 ∼ −λ 4 /(16π 2 )f 2 , where λ is the analogue of the top Yukawa coupling. The required hierarchy can be obtained if λ ≫ g. Inflation can evolve as in the 6d model: we start away from the origin along the X 2 direction and we roll towards the origin, because of the positive mass contribution. For sufficiently big values of X 2 , X 4 is stuck at the origin because of the commutator potential. A: MORE DETAILS ABOUT THE 6 DIMENSIONAL HYBRID MODEL AND ITS DECONSTRUCTION In this Appendix we present more detailed discussion of the 6d hybrid model and its 4d deconstruction in sec. IV B 2. symmetry: a term of the form [ΦΦe ξV ] D will be invariant, provided the compensator undergoes a super-Weyl transformation Φ → e −ξΛ Φ,Φ → e −ξΛΦThis point is often overlooked in the literature about inflation. 1 In the superconformal formalism it is easy to understand why the naive extension of the rigid FI term is not gauge invariant by itself. With a compen- sator chiral multiplet Φ, a rigid FI term would be promoted to [ΦΦξV ] D , which is no longer invariant under V → V + Λ +Λ since [ΦΦΛ] D and [ΦΦΛ] D are non-zero. To write a gauge-invariant generalization of a FI term, Φ must transform under the U(1) Let us first describe the general feature of the little Higgs theories. For specific models Goldstone boson. The little Higgs only learns its PNGB nature in the presence of both sets of couplings when the symmetry is completely broken. Therefore, there is no one-loop quadratically divergent contribution to the little Higgs mass. On the other hand, the quartic coupling can be generated at tree-level combining both sets of couplings. The potential for all little Higgs models has the following formwe refer the reader to the literature [32, 34, 35, 36, 37, 38]. A little Higgs model is based on a chiral Lagrangian from some spontaneously broken global symmetry. This symmetry is also explicitly broken by two (or more) sets of couplings. Each set of couplings preserves a different subset of the global symmetry under which the little Higgs is an exact Nambu- TABLE I : IThe Z 2 parities of various gauge components.A a µ A a 5 A a 6 T 1,2 − − + T 3 + + − component of A 5 and T 1,2 components of A 6 have zero modes. They have a tree level potential from the commutator term in F 2 56 , approximate shift symmetry. The examples are PNGBs and extra components of gauge fields living in extra dimensions. In purely 4d theories, the simplest model based on PNGBs has the difficulties that we need to extrapolate the field theory beyond its regime of validity as the symmetry breaking scale has to be greater than the Planck scale. To avoid this problem we need either extra dimensions or more complicated models in pure 4 dimensions as we discussed in sections III and IV. An approximate symmetry has also been used to have light fields, different from the inflaton itself, during inflation. This is important for the curvaton scenario[18,19] or for moduli fields[20]. Two-field inflation with a PNGB triggering a first order phase transition has been proposed[42], but the problems we are going to discuss are not addressed. For example, if in the UV theory X 1 , X 2 , X 3 , X 4 carry additional gauge charges 1, 3, 7, 5, respectively, the lowest dimensional gauge invariant operators which break the global symmetry are the plaquette operators in (52). 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[ "LOCAL SOBOLEV CONSTANT ESTIMATE FOR INTEGRAL RICCI CURVATURE BOUNDS", "LOCAL SOBOLEV CONSTANT ESTIMATE FOR INTEGRAL RICCI CURVATURE BOUNDS" ]	[ "Xianzhe Dai ", "ANDGuofang Wei ", "Zhenlei Zhang " ]	[]	[]	We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the L 2 Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions. 1991 Mathematics Subject Classification. 53C20.	10.1016/j.aim.2017.11.024	[ "https://arxiv.org/pdf/1601.08191v4.pdf" ]	554,316	1601.08191	0714896784fe8b9e94a73a4d4930d1b20fd9fae3	LOCAL SOBOLEV CONSTANT ESTIMATE FOR INTEGRAL RICCI CURVATURE BOUNDS 30 Nov 2017 Xianzhe Dai ANDGuofang Wei Zhenlei Zhang LOCAL SOBOLEV CONSTANT ESTIMATE FOR INTEGRAL RICCI CURVATURE BOUNDS 30 Nov 2017 We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the L 2 Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions. 1991 Mathematics Subject Classification. 53C20. Introduction Integral curvature is a very natural notion as it occurs in diverse situations, for example, the Chern-Gauss-Bonnet formula, the isospectral problem, and numerous variational problems. Moreover, integral curvature bounds have recently been discovered in various geometric situations, such as the L 2 bound of the curvature tensor for noncollapsed manifolds with bounded Ricci curvature, and the (almost) L 4 bound of the Ricci curvature for the Kähler-Ricci flow as well as the (real) Ricci flow (under certain conditions) [9,17,29,5,25,30,4]. In [23], the important Laplacian comparison and volume comparison are generalized to integral Ricci lower bound. Combining this with D. Yang's estimate [28] on the local Sobolev constant, the Cheeger-Colding-Naber theory has now been successfully extended to integral Ricci curvature bound in the noncollapsed case, with important consequences [26,29]. In the collapsed case a local Sobolev constant estimate was missing. Here we provide the missing piece and extend many of the basic estimates for integral curvature in [26,29] to the collapsed case. For each x ∈ M n let ρ (x) denote the smallest eigenvalue for the Ricci tensor Ric : Then Ric H − p measures the amount of Ricci curvature lying below a given bound, in this case, (n − 1)H, in the L p sense. Clearly Ric H − p,R = 0 iff Ric M ≥ (n − 1)H. It is often convenient to work with the following scale invariant curvature quantity (with H = 0): (1.2) k (x, p, R) = R 2 T x M → T x M,B R (x) ρ p − 1 p , k (p, R) = sup x∈M k (x, p, R) . The main result of the paper is Here the supremum runs over all subdomains Ω ⊂ B r (x) with smooth boundary and ∂Ω ∩ ∂B r (x) = ∅. See Section 2.2 for a discussion of isoperimetric constants. Remark 1.2. The smallness of k(p, 1) is necessary. Namely the result is not true if we only assume that k(p, 1) is bounded; see Section 6 for detail. Also the result is not true when p ≤ n 2 [3]. Remark 1.3. In the presence of the non-collapsing condition vol B r (x) ≥ cr n , our scale invariant curvature quantity k(x, p, r) ≤ c −1/p r 2− n p Ric − p,Br(x) , which is always small when Ric − p,B 1 (x) is bounded and r is small. This has been very nicely applied in [29,30]. Namely when applying to the study of tangent cones, with local volume growth, one only needs to assume that Ric − p,B 1 (x) is bounded. Note also that when k(p, r) is small for some r, it gives control on k(p, r) for all r, see Remark 2.2 for detail. We emphasize that it is very important that the volume dependence here is vol(B r (x)) 1 n . It is of the right scale invariance, and corresponds to the optimal Sobolev constant. We note that a local isoperimetric constant estimate is given indenpendently in a recent paper [24] but with weaker result and under much stronger assumptions. From (2.9) and (2.10), the theorem above immediately gives Corollary 1.5. Under the same assumption as in Theorem 1.1, we have the Cheeger's constant ID ∞ (B r (x)) ≤ 10 2n+4 r and the Sobolev inequality (1.4) Br(x) f n n−1 n−1 n ≤ 10 2n+4 r Br(x) \|∇f \|, for all f ∈ C ∞ 0 (B r (x)) where r ≤ 1. See Definition 2.3 for the definition of Cheeger's constant. By Cheeger's inequality [8], the first eigenvalue λ 1 ≥ 1 4 ID 2 ∞ . Thus we also obtain an eigenvalue lower bound. We emphasize that in the above Sobolev inequality we use the averaged integral (volume normalized). Remark 1.6. Under the pointwise Ricci curvature lower bound, estimates of the type above (namely local or Dirichlet) for Cheeger's constant and isoperimetric constant are proved in [6,2]. For integral Ricci curvature lower bound, D. Yang [28] obtained a local Sobolev constant estimate under the additional assumption that the manifold is noncollaped, see Theorem 2.8. Paeng [21] proved a local Cheeger's constant estimate for integral Ricci curvature under some strong assumption. Remark 1.7. When M is closed, the global (Neumann) normalized isoperimetric constant (see Section 2.2 for definition) for integral Ricci curvature was already obtained in [13], see Theorem 2.9. The proof for global one does not apply to local one here since it uses a result from geometric measure theory which only works for closed manifolds or domains with convex boundary. The local Sobolev inequality enables us to obtain many applications. First we can extend the maximal principle and gradient estimate in [26] to the collapsed case. Namely we have the following maximal principle. Theorem 1.8. Let M be an n-dimensional Riemannian manifold, and p > n/2. There is an ε = ε (n, p) > 0 and C = C (n, p, q) > 1 such that if k(p, 1) ≤ ε and R ≤ 1 then any function u : Ω ⊂ B (x, R) → R with ∆u ≥ f satisfies sup Ω u ≤ sup ∂Ω u + R 2 · C · f − * q,Ω , for any q > n 2 . Here the normalized L q norm f − Also we have the gradient estimate. Theorem 1.9. Let M be an n-dimensional Riemannian manifold, and p > n/2. There is an ε (n, p) > 0 and C (n, p) > 1 such that if k(p, 1) ≤ ε and R ≤ 1 and u is a function on B 1 (x) satisfying ∆u = f, 3 then (1.5) sup B R 2 (x) \|∇u\| 2 ≤ C(n, p)R −2 ( u * 2,B R (x) ) 2 + ( f * 2p,B R (x) ) 2 . With the (relative) local Sobolev constant estimate (1.4), one gets heat kernel upper bound, see e.g. [16, (2.17)]. With this and the volume doubling (2.5), Zhang-Zhu obtained Li-Yau's gradient estimate [32]. Hence one has parabolic Harnack inequality and local Li-Yau heat kernel lower bound, see Theorem 5.5. Consequently we derive the following mean value inequality, extending the one in [29] to the collapsed case. Theorem 1.10. For any integer n and p > n 2 there exist ε = ε (n, p) > 0 and C = C (n, p) > 1 such that the following holds. Given M a complete n-dimensional Riemannian manifold satisfying k(p, 1) ≤ ε, let u be a nonnegative function satisfy- ing ∂ ∂t u ≥ ∆u − f, where f is a nonnegative space-time function, then, for q > n 2 , (1.6) B 1 2 r (x) u(·, 0)d vol ≤ Cu(x, r 2 ) + C(n, p, q) r 2 sup t∈[0,r 2 ] f (t) * q,Br(x) for all x ∈ M, r ≤ 1. With the above tools at our disposal, we can then extend the L 2 Hessian estimate for parabolic approximation of Colding-Naber to integral curvature, see Section 5 for detail. In the noncollapsed case it is established in [29], see also [31]. We expect further applications of our results e.g. to the Cheeger-Colding-Naber theory, which will be discussed in a future paper. Acknowledgment The second author would like to thank Qi Zhang for his interest and helpful conversations. We also would like to thank Christian Rose for pointing out lapses in the argument of heat kernel lower bound in the earlier version. Preliminary In this section we fix notations and recall the previous work [23], [26] that will play a fundamental role here. We also give a review on the isoperimetric and Sobolev constants and their relations, and introduce the normalized version. For functions f on M, the L p norm and normalized L p norm on a ball B(x, r) ⊂ M is denoted f p,B(x,r) = ˆB (x,r) \|f \| p 1 p , f * p,B(x,r) = B(x,r) \|f \| p 1 p . (The notation of the volume form of g is often omitted in this paper.) f p , f * p denote the norm, normalized norm of f on M. 4 2.1. Volume Comparison for Integral Curvature. For simplicity, we state the case when H = 0. Let M n be a complete Riemannian manifold of dimension n. Given x ∈ M, let r(y) = d(y, x) be the distance function and ψ(y) = ∆r − n−1 r + . The classical Laplacian comparison states that if Ric M ≥ 0, then ∆r ≤ n−1 r , i.e., if Ric − ≡ 0, then ψ ≡ 0. In [23] this is generalized to integral Ricci lower bound. Theorem 2.1 (Laplacian and Volume Comparison [23,26]). Let M n be a complete Riemannian manifold of dimension n. If p > n 2 , then (2.1) ψ 2p.B(x,r) ≤ (n − 1)(2p − 1) 2p − n Ric − p,B(x,r) 1 2 . Equivalently (2.2) ψ * 2p.B(x,r) ≤ (n − 1)(2p − 1) 2p − n Ric − * p,B(x,r) 1 2 . Consequently we have the following volume comparison estimate: for any r 2 ≥ r 1 > 0, (2.3) vol(B r 2 (x)) r n 2 1 2p − vol(B r 1 (x)) r n 1 1 2p ≤ C(n, p)r 1− n 2p 2 Ric − p,B(x,r 2 ) 1 2 . In other words, (2.4) vol(B r 1 (x)) vol(B r 2 (x)) 1 2p ≥ r 1 r 2 n 2p 1 − C(n, p) (k(x, p, r 2 )) 1 2 , where C(n, p) is a constant depending on n, p. Hence there exists ε 0 = ε 0 (p, n) > 0 such that, if k(x, p, r 0 ) ≤ ε 0 , then (2.5) vol(B r (x)) vol(B r 0 (x)) ≥ 1 2 r r 0 n , ∀r < r 0 . Remark 2.2. As pointed out in [26,Section 2.3], if k(x, p, r 2 ) ≤ ε 0 for the ε 0 above, then (2.5) implies, (2.6) k(x, p, r 1 ) ≤ 2 1/p r 1 r 2 2− n p · k(x, p, r 2 ) ≤ 2 1/p k(x, p, r 2 ), ∀r 1 ≤ r 2 . Hence k(x, p, r 1 ) → 0 as r 1 → 0 and k(x, p, r 1 ) ≤ ε 0 (p, n) when r 1 ≤ 2 1 n−2p r 2 . On the other hand, when k(p, r 1 ) ≤ ε 0 (p, n), then (2.7) k(p, r 2 ) ≤ 2 n+1 p r 2 r 1 2 k(p, r 1 ) for all r 2 ≥ r 1 . Hence when k(p, r) is small for some r, it gives control on k(p, r) for all r. Note also that if one has a lower bound for the Ricci curvature Ric ≥ (n − 1) H then the quantity k (p, R) will be small for sufficiently small R. Namely general lower bound can be reduced to zero lower bound in the local analysis. Dirichlet and Neumann Isoperimetric and Sobolev Constants. In this subsection we review the definitions of the isoperimetric and Sobolev constants and their relations, and introduce the normalized form. For details, see [18,7], though we use a different convention here. Definition 2.3. For a complete noncompact Riemannian manifold M n or a compact Riemannian manifold M n with ∂M = ∅, for n ≤ α ≤ ∞, the Dirichlet (also referred as local) α-isoperimetric constant of M is defined by ID α (M) = sup Ω vol(Ω) 1− 1 α vol(∂Ω) , where Ω is an open submanifold of M with compact closure and smooth boundary such that ∂Ω ∩ ∂M = ∅. When α = n, ID n (M) is scale invariant. When α = ∞, this is Cheeger's constant [8], which scales like vol 1/n . The Dirichlet α-isoperimetric constant controls the local volume growth: for given a geodesic ball B(x, r) ⊂ M, vol B(x, r) ≥ r α IDα(M ) α for n ≤ α < ∞. Definition 2.4. The Dirichlet α-Sobolev constant of M is defined by SD α (M) = sup f f α α−1 ∇f 1 , where f ranges over C ∞ c (M). Definition 2.5. When M is compact with or without boundary, the Neumann α-isoperimetric constant of M is defined by IN α (M) = sup Γ min{vol(D 1 ), vol(D 2 )} 1− 1 α vol(Γ) , where Γ varies over compact (n − 1)-dim submanifold of M which divide M into two disjoint open submanifolds D 1 , D 2 of M. From the definition, if Ω ⊂ M, ∂Ω ∩ ∂M = ∅, and vol(Ω) ≤ 1 2 vol(M), then (2.8) ID α (Ω) ≤ IN α (M). Definition 2.6. The Neumann α-Sobolev constant of M is defined by SN α (M) = sup f inf a∈R f − a α α−1 ∇f 1 , where f ranges over C ∞ (M). Theorem 2.7 ( [12,8], see also [18,7]). For all n ≤ α ≤ ∞, ID α (M) = SD α (M), IN α (M) ≥ SN α (M) ≥ 1 2 IN α (M). 6 For convenience we consider the normalized Dirichelet and Neumann α-isoperimetric and α-Sobolev constants: ID * α (M) = ID α (M) vol(M) 1/α , SD * α (M) = SD α (M) vol(M) 1/α , IN * α (M) = IN α (M) vol(M) 1/α , SN * α (M) = SN α (M) vol(M) 1/α . Observe that (2.9) ID * α (M) ≥ ID ∞ (M), IN * α (M) ≥ IN ∞ (M), and SD * α (M) = sup f f * α α−1 ∇f * 1 , where f ranges over C ∞ c (M), SN * α (M) = sup f inf a∈R f − a * α α−1 ∇f * 1 , where f ranges over C ∞ (M). By Theorem 2.7, we have (2.10) ID * α (M) = SD * α (M). These normalized quantities are very useful in studying the collapsed case, see below. They are used in [27] in proving a Neumann type maximal principle without volume lower bound. In [28,Theorem 7.4] D. Yang obtained a Dirichlet isoperimetric constant estimate in the non-collapsing case when Ric − * p is small. Namely Theorem 2.8. Given p > n/2 and v > 0, there is an ε (n, p, v) > 0 such that if B 1 (x) ⊂ M n has vol B 1 (x) ≥ v and k (p, 1) ≤ ε, then ID n (B 1 2 (x)) ≤ C (n, p, v). For closed Riemannian manifold M n , Gallot [13,Theorem 3] showed that the normalized Neumann α-isoperimetric constant is bounded from above when diam(M) 2 Ric − * p is small (≤ ǫ(n, p)) for p > n/2, and α > n. Petersen-Sprouse [22] obtained the bound for α = n. Namely Theorem 2.9. Given p > n/2 and D > 0, there is an ε (n, p, D) > 0 such that if diam M n ≤ D and Ric − * p ≤ ε, then IN * n (M) ≤ C (n, p, D). Local isoperimetric constant estimate for closed manifolds For the local analysis, we need local (Dirichlet) Sobolev constant bound. From (2.8), we automatically get a local estimate when the volume of the domain is small relative to the whole manifold. We show that the measure can only have small concentration whenever Ric − Proposition 3.1. Suppose diam(M) = D. There exists ε = ε(n, p) > 0 such that if (3.1) D 2 Ric − * p ≤ ε, then for any a ≤ a 0 where a 0 = a 0 (n) solves (3.2) 1 2 − a 0 1 2 + a 0 = 3 4 1 n we have (3.3) vol(B aD (x)) ≤ 1 2 vol(M), ∀x ∈ M. Proof. For any x ∈ M we choose a dual point x ′ ∈ M with dist(x, x ′ ) = D 2 . Then, for any radius r < D 2 , (3.4) vol(B r (x)) vol(M) ≤ vol(B r (x)) vol(B D 2 +r (x ′ )) ≤ 1 − vol(B D 2 −r (x ′ )) vol(B D 2 +r (x ′ )) . Therefore it suffices to show that for r = aD with a ≤ a 0 , the last term above is greater than or equal to 1 2 . By (2.4) the last term can be estimated as follows (3.5) vol(B D 2 −r (x ′ )) vol(B D 2 +r (x ′ )) 1 2p ≥ D 2 − r D 2 + r n 2p 1 − C(n, p) k(x ′ , p, D 2 + r) 1 2 . If k(x ′ , p, D) = D 2 Ric − * p ≤ ε 0 , by (2.6), k(x ′ , p, D 2 + r) ≤ 2 1/p k(x ′ , p, D) = 2 1/p D 2 Ric − * p . Hence if we assume that (3.6) D 2 Ric − * p ≤ ε ≤ ε 0 , then (3.7) vol(B D 2 −r (x ′ )) vol(B D 2 +r (x ′ )) ≥ D 2 − r D 2 + r n 1 − C(n, p)2 1/2p ε 1 2 2p . Plug in r = aD with a ≤ a 0 , the choice of a 0 implies that D 2 − aD D 2 + aD n ≥ 3 4 . Now set 3 4 1 − C(n, p)2 1/2p ε 1 2 2p ≥ 1 2 . Clearly there exists ε(n, p) such that this holds for all ε ≤ ε(n, p). Combining this with Theorem 2.9 and (2.8), we have Similarly we have a local version of Proposition 3.1 which will be needed in the next section. Theorem 3.3. There exists ε = ε(n, p) > 0 and r 0 = r 0 (n) > 0 such that the following holds. Let (M, g) be a complete noncompact Riemannian manifold satisfying k(p, 1) ≤ ε, then we have (3.8) vol(B r 0 (x)) vol(B 1 (x)) ≤ 1 2 , ∀x ∈ M. Proof. For any x ∈ M, r < 1 3 , choose a point x ′ with d = dist(x, x ′ ) = 1−r 2 ≥ 1 3 . Then we have B r (x) ⊂ B 1 (x)\B d−r (x ′ ) ⊂ B d+r (x ′ ) ⊂ B 1 (x). As above we calculate vol(B r (x)) vol(B 1 (x)) ≤ vol(B r (x)) vol(B d+r (x ′ )) ≤ 1 − vol(B d−r (x ′ )) vol(B d+r (x ′ )) . To estimate the last term recall that (3.9) vol(B d−r (x ′ )) vol(B d+r (x ′ )) 1 2p ≥ d − r d + r n 2p 1 − C(n, p) k(x ′ , p, d + r) 1 2 . Since d + r ≤ 1, by (2.6), if k(x ′ , p, 1) ≤ ε 0 , we have k(x ′ , p, d + r) ≤ 2 1/p k(x ′ , p, 1). Hence when ε ≤ ε 0 , we get vol(B d−r (x ′ )) vol(B d+r (x ′ )) 1 2p ≥ d − r d + r n 2p 1 − C(n, p)2 1/2p ε 1 2 . Now we choose a 0 such that 1 − a 0 1 + a 0 = 3 4 1 n , then for any r ≤ 1 3 a 0 , we have, since d ≥ 1 3 , d − r d + r n ≥ 3 4 . Choose ε ≤ ε 0 such that (3.10) 1 − C(n, p)2 1/2p ε 1 2 2p ≥ 2 3 . Then vol(B r (x)) vol(B 1 (x)) ≤ 1 − d − r d + r n 1 − C(n, p)2 1/2p ε 1 2 2p ≤ 1 − 3 4 · 2 3 = 1 2 . The proof is complete by choosing r 0 = 1 3 a 0 and any 0 < ε ≤ ε 0 satisfying (3.10). Local isoperimetric constant estimate for complete manifolds In this section we first obtain an estimate on the weak Cheeger's constant with an error using Laplacian comparison for integral curvature and an idea of Gromov [14,. This will then enable us to prove Theorem 1.1 by using a covering argument of Anderson [2]. Recall the following lemma of Gromov which is stated for closed manifold in [14], but also works for complete manifolds. and any x 2 ∈ W has a unique minimal geodesic connecting to x 1 which intersects H at some z such that (4.2) dist(x 1 , z) ≥ dist(x 2 , z). Using Laplacian comparison estimate we have where D = sup x∈W dist(x 1 , x) and H ′ is the set of intersection points with H of geodesics γ x 1 ,x for all x ∈ W . Proof. Let Γ ⊂ S x 1 be the set of unit vectors such that γ v = γ x 1 ,x 2 for some x 2 ∈ W . We compute the volume in the polar coordinate at x 1 . Write dv = A(θ, t)dθ ∧ dt in the polar coordinate (θ, t) ∈ S x 1 × R + . Recall that [23] ∂ ∂t A t n−1 ≤ ψ A t n−1 where ψ = max(0, △r(θ, t) − n−1 t ) denotes the error term of Laplacian comparison. We thus have (4.4) A(θ, r) ≤ 2 n−1 A(θ, t) + 2 n−1ˆr t ψ(θ, s)A(θ, s)ds, ∀ r 2 ≤ t ≤ r. For any θ ∈ Γ, let r(θ) be the radius such that exp x 1 (rθ) ∈ H. Then, by assumption, W ⊂ {exp x 1 (rθ)\|θ ∈ Γ, r(θ) ≤ r ≤ 2r(θ)}. Thus, vol(W ) ≤ˆΓˆ2 r(θ) r(θ) A(θ, t)dtdθ ≤ 2 n−1ˆΓˆ2 r(θ) r(θ) A(θ, r(θ)) +ˆt r(θ) ψ(θ, s)A(θ, s)ds dθdt ≤ 2 n−1 DˆΓ A(θ, r(θ))dθ + 2 n−1 DˆΓˆD 0 ψ(θ, t)A(θ, t)dθdt On the other hand, vol(H ′ ) =ˆΓ A(θ, r(θ)) cos α(θ) dθ ≥ˆΓ A(θ, r(θ))dθ where α(θ) is the angle between H and radial geodesic exp x 1 (tθ). Thus, vol(W ) ≤ 2 n−1 D vol(H ′ ) + 2 n−1 D ˆΓˆD 0 ψ 2p Adθdt 1 2p ˆΓˆD 0 Adθdt 1− 1 2p . Through the Laplacian estimate (2.1) we get (4.5) vol(W ) ≤ 2 n−1 D vol(H ′ ) + 2 n−1 D vol(B D (x 1 )) 1− 1 2p ˆB D (x 1 ) \|Ric − \| p dv 1 2p the required estimate. Now we can obtain an estimate on the weak Cheeger's constant with an error. . If k(x, p, 1) ≤ 2 −1/p ε 0 , by (2.6), we have k(x, p, r) ≤ ε 0 for all r ≤ 1. Hence by (2.5), vol(B 2r (x)) ≤ 2 n+1 vol(B). Again if k(x, p, 1) ≤ 2 −1/p 2 −2(2n+3) , then k(x, p, r) ≤ 2 −2(2n+3) for all r ≤ 1. Hence vol(B) ≤ 2 n+2 r vol(H ∩ B 2r (x)) + 2 n 2 −(2n+3) vol(B)r −1 , which gives vol(B) ≤ 2 n+3 r vol(H ∩ B 2r (x)). Therefore choosing ε = min{2 −1/p ε 0 , 2 −1/p 2 −2(2n+3) } suffices. This estimate and volume doubling gives an estimate on the local isoperimetric constant via Vitali Covering Lemma. Proof of Theorem 1.1. First of all we show that the isoperimetric constant estimate (1.3) holds for some small radius r 0 = r 0 (n), under the assumption k(p, 1) ≤ ε 1 for some small constant ε 1 = ε 1 (p, n). By Theorem 3.3, we may assume that ε 1 is chosen such that there exists r 0 = r 0 (n) with vol(B 2r 0 (x)) vol(B 1 10 (x)) ≤ 1 2 , ∀x ∈ M. Now given any y ∈ M, let Ω be a smooth subdomain of B r 0 (y). We may assume that Ω is connected and its boundary H = ∂Ω divides M into two parts Ω and Ω c . For any x ∈ Ω, let r x be the smallest radius such that vol(B rx (x) ∩ Ω) = vol(B rx (x) ∩ Ω c ) = 1 2 vol(B rx (x)). Since Ω ⊂ B 2r 0 (x) and vol(B 2r 0 (x)) ≤ 1 2 vol(B 1 10 (x)), we have r x ≤ 1 10 . Take ε 1 as in Corollary 4.4, then by (4.7) (4.8) vol(B rx (x)) ≤ 2 n+3 r x vol(H ∩ B 2rx (x)B i = B 2rx i (x i ) such that ∪ i B 10rx i (x i ) ⊃ Ω. Moreover, we assume ε 1 is chosen such that k(p, r) ≤ ε 0 for all r ≤ 1, then by the volume doubling property (2.5), vol(B 2rx (x)) vol(B 10rx (x)) ≥ 1 2 · 5 n . Hence i vol(B i ) ≥ 1 2 · 5 n i vol(B 10rx i (x i )) ≥ 1 2 · 5 n vol(Ω). Applying the volume doubling property (2.5) again gives (4.9) i vol(B rx i (x i )) ≥ 1 4 · 10 n vol(Ω). Moreover, since the balls B i are disjoint, combining with (4.8) gives, (4.10) vol(∂Ω) ≥ i vol(B i ∩ H) ≥ 2 −n−3 i r −1 x i vol(B rx i (x i )). These two estimates lead to vol(Ω) n−1 n vol(∂Ω) ≤ 10 n−1 2 n+5 i vol(B rx i (x i )) n−1 n i r −1 x i vol(B rx i (x i )) ≤ 10 2n+4 i vol(B rx i (x i )) n−1 n i r −1 x i vol(B rx i (x i )) ≤ 10 2n+4 sup i vol(B rx i (x i )) n−1 n r −1 x i vol(B rx i (x i )) = 10 2n+4 sup i r n x i vol(B rx i (x i )) 1 n . On the other hand, since dist(y, x i ) ≤ r 0 , we have B r 0 (y) ⊂ B 2r 0 (x i ). Now r x i ≤ 1 10 , applying the volume doubling property (2.5) again, vol(B rx i (x i )) ≥ 10 n r n We next make a scaling argument to show that the estimate (1.3) remains hold for any radius r ≤ 1, under the assumption k(p, 1) ≤ ε 2 for a smaller constant ε 2 = ε 2 (p, n) > 0. Put r 1 = r r 0 ≤ 1 r 0 . After a scaling, it is sufficient to check that k(p, r 1 ) ≤ ε 1 . Choose ε 2 such that ε 2 ≤ ε 0 , so (2.5) holds for all r ≤ 1. Now if r 1 ≤ 1, by (2.6) k(p, r 1 ) ≤ 2 1/p k(p, 1) ≤ 2 1/p ε 2 . On the other hand, if 1 ≤ r 1 ≤ 1 r 0 , then by (2.7), k(p, r 1 ) ≤ 2 n+1 p r 2 1 k(p, 1) ≤ 2 n+1 p r −2 0 ε 2 . Combining the two cases we can choose ε 2 = ε 2 (p, n) as ε 2 = min{2 − 1 p ε 1 , 2 − n+1 p r 2 0 ε 1 , ε 0 }. The theorem is now proved by setting ε = ε 2 . Combining Theorem 1.1 with (2.10), we have 13 Corollary 4.5. If k(p, 1) ≤ ε for the ε in Theorem 1.1, then, (4.11) f * n n−1 ,B 1 (x) ≤ 10 2n+4 ∇f * 1,B 1 (x) , ∀f ∈ C ∞ 0 (B 1 (x)), Applying (4.11) to f 2(n−1) n−2 and using the Hölder inequality gives (4.12) f * 2n n−2 ,B 1 (x) ≤ 2(n − 1) n − 2 10 2n+4 ∇f * 2,B 1 (x) , ∀f ∈ C ∞ 0 (B 1 (x)). This is essential in the applications. By a scaling argument, we have Corollary 4.6. If k(p, 1) ≤ ε for the ε in Theorem 1.1, then, for any r ≤ 1, (4.13) f * n n−1 ,Br(x) ≤ C(n)r ∇f * 1,Br(x) , ∀f ∈ C ∞ 0 (B r (x)), and (4.14) f * 2n n−2 ,Br(x) ≤ C(n)r ∇f * 2,Br(x) , ∀f ∈ C ∞ 0 (B r (x)). Corollary 4.7. If k(p, 1) ≤ ε for the ε in Theorem 1.1, then, for any r ≤ 1, the first eigenvalue of Dirichlet Laplace has lower bound (4.15) λ 1 (B r (x)) ≥ C(n) −1 r −2 . Proof. Suppose △f = −λf for some λ > 0 and f with ffl f 2 dv = 1 and f = 0 on ∂B r (x). Then = Br(x) f 2 dv ≤ Br(x) f 2n n−2 n−2 n ≤ C(n)r 2 Br(x) \|∇f \| 2 = C(n)r 2 λ. Thus λ ≥ C(n) −1 r −2 for any eigenvalue λ > 0. Applications With this new local Sobolev constant estimate many of the results for integral curvature in [26,29] can be easily extended to the collapsed case. In particular, we have maximum principle, gradient estimate for harmonic function and heat kernel, excess estimate, L 2 estimate for the Hessian of the harmonic and parabolic approximation of the distance function. Denote C s (Ω) the normalized local Soboleve constant of Ω ⊂ M n , (5.1) f * 2n n−2 ,Ω ≤ C s (Ω) ∇f * 2,Ω , ∀f ∈ C ∞ 0 (Ω). Note that C s (Ω) scales like diameter. Recall the following maximal principle [26, Corollary 3.2]. 14 Theorem 5.1. Let M be an n-dimensional Riemannian manifold, and p > n/2. For any function u : Ω ⊂ M → R with ∆u ≥ −f , where f is non-negative on Ω, we have sup Ω u ≤ sup ∂Ω u + C(n, p) · C 2 s (Ω) · f * p,Ω . Combining this with (4.14) gives Theorem 1.8. Now we derive the following gradient estimate. \|∇u\| 2 ≤ C(n, p)R −2 · vol B R (x) vol B 3 4 R (x) · ( u * 2,B R (x) ) 2 + ( f * 2p,B R (x) ) 2 · R −2 C 2 s (B R (x)) 2p 2p−n 1 + k(p, R) 2p 2p−n + R −2 C 2 s 1 + R −2 C 2 s k(p, R) n/2 The estimate follows from the standard Nash-Moser iteration, by using the L p integrability of Ric and f 2 for p > n 2 . On the other hand, as we do not assume the harmonicity of u (i.e. f = 0), and Ricci curvature pointwise lower bound, the proof requires several extra estimates and the Laplacian comparison estimate (2.2). This full general version is often needed in applications. Since a proof is not in the literature, we give a detailed proof here. Proof. By scaling we may assume R = 1. Recall the Bochner formula, (5.2) 1 2 △\|∇u\| 2 = \| Hess u\| 2 + ∇u, ∇f + Ric(∇u, ∇u) ≥ ∇u, ∇f − \|Ric − \|\|∇u\| 2 . Put v = \|∇u\| 2 + f 2 * p . Note that when f is constant, one can iterate with v = \|∇u\| 2 and the proof simplifies. For any function η ∈ C ∞ 0 (B 1 (x)) and constant q > 1, computê \|∇(ηv q/2 )\| 2 = −ˆηv q ∆η − 2ˆηv q/2 ∇η, ∇v q/2 −ˆη 2 v q/2 △v q/2 =ˆ(2\|∇η\| 2 − η∆η)v q − 2ˆv q/2 ∇η, ∇(ηv q/2 ) −(1 − 2 q )ˆ\|∇(ηv q/2 ) − v g/2 ∇η\| 2 − q 2ˆη 2 v q−1 △v. By regrouping, \|∇(ηv q/2 )\| 2 = q 2(q − 1)ˆ( (1 + 2 q )\|∇η\| 2 − η∆η)v q − 1 q − 1ˆv q/2 ∇η, ∇(ηv q/2 ) − q 2 4(q − 1)ˆη 2 v q−1 △v ≤ 1 2ˆ\| ∇(ηv q/2 )\| 2 + 1 2 q 2 + q − 1 (q − 1) 2ˆ\| ∇η\| 2 v q − q 2(q − 1)ˆη v q ∆η − q 2 4(q − 1)ˆη 2 v q−1 △v. Hence, (5.3) \|∇(ηv q/2 )\| 2 ≤ q 2 + q − 1 (q − 1) 2ˆ\| ∇η\| 2 v q − q q − 1ˆη v q ∆η − q 2 2(q − 1)ˆη 2 v q−1 △v. Now plugging (5.2) into (5.3), we havê \|∇(ηv q/2 )\| 2 ≤ q 2 + q − 1 (q − 1) 2ˆ\| ∇η\| 2 v q − q q − 1ˆη v q ∆η + q 2 q − 1ˆη 2 v q \|Ric − \| − q 2 q − 1ˆη 2 v q−1 ∇u, ∇f . For the last term, we havê η 2 v q−1 ∇u, ∇f = −ˆη 2 v q−1 f 2 − 2ˆηf v q−1 ∇u, ∇η − (q − 1)ˆη 2 f v q−2 ∇u, ∇v ≥ −ˆ6η 2 v q−1 f 2 −ˆ\|∇η\| 2 v q − 2(q − 1) q 2 · 1 8ˆη 2 \|∇v q/2 \| 2 ≥ − 2(q − 1) q 2 · 1 4ˆ\| ∇(ηv q/2 )\| 2 −ˆ6η 2 v q−1 f 2 − (1 + q − 1 2q 2 )ˆ\|∇η\| 2 v q . To control ∆η, we choose a more specific cur-off function. For 0 < r < 1, let ϕ ∈ C ∞ 0 (R) be a cut-off function such that 0 ≤ ϕ ≤ 1, ϕ(t) ≡ 1 for t ∈ [0, r], ϕ(t) ≡ 0 for t ≥ 1, and ϕ ′ ≤ 0. Then define (5.4) η(y) = ϕ(d(x, y)), where d(x, y) is the distance function from x. Thus \|∇η\| = \|ϕ ′ \|, and ∆η = ϕ ′′ + ϕ ′ ∆d = ϕ ′′ + ϕ ′ (∆d − n − 1 d + n − 1 d ) ≥ ϕ ′′ + ϕ ′ (ψ + n − 1 d ) ≥ −\|ϕ ′′ \| − \|ϕ ′ \| r − \|ϕ ′ \|ψ, where ψ = (∆d − n−1 d ) + . Therefore we have, for q ≥ n n−2 , \|∇(ηv q/2 )\| 2 ≤ C(n)qˆ \|ϕ ′′ \| + \|ϕ ′ \| r + \|ϕ ′ \|ψ ηv q + \|ϕ ′ \| 2 v q + η 2 f 2 v q−1 + η 2 \|Ric − \|v q . Notice that this formula remains valid for q = 1. In fact \|∇(ηv 1/2 )\| 2 = v 1/2 ∇η + η \|∇u\| v 1/2 ∇\|∇u\| 2 ≤ 2v\|∇η\| 2 + 2η 2 \| Hess u\| 2 , andˆη 2 \| Hess u\| 2 = −ˆ∇ i u(2η∇ j η∇ i ∇ j u + η 2 ∇ i △u + η 2 R ij ∇ j u) ≤ 1 2ˆη 2 \| Hess u\| 2 + 3ˆ\|∇η\| 2 v +ˆη 2 f 2 +ˆη 2 \|Ric − \|v. Denote µ = n n−2 . Applying the Sobolev inequality (5.1), we obtain for q ≥ n n−2 and q = 1, (5.5) B 1 (x) (η 2 v q ) µ 1/µ ≤ C 2 s (B 1 (x))C(n)q B 1 (x) \|ϕ ′′ \| + \|ϕ ′ \| r + \|ϕ ′ \|ψ ηv q + \|ϕ ′ \| 2 v q + f 2 η 2 v q−1 + \|Ric − \|η 2 v q . The integration involving Ricci curvature can be estimated as follows. For p > n 2 , B 1 (x) \|Ric − \|η 2 v q ≤ Ric − * p · B 1 (x) (η 2 v q ) p p−1 p−1 p ≤ Ric − * p B 1 (x) η 2 v q p−1 p a · B 1 (x) (η 2 v q ) µ (1−a) p−1 p ≤ Ric − * p ǫ B 1 (x) (η 2 v q ) µ 1 µ + ǫ − (1−a)µ a · B 1 (x) η 2 v q . where a = a(n, p) = 2p−n 2(p−1) > 0 is determined via a + (1 − a)µ = p p − 1 . Here we used Young's inequality xy ≤ ǫx γ + ǫ − γ * γ y γ * , ∀x, y ≥ 0, γ > 1, (γ * ) −1 + γ −1 = 1, where γ = p (1 − a)(p − 1)µ , γ * = p (p − 1)a . By setting ǫ = 4C(n)qC 2 s ) Ric − * p −1 , we conclude C(n)qC 2 s B 1 (x) η 2 \|Ric − \|v q (5.6) ≤ 1 4 ffl B 1 (x) (η 2 v q ) µ 1 µ + C(n, p) qC 2 s Ric − * p 2p 2p−n · ffl B 1 (x) η 2 v q . For the term ffl B 1 (x) η 2 f 2 v q−1 , since v ≥ f 2 * p , we have B 1 (x) η 2 f 2 v q−1 ≤ 1 f 2 * p B 1 (x) η 2 f 2 v q ≤ B 1 (x) (η 2 v q ) p p−1 p−1 p . Now the same argument as above with ǫ = (4C(n)C 2 s )q) −1 gives C(n)C 2 s q B 1 (x) η 2 f 2 v q−1 (5.7) ≤ 1 4 B 1 (x) (η 2 v q ) µ 1 µ + C(n, p) C 2 s q 2p 2p−n · B 1 (x) η 2 v q . For the term with ψ, using the Hölder inequality and the Laplacian comparison estimate (2.2), B 1 (x) ψη\|ϕ ′ \|v q ≤ ψ * 2p · ηϕ ′ v q * 2p 2p−1 ≤ C(n, p) Ric − * p 1/2 · ηϕ ′ v q * 2p 2p−1 . Note that for b = p(n−2) n(2p−1) < 1, ηϕ ′ v q * 2p 2p−1 = B 1 (x) η 2 v q bµ \|ϕ ′ \| 2 v q p 2p−1 2p−1 2p ≤ B 1 (x) η 2 v q µ b B 1 (x) \|ϕ ′ \| 2 v q np np+2p−n np+2p−n n(2p−1) 2p−1 2p ≤ B 1 (x) η 2 v q µ b B 1 (x) \|ϕ ′ \| 2 v q p 2p−1 2p−1 2p ≤ ǫ B 1 (x) η 2 v q µ 1/µ + 1 4ǫ B 1 (x) \|ϕ ′ \| 2 v q . Here we used the fact that np np+2p−n < 1 since p > n/2. Choose ǫ = (4C(n)C 2 s qC(n, p) Ric − * p 1/2 ) −1 , we have C(n)C 2 s q B 1 (x) ψη\|ϕ ′ \|v q (5.8) ≤ 1 4 B 1 (x) (η 2 v q ) µ 1 µ + (qC(n)C 2 s ) 2 C 2 (n, p) Ric − * p B 1 (x) \|ϕ ′ \| 2 v q . Plugging the three estimates (5.6), (5.7), (5.8) into the inequality (5.5) gives B 1 (x) (η 2 v q ) µ 1/µ ≤ 4C 2 s C(n)q B 1 (x) \|ϕ ′′ \| + \|ϕ ′ \| r ηv q + 1 + qC 2 s C 2 (n, p) Ric − * p \|ϕ ′ \| 2 v q +C(n, p) C 2 s q 2p 2p−n 1 + ( Ric − * p ) 2p 2p−n B 1 (x) η 2 v q . Define q k = µ k , k ≥ 0, and r k = ( 3 4 − k i=0 2 −i−3 ). Choose cut-off functions η k = ϕ k • d ∈ C ∞ 0 (B r k (x)) such that η k ≡ 1, on B r k+1 (x); \|ϕ ′ k \| ≤ 2 k+5 , \|ϕ ′′ k \| ≤ 2 2k+10 . Then substituting η k into the estimate and running the iteration for any k ≥ 0 we get v * ∞,B 1 2 (x) ≤ C(n, p)A n/2 v * 1,B 3 4 (x) , where A = C 2 s (1 + C 2 s Ric − * p ) + C 4p 2p−n s 1 + ( Ric − * p ) 2p 2p−n . Finally observe that, by integrating by parts, for η ∈ C ∞ 0 (B 1 (x)) with η ≡ 1 in B 3 4 (x), and \|∇η\| ≤ 5, we have B 3 4 (x) \|v\| ≤ vol B 1 (x) vol B 3 4 (x) B 1 (x) η 2 (\|∇u\| 2 + f 2 * p ) ≤ vol B 1 (x) vol B 3 4 (x) f 2 * p + B 1 (x) η 2 (u 2 + f 2 ) + 8 B 1 (x) \|∇η\| 2 u 2 ≤ 201 · vol B 1 (x) vol B 3 4 (x) · ( u * 2 ) 2 + 2( f * 2p ) 2 . This gives the gradient estimate. Combining (2.5) and (4.14) with Theorem 5.2 gives Theorem 1.9. Later on we will need Harnack inequality for harmonic function. Hence we also give a gradient estimate for ln u as in Cheng-Yau's gradient estimate [10]. In the proof we need Li-Schoen's trick of bounding high power by lower power [19]. sup B R 2 (x) \|∇ ln u\| 2 ≤ C n, p, R −2 C 2 s , k(p, R) R −2 vol B R (x) vol B 4 5 R (x) . Proof. By scaling we may assume R = 1. Let h = ln u, v = \|∇h\| 2 . Then ∆h = −v. From the Bochner formula, 1 2 △\|∇h\| 2 = \| Hess h\| 2 + ∇h, ∇∆h + Ric(∇h, ∇h) ≥ v 2 n − ∇h, ∇v − \|Ric − \|v. For any η ∈ C ∞ 0 (B 1 (x)), l ≥ 0, multiply above by v l η 2 and integrate on B 1 (x) gives, (5.9) 1 2ˆv l η 2 △v ≥ˆη 2 v l+2 n −ˆv l η 2 ∇h, ∇v −ˆv l+1 η 2 \|Ric − \|. We computê v l η 2 ∇h, ∇v = −ˆv l+1 η 2 ∆h − lˆv l η 2 ∇h, ∇v − 2ˆv l+1 η ∇h, ∇η . Henceˆv l η 2 ∇h, ∇v = 1 l + 1ˆv l+2 η 2 − 2 l + 1ˆv l+1 η ∇h, ∇η ≤ 2 l + 1ˆv l+2 η 2 + 1 l + 1ˆv l+1 \|∇η\| 2 . (5.10) By (5.3), (l + 1) 2 2lˆv l η 2 △v ≤ −ˆ ∇ ηv l+1 2 2 + (l + 1) 2 + l l 2ˆv l+1 \|∇η\| 2 − l + 1 lˆη v l+1 ∆η (5.11) Plugging (5.11) and (5.10) into (5.9), we have −ˆ ∇ ηv l+1 2 2 ≥ (l + 1) 2 l 1 n − 2 l + 1 ˆv l+2 η 2 + l + 1 lˆη v l+1 ∆η − 2l 2 + 4l + 1 l 2ˆv l+1 \|∇η\| 2 − (l + 1) 2 lˆv l+1 η 2 \|Ric − \|. When l ≥ 2n − 1, choose η as in (5.4), we havê ∇ ηv l+1 2 2 ≤ C(n)lˆ \|ϕ ′′ \| + \|ϕ ′ \| r + \|ϕ ′ \|ψ ηv l+1 + v l+1 \|ϕ ′ \| 2 + v l+1 η 2 \|Ric − \| 20 Use Sobolev inequality (5.1) and estimate as in (5.6), (5.8) and iterate from l = 2n − 1 as in Theorem 5.2, we have (5.12) v * ∞,B 1 2 (x) ≤ C(n, p)A n/2 v * 2n−1,B 3 4 (x) , where A = C 2 s (1 + C 2 s Ric − * p ) + C 4p 2p−n s 1 + ( Ric − * p ) 2p 2p−n . Since we have volume doubling, by the proof of Theorem 2.1 in [19], we can lower the power 2n − 1 in (5.12) by adjusting the size of the balls. Namely we have v * ∞,B 1 4 (x) ≤ C n, p, C 2 s , Ric − * p v * 1,B 4 5 (x) . For the L 1 bound, since v = −∆h, B 1 (x) η 2 v = −ˆB 1 (x) η 2 ∆h = 2ˆB 1 (x) η ∇η, ∇h ≤ 1 2ˆB 1 (x) η 2 v + 2ˆB 1 (x) \|∇η\| 2 , where η ∈ C ∞ 0 (B 1 (x)) is a cut-off function with η = 1 on B 4 5 (x) and \|∇η\| ≤ 6. Hence v * 1,B 4 5 (x) ≤ 144 vol B 1 (x) vol B 4 5 (x) . With Theorem 1.9 one can prove as in [26,Theorem 6.4] the following. Lemma 5.4. For any integer n and p > n 2 there exist ε and C such that the following holds. Let M be a complete n-dimensional Riemannian manifold satisfying k(p, 1) ≤ ε. For any metric ball B r (x) with ∂B r (x) = ∅, r ≤ 1, there exists φ ∈ C ∞ 0 (B r (x)) satisfying 0 ≤ φ ≤ 1, \|∇φ\| 2 + \|△φ\| ≤ Cr −2 . With the (relative) local Sobolev constant estimate (1.4), one gets heat kernel upper bound, see e.g. [16, (2.17)]. With this and the volume doubling (2.5), Zhang-Zhu obtained Li-Yau's gradient estimate [32]. Hence one has parabolic Harnack inequality. With this we have the local heat kernel lower bounds as in [11,Lemma 2.3]. Namely, we have Theorem 5.5. Let M be an n-dimensional Riemannian manifold, and p > n/2. There is an ε (n, p) > 0 and C (n, p) > 1 such that if k(p, 1) ≤ ε, then for any real number s, 0 < r < 1, x ∈ M and nonnegative solution u of the heat equation in Q = (s − r 2 , s) × B r (x), sup Q − u ≤ C inf Q + u, where Q − = (s − 3 4 r 2 , s − 1 2 r 2 ) × B 1 2 r (x), Q + = (s − 1 4 r 2 , s) × B 1 2 r (x). With the above tools, we can extend Colding-Naber's L 2 Hessian estimate for the parabolic approximation of the distance function to the integral curvature setting without essentail difficuties. In the noncollapsed case it is done in [29], see also [31]. In what follows, we always assume p > n 2 and M is a complete n-dimensional Riemannian manifold satisfying k(p, 1) ≤ ε(n, p) for the ε so the results above all hold. Fix two points y − , y + in M n , the excess is e(x) = d(y − , x) + d(y + , x) − d(y − , y + ). Define b + (x) = d(y + , x) − d(y − , y + ), b − (x) = d(y − , x). Hence e(x) = b + (x) + b − (x). Note that ∆b ± (x) ≤ n − 1 d(x, y ± ) + ψ ± , where ψ ± = ∆d(y ± , x) − n−1 d(x,y ± ) + is the error term of the Laplacian comparison. Denote d 0 = d(y − , y + ). Without loss of generality, assume d 0 ≤ 1. Denote by A r 1 ,r 2 = A r 1 d 0 ,r 2 d 0 ({y − , y + }) the annulus for the set {y − , y + }, with 0 < r 1 < r 2 . Then Corollary 5.9 and the Laplacian comparison estimate (2.2) gives Theorem 5. 10. Fix some small postive constant δ > 0. There existǭ =ǭ(n, p, δ) and C = C(n, p, δ) such that for all 0 < ǫ <ǭ, x ∈ Aδ 4 ,16 , B ǫd 0 (x) e(y)dy ≤ C e(x) + (ǫd 0 ) 2 ( ψ − * 2p,d 0 + ψ + * 2p,d 0 ) ≤ C e(x) + ǫ 2 d 0 . In particular, if e(x) ≤ ǫ 2 d 0 , then e(y) ≤ Cǫ 1+ 1 n+1 d 0 , ∀y ∈ B 1 2 ǫd 0 (x). Remark 5.11. We obtain the optimal integral bound for the excess as in the pointwise Ricci lower bound case [ and \|∇φ\| 2 + \|∆φ\| ≤ C(n, p, δ). Define the parabolic approximation functions b ±,t and e t by b ±,t (x) =ˆH(x, y, t)φ(y)b ± (y)dvol(y) 24 and e t (x) =ˆH(x, y, t)φ(y)e(y)dvol(y). Then e t = b +,t + b −,t . Following [29], we have following estimates for the approximates, which play important role in the Cheeger-Colding-Naber local theory for Gromov-Hausdorff limits. Theorem 5.12. There exists C = C(n, p, δ) such that for all 0 < ǫ ≤ǭ(n, p, δ), any x ∈ Aδ 2 ,4 with e(x) ≤ ǫ 2 d 0 and any ǫ-geodesic σ connecting y − , y + , there exists r ∈ [ 1 2 , 2] with 1. b ±,rǫ 2 d 2 0 − b ± ≤ Cd 0 (ǫ 2 + ǫ 2− n 2p ). 2. ffl B ǫd 0 (x) \|∇b ±,rǫ 2 d 2 0 \| 2 − 1 ≤ C(ǫ + ǫ 1− n 2p ). 3. ffl (1−δ)d 0 δd 0 ffl B ǫd 0 (σ(s)) \|∇b ±,rǫ 2 d 2 0 \| 2 − 1 ≤ C(ǫ 2 + ǫ 2− n p ). 4. ffl (1−δ)d 0 δd 0 ffl B ǫd 0 (σ(s)) Hess b ±,rǫ 2 d 2 0 2 ≤ C(1+ǫ − n p ) d 2 0 . We will only show the first lemma here to indicate the difference. Lemma 5.13. There exists a constant C = C(n, p, δ) such that ∆b ±,t , ∆e t ≤ C 1 d 0 + t − n 4p for t < 1. Proof. Since, for x ∈ A δ 16 ,16 , ∆(φb + ) = b + ∆φ + 2 ∇φ, ∇b + + φ∆b + ≤ Cd −1 0 + ψ + , we have ∆b +,t (x) =ˆA H(x, y, t)∆ y (φ(y)b + (y))dvol(y) ≤ C d 0 +ˆA δ 16 ,16 H(x, y, t)ψ + dvol(y). 25 Using the upper bound for H(x, y, t) and argue as in Proposition 5.7, we havê A δ 16 ,16 H(x, y, t)ψ + dvol(y) ≤ ψ + 2p H(x, y, t) 2p 2p−1 ≤ C(n, p) ψ + * 2p t − n 4p . These give the estimate; the other terms are exactly the same. Necessity of smallness of integral Ricci By exploring Yang's counter-exmaple [28], we point out that the smallness of integral Ricci curvature, (1.2), is a critical condition in order to get the L p version of Cheeger-Colding theory. For any k > 1, let M = (−1, 1) × T n−1 be a portion of a complete manifold with a family of warped product metric (6.1) g ǫ = dr 2 + (ǫ 2 + r 2 ) k g F where T is a compact torus with flat metric g F and ǫ > 0 is the parameter. A direct calculation gives the sectional curvature K( ∂ ∂x i , ∂ ∂x j ) = −k 2 r 2 (ǫ 2 + r 2 ) −2 , K( ∂ ∂x i , ∂ ∂r ) = −k(ǫ 2 + r 2 ) −1 − k(k − 2)r 2 (ǫ 2 + r 2 ) −2 , and the Ricci curvature Ric( ∂ ∂x i , ∂ ∂x i ) = − (n − 2)k 2 r 2 + k(k − 2)r 2 + k(ǫ 2 + r 2 ) (ǫ 2 + r 2 ) −2 , Ric( ∂ ∂r , ∂ ∂r ) = −(n − 1) k(k − 2)r 2 + k(ǫ 2 + r 2 ) (ǫ 2 + r 2 ) −2 , Ric( ∂ ∂x i , ∂ ∂r ) = 0, where (x 1 , · · · , x n−1 ) is the local normal coordinate of T n−1 . Fix one p ∈ ( n 2 , k(n−1)+1 2 ). In the following calculation, ≈ means equivalence up to a multiplication by a constant depending only on n and p. The first observation is (6.2) \|Rm gǫ \| ≈ \|Ric gǫ \| ≈ k 2 r 2 (ǫ 2 + r 2 ) −2 + k(ǫ 2 + r 2 ) −1 . f (x, t)(ǫ 2 + t 2 ) (n−1)k 2 dv g F dt. 26 Applying to the curvature function we have, whenever ǫ << r, B * r \|Rm gǫ \| p ≈ k 2pˆr 0ˆT (ǫ 2 + t 2 ) (n−1)k 2 −p dv g F dt ≈ k 2p k(n − 1) + 1 − 2p vol(T )r k(n−1)+1−2p ≈ k 2p−1 vol(T )r k(n−1)+1−2p , (6.4) whenever k >> 1 is sufficiently large. Then notice that vol(B * r ) ≈ 1 k(n − 1) + 1 vol(T )r k(n−1)+1 ≈ k −1 vol(T )r k(n−1)+1 . Thus we have, whenever ǫ << r, (6.5) r 2 B * r \|Rm gǫ \| p 1 2p ≈ k 2 . In particular, the formula remains hold on the limit space, (6.6) r 2 Br(o) \|Rm g 0 \| p 1 2p ≈ k 2 , ∀r > 0. The family of manifolds (M, g ǫ ) has polynomial volume growth. Hence a generalized local Sobolev constant and Poincaré constant are still bounded 1 . However, since the limit collapse at the origin, the volume comparison of geodesic spheres fails. Furthermore, to guarantee the splitting property on the tangent cones of the limit space we eventually need that → 0 as r → 0, which never hold. Therefore, the boundedness of L p norm of curvature tensor is not sufficient to extend Cheeger-Colding theory. and Ric H − (x) = ((n − 1)H − ρ(x)) + = max {0, (n − 1)H − ρ(x)},the amount of Ricci curvature lying below (n − 1)H. Let Theorem 1. 1 . 1For p > n/2, there exists ε = ε(p, n) > 0 such that if M n has k(p, 1) ≤ ε, then for any x ∈ M, r ≤ 1 with ∂B 1 (x) = ∅, the normalized Dirichlet isoperimetric constant has the estimate(1.3) ID * n (B r (x)) ≤ 10 2n+4 r, where ID * n (B r (x)) = vol(B r ( Theorem 3. 2 . 2Given p > n/2 and D > 0, there is an ε (n, p, D) > 0, r 0 = r 0 (n) such that if diam M n ≤ D and Ric − * p ≤ ε, then ID * n (B r (x)) ≤ C (n, p, D) for all x ∈ M and r ≤ r 0 . Lemma Lemma 4. 2 . 2Let H, W and x 1 be as in above lemma. Then W ) ≤ 2 n−1 D vol(H ′ ) + vol(B D (x 1 )) Ric − * 1 2 p,B D (x 1 ) Corollary 4. 3 . 3Let H be any hypersurface dividing M into two parts. For any ball B = B r (x) we have min vol(B ∩ M 1 ), vol(B ∩ M 2 ) ≤ 2 n+1 r vol(H ∩ B 2r (x)) + vol(B 2r (x)) Ric − * 1 2 p,B 2r (x) .(4.6) Proof. Put W i = B ∩ M i in above lemma and notice that D ≤ 2r and H ′ ⊂ H ∩ B 2r (x). Corollary 4. 4 . 4Given a hypersurface H dividing M n into two parts, there exists ε = ε(p, n) such that if k(x, p, 1) ≤ ε, then for a metric ball B = B r (x), r ≤ 1 2 , which is divided equally by H, we have vol(B r (x)) ≤ 2 n+3 r vol(H ∩ B 2r (x)).(4.7) Proof. The previous corollary gives vol(B) ≤ 2 n+2 r vol(H ∩ B 2r (x)) + vol(B 2r (x)) B 2r (x) \|Ric − \| p dv 1 2p Theorem 5 . 3 . 53Assume as in above theorem. Let u be a positive harmonic function in B R (x), then r = {(t, x) ∈ M\| − r < t < r}.Then, for any function f ,B * r f dv gǫ =ˆr −rˆT 4.1 ([14]). Let M n be a complete Riemannian manifold and H be any hypersurface dividing M into two parts M 1 , M 2 . For any Borel subsets W i ⊂ M i , there exists x 1 in one of W i , say W 1 , and a subset W in the other one, W 2 , such that(4.1) vol(W ) ≥ 1 2 vol(W 2 ) ).The domain Ω has a covering Ω ⊂ x∈Ω B 2rx (x). By Vitali Covering Lemma, cf. [20, Section 1.3], we can choose a countable family of disjoint balls 11, Theorem 2.6], compare [29, Corollary 2.19], [31, Lemma 4.9]. For the pointwise estimate, note that Abresch-Gromoll's original estimate gives ǫ 1+ 1 n−1 [1]. As in [11], one can extend Lemma 5.4 to annulus so we have the cut-off function φ such that φ = 1 in Aδ 4 ,8 ; φ = 0 outside A δ 16 ,16 It should be understood as the [k(n−1)+1]-Sobolev inequality, but not the n-Sobolev inequality we have proved. Actually, the calculation shows the n-Sobolev constant is not bounded in Yang's example. 21The heat kernel H(x, y, t) satisfies the two-sided Gaussian boundfor all t ∈ (0, 1) and x, y ∈ M.For our purpose, we need two-sided bound on the Dirichlet heat kernels of the balls. Let H B (x, y, t) be the Dirichlet heat kernel of the ball B ρ (x) with t ∈ (0, 1) and ρ ≥ √ t. Note that H B (x, y, t) ≤ H(x, y, t). With the local volume doubling and Poincare inequality, by[16, (3.4)], there exist constants (depending only on the constants from the volume doubling and Poincare inequality) a, τ (small), A(large) and c such thatBy making ε smaller (and a rescaling argument as at the end of the proof of Theorem 1.1) we obtain Theorem 5.6 (Dirichlet Heat kernel upper and lower bounds). For any integer n and p > n 2 there exist ε and C such that the following holds. Let M be a complete n-dimensional Riemannian manifold satisfying k(p, 1) ≤ ε. Let H Br(x) (x, y, t) be the Dirichlet heat kernel of the ball B r (x). ThenThis Dirichlet heat kernel upper and lower bounds give the quantitive mean value inequality.Proposition 5.7. Under the assumption above, let u be a nonnegative function satisfyingwhere f is a nonnegative space-time function. Then, for q > n 2 ,for all x ∈ M, r ≤ 1.Remark 5.8. For our application it's crucial that the norm of f is a normalized local norm instead of the global norm in[29]. It recovers Lemma 2.1 in[11], where it is proven when f is constant. The key here is to use Dirichlet heat kernel of balls.22Proof.By the upper bound of H Br(x) (x, y, r 2 − t), we have, for q > 1,Br(x). In the last step we used the volume doubling property (2.5).u(y, 0)d vol(y) − C(n, p, q)r 2 supHere we used the lower bound for H Br(x) (x, y, r 2 ) on B 1 2 r (x) and q > n 2 . Corollary 5.9. Assume as above. Let u be a nonnegative function satisfying ∆u ≤ f.Then, for q > n 2 ,for all x ∈ M, r ≤ 1.This is the L 1 Harnack inequality. For the Euclidean case, see e.g.[15,Theorem 4.15]. We would like to thank Ruobin Zhang for this reference. On complete manifolds with nonnegative Ricci curvature. U Abresch, D Gromoll, J. Amer. Math. Soc. 32U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc. 3 (1990), no. 2, 355 -374. The L 2 structure of moduli spaces of Einstein metrics on 4-manifolds. M T Anderson, GAFA. 2M. T. Anderson, The L 2 structure of moduli spaces of Einstein metrics on 4-manifolds, GAFA., 2 (1992), 29-89. Finiteness of π 1 and geometric inequalities in almost positive Ricci curvature. E Aubry, Ann. Sci.École Norm. Sup. 4E. Aubry, Finiteness of π 1 and geometric inequalities in almost positive Ricci curvature, Ann. Sci.École Norm. Sup. (4) 40 (2007), no. 4, 675 -695. R H Bamler, arXiv:1512.08527Compactness properties of Ricci flows with bounded scalar curvature. R. H. Bamler, Compactness properties of Ricci flows with bounded scalar curvature, arXiv:1512.08527. Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature. R H Bamler, Q S Zhang, Adv. Math. 319R. H. Bamler and Q. S. Zhang, Heat kernel and curvature bounds in Ricci flows with bounded scalar curvature, Adv. Math. 319 (2017), 396 -450. A note on the isoperimetric constant. P Buser, Ann. Sci.École Norm. Sup. 4P. Buser, A note on the isoperimetric constant, Ann. Sci.École Norm. Sup. (4) 15 (1982), no. 2, 213-230. I Chavel, Riemannian Geometry: A Morden Introduction, Cambridge studies in advanced mathematics 98. Cambridge University PressI. Chavel, Riemannian Geometry: A Morden Introduction, Cambridge studies in advanced mathematics 98, Cambridge University Press, 2006. A lower bound for the smallest eigenvalue of the Laplacian. J Cheeger, Problems in Analysis, R. 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Rose, Heat kernel upper bound on Riemannian manifolds with locally uniform Ricci curva- ture integral bounds, J. Geom. Anal. 27 (2017), no. 2, 1737-1750. M Simon, arXiv:1504.02623Some integral curvature estimates for the Ricci flow in four dimensions. M. Simon, Some integral curvature estimates for the Ricci flow in four dimensions, arXiv:1504.02623 Analysis and geometry on manifolds with integral Ricci curvature bounds. P Petersen, G F Wei, Trans. AMS. IIP. Petersen and G.F. Wei, Analysis and geometry on manifolds with integral Ricci curvature bounds. II, Trans. AMS., 353 (2000), 457-478. A Neumann Type Maximum Principle for the Laplace Operator on Compact Riemannian Manifolds. G Wei, R Ye, Journal of Geometric Analysis. 193G. Wei and R. Ye, A Neumann Type Maximum Principle for the Laplace Operator on Compact Riemannian Manifolds, Journal of Geometric Analysis 19, no. 3 (2009), 719-736. D Yang, Convergence of Riemannian manifolds with integral bounds on curvature. 25D. Yang, Convergence of Riemannian manifolds with integral bounds on curvature. I, Ann. Scient.Èc. Norm. Sup., 25 (1992), 77-105. Regularity of Kähler-Ricci flows on Fano manifolds. G Tian, Z L Zhang, Acta Math. 216G. Tian and Z.L. Zhang, Regularity of Kähler-Ricci flows on Fano manifolds, Acta Math., 216 (2016), 127-176. Convergence of Kähler-Ricci flow on lower dimensional algebraic manifolds of general type. G Tian, Z L Zhang, Int. Math. Res. Not. IMRN. 21G. Tian and Z.L. Zhang, Convergence of Kähler-Ricci flow on lower dimensional algebraic manifolds of general type, Int. Math. Res. Not. IMRN 2016, no. 21, 6493-6511. Q Zhang, M Zhu, arXiv:1511.00791Li-Yau gradient bounds under nearly optimal curvature conditions. Q. Zhang and M. Zhu, Li-Yau gradient bounds under nearly optimal curvature conditions, arXiv:1511.00791. Li-Yau gradient bound for collapsing manifolds under integral curvature condition. Q Zhang, M Zhu, Proc. Amer. Math. Soc. 1457Q. Zhang and M. Zhu, Li-Yau gradient bound for collapsing manifolds under integral curvature condition, Proc. Amer. Math. Soc. 145 (2017), no. 7, 3117-3126. . (xianzhe Dai ; Department Of Mathematics, Ecnu, Ucsb Shanghai, Santa Barbara, C A , email: [email protected] Barbara CA 93106; ChinaGuofang Wei) Department of Mathematics, University of California ; Zhenlei Zhang) Department of Mathematics, Capital Normal Universityemail:[email protected]. email: [email protected](Xianzhe Dai) Department of Mathematics, ECNU, Shanghai and UCSB, Santa Bar- bara CA 93106, email:[email protected] (Guofang Wei) Department of Mathematics, University of California, Santa Bar- bara CA 93106, email: [email protected] (Zhenlei Zhang) Department of Mathematics, Capital Normal University, China, email: [email protected]	[]
[ "Privacy Concerns Raised by Pervasive User Data Collection From Cyberspace and Their Countermeasures", "Privacy Concerns Raised by Pervasive User Data Collection From Cyberspace and Their Countermeasures" ]	[ "Yinhao Jiang \nSchool of Computing and Mathematics\nCharles Sturt University\nNSWAustralia\n\nCyber Security Cooperative Research Centre\nJoondalupWAAustralia\n", "Ba Dung Le \nSchool of Computing and Mathematics\nCharles Sturt University\nNSWAustralia\n\nCyber Security Cooperative Research Centre\nJoondalupWAAustralia\n", "Tanveer Zia \nSchool of Computing and Mathematics\nCharles Sturt University\nNSWAustralia\n\nCyber Security Cooperative Research Centre\nJoondalupWAAustralia\n\nCenter of Excellence in Cybercrime and Digital Forensics\nNaif Arab University for Security Sciences\nSaudi Arabia\n", "Praveen Gauravaram \nTata Consultancy Services\nAustralia\n" ]	[ "School of Computing and Mathematics\nCharles Sturt University\nNSWAustralia", "Cyber Security Cooperative Research Centre\nJoondalupWAAustralia", "School of Computing and Mathematics\nCharles Sturt University\nNSWAustralia", "Cyber Security Cooperative Research Centre\nJoondalupWAAustralia", "School of Computing and Mathematics\nCharles Sturt University\nNSWAustralia", "Cyber Security Cooperative Research Centre\nJoondalupWAAustralia", "Center of Excellence in Cybercrime and Digital Forensics\nNaif Arab University for Security Sciences\nSaudi Arabia", "Tata Consultancy Services\nAustralia" ]	[]	The virtual dimension called 'Cyberspace' built on internet technologies has served people's daily lives for decades. Now it offers advanced services and connected experiences with the developing pervasive computing technologies that digitise, collect, and analyse users' activity data. This changes how user information gets collected and impacts user privacy at traditional cyberspace gateways, including the devices carried by users for daily use. This work investigates the impacts and surveys privacy concerns caused by this data collection, namely identity tracking from browsing activities, user input data disclosure, data accessibility in mobile devices, security of delicate data transmission, privacy in participating sensing, and identity privacy in opportunistic networks. Each of the surveyed privacy concerns is discussed in a well-defined scope according to the impacts mentioned above. Existing countermeasures are also surveyed and discussed, which identifies corresponding research gaps. To complete the perspectives, three complex open problems, namely trajectory privacy, privacy in smart metering, and involuntary privacy leakage with ambient intelligence, are briefly discussed for future research directions before a succinct conclusion to our survey at the end.	null	[ "https://arxiv.org/pdf/2202.04313v1.pdf" ]	246,680,233	2202.04313	0bc140546b6d0283929e1da23fb2942cbd5a4e43	Privacy Concerns Raised by Pervasive User Data Collection From Cyberspace and Their Countermeasures 9 Feb 2022 Yinhao Jiang School of Computing and Mathematics Charles Sturt University NSWAustralia Cyber Security Cooperative Research Centre JoondalupWAAustralia Ba Dung Le School of Computing and Mathematics Charles Sturt University NSWAustralia Cyber Security Cooperative Research Centre JoondalupWAAustralia Tanveer Zia School of Computing and Mathematics Charles Sturt University NSWAustralia Cyber Security Cooperative Research Centre JoondalupWAAustralia Center of Excellence in Cybercrime and Digital Forensics Naif Arab University for Security Sciences Saudi Arabia Praveen Gauravaram Tata Consultancy Services Australia Privacy Concerns Raised by Pervasive User Data Collection From Cyberspace and Their Countermeasures 9 Feb 2022Preprint submitted to Elsevier February 10, 2022User privacyWeb privacy protectionWearable devicesOpportunistic network privacy * Corresponding author The virtual dimension called 'Cyberspace' built on internet technologies has served people's daily lives for decades. Now it offers advanced services and connected experiences with the developing pervasive computing technologies that digitise, collect, and analyse users' activity data. This changes how user information gets collected and impacts user privacy at traditional cyberspace gateways, including the devices carried by users for daily use. This work investigates the impacts and surveys privacy concerns caused by this data collection, namely identity tracking from browsing activities, user input data disclosure, data accessibility in mobile devices, security of delicate data transmission, privacy in participating sensing, and identity privacy in opportunistic networks. Each of the surveyed privacy concerns is discussed in a well-defined scope according to the impacts mentioned above. Existing countermeasures are also surveyed and discussed, which identifies corresponding research gaps. To complete the perspectives, three complex open problems, namely trajectory privacy, privacy in smart metering, and involuntary privacy leakage with ambient intelligence, are briefly discussed for future research directions before a succinct conclusion to our survey at the end. Introduction Technologies, especially computing and communication technologies have significantly influenced our busy daily lives and created a virtual dimension to our lives. This virtual space, we call "cyberspace", is usually described as a web of knotting communicating handsets and computers knotted together via a variety of networks. In cyberspace, instances can include an individual's avatars, smart devices, cloud computing portals, network gateways and so on. The interconnections between instances in cyberspace offer convenient platforms for information exchange, e.g., the internet. With the explosive growth of the internet from the end of the last century, we have experienced a fundamental transformation regarding how information can be created, acquired, disseminated, and used. The internet continues its development by connecting embedded sensors, electronic tags on goods or freights, networked cameras, smart phones and vehicles, and almost all the things that are used on a frequent/daily basis. This latest development, including the Internet of Things (IoT) and wireless sensor networks, has raised a new concept of how people approach the cyberspace. Considering more items are being connected to networks, most people now rely heavily on this computing environment and advanced data analytics. The resulting benefits encourage more connectivity to even more items and more connectivity opportunities from different types of networks towards a 'hyper-connected world' [1] where people can freely interact between the physical world and cyberspace with other people and resources. However, these connected items also collect personal information and feed the sensitive data for further analysis. New surveillance to private activities and unexpected personal information leakage have been generated, which creates security uncertainties and reshapes the digital personal privacy landscape. The concept of privacy in cyberspace was established in 1997 [2], helping the cultivation and fast development of the Internet. Many regulations and bills have been proposed and issued for the protection of privacy. A common background that cyberspace serves users' calls as a distant and abstract oracle was generally recognised. This may be the fact how cyberspace has been considered in the last two decades: we use a computer or handset to connect to cyberspace; we enjoy the provided services with full control; and cut the connection by simply closing the websites. Fortunately or not, due to the new connected items in users' proximity, cyberspace no longer remains passive to users' calls. Thus, we need to examine how cyberspace activly collect users' information and how this information is used or exchanged in cyberspace. It becomes essential for us to learn what new privacy concerns and issues may happen in cyberspace and what the corresponding countermeasures are. Novel technologies have changed how user information gets collected from users' manual input to more pervasively collection. The pervasive user data collection is conducted by both connected software and hardware that monitors users' activities in cyberspace and the physical world. For users' activities in cyberspace, advertisement cookies, Flash content and hyperlinked images quietly collect what users are browsing, what applications users are focusing on and the data indicating user preferences. Between cyberspace and the physical world, smart devices and wearable gadgets including mobile phones, smart watches and fitness trackers have been equipped with more resources so that they are capable of recording, storing and processing health data, daily routines, and other activity data. Furthermore, for users' activities in the physical world, ubiquitous computing devices, especially the mobile phones, are now capable of collecting and uploading captured images, sounds, voices and videos, which may cause involuntary consent. The change of pervasive data collection triggers many new privacy concerns, and countermeasures are needed to prevent user privacy from slipping into a turmoil. The rest of this paper is organised as follows. In Section 2, we discuss privacy concerns at the traditional borders with cyberspace and survey two featured scenarios of user activities in cyberspace, namely web privacy protection and user input disclosure, including existing technologies. In Section 3, privacy problems between cyberspace and the physical world are investigated. This focuses on safe data accessibility and delicate transmission security. In Section 4, we explore user activities in the physical world involving pervasive computing where user privacy in participatory sensing and opportunistic networks riase user privacy concerns. Open issues are discussed in Section 5 considering privacy protection for complex applications. The conclusions of this paper are outlined in Section 6. Privacy in data exhaust tracing First we take a close look at the traditional border between the physical world and cyberspace, e.g., the gateways from web browsers. With the rise of machine learning applications, big data analysis has been developed on the data exhaust from people's daily we browsing. The advertising industry, usually the third-party domains connected with publishers' websites, uses tools like cookies that put different identifiers on users so that browsing data exhaust can be traced and used to reconstruct individual browsing history. Fig 1 illustrates that different tracking entities collect data exhaust underneath normal web browsing, using cookies [3], flash [4] or browser fingerprinting [5], and analyse the data with cookie synchronising technology [6]. With the analysis, advertisers evaluate users' features including behavioural targeting, frequency capping, re-targeting and conversion tracking, to display their advertisements. At the same time, the publishers also tailor information suiting users' conditions and predicted requirements based on evaluated preferences that the users have never set manually. The data exhaust collection and the following data analysis have brought incontrovertible benefits for both parties. While users enjoy the automatically personalised responses from a variety of websites and network services, publishers boast an approximately 52% increased revenue from third-party cookie usage [7]. However, this trend of data exhaust collection also triggers two major privacy concerns: identity tracking from browsing exhaust and private data disclosure. In the remaining part of this section, we survey these two privacy concerns and discuss solutions that can mitigate these privacy risks. Identity tracking from browsing exhaust At the traditional boundary between the physical world and cyberspace, advertising revenue has prompted the development of tracking individual browsing history, shopping behaviours and purchase habits from browsing exhaust. When a website is accessed, many entities other than the publisher domain are also connected. The user generally classifies all the entities into two categories: first and third parties, where the publisher, the entity that the user visits is the first party,and other connected stakeholders, such as advertisers are the third parties that provide the publisher with a variety of services. Generally, the adopted technologies from the third-parties are various cookies, local data storage [4] or fingerprinting [5], with which third parties put user identifications in temporary files and track from different publishers (or combine with HTTP referrer fields) to build a comprehensive data dump that will be later analysed for further evaluations or even exchanged with other third parties via cookie synchronising for an improved data collection. Thus, it becomes important that users' private information can be protected from ambient collection by third-parties. Web privacy protection: browser extensions Different techniques have been introduced to protect users from rowsing exhaust tracking.. Network monitor techniques include DNS filtering and network proxy, which are effective in specific scenarios but have apparent shortcomings against encrypted transmissions and individual URLs [8]. A popular type of protection comes from browser extensions that can reliably distinguish third-party content from a publisher's web pages and block unwanted content including encrypted web traffic. In the following section, we describe several popular and featured browser extensions in detail. This information is summarised in Table 1. We classify selected tracking protection extensions by their blocking techniques: crowd-sourced lists, centralised maintenance, algorithm-based and machine learning based blocking. For extensions that are based on crowdsourced lists, there are many famous and familiar names such as Ad-Blocker [9], AdBlock [10], Adblock Plus [11] and uBlock [12]. These extensions have drawn attentions in the community since the early days of the battle against third-party advertisements. As a result, a back-bone blocking list EasyList has been contributed and maintained by the community. At the time of writing, EasyList consists of over 17000 third-party advertisers, 13500 general, third-party and specific URL patterns as well as 31000 advertisement elements filters [13]. With EasyList, extensions like Ad-Blocker use filter rules to block ads from being loaded to help restrain user tracking. In addition to EasyList, extensions based on rules also adopt other filter lists, including anti-circumvention lists or third-party tracker filter lists. The anti-circumvention lists help advertisement-blocking extensions fight against detection and circumvention to the extensions and re-insertion of ads. The third-party tracker filter lists help against tracking from companies and organisations that does not directly insert advertisements. The other popular type of advertisement tracking blocking relies on centralised maintenance including extensions like Ghostery [14], Disconnect [15] and Blur [16]. Due to the centralised control, the extension companies set the blocking rules, and they typically have considerably fewer rules than crowdsourced lists [8]. Although these extensions allow users to customise some blocking rules, certain network requests are defined as necessary third-party content, which cannot be blocked. Besides the difference in blocking ads, commercialised versions are provided with extra functionalities: Ghostery Insights [17] offers time-lined analysis and loading performance details; Disconnect Premium [18] offers an optional VPN, full IP masking and data encryption; Blur Premium [19] offers further protections to personal information. Unfortunately, these new features requires not only financial support but also permission to collect user information. An alternative type of protection extension, different to filter lists, uses algorithms to automatically decide whether a third-party's content needs to be blocked. A popular example is an extension called Privacy Badger [20], which counts how many websites a third-party organisation uses to track a user and blocks content from loading if the count for that organisation reaches three [21]. In addition to blocking trackers from ads, Privacy Badger can also detect canvas-based browser fingerprinting and block tracking from the third-party domains [21]. A newly emerged ad-blocking tool adopts machine learning techniques based on a perceptual study from the ads'loading content. [22,23,24] have introduced a new concept of perceptual ad-blocking, which seeks to improve resilience against ad obfuscation and minimise the manual effort needed to create ad-blockers. For traditional ad-blocking relying on crowd-sourced lists or ones based on centralised maintenance, two downsides have been identified: 1) the consistency of filter lists requires constant synchronisation with the latest versions; 2) different strategies have been developed for evading crowd-sourced lists (like EasyList) such as changing domains, moving resources to the publishers, removing ad keywords from URLs, and removing image dimensions from URLs. Thus, it became an arms race between ads and tracker blocking tools and third-party domains. Researchers claim Machine learning On image pattern [24] On screenshot [23] On rendering engine [28] On behaviour pattern the novel approach of using perceptual signals effectively reduces the arms race with web publishers and ad networks [25]. Storey et al. [22] founded their perceptual ad-blocking based on a legal requirement for the recognisable display of ads by humans. Based on the legal requirement, Storey et al.'s Ad-Highlighter [26] focuses on learning captured visual and behavioural information that can be used to distinguish ads, e.g., the text "Sponsored", the ad's circled "i" information icon, or an ad network logo. However, their foundation is not as solid as they thought since the markup information, visual or behavioural, can be further rendered to an invisible state. To overcome the challenge of this specific rendering, [24] introduced the project Sentinel, a machine learning version of Adblocker Plus that uses an objectdetection neural network to locate ads in raw website page screenshots [27]. To further exploit rendered web pages, [23] introduced a new technique to achieve the goal. In their work, a deep-learning based ad-blocker module is embedded into Chromium's rendering engine so that images of ads can be detected directly [23]. Besides many experimental adoptions of machine learning for perceptual ad-blocking, [28] showed a different way of using machine learning based classification to block ads. Iqbal et al. [28] introduced AdGraph, which applies machine learning approaches on graph representations built from web pages considering aspects such as the HTML structure, network requests, and JavaScript behaviour. A modification based on AdGraph was implemented on Chromium, resulting in high accuracy and fewer computational overheads compared to traditional ad-blocking extensions. User input data disclosure Another important change that happens at the border, which differs from the passive collecting and tracking activities, is how users' input gets recorded and utlised. This change is associated with many questions simultaneously: how much personally identifiable input has been collected, how is the input data with sensitive information transmitted, how much sensitive information will be disclosed, would any critical information get leaked and would the user be re-identified from authorised analysis services, etc. To cope with these questions, a relatively traditional approach, called differential privacy, was proposed that embeds extra noise and increases the entropy so that collected input data can be possibly denied by users and thus protected [29]. However, many problems remain for applying differential privacy technology to large-scale sensitive input data stored centrally [30]. Conventional differential privacy is applied after data are collected in a centralised way and conducted using a quantitative approach by a trusted party. This brings uncertainties for users inclined towards independent intuitive methods when private information is involved. To address the uncertainties, the recent topic of local differential privacy (LDP), where users randomly perturb their inputs to provide plausible deniability of their data without the need for a trusted party, has come to the fore. Local differential privacy The goal of differential privacy is to process a dataset in such a way that it is not possible to determine if a certain entry has been removed. The distribution remains indistinguishable in a preset range even if a certain entry got removed [29]. When an algorithm achieves the goal with the preset range, it instantly guarantees that no observer can determine if a particular individual participated or if the data from this subject has been used [31]. For LDP, users interact with an untrusted aggregator such that the aggregator learns statistical information about the distribution of the private value in the user population, while the information leakage for each individual is bounded. An important property for local differential privacy is that noise perturbation is conducted at the user end, so the collected data is not original. This property has brought significant applications in user data collection including Google's RAPPOR [32] and Apple's Learning with privacy [33]. The foundation of most LDP realisations is an easily understandable concept extracted from Random Response [34] that with a probability the collected data is the true value. For different collecting styles and data use purposes, developments from Google, Apple and Microsoft are combined with other techniques targeting longitudinal collections, itemsets mining and repeated numeric value collection. Google's RAPPOR [32] uses a unary encoding that reports on a permanent randomised response for a question asked. The technique ensures privacy, keeps information utility and helps with repeated data collection. In the case of a survey with a large domain, they further adopted Bloom filters [35] for efficient element encoding. The RAPPOR technique has initialised a series of research developments in heavy hitter identification with LDP [36,37,38], which can be seen as an extended problem of frequency estimation [39]. After Google's efforts with heavy hitters, Apple's design focuses on the problem of frequent itemsets mining on a large scale. They first used discrete Fourier Transformation to cope with privacy sensitivity and noise adding, which successfully handles the sparsity of the input vectors. They then adopted sketching algorithms that reduce the dimensionality of the domain helping to learn the processed data. Apple's implementation successfully demonstrated an application of centralised differential privacy in the LDP setting. Several following works started a research trend on itemset mining [40] and marginal release [41] focusing on utilising LDP data. Although LDP has attracted much research from academia and industry, current research focuses on how to get useful information from collected LDP data. An equally important question in LDP is how to implement LDP for other data collecting scenarios considering survey domain and approaches with specific characteristics. An example is Microsoft's Telemetry collection [42], which challenges a private numerical value scenario with very small but frequent changes. Other identified special survey domain includes graph data analysis [43], language data analysis [44]; and survey approaches considering multiple rounds of interactions [45] and prior knowledge [46]. Discussion We have surveyed protections on browsing exhaust and user response disclosure for data collection at the traditional border between the physical world and cyberspace. For browsing exhaust protection, there is an ongoing arms race between web browser extensions and exhaust tracking techniques. Core concepts for the developing browser extensions have evolved from elementary rule-based filters to perceptual blocking involving machine learning technologies, where further research can be focused. For private user response disclosure, LDP has shown its promising applications from Google and Apple's implementation. Despite the current research streams on LDP data utilisation, the realisation of LDP on special survey domains and approaches requires more research. Privacy in personal data hub After close inspection of the changes happening at the border, we now investigate privacy concerns raised with the expanding industry of smart devices that collect private user data, store the data locally for access from applications, and transmit the data to the internet for further analysis and advanced services. Smart devices and wearable gadgets can provide instant connected experiences and services with novel functionality. At the same time, they also collect data and communicate with each other and the internet for further and better user benefits. As showed in Fig. 2, the data collected by existing smart devices and wearable equipment concentrate on user status information that is considered highly private. The collected user status information includes activity, location and health information, which can be used by different applications in smart devices and analytics from cloud computing. Thus, the smart devices and wearable equipment form a personal data hub that collects, shares and distributes data from users, which leads to questions about user privacy protections. In such a personal data hub system, there are two important questions about user privacy: how to control the access of data stored locally and how to guarantee data security during transmission. Although these two questions also exist in centralised data hub systems, the collected data in personal data hub systems mostly consists of personal information. Also, the devices carried by users do not have consistent computational resources due to heterogeneous manufacturer designs. In the following section, we examine these two questions and state-of-art solutions. Safe data accessibility With smart devices and wearable gadgets playing important roles in people's health, an essential privacy problem is the collected data. These devices have a wide range of specifications regarding computing processors, memory and communication power. This brings them function space for third-party applications with direct connections to the internet. Thus, multiple entities could try to access these devices for their collected data, including malicious adversaries. In addition to potential external attacks, the collected data may contain highly sensitive information like health records and location coordinates, which could lead to misuse between applications on wearable gadgets and damage user privacy. Access control enforcement for the wearable equipment The technology of access control enforcement plays a core protection role in many IoT network systems, since it directly answers the privacy issue of accessibility mentioned above. It applies a range of selective policies, setting the criteria of who can access the data. The main purpose of an access control enforcement mechanism is to block unauthorised and random queries towards a protected data repository. Besides the passive protection, rather than blocking arbitrary connections, it sets up a bottom line against insider attacks or general platform sharing with an efficient privilege update and revocation mechanism. Access control mechanisms for IoT systems have drawn much research attention and several works have been proposed as effective and practical solutions for wearable technology in different scenarios. Since access control enforcement has a wide research scope, in the following section we survey a few typical works and focus on wearable gadgets and connected healthcare devices. One research focus required for wearable gadgets is to develop contextaware access control with a more expressive policy. In 2010, Garcia-Morchon and Wehrle [47] proposed a modular context-aware access control mechanism that allows a system administrator to compose each module with a well-defined goal so that access policies for different required functionalities can be assigned to different modules. Ray et al. [48] tried to improve the expressiveness using attribute-based access control from the NIST NGAC framework and achieved the first conceptual prototype for an IoT infrastructure. Later in the same year, Salama et al. [49] successfully combined public key infrastructure and attribute-based access control for a multi-level access control on patient healthcare monitoring. Another research focus for wearable gadgets and connected healthcare devices is usability. This feature is neglected by most of the existing access control works since an administrative model is generally assumed for access control scenarios. However, especially for wearable gadgets, there is no administrative staff for these private devices and the users are the ones who configure, manage and protect the devices and resources. Thus, for the users who often lack the necessary security knowledge, an easy-to-use interface and enhanced presentation need to be provided for policy configuration [50]. In 2011, Kim et al. [51] proposed the first access control mechanism that provides a full solution to usability. Their newly introduced automated Clairvoyant access right assignment mechanism can suggest suitable access control policies. Unfortunately, their work is designed for smart home scenarios where its inherent overprivilege property can be tolerated [50]. To address this issue of overprivilege, Tian et al. [52] proposed an automated access policy generation based on checking the functionality and behaviour of the entity that asks for the access. Their access control mechanism was oriented for smartphone applications accessing local resources, which can be extended to other IoT systems like accessing data in wearable gadgets. After an appropriate access policy is generated, it is then provided to the user for review. Other research focuses include distributed environments [53,54], dynamic access control [55], scalability [56] and multilateral security [57]. These works will be compared with aforementioned works from other focuses in Table 2. Security in delicate transmission Besides guarding the access of the data stored locally, the problem of protecting sensitive data during transmission has brought many challenges to connected devices, especially resource-restrained healthcare devices. Solutions to this problem have been focused on the field of encryption and its efficiency optimisation. While conventional encryption schemes like the Advanced Encryption Standard work fine on wearable gadgets with reasonable [47] Modular context-aware access control [48] NIST NGAC framework [49] Multi-level access control [52] Automated access policy generation [53] BiLayer access control model [54] Virtual patient record alternative [55] Indeterminacy-tolerant access control [56] Healthcare RFID tag access control [57] Multi-level and multilateral security processing power and memory capabilities [58], they meet various bottlenecks when applied to monitoring devices and small sensors. To make the encryption approach compatible with these devices, lightweight encryption methods are proposed focusing on reducing computational overheads and increasing scheme efficiency. Lightweight encryption in healthcare devices Lightweight symmetric encryption can provide encryption requirements from connected healthcare devices, especially implantable medical devices like pacemakers where other protections are difficult to implement. Connected healthcare devices are usually computationally weak and restrained by battery life, and implantable medical devices often are additionally restricted with a minimal physical size that leads to implementation constraints in hardware [59,60,61]. With these limitations, some features/properties in lightweight encryption become rather more acceptable and welcome. These features include implementation flexibility, smaller block size, encryption rounds saving, and restricted versatility. • Implementation Flexibility-For the implementation of encryption on resource-restrained devices, the trade-off is only determined when applied to a specific scenario [62]. Thus, when a feature is specifically needed for a deployment scenario, the encryption algorithm should be optimised with acceptable sacrifice to other aspects. • Lower Size-For healthcare devices that have a small physical size and need to run for an extended period with limited battery, the design of an encryption algorithm may need to prioritise resource limitations. In this case, a smaller size of block size or internal state becomes acceptable. • Less Rounds-For healthcare devices, a particular nature is that its total amount of output messages is considered relatively fewer. For example, a pacemaker working for ten years outputs less than 2 30 pairs of plaintext and ciphertext, which may lead to a relax of the total number of primitive rounds while retaining approximate the same security level [63]. • Limited Versatility-The healthcare device where the encryption algorithm is to be implemented is usually function and operation focused, which makes encryption algorithms that have limited versatility rather welcome. Considering the above implementation difficulties, security requirements, and feature preferences, our survey on lightweight symmetric encryption focuses on the algorithms that have a small block size or internal state, and can manage short keys. Most of the candidate algorithms lie in block ciphers and stream ciphers due to the restrained resource. For hash function based algorithm, only PHOTON [64] and Spongent [65] have ideally small internal state size. A summary of surveyed algorithms is showed in Table 3. Discussion For private data collected stored in many smart devices including wearable equipment that builds a personal data hub, we selectively surveyed privacy concerns on how the collected data can be accessed and how the sensitive data is transmitted by resource-restrained devices. Existing access control approaches help with the general purpose of controlling accessibility. However, most research works have not considered the usability that presents an essential requirement for personal scenarios. Another field in access control for future research is how to delicately assign accessibility according to the sensitivity of the collected data. An example would be location information in residential areas, compared to public places, should be considered highly private and not suitable to be accessed by most applications. In terms of protection during transmission, lightweight encryption has showed practical promise in many IoT devices. For healthcare devices, which could benefit from the seamless 5G network in the near future, characterised lightweight encryption schemes are expected to fit the challenging privacy scenarios. Privacy in active data collection We now explore on privacy issues associated with the various uses of connected devices, e.g., to capture and record thescenery, people, activities, and other phenomena in the physical world. People carry connected devices equipped with different types of sensors and use them in their daily lives to retrieve information on a more and more frequent basis. Fig. 3 illustrates the paradigm of how data collected via smart phones from the physical world may also be easily transmitted to different entities and services in cyberspace. In parallel with this active capturing and sharing, services have been developed based on a new environmental data collection model that individuals collect data on behalf of the service provider. These participators contribute their smart phones (or other smart devices) as sensors for applications helping collect environmental data such as surrounding noise, traffic conditions, thermal columns, etc.. With an estimated 3 billion people having smart phones, this new model, known in the literature as participatory sensing [73,74], can accomplish large-scale sensing. However, personal information may get uploaded as media like photos and videos, which carry time and location information, or through being leaked from access by applications with permission. Both these cases of data collecting via connected devices share common privacy concerns for user location information. At the same time, active data collecting encourages people to expand their collecting area to barren lands or distant villages in the mountains, valleys where network infrastructure is not available so that self-organising opportunistic networks need to be relied on. In this scenario, data collecting behaviours trigger the use of opportunistic networks and expose people's identities to untrusted authentications that raise identity privacy concerns, especially for participatory sensing where participants need records of sensing activities with identity for rewards. Below we take participatory sensing and opportunistic networks as two typical scenarios to survey related privacy concerns. Participatory sensing and collector protection Participatory sensing rushes onto the scene as a development of users' socialising data dissemination habits from users' activities on social media to remote data collection via multiple sensors on smart devices for services. When data is captured in social activities or participatory sensing, information about location, time and identity is simultaneously recorded and bound with the media data for uploading [75]. This bundle uploading happens at most user activities in cyberspace, from social media to ubiquitous participatory sensing. However, personal information that users think they have deleted adheres to its primary data in cyberspace [76], which then will be extracted, categorised, and analysed for long-term use. Location privacy in participatory sensing Location privacy is one of the most important concerns for participants in the participatory sensing activities because revealing location details within the sensing communication framework and to external entities may endanger personal privacy. Nevertheless, the location data utility challenges the method used to protect location privacy, as low quality in location data can damage the significance of participatory sensing. Thus, the trade-off between location quality and location privacy has drawn attention from researchers. Many techniques have been studied and implemented for protecting location privacy while maintaining data utility. Most of the techniques can be traced back to the following three influential methods: • Dummy locations-The work proposed in [77], brings the concept of dummy locations accompanying the true location of the user together in the query for services. Their technique focuses on scenarios of an individual's location information for services. Since the location information is from a user alone, the conventional technique of reducing accuracy [78] may not guarantee the user's privacy, while dummy locations make service providers unable to distinguish where the user truly is. • Obfuscation-The work in [79] proposed a novel approach of degrading location information with negotiation algorithms so that users privacy can be protected as location information obfuscated while the quality of service remains due to the negotiations. The obfuscation has since been developed as a major approach for location privacy; most of the following works have been carried out via the technique of perturbation [80] or generalisation [81]. • k-anonymity-One of the most promising location privacy preserving approaches was originated from the k-anonymity concept [82] by Gruteser and Grunwald [78], which conceals the location information of a user within a group of k-1 other users. The k-anonymity approach is considered effective for location privacy in general scenarios where the other k-1 users are not difficult to find. However, it does not perform well on continuous location information transmission and sometimes has too low accuracy resulting in service failure [83]. Privacy in opportunistic networks When people travel between cities, states or rural regions, they need to connect to different networks for services. A surfacing type of network, named opportunistic networks (OppNets), plays an important role in maintaining the connections in a simple, easy, and convenient way. Opportunistic networks usually are built on convenient communications technologies for short-term connectivity when there is limited or no existing traditional network infrastructure. Although these networks may only provide connections with limited bandwidth, high latency and packet-loss rate, applications still upload collected data. Since opportunistic networks are provided as a necessary alternative, their nature of expanding from node to node makes them self-organising, which unfortunately come with some undesired features. OppNets use store-carry and forward mechanism to connect and extend the network because the path between the source and destination does not exist [84,85,86]. For messages or packets through intermediate nodes, any of these intermediate nodes can turn malicious and exploit received information. Thus, countermeasures are needed to protect user privacy considering the forwarded messages may contain personal information, especially during the authentication procedure. Anonymous authentication for OppNets The node authentication procedure plays an essential role in OppNets: it confirms a node's credentials and prevents unauthorised nodes from joining/accessing the network; it also authenticates the integrity of the packet received by nodes. As mentioned above, the path between two distant nodes in OppNets does not exist, so it is a challenging task for researchers to construct efficient authentication algorithms and solutions validating connectivity between nodes. It adds further difficulties for authentication algorithms to achieve anonymity in a dynamic OppNets environment. So far in the literature, only a limited range of anonymous authentication algorithms succeed in OppNets. These are compared in Table 4. In 2012, Carver and Lin utilised group-oriented broadcast encryption based on pairing and identity-based signature, and constructed an authentication scheme for OppNets based on Bluetooth and 3G communications [87]. In their work, the forwarding packet does not require knowledge of recipients, so the privacy of users is partially protected. The sender's information, however, will be acknowledged after authentication. In addition, their work relies on a trusted third party for key generation and group categorisation, which leads to a critical risk on system security and user privacy if the third-party is compromised [88]. Ref. Technique Feature [87] Broadcast encryption Partial privacy [89] Symmetric & asymmetric encryption Hashed user ID [84] RSA encryption Dynamic user ID [90] Hash functionality Point operation Encryption free In 2015, Guo et al. proposed an authentication protocol for OppNets that fully protects user privacy [89]. The authentication is designed for short-term and limited-time wireless network environment where node registration is performed by a selected super node [89]. In their protocol, both symmetric and asymmetric encryption are adopted as countermeasures for security threats and attacks while privacy is achieved by protecting user identities with hash functions. Later in 2017, Kumar et al. proposed a security algorithm based on [89] that fulfils the authentication requirements in OppNets and protects user privacy. Different from [89], Kumar et al.'s work utilises dynamic user identities for the key exchange mechanism and RSA encryption for message transmission, so both user and data privacy can be protected. Besides the aforementioned works that heavily rely on symmetric and asymmetric encryption, Kuo et al. proposed an efficient and secure anonymous authentication scheme utilising hash functionality and point operations [90]. Although their work was not motivated for OppNets, the roaming authentication feature can be applied to OppNets environments with enhanced performance and additional security properties. Discussion The existence of users and data generated by users raises concerns about protecting user privacy during active data collecting activities. From these concerns, we survey the problems of location privacy in participatory sensing and anonymous authentication in OppNets. To protect a participant's location data, a compromise in the quality of location data is usually the trade-off, although many efforts have been made to mitigate the effect. Concerning identity privacy in OppNets, existing solutions heavily rely on encryption techniques, which can be expensive considering heterogeneous devices. Encryption free anonymous authentication requires more research as it potentially has more application scenarios. Future vision on complex privacy problems There are many complex privacy problems already identified that soon could have a considerable impact on industry as well as our daily lives. We deliver our future vision on three of these privacy concerns, i.e., trajectory privacy, privacy in smart metering, and involuntary information leakage with ambient intelligence. Trajectory privacy When we apply the traditional scenario of cookie privacy concerns to mobile applications, users' trajectories become at risk due to location information embedded in cookie logs. Cookie logs in cyberspace may contain high-quality user location information, which can be collected directly using GPS coordinates with a user's fast consent to an unexplained location service permission requirement, or indirectly collected with location tags from a local network or service provider in the physical world. This potential privacy breach should be categorised to a more dangerous level than web browsing history or personal preference logs. More detailed physical activities, routine habits or even mental status can be inferred by analytical work on user trajectories. The infamous Uber travel history leakage lawsuit in 2017 [91] is a relevant example. [92] developed a privacy analysis system on user login records and physical context information, and deepened the understanding of user physical-world privacy leakage via cyberspace privacy leakage. It becomes clear that user trajectories can be discovered and confirmed when third parties analyse their cookie logs as users move and browse in their daily lives let alone potential exogenous records of GPS coordinates. These cookie logs may further be exchanged with other analytics companies for centralised analysis connecting with other web activities, exposing exposes private physical trajectory to more entities. Compared with other private data, physical trajectory is more effective on re-identification with auditing relevant activity logs at locations and comparing differential timelines. The balance between utility and privacy with location data has drawn much research attention. However, for this physical world trajectory leakage via user cyberspace data, further research efforts are required. Privacy in smart metering Another emerging privacy concern is in smart metering in smart energy supply networks, which is considered as the next evolutionary step in the industry [93]. For smart energy supply, smart metering helps evaluate the status of a smart energy grid and contribute detailed usage information for efficient resource distribution. To achieve this, many smart meters and sensors are needed between consuming points and monitoring centres [94,95,96] as well as the networks used for sending and receiving precise metering data. Thus, privacy concerns arise as the consumption usage data are transmitted and stored in plaintext. Since the consumption usage data include private activities of consumers, this information, which is invaluable to service providers [97], together with identity/location tags and physical address information, forms a large attack surface drawing adversaries' attention. For this privacy concern, a key obstacle is the limited resource the smart meters have to perform strong cryptography [97]. Therefore, it remains a challenge for future research focusing on cryptography-based mechanisms that must provide confidentiality, while minimising resource consumption. Involuntary privacy leakage with ambient intelligence Ambient intelligence makes environments sensitive to users as sensors deliver the change of the state of an environment for computing faster and better services to a user [98]. As smart devices play more important roles in our daily lives, a mobile ambient intelligence is subtly forming with different functionalities. Beneficial services and better user experience provided by smart phones as well as other wearable gadgets encourage users to give the green light to data collection from smart devices, which implicitly offers a convenient way for private information to be collected and leaked. Since these smart devices are connected to the internet acting as the gate between the physical world and cyberspace, potential attacks on collected private data may come from both dimensions. An example of an attack from cyberspace can be seen from [99] that reported an incident of smart watches leaking its users' real-time location data, which is described as "pretty common" by Manuel. Although attacks from cyberspace focusing on software bugs can be fixed in hours, privacy breaches can occur due to incomplete test procedures, even if there are many effective security approaches in place [99]. Now we consider potential attacks from the physical world, which can be unintentional or malicious. For unintentional attacks, an example would be a stranger's face being captured accidentally by other people in an event, who then upload the photo to social media networks. Although the owner of the photo does not know who the stranger is, machine learning algorithms behind social media can potentially identify the stranger. This information would include details of the time the photo was taken as well as where the photo was taken. With more and more resources that smart phones and other smart devices are equipped with, people can now capture and record almost every detail of what they see, hear and experience in their daily lives. So the fate of information about those captured also lies in their hands. This leads to two questions for future research: would smart devices be able to realise what kind of environment they are in and apply the correct privacy strategy, and could smart devices notify their owners about what kind of environment they are currently in and warn them to behave with caution. Conclusion User data is being increasingly collected by new pervasive technologies for further data analytics in cyberspace, which raises many privacy concerns. In this paper, we comprehensively surveyed and studied these concerns that are classified as data exhaust tracing, personal data hub and active data collection. Among them, six privacy concerns including identity tracking from browsing exhaust, security in delicate transmission, and privacy in opportunistic networks, are featured and defined with a real-world application scenario. The privacy concerns are studied in the scenarios for their properties and the differences compared to similar but well-known problems are highlighted. With these highlighted differences, we surveyed existing technologies for countermeasures, compared their methodologies and performance, and discussed implementation difficulties and research gaps. With these impending and overhanging concerns analysed, we further analysed the privacy in complex problems of user trajectory, smart metering and ambient intelligence. In these complex problems, privacy concerns involve elements from many aspects, which present many challenges with complicated difficulties. Possible research suggestions and further directions are discussed. Figure 1 : 1Analysing and tracking activities at the border. User preference information gets collected, exchanged and analysed from browsers. Figure 2 : 2Personal data hub paradigm. Smart devices and healthcare devices collect personal data for cyberspace entities to provide advanced services and connected experiences. Figure 3 : 3Information leakage during active data collecting behaviours. People use smart devices collect and upload different types of data that simultaneously gets collected by cyberspace. Table 1 : 1Surveyed Web Browser ExtensionsRef. Technique Feature [10] Rule-based filter Crowd-sourced list [9] [11] [12] [14] Centralised maintenance [15] [16] [20] Algorithm-based Detect browser fingerprinting [21] [22] Table 2 : 2Comparison among access control enforcement for wearable equipmentRef. Feature Table 3 : 3Comparison Among Suitable Lightweight Encryption SchemesBlock Ciphers Name Ref. Key Block Rounds Joltik [66] 64/80/96/128 64 24/32 Mantis [67] 128 64 14 Skinny [67] 64-384 64/128 32-56 Qarma [68] 128/256 64/128 16/24 T-TWINE [69] 80/128 64 36 Stream Ciphers Name Ref. Key IV IS A2U2 [70] 61 64 95 Sprout [71] 80 70 89 Plantlet [72] 80 90 110 Hash Name Ref. Digest Block IS PHOTON [64] 80-256 16/32/64 100-288 Spongent [65] 80-256 8/16 Table 4 : 4Comparison Among Anonymous Authentication for OppNets AcknowledgementThe work has been supported by the Cyber Security Research Centre Limited whose activities are partially funded by the Australian Government's Cooperative Research Centres Programme. M Conti, A Passarella, S K Das, The internet of people (iop): A new wave in pervasive mobile computing. 41M. Conti, A. Passarella, S. K. Das, The internet of people (iop): A new wave in pervasive mobile computing, Pervasive and Mobile Computing 41 (2017) 1-27. Whatever works": The american public's attitudes toward regulation and self-regulation on consumer privacy issues, US Dep't Of Commerce, Privacy And Self-Regulation In The Information Age. A F Westin, A. F. 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[ "RULED AND QUADRIC SURFACES OF FINITE CHEN-TYPE", "RULED AND QUADRIC SURFACES OF FINITE CHEN-TYPE" ]	[ "Hassan Al-Zoubi ", "Stylianos Stamatakis ", "Hani Almimi " ]	[]	[]	In this paper, we study ruled surfaces and quadrics in the 3dimensional Euclidean space which are of finite III-type, that is, they are of finite type, in the sense of B.-Y. Chen, with respect to the third fundamental form. We show that helicoids and spheres are the only ruled and quadric surfaces of finite III-type, respectively.2010 Mathematics Subject Classification. 53A05.	null	[ "https://arxiv.org/pdf/1710.07619v2.pdf" ]	119,652,090	1710.07619	428baecb30cae044d6f7a3fdce75b7e5b1e3c90d	RULED AND QUADRIC SURFACES OF FINITE CHEN-TYPE 4 Feb 2022 Hassan Al-Zoubi Stylianos Stamatakis Hani Almimi RULED AND QUADRIC SURFACES OF FINITE CHEN-TYPE 4 Feb 2022 In this paper, we study ruled surfaces and quadrics in the 3dimensional Euclidean space which are of finite III-type, that is, they are of finite type, in the sense of B.-Y. Chen, with respect to the third fundamental form. We show that helicoids and spheres are the only ruled and quadric surfaces of finite III-type, respectively.2010 Mathematics Subject Classification. 53A05. Introduction Let M n be a (connected) submanifold in the m-dimensional Euclidean space E m . Let x, H be the position vector field and the mean curvature field of M n respectively. Denote by ∆ I the second Beltrami-Laplace operator corresponding to the first fundamental form I of M n . Then, it is well known that [21] ∆ I x = −nH. From this formula one can see that M n is a minimal submanifold if and only if all coordinate functions, restricted to M n , are eigenfunctions of ∆ I with eigenvalue λ = 0. Moreover in [32] T. Takahashi showed that the submanifold M n for which ∆ I x = λx, i.e., for which all coordinate functions are eigenfunctions of ∆ I with the same eigenvalue λ ∈ R, are precisely either the minimal submanifold with eigenvalue λ = 0 or the minimal submanifold of hyperspheres S m−1 with eigenvalue λ > 0. Although the class of finite type submanifolds in an arbitrary dimensional Euclidean spaces is very large, very little is known about surfaces of finite type in the Euclidean 3-space E 3 . Actually, so far, the only known surfaces of finite type corresponding to the first fundamental form in the Euclidean 3-space are the minimal surfaces, the circular cylinders and the spheres. So in [23] B.-Y. Chen mentions the following problem Problem 1. Determine all surfaces of finite Chen I-type in E 3 . In order to provide an answer to the above problem, important families of surfaces were studied by different authors by proving that finite type ruled surfaces, finite type quadrics, finite type tubes, finite type cyclides of Dupin and finite type spiral surfaces are surfaces of the only known examples in E 3 . However, for another classical families of surfaces, such as surfaces of revolution, translation surfaces as well as helicoidal surfaces, the classification of its finite type surfaces is not known yet. For a more details, the reader can refer to [24]. In this context, Chen and Piccini in [25], introduced in the same way the theory of submanifolds of finite type Gauss map. A special case for E 3 one can ask Problem 2. Classify all surfaces in E 3 with finite type Gauss map. Results concerning this problem can be found in ( [2], [15], [16], [17]). Later in [28] O. Garay generalized T. Takahashi's condition studied surfaces in E 3 for which all coordinate functions (x 1 , x 2 , x 3 ) of x satisfy ∆ I x i = λ i x i , i = 1, 2, 3, not necessarily with the same eigenvalue. Another generalization was studied in [27] for which surfaces in E 3 satisfy the condition ∆ I x = Ax + B ( ‡) where A ∈ R 3×3 ; B ∈ R 3×1 . It was shown that a surface S in E 3 satisfies ( ‡) if and only if it is an open part of a minimal surface, a sphere, or a circular cylinder. Surfaces satisfying ( ‡) are said to be of coordinate finite type. In the thematic circle of the surfaces of finite type in the Euclidean space E 3 , S. Stamatakis and H. Al-Zoubi in [30] restored attention to this theme by introducing the notion of surfaces of finite type corresponding to the second or the third fundamental forms of S in the following way: A surface S is said to be of finite type corresponding to the fundamental form J, or briefly of finite J-type, J = II, III, if the position vector x of S can be written as a finite sum of nonconstant eigenvectors of the operator ∆ J , that is if x = x 0 + k i=1 x i , ∆ J x i = λ i x i , i = 1, ..., k, (1.1) where x 0 is a fixed vector and x 1 , ..., x k are nonconstant maps such that ∆ J x i = λ i x i , i = 1, ..., k. If, in particular, all eigenvalues λ 1 , λ 2 , ..., λ k are mutually distinct, then S is said to be of J-type k, otherwise S is said to be of infinite type. When λ i = 0 for some i = 1,..., k, then S is said to be of null J-type k. In general when S is of finite type k, it follows from (1.1) that there exist a monic polynomial, say R(x) = 0, such that R(∆ J )(x − c) = 0. Suppose that R(x) = x k + σ 1 x k−1 + ... + σ k−1 x + σ k , then coefficients σ i are given by σ 1 = −(λ 1 + λ 2 + ... + λ k ), σ 2 = (λ 1 λ 2 + λ 1 λ 3 + ... + λ 1 λ k + λ 2 λ 3 + ... + λ 2 λ k + ... + λ k−1 λ k ), σ 3 = −(λ 1 λ 2 λ 3 + ... + λ k−2 λ k−1 λ k ). σ k = (−1) k λ 1 λ 2 ...λ k . Therefore the position vector x satisfies the following equation, (see [21]) (∆ J ) k x + σ 1 (∆ J ) k−1 x + ... + σ k (x − c) = 0. (1.2) In [4] Ruled surfaces were studied regarding the second fundamental form, another classes of surfaces were investigated in ( [5], [8], [9]), meanwhile similar study was done but for the Gauss map of the surface as one can see in [6]. In this paper we contribute to the solution of this problem by investigating the ruled surfaces and the quadric surfaces in E 3 . On the other hand it is also interesting studying surfaces in the three-dimensional Euclidean space of coordinate finite type or coordinate finite type Gauss map with respect to the second or third fundamental form. result concerning this can be found in ( [1], [3], [7], [10], [11], [12], [14], [29], [31]). Our main results are the following Theorem 1. The only ruled surfaces of finite III-type in the three-dimensional Euclidean space are the helicoids. Theorem 2. The only quadric surfaces of finite III-type in the three-dimensional Euclidean space are the spheres. Proof of Theorem 1 In the three-dimensional Euclidean space E 3 let S be a ruled C r -surface, r ≥ 3, of nonvanishing Gaussian curvature defined by an injective C r -immersion x = x(s, t) on a region U : = I × R (I ⊂ R open interval) of R 2 . 1 The surface S can be expressed in terms of a directrix curve Γ : σ = σ(s) and a unit vector field ρ(s) pointing along the rulings as follows S : x(s, t) = σ(s) + t ρ(s), s ∈ I, t ∈ R. (2.1) Moreover, we can take the parameter s to be the arc length along the spherical curve ρ(s). Then we have σ ′ , ρ = 0, ρ, ρ = 1, ρ ′ , ρ ′ = 1, where the differentiation with respect to s is denoted by a prime and , denotes the standard scalar product in E 3 . It is easily verified that the first and the second fundamental forms of S are given by I = n ds 2 + dt 2 , II = m √ n ds 2 + 2A √ n ds dt, where n = σ ′ , σ ′ + 2 σ ′ , ρ ′ t + t 2 , m = (σ ′ , ρ, σ ′′ ) + [(σ ′ , ρ, ρ ′′ ) + (ρ ′ , ρ, σ ′′ )] t + (ρ ′ , ρ, ρ ′′ ) t 2 , A = (σ ′ , ρ, ρ ′ ) . If, for simplicity, we put ζ := σ ′ , σ ′ , η := σ ′ , ρ ′ , µ := (ρ ′ , ρ, ρ ′′ ) , ν := (σ ′ , ρ, ρ ′′ ) + (ρ ′ , ρ, σ ′′ ) , ξ := (σ ′ , ρ, σ ′′ ) , we have n = t 2 + 2η t + ζ, m = µ t 2 + ν t + ξ. For the Gauss curvature K of S we find K = − A 2 n 2 . The second Beltrami differential operator with respect to the third fundamental form is defined by △ III f = −1 √ e ∂ √ e e ij ∂f ∂u i ∂u j , where f is a sufficient differentiable function on S and e := det(e ij ). After a long computation it can be expressed as follows: △ III = − n A 2 ∂ 2 ∂s 2 + 2n m A 3 ∂ 2 ∂s∂t − n 2 A 2 + n m 2 A 4 ∂ 2 ∂t 2 + n s 2A 2 + n m t A 3 − m n t 2A 3 ∂ ∂s + n m s A 3 − m n s 2A 3 − m nA ′ A 4 − n n t 2A 2 + m 2 n t 2A 4 − 2n m m t A 4 ∂ ∂t = P 1 ∂ 2 ∂s 2 + P 2 ∂ 2 ∂s∂t + P 3 ∂ ∂s + P 4 ∂ ∂t + P 5 ∂ 2 ∂t 2 , (2.2) where n t = ∂n ∂t , n s = ∂n ∂s , m t = ∂m ∂t , m s = ∂m ∂s and P 1 , . . . , P 5 are polynomials in t with functions in s as coefficients and deg(P i ) ≤ 6. More precisely we have P 1 = − 1 A 2 t 2 + 2η t + ζ , P 2 = 2 A 3 µ t 4 + (2η µ + ν) t 3 + (2η ν + ξ + ζ µ) t 2 + (2η ξ + ζ ν) t + ζ ξ , P 3 = 1 A 3 µ t 3 + 3η µ t 2 + (η ν − ξ + 2ζ µ + η ′ A) t + 1 2 ζ ′ A − η ξ + ζ ν , P 4 = 1 A 4 − 3µ 2 t 5 + µ ′ A − µ A ′2 t 4 + ν ′ A − ν A ′ + 2η µ ′ A − 2η µA ′ − η ′ µ A −A 2 − 10η µ ν − 2µ ξ − ν 2 − 4ζ µ 2 t 3 + ζ µ ′ A − ζ µA ′ − 1 2 ζ ′ µ A + 2η ν ′ A − 2η ν A ′ − η ′ νA −ξA ′ + ξ ′2 − 3η ν 2 − 6η µ ξ − 6ζ µ ν t 2 + ζ ν ′ A − ζ ν A ′ − 1 2 ζ ′′ A − 2η ξ A ′ − η ′ ξ A −ζA 2 − 2η 2 A 2 − 2ζ ν 2 + ξ 2 − 2η ν ξ − 4ζ µ ξ t + ζ ξ ′ A − ζ ξ A ′ − 1 2 ζ ′ ξ A + η ξ 2 − ζ η A 2 − 2ζ ν ξ , P 5 = − 1 A 4 µ 2 t 6 + 2µ ν + 2η µ 2 t 5 + 2µ ξ + ν 2 + 4η µ ν + ζ µ 2 + A 2 t 4 + 2ν ξ + 4η µ ξ + 2η ν 2 + 2ζ µ ν + 4η A 2 t 3 + ξ 2 + 4η ν ξ + 2ζ µ ξ + ζ ν 2 + 4η 2 A 2 + 2ζA 2 t 2 + 2η ξ 2 + 2ζ ν ξ + 4η ζ A 2 t + ζ ξ 2 + ζ 2 A 2 . Applying (2.2) on the position vector (2.1) of the ruled surface S we find △ III x = P 1 σ ′′ + P 2 ρ ′ + P 3 σ ′ + P 4 ρ + (P 1 ρ ′′ + P 3 ρ ′ )t. We write this last expression of △ III x as a vector Q 1 (t) whose components are polynomials in t with functions in s as coefficients as follows: Q 1 (t) = 1 A 4 − 3µ 2 ρt 5 + µ ′ A − µA ′2 ρ + 3µAρ ′ t 4 + µAσ ′2 ρ ′′ + 2νA + 7ηµA ρ ′ + ν ′ A − νA ′ + 2ηµ ′ A − 2ηµA ′ − η ′ µA −A 2 − 10ηµν − 2µξ − ν 2 − 4ζµ 2 ρ t 3 + ζµ ′ A − ζµA ′ − 1 2 ζ ′ µA + 2ην ′ A − 2ηνA ′ − η ′ νA −ξA ′ + ξ ′2 − 3ην 2 − 6ηµξ − 6ζµν ρ +3ηµAσ ′2 ρ ′′ − A 2 σ ′′ + η ′ A + 5ην + 4ζµ + ξ Aρ ′ t 2 + ζν ′ A − ζνA ′ − 1 2 ζ ′ νA + 2ηξ ′ A − 2ηξA ′ − η ′ ξA −ζA 2 − 2η 2 A 2 − 2ζν 2 + ξ 2 − 2ηνξ − 4ζµξ ρ − 2ηA 2 σ ′′ + 1 2 ζ ′ A + 3ζν + 3ηξ Aρ ′2 ρ ′′ + (ην − ξ + 2ζµ + η ′ A) Aσ ′ t +(ζξ ′ A − ζξA ′ − 1 2 ζ ′2 − ζηA 2 − 2ζνξ)ρ + 1 2 ζ ′ A − ηξ + ζν Aσ ′ + 2ζξAρ ′2 σ ′′ . Notice that deg(Q 1 ) ≤ 5. Furthermore deg(Q 1 ) = 5 if and only if µ = 0, otherwise deg(Q 1 ) ≤ 3. Before we start the proof of the first theorem we give the following Lemma which can be proved by a straightforward computation. Lemma 1. Let g be a polynomial in t with functions in s as coefficients and deg(g) = d. Then △ III g = g, where g is a polynomial in t with functions in s as coefficients and deg( g) ≤ d + 4. We suppose that S is of finite III-type k. It is well known that there exist real numbers c 1 , . . . , c k such that △ III k+1 x + c 1 △ III k x + · · · + c k △ III x = 0,(2.3) see [21]. By applying Lemma 1, we conclude that there is an E 3 -vector-valued function Q k in the variable t with some functions in s as coefficients, such that △ III k x = Q k (t), where deg(Q k ) ≤ 4k + 1. Now, if k goes up by one, the degree of each component of Q k goes up at most by 4. Hence the sum (2.3) can never be zero, unless of course △ III x = Q 1 = 0. (2.4) On account of the well known relation △ III x = ∇ III 2H K , n − 2H K n, where H, n and ∇ III denote the mean curvature, the unit normal vector field and the first Beltrami-operator with respect to III, see [30], from (2.4) we result that S is minimal, and that S is a helicoid. Proof of Theorem 2 Let now S be a quadric in E 3 . Then S is either a ruled surface or one of the following two kinds, see [26], z 2 − a x 2 − b y 2 = c, a, b, c ∈ R, a b = 0, c > 0, (3.1) or z = a 2 x 2 + b 2 y 2 , a, b ∈ R, a, b > 0. (3.2) If S is a ruled surface of finite III-type, then, according to theorem 1, S is a helicoid. In this section we will first show that a quadric of the kind (3.1) is of finite III-type if and only if a = −1 and b = −1, that is, if and only if S is a sphere. Next we will show that a quadric of the kind (3.2) is of infinite type. 3.1. Quadrics of the first kind. A part of a quadric of this kind can be parametrized by x(u, v) = u, v, c + a u 2 + b v 2 , c + a u 2 + b v 2 > 0. (3.3) We put for simplicity c + a u 2 + b v 2 : = ω. The third fundamental form of S becomes III = a 2 ω T 2 C(u, v)du 2 − 2 a b ω T 2 B(u, v)du dv + b 2 ω T 2 A(u, v)dv 2 , where T = c + a(a + 1)u 2 + b(b + 1)v 2 , A(u, v) = a 2 u 2 v 2 + (a u 2 + c) 2 + a 2 u 2 ω, B(u, v) = u v c(a + b) + a b(u 2 + v 2 + ω) , C(u, v) = b 2 u 2 v 2 + (b v 2 + c) 2 + b 2 v 2 ω. Then the second Beltrami operator △ III of S can be expressed as follows: △ III = − T a 2 b 2 c 2 b 2 A ∂ 2 ∂u 2 + 2a b B ∂ 2 ∂u∂v + a 2 C ∂ 2 ∂v 2 − T a 2 b 2 c 2 b b ∂A ∂u + a ∂B ∂v ∂ ∂u + a a ∂C ∂v + b ∂B ∂u ∂ ∂v + T a 2 b 2 c 2 a b 2 ω (u A + v B) ∂ ∂u + a 2 b ω (u B + v C) ∂ ∂v + 1 a 2 b 2 c 2 a b 2 ((a + 1) u A + (b + 1) v B) ∂ ∂u +a 2 b ((b + 1) v C + (a + 1) u B) ∂ ∂v . (3.4) We note that b ∂A ∂u + a ∂B ∂v = a u 5a b(a + 1)u 2 + 5a b(b + 1)v 2 + c(3a b + 5b + a) , a ∂C ∂v + b ∂B ∂u = b v 5a b(a + 1)u 2 + 5a b(b + 1)v 2 + c(3a b + 5a + b) , u A + v B = c + a(a + 1)u 2 + a(b + 1)v 2 uω, u B + v C = c + b(a + 1)u 2 + b(b + 1)v 2 vω, (a + 1) u A + (b + 1) v B = c(a + 1) + a(a + 1)u 2 + a(b + 1)v 2 u T, (b + 1) v C + (a + 1) u B = c(b + 1) + b(a + 1)u 2 + b(b + 1)v 2 v T. Hence (3.4) becomes △ III = − a(a + 1) 2 u 5 c 2 u ∂ 2 ∂u 2 + 3 ∂ ∂u − b(b + 1) 2 v 5 c 2 v ∂ 2 ∂v 2 + 3 ∂ ∂v +f 1 (u, v) ∂ 2 ∂u∂v + f 2 (u, v) ∂ 2 ∂u 2 + f 3 (u, v) ∂ 2 ∂v 2 +f 4 (u, v) ∂ ∂u + f 5 (u, v) ∂ ∂v ,(3.5) where f 1 (u, v) = −2uv a(a + 1) 2 c 2 u 4 + (a + 1)(a + ab + 2b) bc u 2 + a + b + ab ab −2uv b(b + 1) 2 c 2 v 4 + (b + 1)(b + ab + 2a) ac v 2 −2uv (a + 1)(b + 1)(a + b) c 2 u 2 v 2 , f 2 (u, v) = − (a + 1)(a + 3) c u 4 − (2a + 3) a u 2 − c a 2 − (a + 1)(b + 1)(a + b) c 2 u 4 v 2 − b(b + 1) 2 c 2 u 2 v 4 − (b + 1)(a + ab + 2b) ac u 2 v 2 − b(b + 1) a 2 v 2 , f 3 (u, v) = − (b + 1)(b + 3) c v 4 − (2b + 3) b v 2 − c b 2 − (a + 1)(b + 1)(a + b) c 2 u 2 v 4 − a(a + 1) 2 c 2 u 4 v 2 − (a + 1)(2a + ab + b) bc u 2 v 2 − a(a + 1) b 2 u 2 , f 4 (u, v) = − (a + 1)(a + 6b + 2ab) bc u 3 − (2ab + a + 3b) ab u − 3(a + 1)(b + 1)(a + b) c 2 u 3 v 2 − 3b(b + 1) 2 c 2 uv 4 − (b + 1)(4a + 2ab + 3b) ac uv 2 , f 5 (u, v) = − (b + 1)(6a + b + 2ab) ac v 3 − (2ab + 3a + b) ab v − 3(a + 1)(b + 1)(a + b) c 2 u 2 v 3 − 3a(a + 1) 2 c 2 u 4 v − (a + 1)(3a + 2ab + 4b) bc u 2 v. Here again the functions f i , i = 1, . . . , 5, are polynomials in u and v with deg(f i ) ≤ 6. We consider a function g(u) ∈ C ∞ (U ). By means of (3.5), we find △ III g = − a(a + 1) 2 u 5 c 2 u ∂ 2 g ∂u 2 + 3 ∂g ∂u + f 2 (u, v) ∂ 2 g ∂u 2 + f 4 (u, v) ∂g ∂u . (3.6) If we put v = 0, then the functions f 2 and f 4 are polynomials in u of degree less than or equal 4. Now we prove the following Lemma 2. The relation △ III k u = (−1) k 2k i=1 (2i − 1) a k (a + 1) 2k u 4k+1 c 2k + P 4k (u, v), holds true, where deg(P 4k (u, 0)) ≤ 4k. Proof. The proof goes by induction on k. For k = 1 the formula follows immediately from (3.6) applied to g = u. Suppose the Lemma is true for k − 1. Then △ III k−1 u = (−1) k−1 2k−2 i=1 (2i − 1) a k−1 (a + 1) 2k−2 u 4k−3 c 2k−2 + P 4k−4 (u, v). Taking into account (3.6) we obtain △ III k u = △ III △ III k−1 u = − a(a + 1) 2 u 5 c 2 (−1) k−1 2k−2 i=1 (2i − 1) a k−1 (a + 1) 2k−2 c 2k−2 u ∂ 2 ∂u 2 u 4k−3 + 3 ∂ ∂u u 4k−3 − a(a + 1) 2 u 5 c 2 u ∂ 2 ∂u 2 (P 4k−4 ) + 3 ∂ ∂u (P 4k−4 ) − (−1) k 2k−2 i=1 (2i − 1) a k−1 (a + 1) 2k−2 c 2k−2 f 2 (u, v) ∂ 2 ∂u 2 u 4k−3 − (−1) k 2k−2 i=1 (2i − 1) a k−1 (a + 1) 2k−2 c 2k−2 f 4 (u, v) ∂ ∂u u 4k−3 +f 2 (u, v) ∂ 2 ∂u 2 (P 4k−4 ) + f 4 (u, v) ∂ ∂u (P 4k−4 ) = (−1) k 2k i=1 (2i − 1) a k (a + 1) 2k u 4k+1 c 2k + P 4k (u, v), where P 4k (u, v) = − a(a + 1) 2 u 5 c 2 u ∂ 2 ∂u 2 (P 4k−4 ) + 3 ∂ ∂u (P 4k−4 ) − (−1) k 2k−2 i=1 (2i − 1) a k−1 (a + 1) 2k−2 c 2k−2 f 2 (u, v) ∂ 2 ∂u 2 u 4k−3 − (−1) k 2k−2 i=1 (2i − 1) a k−1 (a + 1) 2k−2 c 2k−2 f 4 (u, v) ∂ ∂u u 4k−3 +f 2 (u, v) ∂ 2 ∂u 2 (P 4k−4 ) + f 4 (u, v) ∂ ∂u (P 4k−4 ) . (3.7) Since deg (P 4k−4 (u, 0)) ≤ 4k − 4, deg(f 2 (u, 0)) ≤ 4 and deg(f 4 (u, 0)) ≤ 4, from (3.7) we find that deg(P 4k (u, 0)) ≤ 4k. By applying now (3.5) on a function h(v) ∈ C ∞ (U ) we find △ III h = − b(b + 1) 2 v 5 c 2 v ∂ 2 h ∂v 2 + 3 ∂h ∂v + f 3 (u, v) ∂ 2 h ∂v 2 + f 5 (u, v) ∂h ∂v . If we put u = 0, then the functions f 3 and f 5 are polynomials in v of degree less than or equal 4. Proceeding analogously as in Lemma 2, we prove the following Lemma 3. The relation △ III k v = (−1) k 2k i=1 (2i − 1) b k (b + 1) 2k v 4k+1 c 2k + Q 4k (u, v) holds true, where deg(Q 4k (0, v)) ≤ 4k. We suppose now that S is of finite III-type k. Then there exist real numbers c 1 , . . . , c k such that △ III k+1 x + c 1 △ III k x + . . . + c k △ III x = 0.△ III k+1 u + c 1 △ III k u + · · · + c k △ III u = 0, (3.9) △ III k+1 v + c 1 △ III k v + · · · + c k △ III v = 0. (3.10) From Lemma 2 and the relation (3.9) we obtain that there exists a polynomial P 4k+4 (u, v) of degree at most 4k + 4 such that (−1) k+1 2k+2 i=1 (2i − 1) a k+1 (a + 1) 2k+2 u 4k+5 c 2k+2 + P 4k+4 (u, v) = 0. (3.11) If we put v = 0 in (3.11), then we get a nontrivial polynomial in u with constant coefficients. Since a = 0 the relation (3.11) implies a = −1. Similarly, from Lemma 3 and the relation (3.10) we obtain that there exists a polynomial Q 4k+4 (u, v) of degree at most 4k + 4 such that (−1) k+1 2k+2 i=1 (2i − 1) b k+1 (b + 1) 2k+2 v 4k+5 c 2k+2 + Q 4k+4 (u, v) = 0. (3.12) Putting u = 0 in (3.12), we get again a nontrivial polynomial in v with constant coefficients. Since b = 0, we obtain from (3.12) b = −1. Hence S must be a sphere. 3.2. Quadrics of the second kind. A quadric surface of this kind can be parametrized by x(u, v) = u, v, a 2 u 2 + b 2 v 2 . (3.13) Then the third fundamental form of S is the following III = a 2 g 2 (1 + b 2 v 2 )du 2 − 2 a 2 b 2 g 2 u v du dv + b 2 g 2 (1 + a 2 u 2 )dv 2 , where g : = det (g ij ) = 1 + a 2 u 2 + b 2 v 2 is the discriminant of the first fundamental form I = (1 + a 2 u 2 )du 2 + 2a b u v du dv + (1 + b 2 v 2 )dv 2 of S. Hence the Beltrami operator △ III of S takes the following form △ III = − g(1 + a 2 u 2 ) a 2 ∂ 2 ∂u 2 − g(1 + b 2 v 2 ) b 2 ∂ 2 ∂v 2 −2u v g ∂ 2 ∂u∂v − 2u g ∂ ∂u − 2v g ∂ ∂v , We put v = 0 in (3.17). Then the left member of the equation (3.17) is a nontrivial polynomial in u with constant coefficients. This polynomial can never be zero, unless a = 0. Similarly, if we put u = 0 in (3.18), then the left member of (3.18) is a nontrivial polynomial in v with constant coefficients. This implies b = 0. This is clearly impossible since a, b > 0. on the coordinate functions x 1 = u and x 2 = v of the position vector (3.3) of the quadric S we obtain The reader is referred to[?] for definitions and formulae on ruled surfaces. which can be written aswhereNotice that the functions f i , i = 1, . . . , 5, are polynomials in u and v with deg(f i ) ≤ 4. By applying the operator △ III on a function g(u) ∈ C ∞ (U ) we find by means of (3.14)If we put v = 0, then the functions f 2 and f 4 are polynomials in u of degree less than or equal 2.Using(3.15) and by induction on k we can prove the following Lemma 4. The relationholds true, where deg(P 2k (u, 0)) ≤ 2k.By applying (3.14) on a function h(v) ∈ C ∞ (U ) we getIf we put u = 0, then the functions f 3 and f 5 are polynomials in v of degree less than or equal 2. In the same way the following Lemma can be proved Lemma 5. The relationNow, if the quadric S is of finite III−type k, then again the relations (3.8), (3.9) and (3.10) are valid. 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Carolin. 61 (2020), 237-256. On surfaces of finite Chen-type. S Stamatakis, H Al-Zoubi, Results. Math. 43S. Stamatakis, H. Al-Zoubi, On surfaces of finite Chen-type, Results. Math., 43 (2003), 181-190. Surfaces of revolution satisfying △ III x = Ax. S Stamatakis, H Al-Zoubi, Journal for Geometry and Graphics. 14S. Stamatakis, H. Al-Zoubi, Surfaces of revolution satisfying △ III x = Ax, Journal for Ge- ometry and Graphics, 14 (2010), 181-186. Minimal immersions of Riemannian manifolds. T Takahashi, J. Math. Soc. Japan. 18Department of Mathematics, Al-Zaytoonah University of JordanAmman. Jordan 11733 Email address: [email protected]. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380-385. Department of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Am- man, Jordan 11733 Email address: [email protected]	[]
[ "Analysis of an M/G/1 system for the optimization of the RTG performances in the delivery of containers in Abidjan Terminal", "Analysis of an M/G/1 system for the optimization of the RTG performances in the delivery of containers in Abidjan Terminal" ]	[ "Bakary Koné [email protected] \nUniversité Cheikh Anta Diop de Dakar\nSenegal\n", "Salimata Gueye Diagne \nUniversité Cheikh Anta Diop de Dakar\nSenegal\n", "Déthié Dione [email protected] \nUniversité Cheikh Anta Diop de Dakar\nSenegal\n", "Coumba Diallo \nUniversité Cheikh Anta Diop de Dakar\nSenegal\n" ]	[ "Université Cheikh Anta Diop de Dakar\nSenegal", "Université Cheikh Anta Diop de Dakar\nSenegal", "Université Cheikh Anta Diop de Dakar\nSenegal", "Université Cheikh Anta Diop de Dakar\nSenegal" ]	[]	In front of major challenge to increase its productivity while satisfying its customer, it is today important to establish in advance the operational performances of the RTG Abidjan Terminal. In this article, by using an M/G/1 retrial queue system, we obtained the average number of parked delivery trucks and as well as their waiting time. Finally, we used Matlab to represent them graphically then analyze the RTG performances according to the traffic rate.	null	[ "https://arxiv.org/pdf/2009.01659v1.pdf" ]	115,242,767	2009.01659	32d34ca85365927cfc9563621f5f465b810281fb	Analysis of an M/G/1 system for the optimization of the RTG performances in the delivery of containers in Abidjan Terminal Bakary Koné [email protected] Université Cheikh Anta Diop de Dakar Senegal Salimata Gueye Diagne Université Cheikh Anta Diop de Dakar Senegal Déthié Dione [email protected] Université Cheikh Anta Diop de Dakar Senegal Coumba Diallo Université Cheikh Anta Diop de Dakar Senegal Analysis of an M/G/1 system for the optimization of the RTG performances in the delivery of containers in Abidjan Terminal RTGEmbedded Markov chainRetrial QueueStationary DistributionPerformance Measures In front of major challenge to increase its productivity while satisfying its customer, it is today important to establish in advance the operational performances of the RTG Abidjan Terminal. In this article, by using an M/G/1 retrial queue system, we obtained the average number of parked delivery trucks and as well as their waiting time. Finally, we used Matlab to represent them graphically then analyze the RTG performances according to the traffic rate. Introduction In today's maritime world, marked by increasing modernity, containers terminals do not only have the obligation to straighten their infrastructures, but also they new ones have to improve to keep their customer and acquire. This task is not easy given the deep international competition. A task Terminals with containers show awareness by trying to satisfy ever increasing their customer. This report was made at Container Terminal of Abidjan (Abidjan Terminal), where one of the priorities is to satisfy the customer in particular trucks in mission of delivery to the city [12]. A vast distribution platform of containers of any singles and the modern installations, allow Abidjan Terminal to receive and to handle a high volume of imported containers and a more and more demanding customer. Given its important role in the Ivory Coast maritime exchanges, it is important to analyze the RTG performances which establish its most used porticoes of courses in the operations of containers deliveries [11]. By using the queue theories, several researchers have already carried out studies which analyses the performances in various domains. It is the case of Mohamed Boualem and al. (2013) who deal retrial queue system and Bernoulli feedback [6]. They considered the type M/G/1 retrial queue and feedback system. By using the same analysis, they obtained important performances measures. The researchers Yang and Templeton (1987), as well as Yang and al. (1994) also used the M/G/1 retrial model in the case of exponential and general distributions of the inter-retrial time. They proved that the generative function of the number of customers in the M/G/1 retrial system is the product of two generative functions: that of the number of customers in the M/G/1 classic system [9] and the generative function for the number of customers in the M/G/1 retrial system given that the server is free [3]. As for Stidham (2002) another important class of queue deals the systems in which the server remains idle when the queue is empty. So, this time of idleness of the server could be used differently with the aim of improving the efficiency of the system [4]. Aissani (2008) made a complete analysis of the constant retrials in the case of the Quasi-Markovian M/G/1 models. He considers a queue of this type with the politics of constant retrial and policy vacancy of the server, when the retrial time, the service time and the policy vacancy time are arbitrarily distributed. The distribution of the number of customers in the stationary system mode is obtained in terms of generative function. Afterward, he gives an approximation for such a distribution [7]. Researchers Djamil Aissani (2011) consider the example of the M/G/1 classic retrials queue and server vacancies. By taking into account the existence of the stationary mode and an embedded Markov chain they make a stationary analysis of the system. They also obtained formula for the limit distribution of the server state, the stochastic decomposition [8] and some performance measures [1]. In this paper, we use the theories of retrials queues to clear and analyze the RTG performances in Abidjan Terminal. This question was not approached in the literature, which is the main contribution of this article. We consider trucks parked in the zone under RTG and their treatment ordered as a Quasi-Markovian M/G/1 queue system with a period of the RTG vacancy. This analysis allows to establish in advance the operational performances of porticoes, to identify the critical elements or, again, to look at the effects of a modification of the operating conditions. We use the method of the generative functions, then present a more detailed analysis with the calculation of the generative functions. The rest of the document is organized as follows: the second section deals with the mathematical description of the model. In the third section, we present an embedded Markov chain and an existence condition of the stationary state of the embedded Markov chain. In the fourth section, we release some clear performance measures. That is going to allow us to release in the fourth section a digital test, and finally a conclusion and perspectives are proposed in the last section. Delivery and operational function of the RTG As well as its role of storage space, the zone under RTG constantly receives containers in import, export and transshipment. But in this section we will present the process of containers delivery. Delevery process The driver arrives at the sentry box of the Terminal provided with a delivery slip. After routine check of the number of the container, the sentry agent affects on Oscar (Operating Software Container and adjournment is a management system of exploitation of the operations of a Container terminal) a token with a number to this container then confirms (putting in delivery). Then he returns the token to the driver. The latter leads the truck on the park and parks at the address of the container indicated on the token. The RTG operator having received the mission loads the truck after checking its numbers (container and the token) (mission in processes). The loaded truck returns to the exit sentry box for the end of the delivery procedure. The RTG operational function The RTG has a mission of execution which handling operations are planned and checked since the control room by the piloting. Indeed, according to the number of containers to be delivered and of their localizations the import piloting proceeds to a division of the zone under RTG on the graphic software OSCAR in small zones (working zone) corresponding to the available RTG. Then, he allocates to every working zone a RTG which receives and executes the missions in its zone. Finally the piloting follows and coordinates the movements of the RTG in the execution of the various operations of the RTG since the control room. Research Question The zone under RTG which constitutes our workspace is formed by 18 blocks parallel to the linear band of the Terminal Quays and the containing each several bays of various types of containers. Containers are piled in standard ISO there, according to their statuses at using addresses to allow their localizations. During the delivery phase of containers, trucks make their entrance by the procedure and park at this address of the container. The RTG operator who covers the zone receives immediately the number of the token from a small screen placed in his cabin. This list reached him in arrival order of trucks. So, after checking the information, (the numbers of the container and that of the token) the RTG operator loads the container. Contrary to arrival spontaneous of the trucks, the RTG being in lower number cannot handle all the containers at reasonable time. The loading is done at a disproportional rhythm in the flows of parked trucks. In order to increase the operational and competitive qualities of the Abidjan Terminal, it is important to analyze the performance of its machines to make it more attractive. In this article, by using the generative functions, we analyze the RTG performances which allows to identify the critical elements or, again, to grasp the effects of a modification of the operating conditions. Modeling problem It is composed of several stages: Model description Delivery trucks make their orderly entry and park at the precise addresses of containers. The zone can contain several trucks which makes its structure look like a classic queue where the server is the RTG. The parked trucks are the considered first category or the second category customers according to their waiting times. While a truck parks and is not served for the moment the service of piloting makes retrials to the RTG operator so that this truck is loaded. Once served, it immediately leaves the Terminal with the container. • The primary trucks are those which are immediately loaded upon their arrival. • The secondary trucks are the ones which wait for a whole. These trucks are loaded further to one or several retrials made by the piloting. Indeed, to avoid of lengthy waiting, the piloting follows and coordinates the movements of machines on the Park. He constantly reminds the RTG operators to handle first and foremost the parked trucks having spent a long time waiting. This principle of reminder is defined according to the number of parked trucks and the duration of their waiting time. That increases in intensity when the zone of the RTG is saturated. The intervals the retrials time are they follow an exponential distribution rate kθ where (k ∈ N is the number of waiting trucks and θ > 0 is the debit of load of a truck by reminder) (Boualem and al.2013). All the trucks entering the system are served in a random order independent from their arrival time. The service time distributed according to an any distribution which the density b(t) is transformed by Laplace-Stieltjes B(t) for t at a given moment. That means of τ s i is the service which is between the i th and (i + 1) th service, then P (τ s i ≤ t) = B(t). If we are in the i th stage, the RTG executes the i th service there which corresponds to the loading of the i th delivery truck. The time of (i+1) th begins just at the end of the loading of the i th truck. It's established by the travel time made by the RTG between these two trucks and by the unproductive movements time as well as the loading of the container. The first two moments finished of this distribution are β 1 and β 2 , respectively. We suppose that the RTG is free every time the system becomes empty(with no trucks in the zone) Either t represents any moment. The state of the system at this arbitrary moment is given by: M (t) = {C(t), N (t), ζ(t), } With, C(t) = 1 If the RTG is engaged 0 Otherwise N (t) represents the number of trucks parked at a given moment t τ (t) represents the duration of the RTG service to handle the truck if C(t) = 1. Embedded Markov chain The Poisson distribution of an M/G/1 queue which sequences can be described there terms of a sequence alternated by various periods. So, the RTG works in a continuous way and only becomes free when all the parked trucks have been served. It resumes the handling when a truck arrives. We can summarize these parameters in the table below. service N (ζ i ) The total number of waiting trucks parked after the i th service. Let us consider {ζ i , i ∈ N } an increasing continuation of moment of the completion of a service time. Or N (ζ i ) the number of trucks parked waiting in the system just after the loading of the i th truck. We then define a stochastic process by the sequence of the random variables: Q i = {N (ζ i ); i = 0, 1, 2, 3, · · · } where ζ i is the exact moment of the end of the treatment of the i th truck. Let us pose: N (ζ i )N i And A i+1 the number of parked trucks while (i + 1) th truck is being served. Then, the random variables A i are independent between them; their common distribution is given by: P (A i+1 = k) = q k = +∞ 0 (λt) k k! e −λt b(t)dt; q k ≥ 1, (k = 1, 2, ...)(1) Its generative function is: A(z) = +∞ k=0 q k z k = B(λ(1 − z)) The fundamental equation of embedded Markov chain When the RTG is handling (i + 1) th truck, if A i+1 arrive, then at the end of his load we shall have: N i+1 = N i − 1 + A i+1 si N i ≥ 1 A i+1 si N i = 0 Hence the fundamental equation of embedded Markov chain is given by: N i+1 = N i − δ N i + A i+1(2) The variable of Bernoulli δ N i is defined by: δ N i = 1 If the (i + 1) th Served truck is from the second category 0 Otherwise depend on N i . It's translated by in most 1 truck left and A i+1 trucks arrived at the end of (i + 1) th service. Conditional distribution For k trucks waiting in the stage i, the following state is determined according to the value taken by δ N i . Then the conditional distribution is given by (Boulem 2011) P (δ N i = 1/N i = k) = kθ λ + kθ P (δ N i = 0/N i = k) = λ λ + kθ(3) With θ the flow of truck loaded from the second category. Transition rate. The probability that there is j trucks parked in (i + 1) th stage knowing that it had k the i th is noted: ∀ j ≥ 0 and 0 ≤ k ≤ j, we have: P kj = P (N i+1 = j/N i = k)(4) Which gives: P kj = P (A i+1 = j−k)P (δ N i = 1/N i = k)+P (A i+1 = j−k+1)P (δ N i = 0/N i = k) According to (1) and (4) we have: P kj = P (N i+1 = j/N i = k) = q j−k kθ λ+kθ + q j−k+1 λ λ+kθ , 0 ≤ k ≤ j 0 Otherwise(5) We remind that ζ i is the precise moment in which the treatment of the i th truck parked in the line ended. We verify that this continuation of random variable is an actually Markov chain in discrete time. Yet A i the number of parked trucks while the i th truck is being served. Existence condition of the stationary distribution of the number of trucks The chain is Markovian because the state of the system in the stage (i + 1), depends on the state i. So that the line does not lengthen infinitely, the average number of arrivals during the duration of a service must be strictly lower than 1. Then, we have: E(A i ) = i k=0 +∞ 0 P (A i = k)b(t)dt = i k=0 k +∞ 0 (λt) k k! e −λt b(t)dt = +∞ 0 e −λt b(t)λt{ +∞ k=0 (λt) i k! }dt = λ +∞ 0 tb(t)dt = λβ 1 With β 1 the moment of order 1 of the distribution. Hence the existence of the stationary distribution is conditioned for: ρ = λβ 1 ≤ 1 (6) The generative function of the trucks number waiting in the RTG zone It corresponds to the generative function of N i and thus is the transformed z of its function of density which is defined by: f i (z) = E(z k ) = +∞ k=0 z k P [N i = k] Yet N i+1 = N i + δ N i + A i+1 f i+1 = E[z N i −δ N i +A i+1 ] As A i and N i are independent then, f i+1 = E[z N i −δ N i +A i+1 ] = E[z N i −δ N i ]E[z A i+1 ] Given that the condition of stationarity is then verified A(z) = E(z A i ) independent of i. Thus we have: f i+1 = E[z N i −δ N i ]E[z A i+1 ] = E[z N i −δ N i ]A(z) = [ +∞ k=0 z k−δ k P (N i = k)]A(z) = [ +∞ k=0 z k P (N i = k) × P (δ N i = 0/N i = k) + +∞ k=1 z k−1 P (N i = k) × P (δ N i = 1/N i = k)]A(z) = +∞ k=0 z k P (N i = k) λ λ + kθ + +∞ k=1 z k−1 P (N i = k) kθ λ + kθ In the stationary state, lim i −→+∞ f i+1 (z) = lim i −→+∞ f i (z) = f (z)f (z) = +∞ k=0 z k π k λ λ + kθ + +∞ k=1 z k−1 π k kθ λ + kθ(7) We pose: ψ(z) = +∞ k=0 π k z k λ + kθ Then ψ (z) = +∞ k=1 kπ j z k−1 λ + kθ So, the equation (10) allows us to obtain the following relation: f (z) = A(z)[λψ(z) + θψ (z)](8) In a different way, we notice that: f (z) = +∞ k=0 π k z k = +∞ k=0 π k z k λ + kθ λ + kθ = λψ(z) + θψ (z) Hence the relation: f (z) = λψ(z) + θψ (z)(9) Consequently, the use of the equations (11) and (12) entails: λψ(z) + θψ (z) = A(z)[λψ(z) + θψ (z)] θψ (z)[A(z) − z] = λψ(z)[1 − A(z)](10) The function h(z) = A(z) − z positive, increasing for z ∈ [0, 1] and ρ < 1, z < A(z) < 1 [2] so h(1) = 1 h (z) = A (z) − z , and h (1) = ρ − 1 < 0. Besides by using the results (4) and (5), we have: ψ (z) = λ θ { 1 − A(u) A(u) − u }ψ(z) For ρ < 1, The resolution of this differential equation gives us: ψ(z) = ψ(1)exp{ λ θ z 1 (1 − A(u)) A(u) − u d u } (11) Yet f (z) = λψ(z) + θψ (z) Therefore f (z) = λψ(z) + {θ λ θ z (1 − A(z)) A(z) − z }ψ(z) = λψ(z)A(z) 1 − z A(z) − z The traffic intensity is ρ = λβ 1 and f (1) = B(0) = 1. So a simple calculation gives us 1−z A(z)−z = h (1) = ρ − 1 then ψ(1) = 1−ρ λ . By replacing ψ(1) by its expression we obtain the generative function of the stationary distribution of the number of trucks waiting is: f (z) = (1 − ρ)B(λ − λz)(1 − z) B(λ − λz) − z × exp{ λ θ z 1 [(1 − A(u))] A(u) − u d u }(12) The RTG performance measures In this section, we establish the explicit expressions of the RTG performance measures during the operations of delivery of containers in this form. Proposition 5.1 In the stationary state, the average number of trucks in waiting in the RTG zone and the average waiting time of the delivery trucks of containers are respectively expressed by: N = ρ + λρ θ(1 − ρ) + λ 2 β 2 2(1 − ρ) (13) W = β 1 + λβ 2 2(1 − ρ) + ρ θ(1 − ρ)(14) Demonstration • Let us determine the average number of trucks waiting in the RTG zone. In the stationary state, the average number of trucks being in the RTG zone corresponds in numbers of trucks which were already parked and those which have just arrived. It thus expresses itself by: N = lim i −→+∞ E(N i+1 ) = lim i −→+∞ E(N i ) Because f (z) of its generative function, This allows to say that N = f (1) With f (z) the derivative of the generative function. So we calculate f (z), then we replace z by (1) to obtain the different performances. Calculation of the derivative of the generative function f (z) = (1 − ρ)B(λ − λz)(1 − z) B(λ − λz) − z × exp{ λ θ z 1 [(1 − A(u))] A(u) − u d u } We pose: G(u) = [(1−A(u))] A(u)−u , where 1 ≤ u ≤ z T (z) = (1 − ρ)B(λ − λz)(1 − z); P (z) = exp{ λ θ z 1 g(u)du} and Q(z) = B(λ − λz) − z So T (1) = 0, P (1) = 1, Q(1) = 0 (15) T (z) = (1 − ρ)B 1 (λ − λz)(1 − z) + (ρ − 1)B(λ − λz) P (z) = λ θ × G(z)exp{ λ θ z 1 g(u)du} Then T (1) = (ρ − 1), P (1) = − λ θ × ρ 1 − ρ , Q (1) = (ρ − 1)(16) Let us now notice that: f (z) = T (z)P (z) Q(z) Because we have: 1 − z B(λ − λz) − z = h (1) = 1 Then f (1) = 1 (17) Therefore f (z)Q(z) = T (z)P (z) f (z)Q(z) + f (z)Q (z) = T (z)P (z) + T (z)P (z) By replacing each function by its expression Z = 1, we have an indefinite shape. We raise it by the application of the hospital theorem. The second derivative gives: f (z)Q(z)+2f (z)Q (z)+f (z)Q (z) = T (z)P (z)+2T (z)P (z)+T (z)P (z) The equations (13) allow to obtain: f (z) = T (z)P (z) + 2T (z)P (z) − f (z)Q (z) 2Q (z)(18) Yet T (z) = 2ρ(ρ − 1) By using the formula of the moments: β k = (−1) k β k (0) β 1 = −β 1 (0), β 2 = β 2 (0) The equations (14) and (15) entail: f (1) = 2ρ(ρ − 1) + 2 (ρ−1)×λ×ρ θ×(1−ρ) − λ 2 β 2 2(ρ − 1)(19) In conclusion, • The average number of trucks waiting in the zone of the RTG is given by: N = ρ + λρ θ(1 − ρ) + λ 2 β 2 2(1 − ρ) This expression is the total sum of the trucks which were parked just at the beginning and those which arrived during (i + 1) th service, the average number of trucks which parks during the duration of its service is ρ. • Let us determine the average waiting time of the delivery trucks of containers Let us apply the formula of Little. So this number expresses itself by: W = N λ Which gives: W = β 1 + λβ 2 2(1 − ρ) + ρ θ(1 − ρ) Simulation and Analysis of the results To simulate our results, we limit our work to an RTG zone. We analyze the influence of the traffic rate on the stationary characteristics of the system to obtain a tendency of the variations of the number of trucks and their waiting time. For that purpose, we consider the variation of the traffic in the M/G/1 system. During a service we have one (1) truck loaded hence the service level is 1 β 1 1. So, the rate of retrials is 1, 4 and corresponds to the retrial average made by the piloting [10]. By using the Matlab software we obtain the influence of the traffic rate on the numbers of waiting trucks and their waiting times. The obtained results are represented in a form of curved graphs below. We notice that there is a big dependency between the variations of the performance measures and the volume of traffic as shown in graph 2. This dependency is very sensitive for both cases of performances which we studied. The waiting time of trucks increases gradually according to the traffic, but as for the number of trucks, it evolves in an exponential way from 0.6 arrival. For that purpose, you only have to increase the fluidity of the traffic so that the number of trucks in the system or the waiting time increases. Depending on the arrival flow, the number of trucks increases. The sensibility is more noticed when the arrival rate is close to one (1). Figure 3: Evolution of time waiting in the system We can say that the unproductive movements contribute to the slowing down of the operational flow of the Abidjan Terminal a lot. The RTG spend a lot of time in the moving and the removing of containers. The numbers of parked trucks and the waiting time when they spend the park depend considerably on the traffic rate. The management of these two fundamental characteristics depends on the evolution of the traffic rate. The higher it is, the longer waiting time and delivery trucks spend on the park as indicates in figure 3. Conclusion In this article, we made our works on the operation zone of an RTG in Abidjan Terminal. By using the method of the generative functions, the theory of quasi Markovian retrial queue allowed us to obtain the performances such as the number of trucks and their waiting time. We used the software of programming 13.10 version of the Matlab for graphical representations to analyze and release the sensitivity of the operations processing regarding to delivery trucks in the Terminal. The obtained results allowed us to say that the waiting times of delivery trucks depend widely on the traffic rate. The operational performances of the RTG depend widely on the traffic rate. In perspective, we plan to estimate the unproductive movements then the probability of breakdowns of the RTG by using the same factor of sensitivity. Figure 1 : 1Map of Abidjan Terminal and the stationary distribution as defined by: π k = lim i −→+∞ P (N i = k) By using them in the equation, we obtain: Figure 2 : 2Evolution of the trucks number in the system Table 1 : 1The elapsed time by the RTG for the i thParameter setting Parameters Definitions i The stage in which a service was made by the RTG. A service corresponds to a stage which amounts to a truck loaded. k, j The number of trucks in waiting. ζ i The exact time from which the i th service is ended. τ s i New results in the theory of repeated orders queueing systèms. H Choo, Conolly, Journal of applied probability. 16Choo. H. Q and Conolly. B (1979). New results in the theory of re- peated orders queueing systèms. Journal of applied probability 16 : 631 − 640 waiting time distribution for Queues with dynamic priorities Naval Research logistics quarterly. Jackson J , 9Jackson J. R (1961), waiting time distribution for Queues with dy- namic priorities Naval Research logistics quarterly, 9 : 31 − 68 Queues with dynamic priorities Management science. Jackson J , 1Jackson J. R (1962), Queues with dynamic priorities Management science, 1 : 18 − 34 Approche régénérative de la file d'attente M/G/1 avec rappels classiques et vacances exhaustives du serveur. B Mohamed, D Natalia, A Djamil, Journal européen des systèmes automatisé. 45Mohamed B. Natalia. D. Djamil A. (2011),Approche régénérative de la file d'attente M/G/1 avec rappels classiques et vacances exhaustives du serveur. Journal européen des systèmes automatisé, 45 : 253 − 267 Analyse des performances du système M/G/1 avec rappels et Bernoulli feedback. Natalia. D. Djamil A. Mohamed B.47Natalia. D. Djamil A. Mohamed B. (2013), Analyse des performances du système M/G/1 avec rappels et Bernoulli feedback. Journal eu- ropéen des systèmes automatisé, 47 : 181 − 193 Problèmes de convergence, optimisation d'algorithmes et analyse stochastique de systèmes de files d'attente avec rappels. Khadidja Arrar Nawel, ARRAR Nawel Khadidja, (2012) Problèmes de convergence, opti- misation d'algorithmes et analyse stochastique de systèmes de files d'attente avec rappels. https://tel.archives-ouvertes.fr/tel-00829089, PP: 35 − 51 Finite source Markovian inventory system with bonus service for certain customers. K Jeganathan, Int. J. of Adv. in Aply. Math. and Mech. 23K. Jeganathan, Finite source Markovian inventory system with bonus service for certain customers, Int. J. of Adv. in Aply. Math. and Mech. 2(3) (2015) 134 -143. Stochastic inventory system with two types of services. N Anbazhagan, B Vigneshwaran, K Jeganathan, Int. J. of Adv. in Aply. Math. and Mech. 21N. Anbazhagan, B. Vigneshwaran, K. Jeganathan, Stochastic inven- tory system with two types of services, Int. J. of Adv. in Aply. Math. and Mech. 2(1) (2014) 120 -127. Supply-and-demand model involving fuzzy parameters. P K Pandit, Int. J. of Adv. in Aply. Math. and Mech. 12P. K. Pandit, Supply-and-demand model involving fuzzy parameters, Int. J. of Adv. in Aply. Math. and Mech. 1(2) (2013) 103 -115.	[]
[ "Delay Optimal Scheduling for Energy Harvesting Based Communications", "Delay Optimal Scheduling for Energy Harvesting Based Communications" ]	[ "Juan Liu ", "Senior Member, IEEEHuaiyu Dai ", "Senior Member, IEEEWei Chen " ]	[]	[]	Green communications attract increasing research interest recently. Equipped with a rechargeable battery, a source node can harvest energy from ambient environments and rely on this free and regenerative energy supply to transmit packets. Due to the uncertainty of available energy from harvesting, however, intolerably large latency and packet loss could be induced, if the source always waits for harvested energy. To overcome this problem, one Reliable Energy Source (RES) can be resorted to for a prompt delivery of backlogged packets. Naturally, there exists a tradeoff between the packet delivery delay and power consumption from the RES. In this paper, we address the delay optimal scheduling problem for a bursty communication link powered by a capacity-limited battery storing harvested energy together with one RES. The proposed scheduling scheme gives priority to the usage of harvested energy, and resorts to the RES when necessary based on the data and energy queueing processes, with an average power constraint from the RES. Through twodimensional Markov chain modeling and linear programming formulation, we derive the optimal threshold-based scheduling policy together with the corresponding transmission parameters. Our study includes three exemplary cases that capture some important relations between the data packet arrival process and energy harvesting capability. Our theoretical analysis is corroborated by simulation results.Index Terms-Energy harvesting, packet scheduling, Markov chain, queueing delay, delay-power tradeoff.	10.1109/jsac.2015.2391972	[ "https://arxiv.org/pdf/1308.5053v1.pdf" ]	6,577,008	1308.5053	b76aa78b768a2d0f81e1a96aea21ea89d853151e	Delay Optimal Scheduling for Energy Harvesting Based Communications 23 Aug 2013 Juan Liu Senior Member, IEEEHuaiyu Dai Senior Member, IEEEWei Chen Delay Optimal Scheduling for Energy Harvesting Based Communications 23 Aug 20131 Green communications attract increasing research interest recently. Equipped with a rechargeable battery, a source node can harvest energy from ambient environments and rely on this free and regenerative energy supply to transmit packets. Due to the uncertainty of available energy from harvesting, however, intolerably large latency and packet loss could be induced, if the source always waits for harvested energy. To overcome this problem, one Reliable Energy Source (RES) can be resorted to for a prompt delivery of backlogged packets. Naturally, there exists a tradeoff between the packet delivery delay and power consumption from the RES. In this paper, we address the delay optimal scheduling problem for a bursty communication link powered by a capacity-limited battery storing harvested energy together with one RES. The proposed scheduling scheme gives priority to the usage of harvested energy, and resorts to the RES when necessary based on the data and energy queueing processes, with an average power constraint from the RES. Through twodimensional Markov chain modeling and linear programming formulation, we derive the optimal threshold-based scheduling policy together with the corresponding transmission parameters. Our study includes three exemplary cases that capture some important relations between the data packet arrival process and energy harvesting capability. Our theoretical analysis is corroborated by simulation results.Index Terms-Energy harvesting, packet scheduling, Markov chain, queueing delay, delay-power tradeoff. I. INTRODUCTION Energy harvesting can provide renewable free energy supply for wireless communication networks. With the help of solar cells, thermoelectric and vibration absorption devices, and the like, communication devices are able to gather energy from surrounding environments. Energy harvesting can also help reduce carbon emission and environmental pollution, as well as the consumption of traditional energy resources [1]- [3]. In practice, harvested energy arrives in small units at random times and the storage battery usually has limited capacity [4]. Hence, wireless communication systems exclusively powered by energy harvesting devices may not guarantee the users' quality of service. To provide dependable communication service, reliable energy resources can serve as backup in the case of energy shortage. In this way, efficient mixed usage of the harvested energy and reliable energy provides a key solution to robust wireless green communications [5], an emerging area of critical importance to future wireless development. In wireless networks, energy efficient transmission has been an ever-present important issue [6]- [8]. Subject to the randomness and causality of energy harvesting, the optimal transmission problem has been investigated for an energy harvesting wireless link with batteries of either finite or infinite capacity in [4], [9], [10]. In these works, the authors assumed that the energy harvesting profile (i.e., the arrival times and associated amount of harvested energy) is known before the transmission starts. This line of work has been extended to wireless fading channels [11], broadcast channels [12] and two-hop networks [13]. Some other recent works have focused on developing efficient transmission and resource allocation algorithms with different objectives and energy recharging models. For example, a save-then-transmit protocol was proposed in [14] to minimize the delay constrained outage probability by using two alternating batteries, where the battery charging rate is modeled as a random variable. In [15], a cross-layer resource allocation problem was studied for wireless networks powered by rechargeable batteries, where the amount of replenished energy is assumed to be independent and identically distributed in each time slot. In [16], an optimal energy allocation problem was studied for a wireless link with time varying channel conditions and energy sources. A line of work pertinent to our study focuses on the queueing performance analysis for optimal energy management policies. In particular, different sleep/wake-up strategies in a solar-powered wireless sensor network were studied in [17]. Energy management policies were proposed in [18] to maximize the stable throughput and minimize the mean delay for energy harvesting sensor nodes. While a node can harvest an infinite amount of energy in the long run, harvested energy actually arrives at random times. Due to the energy causality constraint, the node should accumulate a sufficient amount of energy before each packet transmission. Hence, the waiting time could be undesirably long and some packets might be dropped due to delay violation. Intuitively, this situation can be greatly relieved if one Reliable Energy Source (RES) can be used to transmit backlogged packets when needed. At the other extreme, the problem becomes trivial if the system can always transmit using the reliable energy. Hence, there exists a tradeoff between the packet delivery delay and the energy consumption from the reliable source. In this paper, we investigate the delay optimal scheduling policy for a communication link powered by a capacity-limited battery storing harvested energy and one RES. In our system, the source will first seek energy supply from the capacity limited energy harvesting battery whenever available, and resort to the RES when necessary, but with an average power constraint. In particular, subject to the bursty energy harvesting profile, it transmits with one of the energy supplies according to the data queue status and the energy storage status at the battery. Under the constraint of the average power consumption from the RES, we study the delay optimal scheduling problem, taking into account the match and mismatch between the energy harvesting capabilities and data packet arrival. To analyze the proposed scheme, we formulate a twodimensional discrete-time Markov chain and derive the steadystate probabilities. Based on the Markov chain modeling, we can derive the average delay and the average power consumed from the RES as functions of the steady-state probabilities. Then, by formulating a Linear Programming (LP) problem and analyzing its properties, we are able to characterize the structure of the optimal solution. Moreover, we can obtain an elegant closed-form expression for the optimal solution in the case where each unit of harvested energy can support one data packet transmission. We also develop an algorithm to find the optimal solutions in other cases. From the optimal solution, we can determine the optimal probabilistic transmission parameters. It is found that in the face of a depleted battery, the optimal transmission strategy depends on a critical threshold for the data queue length. In particular, the source relies on the harvested energy supply if the data queue length is below the critical threshold, and resorts to the RES otherwise. Our theoretical analysis is verified by simulations. The rest of this paper is organized as follows. Section II introduces the system model and the stochastic scheduling scheme. In Section III, a two-dimensional Markov chain model is constructed for the data and energy packet queueing system. Section IV formulates an LP problem for our scheduling objective. By analyzing the properties of the LP problem, we derive the optimal steady-state probabilities and then determine the optimal transmission parameters in Section V. Section VI demonstrates the simulation results and Section VII concludes this paper. For better illustration and in the interest of space, most proofs for our results are put in the appendices. II. SYSTEM MODEL A. System Description We consider a communication link which is powered mainly by a battery storing the harvested energy and further by the RES when necessary, as shown in Fig. 1. The RES refers to any reliable energy source, either traditional (such as power grid) or newly developed. The source node (e.g. base station) employs a buffer to store the backlogged packets randomly generated from higher-layer applications. Suppose that the data packets arrive at the source buffer according to a Bernoulli arrival process [19] with probability η 1 . This simple yet widely adopted traffic model allows tractable analysis, and provides insights for further study. The system is assumed to be timeslotted, and at the beginning instant of each slot, k 1 ∈ N data packets arrive at the data queue with capacity Q 1 . In this work, Q 1 is treated as sufficiently large (so no data overflow will incur) and fixed. Let q 1 [t] ∈ Q 1 = {0, 1, 2, · · · , Q 1 } be the length of the data queue at the end of slot t, updated as q 1 [t] = min{q 1 [t − 1] + a 1 [t] − v 1 [t], Q 1 },(1) where a 1 [t] ∈ {k 1 , 0} and v 1 [t] ∈ {1, 0} denote the number of data packets arriving and served in each time slot t, respectively. Without loss of generality, it is assumed that at most one packet is transmitted in each slot due to the capacity limitation of the communication link. Extension to multi-packet transmission will be considered in future work. The harvested energy is generally sporadically and randomly available, and we adopt a probabilistic energy harvesting model similar to [20]. Assume that e s Joule harvested energy arrives at the beginning of a time slot with probability η 2 , which can be used to transmit k 2 packets. That is e s = k 2ẽs , whereẽ s (Joule) denotes the amount of energy needed for transmission of one data packet, and k 2 (≥ 1) is rounded down to the nearest integer. We will consider several interesting combinations of k 1 and k 2 in this study, and leave the case k 2 < 1 to future study. The harvested energy is stored in the battery with the maximum capacity E Joule, and discarded when the battery is full. The battery storage is modeled as an energy queue with a finite capacity Q 2 = ⌊E/ẽ s ⌋, where one unit of transmission energyẽ s is viewed as one energy packet. Let a 2 [t] and v 2 [t] be the number of energy packets received and consumed in each slot t, respectively. At the end of time slot t, the length of the energy queue q 2 [t] ∈ Q 2 = {0, 1, 2, · · · , Q 2 } is updated as q 2 [t] = min{q 2 [t − 1] + a 2 [t] − v 2 [t], Q 2 }.(2) It is assumed that the packet and energy arrival processes are independent, and the newly harvested energy can be used for data transmission in the same slot. 1) Case 1: q[t − 1] = (0, j) (j > 0) In this case, the source can transmit a newly arriving data packet using the harvested energy from the battery in the current time slot t, and the service process can be expressed as v[t] = (1, 1) w.p.1, a[t] = (k 1 , ·), (0, 0) a[t] = (0, ·),(3) where w.p. means 'with the probability'. The notation of (k 1 , ·) is used to denote both (k 1 , k 2 ) and (k 1 , 0). 2) Case 2: q[t − 1] = (i, j) (i > 0, j > 0) In this case, the source can transmit a backlogged packet with the harvested energy. The service process is expressed as v[t] = (1, 1) w.p.1, a[t] = (·, ·).(4) 3) Case 3: q[t − 1] = (0, 0) In this case, when both data and energy packets arrive, the source will transmit with the energy harvested; When new data packets arrive in the absence of energy harvesting, the source shall use the energy from the RES to transmit with probability g 0 . Hence, the service process can be expressed as v[t] =      (1, 1) w.p. 1 a[t] = (k 1 , k 2 ) (1, 0) w.p. g 0 a[t] = (k 1 , 0) (0, 0) otherwise.(5)4) Case 4: q[t − 1] = (i, 0) (i > 0) In this case, the source will transmit definitely using the harvested energy if it is available in the current slot t. Otherwise, it will transmit using the RES energy with probability g i if a 1 [t] = k 1 and with probability f i if a 1 [t] = 0, respectively. The service process is characterized as v[t] =      (1, 1) w.p. 1, a[t] = (·, k 2 ) (1, 0) w.p. g i a[t] = (k 1 , 0) (1, 0) w.p. f i a[t] = (0, 0).(6) The above four cases include all possible scenarios. C. Average Delay and Power Consumption In a queueing system, the average queuing delay is an important metric [21]. From the above description, the queueing system can be modeled as a discrete-time Markov chain, where each state represents the buffer status. Let (i, j) be the state that the data queue length is i and the energy queue length is j, and π (i,j) denote the steady-state probability of state (i, j). By the Little's law, the average queueing delay is related to the average buffer occupancy, and can be computed as D = 1 k 1 η 1 Q1 i=1 iπ i = 1 k 1 η 1 Q1 i=1 i Q2 j=0 π (i,j) ,(7) where π i = Q2 j=0 π (i,j) (i, j ≥ 0). The average transmission power is also an important performance metric in wireless green communication systems. In this work, we focus on the average power consumption from the RES. Denote by c[t] the power consumed in the tth time slot. If the source transmits using the energy from the RES in time slot t, c[t] = p := 1 Tsẽ s , where T s denotes the transmission time. Otherwise, c[t] = 0. As will discussed below, the source draws one energy packet from the RES depending on the current queueing status q[t]. Let ω q (x) = Pr{c[t] = x\|q[t]} denote the probability that the power consumption c[t] is equal to x (x ∈ {0, p}) conditioned on the queue state q[t]. Using the law of total probability, we obtain the normalized average power consumption (with respect to p) as P = q∈Qp π q · ω q (p),(8) where Q p is the set of states conditioned on which the source may draw the RES energy to transmit one data packet. This normalized quantity can be interpreted as the proportion of the number of time slots in which the source transmits using the power from the RES. From (7) and (8), both the average queueing delay and power consumption are functions of the steady-state probabilities. In this work, we aim to study the delay optimal scheduling policy which minimizesD subject to the average power constraintP ≤ p max by determining the optimal transmission parameters {g * i } and {f * i }. As a key step, we will develop two-dimensional Markov chain models for different combinations of k 1 and k 2 in the next section. III. TWO-DIMENSIONAL MARKOV CHAIN MODELING To analyze the proposed scheduling scheme, we formulate a two-dimensional discrete-time Markov chain for the queueing system, as shown in Fig. 2. ¹ 1;Q1´1 ¹ 0 ¹ 0 ¹ 0 ¹ 2 ¹ 2 0 1;0 + ¹ 1 + ¹ 3 ¹1 + ¹3 ¹1 + ¹3 (0,0) (1,0) (2,0) (0,1) (0,2) (0,Q2-1) (0,(1,1) (1,2) (1,Q2-1) (2,1) (2,2) (2,Q2-1) (Q1-1,1) (Q1-1,2) (Q1-1, Q2-1) (Q1,1) (Q1,2) (Q1, Q2-1) ¹ 0 ¹ 1 ¹ 2 ¹ 0 ¹ 0 ¹ 1 ¹ 2 ¹ 0 ¹ 0 ¹ 1 ¹ 2 ¹ 0 ¹ 0 ¹ 2 ¹ 0 ¹ 0 ¹ 2 ¹ 0 ¹ 0 ¹ 0 ¹ 0 ¹ 0 ¹ 0 (1,Q2) (2,Q2) (Q1-1, Q2) (Q1,Q2) ¹ 1 ¹ 1 ¹ 1 ¹ 2 ¹ 1 ¹ 2 ¹ 1 ¹ 2 ¹ 1 ¹ 2 ¹ 1 ¹ 2 ¹ 1 ¹ 2 ¹ 1 ¹ 2 ¹ 1 ¹ 2 ¹ 1 ¹ 2 ¹ 1 ¹ 2 ¹ 1 ¹ 2 ¹ 1 ¹ 2 ¹ 1 ¹ 2 ¹ 1 ¹ 2 ¹ 1 ¹ 2 1 1 ¹ 2 ¹ 0 ¹ 2 ¹ 0 ¹ 2 ¹ 0´1 ¹ 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ¹1 + ¹3 ¹ 3 ¹ 3 ¹ 3 ¹ 3 ¹ 3 ¹ 3 ¹ 3 ¹ 3 ¹ 3 ¹ 3 ¹ 3 ¹ 3 0 1;0 + ¹1 + ¹3 ¹1 + ¹3 11 ¹1 + ¹3 0 1;1 + ¹ 0 1;1 + ¹3 0 1;Q1¡1 + ¹ 0 1;Q1¡1 + ¹3 ¹ 0 1;Q1 + ¹3 0 1;1 + ¹ 0 1;1 + ¹3 0 1;2 + ¹ 0 1;2 + ¹3 0 1;Q1¡1 + ¹ 0 1;Q1¡1 + ¹3 ¢ ¢ ¢¸1 ;0 ¹ 1;1¸0 1;0 + ¹ 1 (0,0) (1,0) (2,0) (0,1) (0,2) (0,Q2-1) (0,Q2) (Q1- 1,0) (Q1,0) (1,1) (1,2) (1,Q2) (1,Q2-1) (2,1) (2,2) (2,Q2) (2,Q2-1) (Q1- 1,1) (Q1- 1,2) (Q1- 1,Q2) (Q1-1, Q2-1) (Q1,1) (Q1,2) (Q1,Q2) (Q1,Q2- 1) 1 1 1 1 1 1 1 1 ¹ 3 ¹ 3 ¹ 3 ¹ 3 ¹ 3 ¹ 1 ¹ 1 0 1;1 + ¹ 0 1;1 0 1;2 + ¹ 0 1;2 0 1;Q1¡1 + ¹ 0 1;Q1¡1 ¢ ¢ ¢ ¹ 0 1;Q1¡1 ¹1 + ¹3 ¹ 0 ¹ 0 1;Q1 ¹ 0 1;Q1¡1 ¹ 0 1;1 ¹ 0 1;2 ¹ 0 1;3 (1,0) (Q1-1,0) (Q1,¹ 2 ¹ 2 ¹ 1 ¹ 1 ¹ 0 ¹ 3 ¹ 2 ¹ 0 ¹ 3 ¹ 0 ¹ 3 ¹ 0 ¹ 3 ¹ 2 ¹ 2 ¹ 2 ¹ 1 ¹ 1 ¹ 0 ¹ 3 ¹ 2 ¹ 0 ¹ 3 ¹ 0 ¹ 3 ¹ 0 ¹ 3 ¹ 2 ¹ 2 ¹ 2 ¹ 1 ¹ 1 ¹ 0 ¹ 3 ¹ 2 ¹ 0 ¹ 3 ¹ 0 ¹ 3 ¹ 0 ¹ 3 ¹ 2 ¹ 2 ¹ 2 ¹ 1 ¹ 1 ¹ 0 ¹ 3 ¹ 2 ¹ 0 ¹ 3 ¹ 0 ¹ 3 ¹ 0 ¹ 3 ¹ 2 ¹ 1 ¹ 1 ¹ 1 ¹ 1 ¹ 1 ¹ 1 ¹ 1 ¹ 1 (0,0) (0,1) (0,2) (0,Q2-1) (0,Q2) 1;011;1 ¹ 0 ¹ 2 ¹ 2 ¹ 1 ¹ 1´1 1 1 ¹ 0 ¹ 3 ¹ 2 ¹ 0 ¹ 3 ¹ 0 ¹ 3 ¹ 0 ¹ 3 ¹ 2 1;0 (2,0) (2,1) (2,2) (2,Q2-1) (2,Q2) 1;Q1¡1 ¹ 0 ¹ 0 ¹ 0 ¹ 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ¹ 1 ¹ 1 ¹ 1 ¹ 1 ¹ 1 (1,0) (Q1-1,0) (Q1,¹ 0 ¹ 0 ¹ 0 ¹ 0 ¹ 0 1 1 ¹ 3 ¹ 3 ¹ 1 ¹ 3 ¹ 1 ¹ 3 ¹ 3 ¹ 2 ¹ 0 ¹ 1 ¹ 1 ¹ 2 ¹ 0 ¹ 2 ¹ 0 ¹ 2 ¹ 0 ¹ 0 1 11 ¹ 3 ¹ 3 ¹ 1 ¹ 3 ¹ 1 ¹ 3 ¹ 3 ¹ 2 ¹ 0 ¹ 1 ¹ 1 ¹ 2 ¹ 0 ¹ 2 ¹ 0 ¹ 2 ¹ 0 ¹ 0 1 11 ¹ 3 ¹ 3 ¹ 1 ¹ 3 ¹ 1 ¹ 3 ¹ 3 ¹ 2 ¹ 0 ¹ 1 ¹ 1 ¹ 2 ¹ 0 ¹ 2 ¹ 0 ¹ 2 ¹ 0 ¹ 0 1 11 ¹ 3 ¹ 3 ¹ 1 ¹ 3 ¹ 1 ¹ 3 ¹ 3 ¹ 2 ¹ 0 ¹ 1 ¹ 1 ¹ 2 ¹ 0 ¹ 2 ¹ 0 ¹ 2 ¹ 0 ¹ 0 1 11 ¹ 3 ¹ 3 ¹ 1 ¹ 3 ¹ 1 ¹ 3 ¹ 3 ¹ 2 ¹ 0 ¹ 1 ¹ 1 ¹ 2 ¹ 0 ¹ 2 ¹ 0 ¹ 2 ¹ 0 ¹ 0 1 11 ¹ 1 ¹ 1 ¹ 1 ¹ 1 ¹ 1 ¹ 0 1;1 ¹ 0 1;2 ¹ 0 1;3 ¹ 0 1;Q1¡1 ¹ 0 1;Q1 (a) Two-dimensional discrete-time Markov chain for the general case (b) Two-dimensional discrete-time Markov chain for Case I with k1 = k2 = 1 (c) Trimmed Markov chain for Case I with k 1 = k 2 = 1 k 1 = k 2 = 1 (d) Two-dimensional discrete-time Markov chain for Case II with k 1 = 1; k 2 > 1 (e) Two-dimensional discrete-time Markov chain for Case III with k1 > 1; k2 = 1 k 1 = 1 k 2 = 1 (0,0) ¹ 0 ¹ 2 ¹ 1;2 1;1 ¹ 1;Q1¡1 Fig. 2. Two-dimensional discrete-time Markov chain 1 . Let Pr{q[t + 1]\|q[t] } denote the one-step transition probability of the Markov chain, which is homogeneous by the scheme description. For ease of expression, we define four constants as µ 0 = (1 − η 1 )η 2 , µ 1 = (1 − η 1 )(1 − η 2 ), µ 2 = η 1 (1 − η 2 ), µ 3 = η 1 η 2 .(9) We further define two subsets of Q i as: Q L i = {0, · · · , Q i −1}, Q R i = {1, · · · , Q i }, and setη i = 1 − η i , for i = 1, 2. We now describe the one-step transition probabilities in Fig. 2(a) in detail, by grouping them into several types. We start with the four transitions among each square unit, for example, those among (2, 1), (2, 2), (1, 2) and (1, 1) in Fig. 2(a). First, let us examine the transition from (2, 1) to (1, 2), more generally, from (i, j) to (i − 1, min{j + k 2 , Q 2 } − 1). This corresponds to the case that there is no data but energy packet arrival, and one backlogged data packet is delivered, so clearly the corresponding probability is µ 0 . When neither data nor energy packets arrive, one data packet stored in the buffer can be transmitted using one energy packet from the battery if there exists. In this case, the state will transfer from (i, j) to (i − 1, j − 1) (e.g., from (2, 2) to (1, 1) in Fig. 2(a)) with probability µ 1 for all i > 0 and Q 2 > j > 0. When k 1 data packets arrive while no energy is harvested, one data packet will be transmitted using one energy packet if there is energy stored in the battery. That is, the state will transfer from (i, j) to (i + k 1 − 1, j − 1) (e.g., from (1, 2) to (2, 1) in Fig. 2(a)) with probability µ 2 for all Q 2 > j > 0. When data and energy packets arrive simultaneously, one data packet is transmitted using one energy packet. In this case, the state will transfer Fig. 2(a)) with probability µ 3 for j ∈ Q L 2 . The case j = Q 2 requires special treatment, as the battery is full and the newly harvested energy has to be discarded anyway. With the capacity limit in mind, we have from (i, j) to (i + k 1 − 1, min{j + k 2 , Q 2 } − 1) (e.g., from (1, 1) to (2, 2) inPr{(i − 1, Q 2 − 1)\|(i, Q 2 )} =η 1 for i > 0, Pr{(0, Q 2 )\|(0, Q 2 )} =η 1 , and Pr{(i + k 1 − 1, Q 2 − 1)\|(i, Q 2 )} = η 1 for all i. We then consider the first row in Fig. 2(a). When no data packets arrive and k 2 energy packets newly arrive, the state (0, j) will transfer to (0, min{j + k 2 , Q 2 }) with the corresponding transition probability µ 0 for j ∈ Q L 2 . We have mentioned that Pr{(0, Q 2 )\|(0, Q 2 )} =η 1 is due to the capacity limitation of the battery. The state (0, j) remains the same with probability µ 1 (when neither data nor energy packets arrive). We now focus our attention on the group of transition probabilities on the first column of Fig. 2 , {λ 1,i } and {λ ′ 1,i } (i ∈ Q L 1 ), {µ 1,i } and {µ ′ 1,i } (i ∈ Q R 1 ) , which corresponds to the case that there is no storage of harvested energy in the battery, and can be obtained as λ 1,i = Pr{(i + k 1 , 0)\|(i, 0)} = µ 2 (1 − g i ) (i ∈ Q L 1 ), λ ′ 1,i = Pr{(i + k 1 − 1, 0)\|(i, 0)} = µ 2 g i (i ∈ Q L 1 ), µ 1,i = Pr{(i − 1, 0)\|(i, 0)} = µ 1 f i (i ∈ Q R 1 ), µ ′ 1,i = Pr{(i, 0)\|(i, 0)} = µ 1 (1 − f i ) (i ∈ Q R 1 ).(10) In particular, when k 1 data packets arrive while no energy is harvested (which happens with probability µ 2 ), λ 1,i and λ ′ 1,i denote the transition probabilities from state (i, 0) to (i+k 1 , 0) and (i+k 1 −1, 0), respectively, depending on whether one data packet is delivered with the reliable energy in this slot (with probability g i ). When neither data nor energy packets arrive (which happens with probability µ 1 ), µ 1,i and µ ′ 1,i denote the transition probabilities from state (i, 0) to (i − 1, 0) and (i, 0), respectively, depending on whether one data packet is transmitted using the reliable energy (with probability f i ). We order the N = (1 + Q 1 )(1 + Q 2 ) states as (0, 0), · · · , (0, Q 2 ), (1, 0), · · · , (1, Q 2 ), · · · (Q 1 , 0), · · · , (Q 1 , Q 2 ), and let P denote the N ×N transition matrix. We denote by π the 1× N column vector containing steady-state probabilities, and by e the N × 1 column vector with all the elements equal to one. For notational convenience, we also define two sub-vectors of π as: π i = [π (i,0) ; · · · ; π (i,Q2) ] andπ i = [π 0 ; · · · ; π i ], and denote by e i a 1 × (i + 1)(1 + Q 2 ) row vector with all the elements equal to one. Given a set of parameters {g i } and {f i }, the steady-state probabilities π (i,j) can be obtained by solving the linear equations πP = π and πe = 1. Note that the transmission parameters {g i } and {f i } only influence the transition probabilities from the states (i, 0), i ∈ Q 1 . We thus considerP s , a submatrix ofP = P − I, to exclude the state transition starting from states (i, 0). In this way, πP s = 0 present the local balance equations at the states (i, j) (i ≥ 0, j > 0). For ease of expression, we also denote byP i the left-top submatrix of (i + 1)Q 2 dimensions fromP s . In the general case with k 1 ≥ 1 and k 2 ≥ 1, the corresponding Markov chain seems not amenable to analysis. In this paper, we mainly focus on three cases: Case I with k 1 = 1 and k 2 = 1, Case II with k 1 = 1 and k 2 > 1, and Case III with k 1 > 1 and k 2 = 1, respectively. These three exemplary cases nonetheless capture some important relations between the data and energy arrival processes, and serve as the basis for further extensions. In the following, we illustrate the Markov chain for each of the three cases. A. Case I: k 1 = 1 and k 2 = 1 In this case, one data packet and one energy packet arrive in each slot with probabilities η 1 and η 2 , respectively. Accordingly, the simplified Markov chain is shown in Fig. 2(b). Essentially all expressions in the general case carry over with the substitution of k 1 = k 2 = 1. For example, the transition Fig. 2(b), again with probability µ 0 . This applies to the states in the first column as well, and as a result, a new notation is needed for the transition from (i, 0) to (i − 1, 0), which combines µ 0 and the previous µ 1,i : from (i, j) to (i−1, min{j +k 2 , Q 2 }−1) in Fig. 2(a) becomes that from (i, j) to (i − 1, j) inµ 1,i = Pr{(i − 1, 0)\|(i, 0)} = µ 0 + µ 1 f i(11) for all i ∈ Q R 1 . Also, it is worth noting that in the dashed square, neither queue length can ever increase regardless of the arrival processes, as one data packet transmission happens for sure. As a result, the states q[t] with q 1 [t] · q 2 [t] > 0 are transient in the following lemma. Lemma 1. In Case I with k 1 = k 2 = 1 when η 1 < 1 or η 2 < 1, the queue status satisfying q 1 [t]·q 2 [t] > 0 is transient. Proof: Let f (n) q denote the probability that the queue state q[t] will return to itself for the first time after n steps. As shown in Fig. 2 (b), when i · j > 0, f (1) q = Pr{(i, j)\|(i, j)} = µ 3 and f (n) q = 0 for n > 1. Hence, ∞ n=1 f (n) q = µ 3 = η 1 η 2 < 1, when η 1 < 1 or η 2 < 1. From [22], the state q[t] with q 1 [t] · q 2 [t] > 0 is a transient state. This implies that either the data queue or the energy queue will be exhausted, even if they are not empty initially. Hence, when calculating the steady-state probabilities π (i,j) , the two-dimensional Markov chain can be reduced to the onedimensional one, as plotted in Fig. 2(c), which consists of the states (i, 0) and (0, j) for all i ∈ Q 1 and j ∈ Q 2 . B. Case II and Case III In Case II, k 2 energy packets (k 2 > 1) arrive at the battery with probability η 2 per slot. Hence, the length of the energy queue may increase by k 2 or k 2 − 1 (when one energy packet is consumed in the current slot) each time. The resulting twodimensional Markov chain is shown in Fig. 2(d). In Case III, k 1 > 1 data packets arrive with the probability η 1 at each slot, and the two-dimensional Markov chain is illustrated in Fig. 2(d), where the data queue length could increase by k 1 or k 1 − 1 (when one data packet is transmitted using an energy packet harvested or drawn from the RES in the current slot). As shown in Fig. 2(d), the solid lines present the fixed state transitions while the dotted lines indicate state transitions that vary with different k 2 . In particular, the state (i, j) transfers to (i − 1, min{j + k 2 , Q 2 } − 1) with the probability µ 0 and to (i, min{j + k 2 , Q 2 } − 1) with the probability µ 3 , respectively. Similarly, the state (0, j) transfers to (0, min{j+k 2 , Q 2 }) with the probability µ 0 , and to (0, min{j + k 2 , Q 2 } − 1) with the probability µ 3 , respectively. Note that the states (i, Q 2 ) for all i > 0 are transient. Similarly in Fig. 2(e), solid and dotted lines are used to present the fixed state transitions and state transitions that vary with different k 1 , respectively. Similar to Case I, the state transfers from (i, 0) to (i − 1, 0) with the combined transition probabilityμ 1,i = µ 0 + µ 1 f i . For the same reason, the transition probability from (i, 0) to (i + k 1 − 1, 0) is λ 1,i = µ 3 + µ 2 g i = µ 3 + λ ′ 1,i = η 1 − λ 1,i .(12) And the states (i, Q 2 ) for all i > 0 are transient. IV. LP PROBLEM FORMULATION As discussed above, both the average delay and power consumption from the RES are functions of the steady-state probabilities of the corresponding Markov chains, which in turn depend on the transmission parameters {g i } and {f i } to be designed. To seek the optimal scheduling policy, we adopt a two-step procedure [23]: first we formulate an LP problem only depending on the steady-state probabilities, and obtain the corresponding solution; then from the optimal solution of the LP problem, we determine the optimal transmission parameters. Our objective is to minimize the average queueing delay subject to the maximum average power constraint from the RES. The corresponding LP problem can be formulated as minD = 1 k 1 η 1 Q1 i=1 i Q2 j=0 π (i,j) s.t.                       P = Q1 i=0 ξ i · π (i,0) − Q1 i=0 ζ i · π (i,1) ≤ p max ,(a)Θ l (i,π i−1 ) ≤ Q2 j=0 π (i,j) ≤ Θ u (i,π i )(i > 0), (b) π (i,j) ≥ 0, (∀i, j), (c) Q1 i=0 Q2 j=0 π (i,j) = 1, (d) πP s = 0. (e)(13) From the properties of a Markov chain, the last three constraints (c)-(e) are straightforward. The original definition of P (c.f. (8)) in constraint (a) does depend on the transmission parameters; to facilitate derivation, we will give a new expression forP in Lemma 2 below that is only a function of the steady-state probabilities π (i,0) and π (i,1) , i ∈ Q 1 . The influence of the transmission parameters on the problem is encapsulated in the constraint (b), which represents the relationship between the steady-state probabilities π (i,j) due to the varying transmission parameters {g i } and {f i }, as discussed later in Lemma 3. The optimal solution to (13) is denoted by π * (i,j) and the minimum average delay byD * . Lemma 2. In Cases I, II and III, the normalized average power consumption from the RES can be expressed as P = Q1 i=0 ξ i · π (i,0) − Q1 i=0 ζ i · π (i,1) ,(14) where the coefficients ξ i and ζ i are presented in Table I. Proof: The proof is deferred to Appendix A. Remark: By exploiting the local balance equations of states (i, 0) (i ∈ Q L 1 ), we can replace all the items π (i,0) µ 2 g i (i ∈ Q 1 ) and π (i,0) µ 1 f i (i ∈ Q R 1 ) ofP with the items ξ i π (i,0) and ζ i π (i,1) (i ∈ Q 1 ). In this way, the average power consumption P becomes a linear function of the steady-state probabilities π (i,0) and π (i,1) . Thus, the direct dependence ofP on the transmission parameters {g i } and {f i } is removed. Then, we discuss the constraint (13.b). The basic idea is to vary the transmission parameters {g i } and {f i } in the full range of [0, 1], so as to obtain an upper and lower bound for each π i . In this way, we transform the constraints on {g i } and {f i } into the relationship between the steadystate probabilities themselves, which allows us to obtain the optimal solution to (13) in terms of {π (i,j) } first. For ease of illustration, we define several constants as τ = η1 1−η1 , φ = µ2 µ0 , and φ 1 = η1 µ0 . Let us define [x] + = max{0, x}. Lemma 3. In Cases I, II and III, the probability π i satisfies (15) where Θ u (·) and Θ l (·) are presented in Table II. Proof: The proof is deferred to Appendix B. Remark: From the proof of Lemma 3, we have π i = Θ u (i,π i ) at g i−k1 = f i = 0, and π i = Θ l (i,π i−1 ) at g i−k1 = f i = 1, respectively, in all the three cases 2 . This lies in the fact that the transmission parameters {g i } and {f i } determine the relationship between the steady-state probabilities {π (i,j) }, and vice versa. As listed in Table II, Θ u (i,π i ) is a linear function of the steady-state probabilities π (i−k1,0) , · · · , π (i−1,Q2) , π (i,0) , and Θ l (i,π i−1 ) is a linear function of π ([i−k1+1] + ,0) , · · · , π (i−1,Q2) . Θ l (i,π i−1 ) ≤ π i = Q2 j=0 π (i,j) ≤ Θ u (i,π i ) (i > 0), From Lemmas 2 and 3,P , Θ u (i,π i ) and Θ l (i,π i−1 ) are all linear functions of the steady-state probabilities {π (i,j) }. Hence, we can represent them in the form ofP (π) = πa 0 , Q2 j=0 π (i,j) − Θ u (i,π i ) = πa u i (i > 0), and Θ l (i,π i−1 ) − Q2 j=0 π (i,j) = πa l i (i > 0) , where a 0 , a u i and a l i are N × 1 column vectors collecting corresponding coefficients. V. DELAY OPTIMAL SCHEDULING UNDER POWER CONSTRAINT In this section, we discuss the optimal solution to Problem (13) by studying its structure with respect to the steady-state probabilities of the corresponding Markov chains. 2 More rigorously, in Case I, π i = Θ l (i,π i−1 ) holds just when g i−1 = 1 and f i can be arbitrary. Case I with k 1 = k 2 = 1 Case II with k 1 = 1 and k 2 > 1 Case III with k 1 > 1 and k 2 = 1 ξ i ξ 0 = µ 2 ξ i = µ 2 + η 2 (Q 1 − i) (i ∈ Q 1 ) ξ 0 = µ 0 Q 1 − µ 3 + k 1 η 1 ξ i = µ 2 − µ 0 (i ∈ Q R 1 ) ξ i = k 1 η 1 − η 2 (1 ≤ i ≤ Q 1 − k 1 ) ξ i = η 1 (Q 1 − i) − η 2 (Q 1 − k 1 + 1 ≤ i ≤ Q 1 ) ζ i ζ i = 0 (i ∈ Q 1 ) ζ 0 = µ 2 Q 1 ζ 0 = µ 2 (Q 1 − k 1 + 1) ζ i =η 2 (Q 1 − i) + µ 1 (i ∈ Q R 1 ) ζ i = (µ 1 + µ 2 )(Q 1 + 1 − i) − µ 2 k 1 (1 ≤ i ≤ Q 1 − k 1 ) ζ i = µ 1 (Q 1 + 1 − i) (Q 1 − k 1 + 1 ≤ i ≤ Q 1 ) Table II Θu(i,π i ) AND Θ l (i,π i−1 ) FOR CASES I, II, AND III. Case I with k 1 = k 2 = 1 Case II with k 1 = 1, k 2 > 1 Case III with k 1 > 1, k 2 = 1 Θu(i,π i ) φπ (i−1,0) τη 2 π (i−1,0) + π (i,0)η2 i < k 1 π (i,0)η2 + i−1 m=0 Q 2 j=0 τ π (m,j) i ≥ k 1 τη 2 π (i−k 1 ,0) + π (i,0)η2 + i−1 m=i−k 1 +1 Q 2 j=0 τ π (m,j) Θ l (i,π i−1 ) 0 0 i−1 m=[i−k 1 +1] + Q 2 j=0 τ π (m,j) A. Structure of The Optimal Solution For ease of discussion, we first consider a scheduling policy strictly based on the threshold m: the source waits for the harvested energy when the number of backlogged data packets is less than or equal to a certain threshold m and transmits using the reliable energy when the data queue length exceeds m. According to the threshold m, we usep m to measure the amount of power drawn from the RES. Sincep m is sufficient for the application of the scheduling policy based on the threshold m + 1, but not vise versa,p m is non-increasing with the threshold m. We will show that the threshold based scheduling policy turns out to be the optimal and the optimal threshold is determined by the power thresholds {p m }. Theorem 4. The optimal threshold is i * = 0 when p max ≥p 0 , and i * > 0 when k 1 η 1 − k 2 η 2 < p max <p 0 , respectively. Proof: The proof is deferred to Appendix C. We notice that the average queueing delayD = 1 k1η1 Q1 i=1 iπ i is a weighted summation of the steady-state probabilities π i . Thus,D can be reduced, if we assign a larger value to π i with a smaller index i and vice versa. Based on this intuition, we can reveal that the optimal solution to the LP problem (13) corresponds to a threshold based scheduling policy with the optimal threshold i * determined by the maximum allowable power consumption from the RES p max . Theorem 5. The optimal solution π * satisfies π * a 0 ≤ p max , π * a u i = 0 (i = 1, · · · , i * − 1), π * a l i = 0 (i = i * + 1, · · · , Q 1 ), where the optimal threshold is obtained as i * = arg miñ pm≤pmax m.(17) Proof: The proof is deferred to Appendix D. Remark: According to Lemma 3, we have π i = Θ u (i,π i ) or πa u i = 0 when g i−k1 = f i = 0, and π i = Θ l (i,π i−1 ) when g i−k1 = f i = 1, respectively. Therefore, associated with (16) is a threshold based scheduling policy that waits for the harvested energy when the number of backlogged data packets is less than a certain threshold i * , and draws the reliable energy definitely when the harvested energy is not available while the number of backlogged data packets exceeds the threshold (i * if there is no new data packet arrival, and i * − k 1 if there is new data packet arrival). Note that the LP problem (13) has an optimal solution only when the queueing system is stable, i.e., when the service rate is greater than the arrival rate, according to Loynes's theorem [24]. Throughout this paper, the service rate is specialized as the total amount of energy that can be drawn either from the RES or from the battery, p max + k 2 η 2 . Hence, we will discuss the optimal solution to the LP problem (13) under the assumption that p max > k 1 η 1 − k 2 η 2 . B. The Optimal Solution By exploiting the result in Theorem 5, we continue to derive the optimal steady-state probabilities for Case I, and develop an algorithm to obtain the optimal solutions for Case II and Case III, respectively. 1) Case I: In this case, the two-dimensional Markov chain is reduced to a one-dimensional one, where transitions takes place only between adjacent states, as shown in Fig. 2(c). We only need to discuss the optimal steady-state probabilities π * (i,0) and π * (0,j) for all i ∈ Q 1 and j ∈ Q 2 . In the sequel, we first show that the optimal π * (0,j) is a function of π * (0,0) in Lemma 6 and then present the optimal π * (i,0) in Corollary 7. Lemma 6. In Case I, the optimal steady-state probability π * (0,j) is related to π * (0,0) as π * (0,j) = π * (0,0) φ −j , 1 ≤ j ≤ Q 2 − 1, π * (0,0) φ −(Q2−1) φ −1 1 , j = Q 2 .(18) Proof: From the proof of Theorem 5, the optimal probability π * (0,j) is a function of π * (0,0) , as given by (18). From (18), we get π * 0 = Q2 j=0 π * (0,j) = απ * (0,0) , where α = Q2−1 i=0 φ −i + φ −(Q2−1) φ −1 1 = (Q 2 + φ −1 1 ), φ = 1, φ1φ Q 2 +φ−φ1−1 φ Q 2 −1 (φ−1)φ1 , φ = 1. From the results obtained in Theorem 5, we show that the optimal π * (i,0) for all i > 0 are functions of π * (0,0) . Further, taking advantage of the dependance ofP on π * (0,0) , we can derive the closed-form optimal solution π * (i,0) in Corollary 7. Corollary 7. In Case I, when p max ≥p 0 = µ 2 α −1 , we have π * (0,0) = α −1 and π * (i,0) = 0 for all i > 0, respectively. When η 1 − η 2 < p max <p 0 , π * (0,0) = pmax−(µ2−µ0) µ2−α(µ2−µ0) , and π * (i,0) (i > 0) is given by π * (i,0) =      π * (0,0) φ i , i ≤ i * − 1, 1 − απ * (0,0) − π * (0,0) i * −1 i=1 φ i , i = i * , 0, i > i * ,(19) where the optimal threshold i * is obtained as i * = Ω φ (π * (0,0) , 1 − απ * (0,0) )(20) with the function Ω φ (a, b) defined as Ω φ (a, b) := max a i−1 m=1 φ m ≤b i = ⌊ b a ⌋ + 1, φ = 1, ⌊log φ (a+b)φ−b a ⌋, φ = 1. Proof: The proof is deferred to Appendix E. From Eqs. (18), (19) and (20), one can see that the optimal steady-state probabilities π * (i,0) and π * (0,j) , and the optimal threshold i * are solely determined by the maximum average power p max for given η 1 , η 2 and Q 2 . We also show that π * (i,0) = 0 for all i > i * . This indicates that the length of the packet queue never exceeds the threshold i * . Hence, no packet loss will be induced as long as the queue capacity Q 1 is larger than i * . 2) Case II and Case III: In Case II with k 1 = 1 and k 2 > 1 and Case III with k 1 > 1 and k 2 = 1, it is challenging to derive a closed-form optimal solution to the LP problem (13). Based on the result in Theorem 5, we then develop an algorithm to find the optimal solutions for these two cases. In Theorem 5, we show that the optimal solution corresponds to the threshold based transmission scheme. The optimal threshold can be determined by comparing the power constraint p max to the power thresholds {p m } (m ≥ 0). In particular,p m can be computed as p m = π ′ m a 0 ,(21) Algorithm 1 Finding the optimal solution for Cases II and III. 1: Initialization: set Q 1 to be a large constant. 2: if p max ≤ k 1 η 1 − k 2 η 2 then 3: The optimal solution and parameters do not exist. 4 Set m = 1.          πa u i = 0 (i = 1, · · · , m), πa l i = 0 (i = m + 1, · · · , Q 1 ), πP s = 0, πe = 1.(22) Let b ′ = [0, · · · , 0, 1] be a 1 × N row vector, and A ′ m = [a u 1 , · · · , a u m , a l m+1 , · · · , a l Q1 ,P s , e] be an N ×N matrix. The solution to (22) can be expressed as π Corollary 8. The optimal solution to (13) for Cases II and III can be computed as π * = π ′ 0 , if p max ≥p 0 , bA −1 , ifp i * ≤ p max <p i * −1 ,(23) where A = [a 0 , a u 1 , · · · , a u i * −1 , a l i * +1 , · · · , a l Q1 ,P s , e] is an N ×N matrix, and b = [p max , · · · , 0, 1] is a 1×N row vector. Proof: The proof is deferred to Appendix F. Remark: By exploiting the structure of the optimal solution, we can compute the optimal solution π * by solving (1 + Q 1 )(1 + Q 2 ) independent linear equations. Based on the definition of the power thresholds {p m }, π * can be alternatively obtained by solving linear equations (22) when p max =p i * , i.e., π * = π ′ m . In Case II, since π i = Θ l (i,π i−1 ) = 0, we have π * (i,j) = 0 for all i > i * . Andπ * i * can be obtained by solving (1 + i * )(1 + Q 2 ) linear equations:π i * a 0 = p max , π i * a u i = 0 (i = 1, · · · , i * − 1),π i * P i * = 0 andπ i * e i * = 1. Based on the above discussion, we develop an algorithm, i.e., Algorithm 1, to show how to find the optimal solution π * (i,j) to the LP problem for Case II and Case III. The optimal threshold i * is sought iteratively by comparing the maximum allowable power consumption p max with the power thresholds Case I with k 1 = k 2 = 1 Case II with k 1 = 1 and k 2 > 1 Case III with k 1 > 1 and k 2 = 1 0 ≤ i < i * − k 1 0 g * i i = i * − k 1 1 − 1−απ * (0,0) −π * (0,0) i * −1 i=1 φ i π * (0,0) φ i * 1 − π * (i * ,0) µ 0 +η 1 Q 2 j=1 π * (i * ,j) µ 2 π * (i * −1,0) 1 −η 1 π * i * −π * (i * ,0) µ 1 −η 1 i * −1 m=i * −k 1 +1 π * m µ 2 π * (i * −k 1 ,0) i > i * − k 1 1 Case I and Case II Case III with k 1 > 1 and k 2 = 1 0 < i < i * 0 0 f * i i = i * 0 (i * ≥ k 1 ) η 1 i * −1 m=0 π * m −η 1 π * i * +π * (i * ,0) µ 1 π * (i * ,0) µ 1 (i * < k 1 ) i > i * 1 {p m }. Once locating the threshold i * , the optimal steady-state probabilities π * (i,j) can be obtained by solving an LP problem. There are two exceptions: (1) when p max ≤ k 1 η 1 − k 2 η 2 , the queueing system is not stable and the optimal solution does not exist; and (2) when the iteration number exceeds the sufficiently large data queue length Q 1 , we regard the optimal threshold i * as the infinity. Given Q 1 , the algorithm runs at most Q 1 iterations, and in each iteration the computation complexity of solving N linear equations is O (N 3 ). Hence, the computation complexity of this algorithm can be roughly estimated as O( Q 1 (1 + Q 1 ) 3 (1 + Q 2 ) 3 ). For a relatively small Q 2 , the complexity can be approximated as O(Q 4 1 ). By comparison of Case I and Case II, we notice that changing the number of energy packets arriving each time, k 2 , does not change the property of the optimal results. From this perspective, it is feasible to deal with the case when k 1 > 1 and k 2 > 1 using the same method as in Case III. Firstly, we formulate a concrete Markov chain for a pair of such k 1 and k 2 and find the mutual relations between the states. Secondly, we construct an LP problem under the power consumption constraint, which manifests as the constant linear combination of the steady-state probabilities. Finally, we can adopt Algorithm 1 to find the optimal solution and the optimal transmission parameters. C. The Optimal Transmission Parameters By exploiting the local equilibrium equations and the corresponding optimal solution π * , we then obtain the optimal transmission parameters {g * i } and {f * i } for Cases I, II and III. Corollary 9. When p max ≥p 0 , the optimal transmission parameters are given by g * i = 1 (i ≥ 0) and f * i = 0 (i > 0); When k 1 η 1 − k 2 η 2 < p max <p 0 , the optimal transmission parameters {g * i } and {f * i } are listed in Table III. Proof: The proof is deferred to Appendix G. Remark: Note that p max ≥p 0 indicates that the allowable backup energy supply is sufficient so that the source can use the reliable energy whenever it needs. In this scenario, packet delivery is guaranteed in each slot, and there will be no backlogged packets in Case I and Case II, while in Case III, the data queue may still accumulate as each data arrival brings in multiple packets while at most one packet is delivered in each slot. When k 1 η 1 − k 2 η 2 < p max <p 0 , the source should transmit according to the optimal threshold i * . For Case I and Case II, the utilization of power from the RES could happen only in two scenarios: when a new packet arrives but no harvested energy can be used, and the data packet queue length is equal to i * − 1 and i * , respectively. In the former case, the source transmits using the power from the RES with the probability g * i * −1 < 1, while in the latter case, the source will transmit using the energy from the RES definitely with g * i * = 1. In Case III, the source should transmit using the reliable energy as soon as the data queue length exceeds the threshold i * , no matter whether there is a new data arrival. That is, g * i−k1 = 1 and f * i = 1 are set for all i > i * . Once getting the optimal g * i and f * i , we can compute the optimal steadystate probabilities π * (i,j) and the corresponding minimum aver- age delayD * = 1 k1η1 Q1 i=1 i Q2 j=0 π * (i,j) , which depends on the allowable reliable energy p max . Hence,D * is an implicit function of p max . As shown by simulation results in the next section, the average queuing delay monotonically decreases with the increase of the power p max . VI. SIMULATION RESULTS In this section, simulation results are presented to demonstrate the performance of the proposed scheduling scheme and validate our theoretical analysis. In simulations, the packet and energy arrival processes are modeled by generating two Bernoulli random variables with the parameters η 1 and η 2 , respectively, at the beginning of each time slot. The packet transmissions are scheduled according to our proposed policy. And we apply the optimal transmission parameters f * i and g * i listed in Table III to get the optimal delay-power tradeoff curves. Each simulation runs over 10 6 time slots. In the figures, the lines and the marks 'o' indicate theoretical and simulation results, respectively. One can see that theoretical and simulation results match well. Fig. 3 plots the optimal delay-power tradeoff performance for Case I. It is observed from Fig. 3(a) that the minimum aver- age queueing delay monotonically decreases with the increase of the maximum power consumption p max , which contributes to the growth of the service rate. That is, when more power can be drawn from the RES, the packets will be transmitted more quickly and the queueing delay is reduced. One can see that the minimum average queueing delay decreases from infinity to zero, when p max grows from zero top 0 , which is equal to µ2 Q2+φ −1 1 in the case of η 1 = η 2 . Hence, the decreasing rate grows with the increase of the battery capacity Q 2 . This means that a larger Q 2 leads to a much smaller queueing delay, since less harvested energy is wasted due to the limitation of the battery capacity. Fig. 3(b) demonstrates the optimal delay-power performance for different energy arrival rates η 2 . When η 2 ≤ η 1 , the average delay is infinite at p max = 0, since the arrival rate is greater than or equal to the service rate. Therefore, the source should exploit extra energy from the RES to transmit backlogged packets, corresponding to a positive p max . While the source can rely only on the harvested energy to transmit, i.e., p max = 0, when η 2 > η 1 . Fig. 4 shows the optimal delay-power tradeoff performance of the proposed scheme in Case II with k 2 = 2, · · · , 6. The optimal delay-power curve of the case with k 1 = k 2 = 1 is also plotted for comparison. In this experiment, we set η 1 = The normalized average power The minimum average delay Fig. 4. The optimal delay-power performance for Case II with η 1 = 0.5, η 2 = 0.1 and Q 2 = 5. 0.5, η 2 = 0.1 and Q 2 = 5. From this figure, one can see that there exists an optimal delay-power tradeoff for each k 2 . The average queueing delay monotonically decreases with the increase of the maximum allowable power consumption p max from the RES due to the enhanced service rate. For the same reason, a larger k 2 means a higher amount of energy harvested each time, and leads to a much better delay-power tradeoff. It is also observed the delay-power curves of k 2 = 5 and k 2 = 6 are almost identical to each other. This owes to the fact that in the case of k 2 = 6, a part of harvest energy is wasted when recharging the battery with capacity Q 2 = 5. Similarly, we plot the optimal delay-power curves of the proposed scheme for Case III with different k 1 in Fig. 5. We set η 1 = 0.1, η 2 = 0.3, and Q 2 = 5. Similar to Case II shown in Fig. 4, a higher p max induces reduced average queueing delay thanks to the enhanced service rate. The only difference between them is the behavior of the minimum average delaȳ D * . In Case I with k 1 = k 2 = 1, the average queueing delay is equal to zero if there exists sufficient energy whether from the battery or the RES, since one newly arriving data packet can always be delivered immediately. In Case III, however, at most one of k 1 data packets that newly arrive at this slot can be delivered, and the other packets shall wait for the next transmission opportunity. And more packets are queued when the data arrival rate is increased due to the growth of k 1 or η 1 . As shown in Fig. 5,D * increases with the increase of k 1 . k 2 =1 k 2 =2 k 2 =3 k 2 =4 k 2 =5 k 2 =6 VII. CONCLUSIONS In this paper, we investigated the delay optimal scheduling problem over a communication link. The source node can rely on energy supply either from an energy harvesting battery of finite capacity or from the RES subject to a maximum power consumption from the RES. Using the two-dimensional Markov chain modeling, we formulated an LP problem and studied the structure of the optimal solution. As a result, we obtained the optimal scheduling policy through rigorous derivation and algorithm design. It is found that the source should schedule packet transmissions according to a critical threshold on the data queue length. Specifically, the source should always wait for the harvested energy when the data queue length is below the optimal threshold i * , and resort to the RES when the data queue length exceeds the threshold i * while no harvested energy can be exploited. The optimal threshold i * is determined by the maximum allowable power from the RES p max . Simulation results confirmed our theoretical analysis. It was shown that there always exists an optimal delay-power tradeoff and its decreasing rate depends on the energy arrival rate and the battery capacity. In this work, we assume that the Bernoulli data and energy arrival processes generate integral packets probabilistically, and only one data packet is transmitted in each slot. In the future, we will extend the study to the scenario where rateflexible physical-layer transmissions are scheduled based on the randomly available amount of harvested energy and timevarying wireless channel conditions. APPENDIX A. Proof of Lemma 2 By applying the stochastic scheduling scheme described in Section II.2, the source shall resort to the RES only when the harvested energy is not available in the current slot. Thus, the state set Q p (c.f. (8)) is given by {(0, 0), (1, 0), · · · , (Q 1 , 0)}. Recall that the source will draw the reliable energy to transmit with probability g i if new data packets arrive and with probability f i if no data packets arrive, respectively. Hence, the reliable energy consumption at state (i, 0) can be expressed as ω (0,0) (p) = µ 2 g 0 and ω (i,0) (p) = µ 2 g i + µ 1 f i (i > 0), respectively. Consequently, the normalized average power consumed from the RES can be obtained asP = Q1 i=0 π (i,0) ω (i,0) (p) = Q1 i=0 π (i,0) µ 2 g i + Q1 i=1 π (i,0) µ 1 f i . The following result eliminates the dependence ofP on the transmission parameters and presents a unified expression for all three cases. 1) In Case I, from Fig. 2(c), the local equilibrium equation of the Markov chain can be expressed as π (i,0) λ 1,i = π (i+1,0)μ1,i+1 , π (0,j) µ 0 = π (0,j+1) µ 2 ,(24) for all i ∈ {0, · · · , Q 1 − 1} and j ∈ {0, · · · , Q 2 − 2}, and π (0,Q2−1) µ 0 = π (0,Q2) η 1 . From (24), we have π (0,0) λ 1,0 = Q1 i=1 π (i,0) (μ 1,i −λ 1,i ), where π (Q1,0) λ 1,Q1 = 0 is introduced for notational convenience. With λ 1,i = µ 2 (1 − g i ) (c.f. (10)) andμ 1,i = µ 0 +µ 1 f i (c.f. (11)), the normalized average power consumption from the RESP can also be expressed as P =π (0,0) µ 2 g 0 + Q1 i=1 π (i,0) (µ 2 g i + µ 1 f i ) =π (0,0) (µ 2 − λ 1,0 ) + Q1 i=1 π (i,0) (µ 2 − λ 1,i +μ 1,i − µ 0 ) =π (0,0) µ 2 + (µ 2 − µ 0 ) Q1 i=1 π (i,0) . Hence, we get ξ 0 = µ 2 , ξ i = µ 2 − µ 0 for all i ∈ Q R 1 and ζ i = 0 for all i ∈ Q 1 . 2) From Fig. 2(d) for Case II, the local equilibrium equation at state (0, 0) can be expressed as π (1,0) µ 1,1 − π (0,0) λ 1,0 = η 2 π (0,0) − µ 2 π (0,1) − µ 1 π (1,1) . Following a similar procedure, the local equilibrium equation at state (i, 0) (i ≥ 1) is thus π (i+1,0) µ 1,i+1 − π (i,0) λ 1,i = π (i,0) µ 1,i − π (i−1,0) λ 1,i−1 + (µ 0 + µ 3 )π (i,0) − µ 2 π (i,1) − µ 1 π (i+1,1) =η 2 i m=0 π (m,0) − µ 2 i m=0 π (m,1) − µ 1 i+1 m=1 π (m,1) , where the second equality is obtained through recursion of (π (i,0) µ 1,i − π (i−1,0) λ 1,i−1 ). Hence, we can compute the normalized average power consumption as P = Q1 i=0 π (i,0) (µ 2 − λ 1,i ) + Q1 i=1 π (i,0) µ 1 f i =µ 2 Q1 i=0 π (i,0) + Q1 i=1 π (i,0) µ 1,i − π (i−1,0) λ 1,i−1 = Q1 i=0 π (i,0) (µ 2 + η 2 (Q 1 − i)) − µ 2 Q 1 π (0,1) − Q1 i=1 (η 2 (Q 1 − i) + µ 1 )π (i,1) , where the second equality is due to the fact that µ 1,i = µ 1 f i , and the third equality is obtained through the summation of π (i,0) µ 1,i − π (i−1,0) λ 1,i−1 over all i > 0. In this way, we get ξ i = µ 2 + η 2 (Q 1 − i) for all i ∈ Q 1 , ζ 0 = µ 2 Q 1 , and ζ i =η 2 (Q 1 − i) + µ 1 for all i ∈ Q R 1 . 3) In Case III, the corresponding Markov chain is depicted in Fig. 2(e). When i < k 1 − 1, the local balance equation at state (i, 0) is given bỹ µ 1,i+1 π (i+1,0) = π (i,0)μ1,i + π (i,0) (λ 1,i +λ 1,i ) − µ 1 π (i+1,1) =π (0,0) µ 0 + i m=0 π (m,0) η 1 − i+1 m=1 µ 1 π (m,1) , where λ 1,i +λ 1,i = η 1 (c.f. (12)) is applied. When i ≥ k 1 − 1, the local balance equation at state (i, 0) can be expressed as π (i−k1+1,0)λ1,i−k1+1 +μ 1,i+1 π (i+1,0) =π (i,0)μ1,i − λ 1,i−k1 π (i−k1,0) + π (i,0) (λ 1,i +λ 1,i ) − µ 2 π (i−k1+1,1) − µ 1 π (i+1,1) = η 1 i m=i−k1+1 π (m,0) + π (0,0) µ 0 − µ 1 i+1 m=1 π (m,1) − µ 2 i−k1+1 m=0 π (m,1) . Then, we can compute the normalized average power consumption as P = Q1 i=0 π (i,0) µ 2 g i + Q1 i=1 π (i,0) µ 1 f i = Q1−1 i=k1−1 (π (i−k1+1,0)λ1,i−k1+1 +μ 1,i+1 π (i+1,0) ) + k1−2 i=0μ 1,i+1 π (i+1,0) − µ 3 π (0,0) − η 2 Q1 i=1 π (i,0) = Q1 i=0 ξ i π (i,0) − Q1 i=0 ζ i π (i,1) . where the second equality is due to the fact thatλ 1,i = µ 3 + µ 2 g i (c.f. (12)) andμ 1,i = µ 0 + µ 1 f i (c.f. (11)), the last equality is obtained via the summation of π (i−k1+1,0)λ1,i−k1+1 + µ 1,i+1 π (i+1,0) andμ 1,i+1 π (i+1,0) over i ≥ k 1 − 1 and i < k 1 − 1, respectively. As a result, we can compute the coefficients ξ i and ζ i as listed in Table I. B. Proof of Lemma 3 We will prove that for each i, the probability π i satisfies the inequality (15) for Cases I, II and III, respectively. 1) In Case I, we have λ 1,i = µ 2 (1−g i ) andμ 1,i = µ 0 +µ 1 f i , which satisfy 0 ≤ λ 1,i ≤ µ 2 , µ 0 ≤μ 1,i ≤ µ 0 + µ 1 =η 1 , since 0 ≤ g i ≤ 1 and 0 ≤ f i ≤ 1. From the local equilibrium equation (24), we have 0 ≤ π i = π (i,0) ≤ φπ (i−1,0) , since 0 ≤ π (i,0) π (i−1,0) = λ1,i−1 µ1,i ≤ µ2 µ0 = φ. In this case, we have Θ u (i,π i ) = φπ (i−1,0) when g i−1 = 0 and f i = 0, and Θ l (i,π i−1 ) = 0 when g i−1 = 1. 2) From Fig. 2(d) in Case II, the local balance equation π (i−1,0) λ 1,i−1 = π (i,0) (µ 1,i + µ 0 ) + (µ 0 + µ 1 ) Q2−1 j=1 π (i,j) + η 1 π (i,Q2) holds for all i > 0, thus leading to π (i−1,0) λ 1,i−1 = π (i,0) (µ 1,i + µ 0 ) +η 1 Q2 j=1 π (i,j) because of µ 0 + µ 1 =η 1 . Hence, we haveη 1 π i =η 1 Q2 j=0 π (i,j) = π (i−1,0) λ 1,i−1 − π (i,0) (µ 1,i + µ 0 ) +η 1 π (i,0) . Since 0 ≤ g i ≤ 1 and 0 ≤ f i ≤ 1, we get 0 ≤ λ 1,i = µ 2 (1 − g i ) ≤ µ 2 and µ 0 ≤ µ 1,i + µ 0 = µ 1 f i +µ 0 ≤η 1 . With the varying parameter {λ 1,i } and {µ 1,i }, we further have 0 ≤ π i ≤ Θ u (i,π i ), where π i = Θ u (i,π i ) = τη 2 π (i−1,0) + π (i,0)η2 when λ 1,i−1 = µ 2 and µ 1,i + µ 0 = µ 0 (g i−1 = f i = 0), and π i = Θ l (i,π i−1 ) = 0 when λ 1,i−1 = 0 and µ 1,i + µ 0 =η 1 (g i−1 = f i = 1), respectively. 3) In Case III, from Fig. 2(e), the local equilibrium equation between states (i − 1, j) and (i, j) (j ∈ Q 2 ) can be expressed as i−1 m=0 π (m,0) (λ 1,m +λ 1,m ) + (µ 2 + µ 3 ) i−1 m=0 Q2−1 j=1 π (m,j) + η 1 i−1 m=0 π (m,Q2) = π (i,0)μ1,i + (µ 0 + µ 1 ) Q2−1 j=1 π (i,j) +η 1 π (i,Q2) for all i < k 1 . With λ 1,i−1 +λ 1,i−1 = µ 2 + µ 3 = η 1 and µ 0 + µ 1 =η 1 , it can be rewritten as η 1 i−1 m=0 π m = π (i,0)μ1,i +η 1 Q2 j=1 π (i,j) . Since µ 0 ≤μ 1,i = µ 0 + µ 1 f i ≤η 1 , we get Θ l (i,π i−1 ) ≤ η 1 Q2 j=0 π (i,j) ≤ Θ u (i,π i ). Thus, π i = Θ u (i,π i ) = µ1 η1 π (i,0) + η1 η1 i−1 m=0 π m whenμ 1,i = µ 0 (f i = 0), and π i = Θ l (i,π i−1 ) = η1 η1 i−1 m=0 π m whenμ 1,i =η 1 (f i = 0), respectively. Similarly, when i ≥ k 1 , the corresponding local equilibrium equation between states (i − 1, j) and (i, j) is given by η 1 i−1 m=i−k1+1 π m = π (i,0)μ1,i − π (i−k1,0) λ 1,i−k1 + η 1 Q2 j=1 π (i,j) . Since 0 ≤ λ 1,i = µ 2 (1 − g i ) ≤ µ 2 and µ 0 ≤μ 1,i = µ 0 + µ 1 f i ≤η 1 , we get η 1 i−1 m=i−k1+1 π m ≤ η 1 Q2 j=0 π (i,j) ≤ µ 2 π (i−k1,0) + µ 1 π (i,0) + η 1 i−1 m=i−k1+1 π m . Thus, π i = Θ u (i,π i ) = µ2 η1 π (i−k1,0) + µ1 η1 π (i,0) + η1 η1 i−1 m=i−k1+1 π m when λ 1,i−k1 = µ 2 andμ 1,i = µ 0 (g i−k1 = 0, f i = 0), π i = Θ l (i,π i−1 ) = η1 η1 i−1 m=0 π m when λ 1,i−k1 = 0 andμ 1,i =η 1 (g i−k1 = 1, f i = 1), respectively. Combining the above two cases: i < k 1 and i ≥ k 1 , we get Θ u (i,π i ) and Θ l (i,π i−1 ) as listed in Table II. C. Proof of Theorem 4 Subject to the constraint (13.b), we haveD = 1 k1η1 Q1 i=1 iπ i ≥ 1 k1η1 Q1 i=1 iΘ l (i,π i−1 ) . This means that the average queuing delayD, as a weighted summation of π i , can be minimized, if each π i chooses its lower bound Θ l (i,π i−1 ) for all i ≥ 1. From Lemma 3, this happens when all the transmission parameters satisfy g i−k1 = f i = 1 for all i > 0. This corresponds to the scheduling policy based on the optimal threshold i * = 0 and the corresponding average power consumption from the RES isp 0 . Hence, when p max ≥p 0 , the minimum average queueing delaȳ D * is obtained if π * i = Θ l (i,π * i−1 ) or π * a l i = 0 for all i = 1, · · · , Q 1 , and the optimal threshold i * is zero. When k 1 η 1 − k 2 η 2 < p max <p 0 , g i−k1 = f i = 1 does not hold for all i > 0 and hence there must exist i * > 0. D. Proof of Theorem 5 Subject to the constraint Q1 i=0 π i = 1, The average queueing delayD = 1 k1η1 Q1 i=1 iπ i can be minimized, if each π i chooses its lower bound Θ l (i,π i−1 ) for all i ≥ 1, corresponding to the case when p max ≥p 0 shown in the proof of Theorem 4. Here, we focus on studying the optimal solution in the case when k 1 η 1 − k 2 η 2 < p max <p 0 . In this case, P = Q1 i=0 ξ i π * (i,0) − Q1 i=0 ζ i ·π * (i,1) = p max is straightforward. This lies in the fact that the data queue length becomes smaller when more reliable energy can be exploited to transmit backlogged data packets. In the sequel, we will show that when k 1 η 1 − k 2 η 2 < p max <p 0 , the average queueing delay can also be minimized, if the optimal solution π * satisfies (16) (π 0 , π 1 , · · · , π i * −1 are assigned to their upper bounds and π i * +1 , · · · , π Q1 are assigned to their lower bounds) for Cases I, II, and III, respectively. 1) Case I: From Lemma 1, we only need to consider the steady-state probabilities π (i,0) and π (0,j) for all i ∈ Q 1 and j ∈ Q 2 . The constraint Q2 j=0 π (i,j) ≥ Θ l (i,π i−1 ) = 0 naturally holds since π (i,j) ≥ 0, and hence we do not consider the corresponding constraints Q2 j=0 π (i,j) ≥ 0 (i > 0). From the Markov chain in Fig. 2(c), the optimal solution π * (0,j) satisfies µ 2 π * (0,j) = µ 0 π * (0,j−1) for j = 1, · · · , Q 2 − 1 and η 1 π * (0,Q2) = µ 0 π * (0,Q2−1) for j = Q 2 . Hence, the probability that the data queue length is zero is obtained as π * 0 = π * (0,0) + π * (0,0) 0) . And the normalized average power consumption from the RES isP = µ 2 π * (0,0) + (µ 2 − µ 0 ) Q1 i=1 π * (i,0) = µ 2 π * (0,0) + (µ 2 − µ 0 )(1 − απ * (0,0) ), from which we obtain π * (0,0) = pmax−(µ2−µ0) µ2−α(µ2−µ0) . Hence, π * (0,0) depends only on p max . As shown in the proof of Theorem 4, when p max ≥p 0 , we have π * 0 = απ * (0,0) = 1 and thus π * (0,0) = 1 α . When p max <p 0 , there must exist π * (0,0) < 1 α and Q1 i=1 π * (i,0) > 0. By contradiction, we will show that the optimal solution π * (i,0) satisfies (16): π * (i,0) = φπ * (i−1,0) for i < i * , 0 < π * (i,0) < φπ * (i−1,0) for i = i * , and π * (i,0) = 0 for i > i * , since it leads to the minimum average queueing delay. Suppose that there exists another set of steady-state probabilities π (i,0) : 0) ). Hence, the corresponding average Q2−1 j=1 φ −j + π * (0,0) φ Q 2 −1 φ1 = απ * (0,0) , where α = Q2−1 i=0 φ −i + φ −(Q2−1) φ −1 1 . Thus, we have Q1 i=1 π * (i,0) = 1 − π * 0 = 1 − απ * (0,π (i,0) = π * (i,0) = π * (0,0) φ i for 0 ≤ i < m, and 0 < π (m,0) < φπ (m−1,0) for m ≤ i ≤ i 1 . Subject to π 0 + i1 i=1 π (i,0) = π * 0 + i * i=1 π * (i,0) = 1, there must exist π (i,0) < π * (i,0) for m ≤ i < i * and i1 i=i * π (i,0) −π * (i * ,0) = i * −1 i=m (π * (i,0) −π (i,0) ) with i 1 ≥ i * . Thus, we have i * ( i1 i=i * π (i,0) − π * (i * ,0) ) = i * i * −1 i=m (π * (i,0) − π (i,queueing delayD = 1 η1 i1 i=1 i · π (i,0) satisfies D = 1 η1 i * i=1 i · π * (i,0) − 1 η1 i * −1 i=1 i · (π * (i,0) − π (i,0) ) + 1 η1 ( i1−1 i=i * iπ (i,0) − i * π * (i * ,0) ) >D * . As a result, we can obtain the minimum average queueing delay when the optimal solution π * satisfies (16). 2) Case II: Similar to Case I, we have Θ l (i,π i−1 ) = 0, as listed in Table II. From the corresponding Markov chain shown in Fig. 2(d), we have π i = Θ l (i,π i−1 ) = 0 and π (i,j) = 0 for all i > m, if π m = Θ l (m,π m−1 ) holds. We first show that π i = Θ l (i,π i−1 ) does not hold for i ≤ i * . If the solution π satisfies 0 < π i ≤ Θ u (i,π i ) for 1 < i < i 1 and π i1 = Θ l (i 1 ,π i1−1 ) for some i 1 ≤ i * , the corresponding power consumption from the RESP will be larger than p max , since it satisfiesP ≥p i1−1 ≥p i * −1 > p max ≥p i * (p m decreases with the threshold m). This violates the constraint (13.a). Hence, the solution π * should satisfy 0 < π * i ≤ Θ u (i,π * i ) for 0 < i ≤ i 1 , and π * i = Θ l (i,π * i−1 ) = 0 for i ≥ i 1 > i * , respectively. Then, we show that among all the candidate solutions, the solution π * satisfying (16) leads to the minimum queueing delay. According to the condition, we have 0 < π i ≤ τη 2 π (i−1,0) + π (i,0)η2 for 0 < i ≤ i 1 . In this case, the solution π * satisfies π * i = Q2 j=0 π * (i,j) = τη 2 π * (i−1,0) +η 2 π * (i,0) for 0 < i < i * and the minimum average queueing delay isD * = i * i=1 iπ * i . Suppose that there is a solution π that satisfies π i = τη 2 π (i−1,0) +η 2 π (i,0) for 0 < i < m, 0 < π m < τη 2 π (m−1,0) +η 2 π (m,0) for some m ≤ i * − 1, and 0 < π i ≤ τη 2 π (i−1,0) +η 2 π (i,0) for m < i ≤ i 1 . Subject to i * i=0 π * i = i1 i=0 π i = 1, the average queueing delayD = 1 η1 i1 i=1 i · π i satisfies D = 1 η1 i * i=1 i · π * i + 1 η1 ( i * −1 i=1 i · (π i − π * i ) +( i1 i=i * iπ i − i * π * i * )) >D * + 1 η1 i * −1 i=0 (i * − i)(π * i − π i ) =D * + 1 η1 i * −1 i=0 (i * − i)η 2 (π * (i,0) − π (i,0) ) + 1 η1 i * −2 i=0 (i * − i − 1)τη 2 (π * (i,0) − π (i,0) ) ≥D * where the first inequality holds since i1 i=i * iπ i − i * π * i * = i * −1 i=0 i * (π * − π i ) + i1 i=i * +1 (i − i * )π i > i * −1 i=0 i * (π * − π i ), the last two inequalities hold since we obtain π 0 >η 2 π (0,0) and i * −1 i=0 π * (i,0) ≥ i * −1 i=0 π (i,0) based on the property of the Markov chain. Hence, the optimal solution π * should satisfy (16) to minimize the average queueing delay. 3) Case III: In this case, we have Θ l (i,π i−1 ) = τ i−1 m=[i−k1+1] + π m > 0. Note that Q 1 should be sufficiently large to avoid buffer overflow. To decrease the average queueing delayD = 1 k1η1 Q1 i=1 iπ i , π i with large index should be assigned its lower bound τ i−1 m=[i−k1+1] + π m . In this sense, there exists an integer i 1 that satisfies π i = τ i−1 m=[i−k1+1] + π m for all i ≥ i 1 . For the same reason as stated in Case II, i 1 > i * should be satisfied to meet the power consumption (from the RES) constraintP ≤ p max . In the same way, we will compare the delay performances between the optimal solution π * satisfying (16) and a candidate solution π. Suppose that π satisfies π i = Θ u (i,π i ) for 0 < i < m, Θ l (i,π i−1 ) < π m < Θ u (i,π i ) for some m ≤ i * −1, and 0 < π i ≤ Θ u (i,π i ) for m < i ≤ i 1 . We notice that {π * i = τ i−1 m=[i−k1+1] + π * m } (i > i * ) is a decreasing sequence (otherwise the data queue will be unstable). So does the sequence {π i } (i > i 1 ). And π * i = Θ u (i,π * i ) increases with the growth of i for 0 ≤ i ≤ i * . Then, subject to the constraint Q1 i=0 π * i = Q1 i=0 π i = 1, we have π * i ≥ π i for 0 ≤ i ≤ i * and π * i ≤ π i for i * < i ≤ Q 1 , and i * i=0 (π * i − π i ) = Q1 i=i * +1 (π i − π * i ). As a result, the average queueing delayD satisfies D = 1 k1η1 Q1 i=1 iπ * i + 1 k1η1 ( i * i=1 i(π i − π * i ) + Q1 i=i * +1 i(π i − π * i )) >D * + 1 k1η1 (− i * i=1 i * (π i − π * i ) + Q1 i=i * +1 i(π i − π * i )) >D * . In this way, we show that the optimal solution π * should satisfy (16) in Case III. Note that the optimal solution π * corresponds to the threshold based scheduling policy. Naturally, a larger threshold m leads to a larger queueing delay. Meanwhile, a lower power p m is consumed from the RES. The optimal threshold can be obtained by comparing the maximum allowable power consumption p max with the power thresholds {p m } as: i * = arg minp m≤pmax m. E. Proof of Corollary 7 From Theorem 5 and Lemma 3, when p max ≥p 0 , π * i = π * (i,0) = 0 for all i > 0. Then, by substituting (18) into the equation π * (0,0) + Q2 j=1 π * (0,j) = 1, we obtain π * (0,0) = 1 α . Accordingly, the power thresholdp 0 = µ 2 π * (0,0) = µ 2 α −1 because of g 0 = 1. Then, we discuss the optimal solution for the case when η 1 − η 2 < p max <p 0 . From Theorem 5 and its proof, we know that there exists an optimal threshold i * > 0 so that π * (i,0) = φπ * (i−1,0) = π * (0,0) φ i for i < i * and π * (i,0) = 0 for i > i * , where π * (0,0) = pmax−(µ2−µ0) µ2−α(µ2−µ0) . Since π * 0 = απ * (0,0) and thus i * i=1 π * (i,0) = 1 − απ * (0,0) , we obtain π * (i * ,0) = 1 − απ * (0,0) − i * −1 i=1 π * (i,0) = 1 − π * (0,0) (α + i * −1 i=1 φ i ). From the property of the optimal solution π * (i,0) , the optimal threshold i * can be evaluated as i * = Ω φ (π * (0,0) , 1 − απ * (0,0) ), which is the integer that satisfies π * (0,0) i * −1 m=1 φ m ≤ 1 − απ * (0,0) < π * (0,0) i * m=1 φ m . F. Proof of Corollary 8 From Theorem 5 and its proof, the optimal threshold is i * = 0 when p max ≥p 0 and hence the optimal solution π * can be obtained by solving (1 + Q 1 )(1 + Q 2 ) independent linear equations: πa u i = 0 (∀i > 0), πP s = 0, and πe = 1. In this case, we get π * = π ′ 0 according to (22). Whenp i * ≤ p max <p i * −1 , the optimal solution π * satisfies π * a 0 = p max , π * a u i = 0 (i = 1, · · · , i * − 1), π * a l i = 0 (i = i * +1, · · · , Q 1 ), πP s = 0, and πe = 1. Hence, we can obtain π * = bA −1 by solving (1 + Q 1 )(1 + Q 2 ) linear equations. G. Proof of Corollary 9 We will discuss the optimal transmission parameters in Cases I, II and III, respectively. 1) Case I: When p max ≥p 0 , from the local equilibrium equation π * (0,0) λ * 1,0 = π * (1,0)μ * 1,1 = 0, we must have λ * 1,0 = µ 2 (1 − g * 0 ) = 0 and g * 0 = 1, since π * (0,0) = α −1 > 0 and π * (i,0) = 0 for all i > 0. Also, since π * (i,0) λ * 1,i = π * (i+1,0)μ * 1,i+1 = 0 always holds for i > 0, we can set g * i = 1 for all i ≥ 0 and f * i = 0 for i > 0. When η 1 − η 2 < p max <p 0 , from (19), π * (i,0) = π * (0,0) φ i for all i < i * . Thus, = φ. On the other hand, 0 ≤ λ * 1,i ≤ µ 2 and µ 0 ≤μ * 1,i ≤η 1 . Hence, we have λ * 1,i = µ 2 andμ * 1,i+1 = µ 0 for i < i * − 1. Substituting λ * 1,i andμ * Fig. 1 . 1System model. 1 . 1* = m. Compute π * = bA −Exitthe solution to the following linear equations the power thresholds {p m }, we can compute the optimal solution π * as follows. Fig. 3 . 3The delay-power curve for Case I. Fig. 5 . 5The delay-power performance for Case III with η 1 = 0.1, η 2 = 0.3, and Q 2 = 5. As we mentioned above, the source node is encouraged to exploit the harvested energy whenever available, and resort to the backup RES when necessary. To this end, the source should always transmit using the energy stored in the battery or newly arriving energy packet when possible, which corresponds to the case q 2 [t − 1] > 0 or a 2 [t] > 0. When the harvested energy is not available, i.e., q 2 [t − 1] = a 2 [t] = 0, the source schedules the transmission of data packets with the RES energy according to the data queue status q 1 [t−1] and the data packet arrival status a 1 [t]. For generality, we define two sets of parameters: {g i } and {f i } in our scheduling scheme. In particular, with q 1 [t − 1] = i, if there is new data packet arrival in this slot, i.e., a 1 [t] > 0, the source node transmits one data packet with probability g i with the RES energy and holds from transmission with probability 1 − g i , respectively; If no new data packet arrives, i.e., a 1 [t] = 0, it transmits with probability f i and holds with probability 1 − f i , respectively. As discussed later, these parameters {g i } and {f i } shall be optimized to achieve the minimum average queueing delay in different cases.According to the proposed scheduling policy, the service process v[t] depends on the queue status q[t − 1] and the arrival process a[t], as described below.For notational convenience, we set q[t] = (q 1 [t], q 2 [t]) to be the buffer status in the time slot t. Similarly, the arrival and service processes can be characterized by the vectors a[t] = (a 1 [t], a 2 [t]) and v[t] = (v 1 [t], v 2 [t]), respectively. B. Stochastic Scheduling Table I THE ICOEFFICIENTS ξ i AND ζ i FOR CASES I, II, AND III. Table III THE IIIOPTIMAL TRANSMISSION PARAMETERS g * i AND f * i FOR CASES I, II, AND III. The subfigureFig.2(a)is intended for the general case of k 1 ≥ 1 and k 2 ≥ 1 (so the dashed lines are used for transitions); but k 1 = k 2 = 2 can be assumed when checking the transition probabilities given in Section III. 1,i+1into(11)gives g * i = 0 for 0 ≤ i ≤ i * − 2 and f * i = 0 for 1 ≤ i ≤ i * − 1.From π * (i−1,0) λ 1,i−1 = π * (i,0)μ 1,i , we can get λ * 1,i * = 0 and g * i * = 1, since π * (i,0) = 0 for all i > i * . Without loss of generality, we can set g i = 1 and f i = 0 for i > i * to maintain consistency. From π * (i * −1,0) λ 1,i * −1 = π * (i * ,0)μ 1,i * , we havewhich is satisfied whenIn this way, we get f * i = 0 for all i > 0 and g * i as listed inTable III. 2) Case II: Similar to Case I, by exploiting the local equi-and π * (i,j) = 0 for all i > i * , we can obtain g * i = 1 (i ≥ 0) and f * i = 0 (i > 0) when p max ≥p 0 , and f * i = 0 and g * i listed inTable IIIwhen η 1 − k 2 η 2 < p max <p 0 , respectively.3) Case III: Similar to the above two cases, when p max ≥ p 0 , we have g * i = f * i+1 = 1 for all i ≥ 0, since i * = 0. When k 1 η 1 − η 2 < p max <p 0 , the local equilibrium equation at the state (i * , 0) is given by π when i ≥ k 1 . When 0 ≤ i * < k 1 , the local equilibrium equation at the state (i * , 0) can be expressed as π * (i * ,0) (µ 0 +j). From the local equilibrium equations, we can compute g * i * −k1 and f * i * for i * ≥ k 1 and i * < k 1 , respectively, as listed inTable III. An environmental energy harvesting framework for sensor networks. 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[]	[ "Landau Singularity \nGeorges GRUNBERG Centre de Physique Théorique de l'Ecole Polytechnique\n91128Palaiseau CedexFrance\n" ]	[ "Georges GRUNBERG Centre de Physique Théorique de l'Ecole Polytechnique\n91128Palaiseau CedexFrance" ]	[]	Standard power-behaved contributions in QCD arising from non-perturbative effects at low scale can be described, as shown by Dokshitzer, Marchesini and Webber, with the notion of an infrared regular effective coupling. In their approach, a non-perturbative contribution to the coupling, essentially restricted to low scales, parametrizes the non-perturbative power corrections. I argue that their framework naturally allows for another type of power contributions, arising from short distances (hence unrelated to renormalons and the operator product expansion) which appear in the process of removing the Landau singularity present in perturbation theory. A natural definition of an infrared finite perturbative coupling is suggested within the dispersive method. Implications for the tau hadronic width, where O(1/Q 2 ) contributions can be generated, are pointed out.	null	[ "https://arxiv.org/pdf/hep-ph/9705460v1.pdf" ]	14,085,532	hep-ph/9705460	6df773b7bb3afaac7d971ca50997a522df18424a	May 1997 May 1997 Landau Singularity Georges GRUNBERG Centre de Physique Théorique de l'Ecole Polytechnique 91128Palaiseau CedexFrance May 1997 May 1997arXiv:hep-ph/9705460v1 29 Talk presented at the Moriond Conference "QCD and High Energy Hadronic Interactions", Les Arcs, France, March 22-29, 1997. † CNRS UPRA 0014 Standard power-behaved contributions in QCD arising from non-perturbative effects at low scale can be described, as shown by Dokshitzer, Marchesini and Webber, with the notion of an infrared regular effective coupling. In their approach, a non-perturbative contribution to the coupling, essentially restricted to low scales, parametrizes the non-perturbative power corrections. I argue that their framework naturally allows for another type of power contributions, arising from short distances (hence unrelated to renormalons and the operator product expansion) which appear in the process of removing the Landau singularity present in perturbation theory. A natural definition of an infrared finite perturbative coupling is suggested within the dispersive method. Implications for the tau hadronic width, where O(1/Q 2 ) contributions can be generated, are pointed out. The study of power corrections in QCD has been the subject of active investigations in recent years. Their importance for a precise determination of α s has been recognized, and various techniques (renormalons, finite gluon mass, dispersive approach) have been devised to deal with situations where the standard operator product expansion (OPE) does not apply. In this talk (which is a summary of 1) ) , I focuss on the dispersive approach 2) , based on the notion of an infrared (IR) regular 3) QCD coupling, where a non-perturbative contribution to the coupling, essentially restricted to low scales, parametrizes the power corrections. I point out that within this framework, it is very natural to expect the existence of new type of power contributions of ultraviolet (UV) origin, hence not controlled by the OPE, related to the removal of the IR Landau singularity presumably present in the perturbative part of the coupling. Consider the contribution to an Euclidean (quark dominated) observable arising from dressed single gluon virtual exchange, which takes the generic form (after subtraction of the Born term): D(Q 2 ) = ∞ 0 dk 2 k 2 α s (k 2 ) ϕ k 2 Q 2(1) The "physical" coupling α s (k 2 ) is assumed to be IR regular, and thus must differ from the perturbative coupling α P T s (k 2 ) by a "power correction" piece δα s (k 2 ). To determine the various types of power contributions, it is appropriate to disentangle long from short distances "a la SVZ" with an IR cutoff Λ I : D(Q 2 ) = ∞ 0 dk 2 k 2 α P T s (k 2 ) ϕ k 2 Q 2 + Λ 2 I 0 dk 2 k 2 δα s (k 2 ) ϕ k 2 Q 2 + ∞ Λ 2 I dk 2 k 2 δα s (k 2 ) ϕ k 2 Q 2 (2) The first integral on the right hand side of eq.(2) may be identified to the Borel sum D P T (Q 2 ) of perturbation theory. The second integral gives "long distance " power corrections which correspond to the standard OPE "condensates" 4) . If the Feynman diagram kernel ϕ k 2 Q 2 is O ((k 2 /Q 2 ) n ) at small k 2 , this piece contributes an O ((Λ 2 /Q 2 ) n ) correction from a dimension n condensate. The last integral in eq.(2) yields at large Q 2 new power contributions of short distance origin , unrelated to the OPE. If the short distance power corrections are neglected 3) (i.e. if one assumes that δα s (k 2 ) is sufficiently small at large k 2 ), one recovers the standard view 5) that the first correction to the Borel sum is given by the OPE. To determine whether this is the case, one needs a closer look at δα s (k 2 ). One may define: δα s (k 2 ) = δα P T s (k 2 ) + δα N P s (k 2 )(3) where δα N P s represents a "physical", "genuinely non-perturbative"component, which one can assume 2) to be restricted to low k 2 : in accordance with the OPE ideology of 4) , it induces an O ((Λ 2 /Q 2 ) n ) power correction of IR origin, consistent with the OPE, parametrized with the low energy moment: K N P n ≡ ∞ 0 n dk 2 k 2 k 2 Λ 2 n δα N P s (k 2 )(4) (where I extended Λ I to infinity, since the integral is dominated by low k 2 ). On the other hand, the δα P T s piece is "unphysical", its role being to remove the Landau pole in α P T s , and has no a priori reason to be restricted to low k 2 ; generically (see below) one expects for large k 2 : δα P T s (k 2 ) ≃ b P T Λ 2 k 2(5) Besides an (ambiguous) O ((Λ 2 /Q 2 ) n ) power correction of IR origin, parametrized with the low energy moment: K P T n ≡ Λ 2 I 0 n dk 2 k 2 k 2 Λ 2 n δα P T s (k 2 )(6) this piece will induce an (unambiguous) short distance O (Λ 2 /Q 2 ) correction, unrelated to the OPE, from the last integral in eq.(2). In particular, the range Q 2 < k 2 < ∞ contributes: ∞ Q 2 dk 2 k 2 δα P T s (k 2 ) ϕ k 2 Q 2 ≃ A b P T Λ 2 Q 2 (7) where A ≡ ∞ Q 2 dk 2 k 2 Q 2 k 2 ϕ k 2 Q 2 is a number. For instance, the simplest "minimal" regularization of the one loop coupling: α P T s,reg (k 2 ) ≡ 1 β 0 ln(k 2 /Λ 2 ) − 1 β 0 1 k 2 Λ 2 − 1 ≡ α P T s (k 2 ) + δα P T s (k 2 )(8) gives b P T = −1/β 0 . This example has the interesting feature that the time-like discontinuity of the regularized coupling coincides with that of the perturbative coupling, and suggests a general ansatz (which has actually been suggested long ago in QED, and has been recently revived 6) in QCD): α P T s,reg (k 2 ) = k 2 ∞ 0 dµ 2 (µ 2 + k 2 ) 2 α P T ef f (µ 2 )(9) where the perturbative "effective coupling" α P T ef f (µ 2 ) is related to the "spectral density" of the perturbative coupling ρ P T (µ 2 ) ≡ − 1 2πi {α P T s [−(µ 2 + iǫ)]−α P T s [−(µ 2 − iǫ)]} by: dα P T ef f d ln µ 2 = ρ P T (µ 2 )(10) The same dispersion relation was assumed in 2) for the total coupling α s = α P T s,reg + δα N P s (11) i.e.: α s (k 2 ) = k 2 ∞ 0 dµ 2 (µ 2 + k 2 ) 2 α ef f (µ 2 )(12) where α ef f (µ 2 ) is related to the discontinuity of α s by eq.(10). Putting: α ef f (µ 2 ) = α P T ef f (µ 2 ) + δα N P ef f (µ 2 )(13) it follows that the "non-perturbative modification" satifies also: δα N P s (k 2 ) = k 2 ∞ 0 dµ 2 (µ 2 + k 2 ) 2 δα N P ef f (µ 2 )(14) Contrary to α P T s (k 2 ), α P T ef f (µ 2 ) in eq.(9) is likely to be IR finite, which explains 7) that α P T s,reg (k 2 ) differs from α P T s (k 2 ) by power corrections. In the one-loop coupling example, one has: α P T ef f (µ 2 ) = 1 πβ 0 π 2 − arctan 1 π ln µ 2 Λ 2(15) which is indeed IR finite 5) : α P T ef f (µ 2 = 0) = 1 β 0 = α P T s,reg (k 2 = 0)(16) Corresponding to the split eq.(11), one can distinguish in D(Q 2 ) a "regularized perturbation theory" piece: D P T reg (Q 2 ) ≡ ∞ 0 dk 2 k 2 α P T s,reg (k 2 ) ϕ k 2 Q 2(17) and a "non-perturbative" power correction piece: δD N P (Q 2 ) = ∞ 0 dk 2 k 2 δα N P s (k 2 ) ϕ k 2 Q 2(18) Note that D P T reg differs from the Borel sum D P T by "perturbative" power corrections: δD P T (Q 2 ) = ∞ 0 dk 2 k 2 δα P T s (k 2 ) ϕ k 2 Q 2(19) and it is an important issue whether it is possible to disentangle these two types of power corrections. For Minkowskian observables, it is necessary to introduce, instead of eq.(1), a representation in term of α ef f : D(Q 2 ) = ∞ 0 dµ 2 µ 2 α ef f (µ 2 )Ḟ µ 2 Q 2(20) where F is the "characteristic function" 2) , i.e. the O(α s ) Feynman diagram computed with a finite gluon mass 5) µ 2 , andḞ ≡ −dF /d ln µ 2 . D P T reg is directly related to α P T ef f by: D P T reg (Q 2 ) = ∞ 0 dµ 2 µ 2 α P T ef f (µ 2 )Ḟ ( µ 2 Q 2 )(21) Concerning the "perturbative " power corrections, I quote the following result for an analytic small µ 2 behavior of F ( µ 2 Q 2 ) (in which case the power correction can be shown to be of short distance origin). If: F ( µ 2 Q 2 ) − F (0) ≃ −d µ 2 Q 2 (µ 2 ≪ Q 2 )(22) then: δD P T (Q 2 ) ≃ b P T d Λ 2 Q 2 (Q 2 ≫ Λ 2 )(23) However, b P T is difficult to calculate, since it depends on the α s beta-function to all orders (similarly to IR renormalons residues). As an application, consider the hadronic width of the τ lepton. It is usually expressed in term of the quantity R τ , itself related to the total e + e − annihilation cross-section into hadrons R e + e − by: R τ (m 2 τ ) = 2 m 2 τ 0 ds m 2 τ 1 − s m 2 τ 2 1 + 2 s m 2 τ R e + e − (s)(24) In the small µ 2 limit, one finds 2) for the corresponding characteristic function: F τ ( µ 2 m 2 τ ) − F τ (0) ≃ −d τ µ 2 m 2 τ(25) with: d τ = 16 3π (4 − 3ζ(3))(26) which implies a leading 1/m 2 τ power correction of UV origin: δR P T τ (m 2 τ ) ≃ b P T d τ Λ 2 m 2 τ(27) For a numerical estimate, assume the "large β 0 " value b P T = −1/β 0 , and take: Λ = Λ V = 2.3Λ M S to be the Landau pole of the "V-scheme" 5) coupling. Then (for 3 flavors): δR P T τ (m 2 τ ) ≃ −0.934 Λ 2 V m 2 τ(28) wich gives, assuming e.g. α M S s (m 2 τ ) = 0.32: δR P T τ (m 2 τ ) ≃ −0.063. One thus gets a sizable correction with respect to the (principal-value) Borel sum estimate 5) (still in the large β 0 limit): R τ (m 2 τ ) − 1 ≃ 0.227 , or to the experimental value: R τ (m 2 τ ) − 1 ≃ 0.20 (R τ is normalized as R τ = 1 + αs π + ...). This result shows that 1/m 2 τ terms could be at the same level as radiative corrections in τ decay (where standard power contributions of IR origin are estimated to be very small!). Note also that a corresponding 1/Q 2 power correction is absent from R P T e + e − (Q 2 ) (for which the leading power correction (of UV origin) is only 2) O(1/Q 4 )). In conclusion, the removal of the Landau singularity is likely to induce power corrections of UV origin (hence unrelated to, thus not inconsistent with, the OPE), which a priori should be of similar size as higher order radiative corrections. An important issue is to assess whether these corrections will modify in a significant way the standard IR power contributions phenomenology. As a first guess, one might expect them to be relevant in processes where many orders of perturbation theory should be taken into account, such as inclusive τ decay, or to handle the "perturbative tail" of the gluon condensate on the lattice. Similar remarks have recently been put forward by R. Akhoury and V.I. Zakharov (hep-ph/9705318). . G Grunberg, hep-ph/9705290G. Grunberg, hep-ph/9705290. . Yu L Dokshitzer, G Marchesini, B R Webber, Nucl. Phys. 46993Yu.L. Dokshitzer, G. Marchesini and B.R. Webber, Nucl. Phys. B469 (1996) 93. . Yu L Dokshitzer, B R Webber, Phys.Lett. 352451Yu.L. Dokshitzer and B.R. Webber, Phys.Lett. B352 (1995) 451. . M A Shifman, A I Vainshtein, V I Zakharov, Nucl. Phys. 147385M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385. . P Ball, M Beneke, V M Braun, Nucl. Phys. 452563P. Ball, M. Beneke and V.M. Braun, Nucl. Phys. B452 (1995) 563. hep-ph/9704333; A.I Alekseev and B.A. Arbuzov, hep-ph/9704228 and references therein. D V Shirkov, I L Solovtsov, D.V. Shirkov and I.L. Solovtsov, hep-ph/9704333; A.I Alekseev and B.A. Arbuzov, hep-ph/9704228 and references therein. . G Grunberg, ibid. hep-ph/9608375Phys.Lett. 372121G. Grunberg, Phys.Lett. B372 (1996) 121; ibid. hep-ph/9608375; . Yu L Dokshitzer, N G Uraltsev, Phys.Lett. 380141Yu.L. Dok- shitzer and N.G. Uraltsev, Phys.Lett. B380 (1996) 141.	[]
[ "Analytical computation of stray light in nested mirror modules for X-ray telescopes", "Analytical computation of stray light in nested mirror modules for X-ray telescopes" ]	[ "Daniele Spiga \nINAF/Brera Astronomical Observatory\nVia Bianchi 4623807MerateItaly\n" ]	[ "INAF/Brera Astronomical Observatory\nVia Bianchi 4623807MerateItaly" ]	[]	Stray light in X-ray telescopes is a well-known issue. Unlike rays focused via a double reflection by usual grazing-incidence geometries such as the Wolter-I, stray rays coming from off-axis sources are reflected only once by either the parabolic or the hyperbolic segment. Although not focused, stray light may represent a major source of background and ghost images especially when observing a field of faint sources in the vicinities of another, more intense, just outside the field of view of the telescope. The stray light problem is faced by mounting a pre-collimator in front of the mirror module, in order to shade a part of the reflective surfaces that may give rise to singly-reflected rays. Studying the expected stray light impact, and consequently designing a pre-collimator, is a typical ray-tracing problem, usually time and computation consuming, especially if we consider that rays propagate throughout a densely nested structure. This in turn requires one to pay attention to all the possible obstructions, increasing the complexity of the simulation. In contrast, approaching the problems of stray light calculation from an analytical viewpoint largely simplifies the problem, and may also ease the task of designing an effective pre-collimator. In this work we expose an analytical formalism that can be used to compute the stray light in a nested optical module in a fast and effective way, accounting for obstruction effects. * More exactly, we should be speaking of "polar angles" because the reflections on the primary and the secondary occur at slightly different values ϕ1 and ϕ2. However, it can be demonstrated 6 that this difference is negligible for our scopes, so we simply denote the nearly-common value of the two polar angles with ϕ.	10.1117/12.2185414	[ "https://arxiv.org/pdf/1609.09655v1.pdf" ]	119,021,254	1609.09655	60ace013bd3912f959ede1c7b9eef959dd089767	Analytical computation of stray light in nested mirror modules for X-ray telescopes 30 Sep 2016 Daniele Spiga INAF/Brera Astronomical Observatory Via Bianchi 4623807MerateItaly Analytical computation of stray light in nested mirror modules for X-ray telescopes 30 Sep 2016X-ray mirror modulesstray lightanalytical Stray light in X-ray telescopes is a well-known issue. Unlike rays focused via a double reflection by usual grazing-incidence geometries such as the Wolter-I, stray rays coming from off-axis sources are reflected only once by either the parabolic or the hyperbolic segment. Although not focused, stray light may represent a major source of background and ghost images especially when observing a field of faint sources in the vicinities of another, more intense, just outside the field of view of the telescope. The stray light problem is faced by mounting a pre-collimator in front of the mirror module, in order to shade a part of the reflective surfaces that may give rise to singly-reflected rays. Studying the expected stray light impact, and consequently designing a pre-collimator, is a typical ray-tracing problem, usually time and computation consuming, especially if we consider that rays propagate throughout a densely nested structure. This in turn requires one to pay attention to all the possible obstructions, increasing the complexity of the simulation. In contrast, approaching the problems of stray light calculation from an analytical viewpoint largely simplifies the problem, and may also ease the task of designing an effective pre-collimator. In this work we expose an analytical formalism that can be used to compute the stray light in a nested optical module in a fast and effective way, accounting for obstruction effects. * More exactly, we should be speaking of "polar angles" because the reflections on the primary and the secondary occur at slightly different values ϕ1 and ϕ2. However, it can be demonstrated 6 that this difference is negligible for our scopes, so we simply denote the nearly-common value of the two polar angles with ϕ. INTRODUCTION Optical modules for X-ray telescopes are usually double reflection systems, like the widespread Wolter-I design. 1 Reflecting X-rays twice halves the focal length and largely suppresses the coma aberration, enabling more compact spacecrafts and larger field of views, at the sole cost of some reduction of the effective area with respect to a single reflection. When a Wolter-I mirror module is illuminated by an on-axis source at infinite distance, all the rays that are reflected by the parabolic segment also impinge onto the hyperbolic segment. But things may change when the source is off-axis or at finite distance: some X-rays can make a single reflection on the parabola, while others can directly impinge on the hyperbola (Fig. 1). In both cases, they can reach the focal plane without being focused and increase the background or generate "ghost" images. These rays, usually referred to as "stray light", are a well-known problem in X-ray astronomy as they can seriously hamper the observation of faint targets by contamination from intense X-ray objects just outside the telescope field of view, like e.g., the Crab nebula or even the Sun if the detectors are not shielded against the visible or the infrared light. 2 When designing X-ray optical modules, the assessment of the stray light contamination is an important step in order to study possible countermeasures. Mirror modules consist of densely nested mirror shells, hence part of stray rays is blocked by the rear (usually non-reflective) side of the inner shells. However, the mirror nesting cannot be too tight, or also doubly-reflected rays from off-axis sources will be obstructed, at the expense of the effective area for imaged sources within the field of view. For this reason, other solutions have been devised out, like X-ray precollimators (see, e.g. 3,4 ) to prevent X-rays from reaching the optical surfaces from directions that may generate stray rays. These auxiliary items are carefully designed, manufactured and aligned to the mirror aperture, 5 but at the same time to minimize the obstruction of the effective area for double reflection. Because of the complexity of the possible paths followed by X-rays (stray or not) throughout an optical module, the design and the performance verification of an X-ray optical module and of the pre-collimator is usually done via ray-tracing. This task can be computationally intense and time consuming, especially because the design performance has to checked until an optimal solution is found. In contrast, approaching the same problem from an analytical viewpoint would be useful. Not only to compute in an easy and fast way the stray light impact on a given X-ray module design, but also to find the optimal configuration for the mirror module and the precollimator without the need of writing complex ray-tracing routines. In previous papers 6, 7 we had already faced the problem of off-axis effective area computation for mirror shells, using analytical expressions. The results could be extended to the case of shells nested in mirror modules, accounting for the mutual obstruction. 8 The effective area, as a function of the off-axis angle and the X-ray wavelength, can be expressed by integral equations, with results in excellent agreement with the findings of ray-tracing. In addition, the numerical integration of the analytical formulae requires a time that is orders of magnitudes lower than ray-tracing, and without being affected by statistical uncertainties. In this paper we show that the same formalism can be easily extended to the computation of the effective area for stray light for an off-axis source. In Sect. 2 we briefly recall the analytical theory of the effective area. 6,8 This will introduce us to some concepts that we can use to write analytical expressions of the effective area for doubly-reflected rays and, in Sect. 3, of the effective area for stray light. The expressions for the stray light off the primary and the secondary segment are different and, for brevity, we refer to the former as "primary stray light" and to the latter as "secondary stray light". In Sect. 4 we show examples of computation, validating the results by comparison with the findings of a ray-tracing routine. In Sect. 5 we solve the integral equations for the ideal case of a mirror with 100% reflectivity, obtaining algebraic expressions for the stray light geometric area, in a completely similar way as we did for the double-reflection area. 6,8 Results are briefly summarized in Sect. 6. We explicitly remark that the formalism for the focused effective area and for stray light are both developed in double cone approximation. As discussed in detail, 6 we are allowed to do this owing to the shallow incidence angles. While the longitudinal curvature of the Wolter-I profile is crucial to concentrate X-ray to a focus, this affects the effective area only to a small extent. For example, in a double cone geometry the incidence angle for a infinitely distant source on-axis is a constant, α 0 , throughout the entire surface of the primary and the secondary segments. In a real Wolter-I profile, they exhibit a small variation ∆α, related to the curvature of the profile. Fortunately, in grazing incidence geometry it is possible to prove that ∆α/α 0 ≈ L/4f , where L is the length of the single segment of the mirror and f its focal length. In practice, the error we make assuming a constant incidence angle is below a few percent in real cases. It can also be proven 6 that also the estimation of factors affecting the effective area (e.g., the vignetting) are still on the order of the L/f ratio; therefore, the double cone approximation can be safely applied in the effective area computation. In this work we assume that the double cone approximation can be applied to the stray light theory within a relative error of L/2f , which in worst cases amounts to a few percent. Finally, for simplicity we assume the shells to be continuous at the intersection plane. The theory is easy to extend to the case of primary and secondary segments separated by a gap, but we are not reporting it here, in order to avoid a complication of the expressions. THE OFF-AXIS EFFECTIVE AREA OF AN X-RAY MIRROR SHELL We now consider a pair of nested, integral mirror shells, with optical axis parallel to the z-axis (see Fig. 2, A). We limit ourselves to the case of mirror shells having all the same primary/secondary intersection plane, and we define it to be the xy plane (therefore, the theory is not directly applicable to a Wolter-Schwarzschild configuration). When the shells are illuminated by a distant X-ray source, the inner shell may cause an obstruction of rays focused by the outer one, and so reduce its effective area. We therefore refer to the outer shell as "reflective" and to the inner shell as "obstructing". Doing this, we implicitly assume that the reflective shell can be uniquely obstructed by the next shell with smaller diameter in the mirror module. For generality, however, we assume the primary and the secondary segments to have different lengths along the optical axis. We denote these lengths as L 1 , L 2 for the reflective shell, and L * 1 , L * 2 for the obstructing shell. If L 1 = L 2 or L * 1 = L * 2 , then we denote their common value with L and L * , respectively. The radii at the primary-secondary intersection plane are R 0 for the inner side of the reflective shell and R * 0 for the outer side of the obstructing shell. The respective radii at the entrance pupil are denoted as R M and R * M , while the radii at the exit pupil are R m and R * m . Finally, be α 0 and α * 0 the respective on-axis incidence angles on the primary mirror, at the intersection plane ( Fig. 2, B). Should rays impinge from the source on the secondary segment (and this is really unwanted since it is one of the possible sources of stray light), the incidence angles are 3α 0 and 3α * 0 . We assume the source to be in the xz plane, on the side of positive x's, at z = D (either positive or negative, but always \|D\| ≫ L and \|D\| ≫ f ). Finally, we denote with δ = R 0 /D the beam half-divergence seen by the reflective shell. In this section we recall the expressions for the double-reflection effective area of the reflective mirror shell, seen by the source off-axis by a small angle θ > 0, at the X-ray wavelength λ, and accounting for the obscuration by the obstructing shell. 8 We denote it with A D (λ, θ), with the special case of a source at infinity, A ∞ (λ, θ). When we refer to the geometric area (i.e., assuming a coating reflectivity r λ (α) = 1), we simply write A D (θ) (or A ∞ (θ) for the astronomical case). We will see in Sect. 3 that the theory can be easily extended to the computation of a similar effective area for the stray light, i.e., for singly-reflected rays from an off-axis source. This in turn allows one to compute easily the amount of stray light contamination in the field of view of the X-ray telescope if the intensity of the source is known. We follow the aforementioned notation to denote the effective area for stray light, only adding the "SL" superscript. Vignetting coefficients We henceforth recall some useful quantities that enter the computations of the mirror shell effective area. 6,8 The key concept is the vignetting coefficient V , meant as the fraction of a mirror length left clear by a given factor of obstruction. When we refer to the blocked length fraction for the same reason, we denote it as obstruction coefficient (i.e., 1 − V ). 1. A focused ray undergoes two reflections in sequence: the first one on the primary mirror at the grazing angle α 1 and the second one on the secondary at the grazing angle α 2 . In the assumed double-cone approximation, if the source is on-axis we have α 1 = α 2 = α 0 throughout the entire mirror surface. If the source is off-axis by θ, the incidence angles vary with the polar angle ϕ, measured in the xy plane from the x axis * : α 1 (ϕ) = α 0 + δ − θ cos ϕ,(1)α 2 (ϕ) = α 0 − δ + θ cos ϕ.(2) We notice that α 1 + α 2 = 2α 0 and that the behavior of α 1 and α 2 is symmetric with respect to the x-axis. The expressions are valid as far as they are non-negative. 2. We define V , the double reflection vignetting coefficient at the ϕ polar angle as the fraction of primary mirror length from which reflected rays impinge onto the secondary segment. A detailed analysis 6 shows that the V coefficient has the expression V (ϕ) = L 2 α 2 L 1 α 1 .(3) The part of primary mirror that contributes to the double reflection is located on the side of the intersection plane ( Fig. 1). Equation 3 is valid for non-negative α 1 and α 2 , otherwise it should be set to zero. Moreover, V = 1 if L 2 α 2 > L 1 α 1 . 3. The focused beam can be obstructed in three different ways, shown in Fig. 2, A. Obstruction occurs when the incidence angles exceed three angles that characterize the spacing left between the reflective and the obstructing shell, the vignetting parameters (Fig. 2, B): Φ = R 0 − R * M L * 1 + α 0 , Ψ = R 0 − R * 0 L 1 , Σ = R 0 − R * m L * 2 − 3α 0 .(4)If L * 1 = L 1 = L * 2 and f ≫ L 1 , then Φ ≈ Ψ ≈ Σ. 4. The first kind of obstruction occurs when rays are blocked before they can make the first reflection: the primary segment of the obstructing shell casts a shadow on the reflective shell near the intersection plane ( Fig. 2, A). The residual fraction of illuminated primary mirror at ϕ is located on the side of the entrance pupil and expressed by the V 1 coefficient: V 1 (ϕ) = 1 + L * 1 (Φ − α 1 ) L 1 α 1 .(5) The obstruction of this species is always maximum at ϕ ≈ π, where α 1 is larger. 5. The second kind of obstruction may occur after the first reflection, if rays are blocked by the rear side of the obstructing shell at z > 0. Just like V , the fraction of primary segment that is left clear is on the side of the intersection plane and expressed by the V 2 coefficient, V 2 (ϕ) = Ψ α 1 .(6) Also in this case, the obstruction becomes more severe for large values of α 1 . Fortunately, mirror modules can be designed to be completely unaffected by this kind of obstruction 8 at any value of θ. 6. The third kind of obstruction may occur after the second reflection, when the ray impacts on the rear side of the secondary segment of the obstructing shell. In this case, the corresponding fraction of primary mirror V 3 that is not obscured is on the side of the entrance pupil (just like V 1 ) and provided by the expression V 3 (ϕ) = 1 + L * 2 (Σ − α 2 ) L 1 α 1 .(7) This kind of obstruction, unlike the others, becomes important near ϕ = 0, where α 1 is shallower and α 2 is larger. The explicit dependence of V , V 1 , V 2 , and V 3 on ϕ is obtained substituting the expressions of α 1 and α 2 (Eqs. 1 and 2), always with the constraint that 0 ≤ V j ≤ 1, and 0 or 1 outside this range. Hence, there is no obstruction of the first kind whenever α 1 < Φ, of the second kind if α 1 < Ψ, and of the third kind if α 2 < Σ. Expression for the double-reflection effective area Using the vignetting coefficient expressions in Sect. 2.1, the general expression for the double-reflection effective area of an obstructed mirror shell can be derived easily (refer to 8 for the derivation, with a slight change of notation): A D (λ, θ) = 2R 0 π 0 [(Lα) min − (Lα) max ] ≥0 r λ (α 1 )r λ (α 2 ) dϕ,(8) where we have set (Lα) min = min (L 1 α 1 , L 2 α 2 , L 1 Ψ) ,(9)(Lα) max = max [L * 1 (α 1 − Φ), L * 2 (α 2 − Σ), 0] .(10) The focused effective area is therefore completely expressed in terms of the vignetting parameters and the incidence angles, which in turn are a function of ϕ. The brackets [ ] ≥0 mean that the enclosed expression has to be nonnegative, otherwise it is set to zero (as it is when either α 1 < 0 or α 2 < 0). For this reason, the integration of Eq. 8 cannot be carried out separately for the two terms. In the previous formula, r λ (α 1 ), r λ (α 2 ) are the reflectivities of the mirror coating at the X-ray wavelength λ on the primary and the secondary segment, respectively. Since the r λ (α)'s functions are difficult to express in an algebraic form, Eq. 8 is seldom integrated analytically. In contrast, an explicit integration is possible for the geometric area, setting r λ (α) = 1 for any value of λ and α. Of course, Eq. 8 can always be integrated numerically. α 0 = 0 . 2 0 d e g θ = 0 . 1 5 d e g Φ = 0 . 2 5 d e g δ = 0 d e g V 1 R M blocked before the first reflection missed the secondary after the first reflection focused, after double reflection blocked after the second reflection blocked after the first reflection In Eq. 8, the integration is extended to the entire range of polar angles, as we reasonably suppose that the focal spot is entirely included in the detector area. Actually, the integration range is reduced to [0, π] -adding a factor of 2 -because the integrand is symmetric with respect to the x-axis. The effective area of the entire module is simply obtained by summing the contributions of pairs of adjacent shells, playing in sequence the role of reflective and obstructing shells. We have hitherto neglected the obstruction of supporting structures ("spiders"), which are, indeed, always present. To preserve the mirror stiffness, the spiders spokes usually have a thickness that increases in proportion with R 0 , therefore the set of polar angles P they occupy is the same for all the shells in the module. In order to account for the effective area loss, one just has to compute the integral in Eq. 11 zeroing the integrand at the ϕ values occupied by the spider spokes. More formally, we introduce χ P (ϕ), the characteristic function of the P set, and Eq. 8 becomes: V 2 V V 3 R M A D (λ, θ) = 2R 0 π 0 [(Lα) min − (Lα) max ] ≥0 r λ (α 1 )r λ (α 2 ) χ P dϕ,(11) where the [ ] ≥0 brackets have the same meaning as in Eq. 8. ANALYTICAL FORMALISM FOR STRAY LIGHT General expressions Primary stray light From the definition of V (Eq. 3), the fraction of primary that does not make the second reflection at the polar angle ϕ is 1 − V . This means that the infinitesimal geometric area for stray light between ϕ and ϕ + ∆ϕ is L 1 α 1 (1 − V )R 0 ∆ϕ. Therefore, in absence of obstructions, the total effective area element for primary stray light would be R 0 L 1 α 1 (1 − V )r λ (α 1 ) ∆ϕ where r λ (α 1 ) is the primary mirror reflectivity. Indeed, at sufficiently large angles the stray light starts to be blocked at the intersection plane by the rear side of the obstructing shell, i.e., V 2 (Eq. 6). A typical situation is depicted in Fig. 3: the primary mirror segment region that generates the primary stray light (green dots) is the one on the side of the entrance pupil, mostly in the neighborhoods of ϕ ≈ π. The obstruction of second kind, however, concerns exactly the same region (orange dots) and this is the reason why it is mostly harmless for double reflection. Also the obstruction of the first species (yellow dots) contributes to block the primary stray light, and the relative vignetting is described by V 1 (Eq. 5). As V 3 (Eq. 7) describes the vignetting after the second reflection, it is not directly applicable to this case. Nevertheless, also singly-reflected rays can strike on the rear side of the secondary segment of the blocking shell. This case was not considered in the original paper 8 because the obstruction at the exit pupil was therein considered only for doubly-reflected rays. However, the computation can be easily adapted to the primary stray light case. Following passages similar to those in Appendix B.3 in, 8 which we omit here, we obtain a modified form of V 3 : V 3 (ϕ) = L 2 (Σ + α 2 ) L 1 α 1 .(12) Just like the second one, this kind of obstruction occurs near the entrance pupil (Fig. 4, A). Summarizing, the non-obstructed length of the primary mirror at ϕ is the least between V 2 and V 3 . From this term, we have to subtract the maximum between the fraction of doubly-reflected rays, V , and the obstruction at the intersection plane, (1 − V 1 ), to obtain the total vignetting: V tot,1 = min(1, V 2 , V 3 ) − max(V, 1 − V 1 , 0),(13) where we have added a "1" in the first term and a "0" in the second term to avoid negative obstructions. Replacing now Eqs. 3, 5, 6, and 12 into the previous expression, we have V tot,1 = min 1, Ψ α 1 , L * 2 (Σ + α 2 ) L 1 α 1 − max L 2 α 2 L 1 α 1 , L * 1 (α 1 − Φ) L 1 α 1 , 0 .(14) Since the effective area element between ϕ and ϕ + ∆ϕ is V tot,1 L 1 α 1 R 0 ∆ϕ r λ (α 1 ), we finally obtain the general expression of the effective area for the primary stray light: A SL,1 D (λ, θ) = 2R 0 π 0 [(Lα) min,1 − (Lα) max,1 ] ≥0 r λ (α 1 ) χ P dϕ,(15) where the superscripts indicate primary stray light, we have included the shading of structures via the characteristic function χ P (ϕ) (Sect. 2.2), and we have set (Lα) min,1 = min[L 1 α 1 , L 1 Ψ, L * 2 (Σ + α 2 )],(16)(Lα) max,1 = max[L 2 α 2 , L * 1 (α 1 − Φ), 0],(17) provided that α 1 ≥ 0. The condition α 2 ≥ 0 is not necessary because there is no true second reflection, and is anyway fulfilled by the "0" in Eq. 17. The [ ] ≥0 brackets have exactly the same meaning as in Eq. 11. Inspection of Eqs. 16 and 17 shows that a reduction of the stray light from primary mirror of given length and incidence angle can be achieved by reducing (Lα) min , e.g., diminishing Ψ (e.g., via a denser nesting) or Σ. This can be obtained also by an X-ray baffle located at either the intersection plane or the exit pupil. 3 Another method consists of increasing (Lα) max , for example adopting a design with L 2 > L 1 that has also the desired effect of increasing the off-axis effective area for focused rays. In contrast, a reduction of Φ has a minor effect on the primary stray light, since it would chiefly obstruct doubly-reflected rays before blocking the primary stray light (see Fig. 3). This is the reason why X-ray baffles at the entrance pupil are not very effective at suppressing primary stray light. 3 Secondary stray light The treatment of the secondary stray light, i.e., light directly impinging onto the secondary segment, can be derived using exactly the same arguments we used to obtain Eq. 15. The incidence angle, β 2 , clearly differs from Eq. 1 as the mirror slope is 3 times as large: β 2 = 3α 0 + δ − θ cos ϕ.(18) No second reflection is obviously possible for this species of stray light, but for reasons that will be explained later we also define β 1 = θ cos ϕ − α 0 − δ :(19) this angle represents the incidence angle on the rear side of the primary segment of the reflective shell. In this case the concept of vignetting for double reflection cannot be applied; hence the usual expression of the V coefficient (Eq. 3) is meaningless here. There is, however, the obstruction by the upper edge of the primary segment of the reflective shell, which starts to occur only when the incidence angle on the primary segment becomes negative and shades the regions near the intersection plane: developing calculations similar to those in the Appendix A of the already cited work, 6 we obtain a vignetting coefficient very similar to the usual expression for V : V = 1 − L 1 β 1 L 2 β 2 ,(20) and so we keep denoting it with V . In addition, there is the usual obscuration by the upper edge of the obstructing shell, which shades the mirror length near the exit pupil (V 1 ). There is still the obstruction on the same side by the mirror kink at the intersection plane (V 2 ), which however has to be modified to account for blocking the direct illumination, not the reflected one. Finally, there is also the obstruction after the reflection, chiefly involving rays reflected near the intersection plane (V 3 ). The vignetting coefficients can be derived using the same method reported in the Appendix B of a previous paper, 8 and -omitting the proofs -the resulting expressions are V 1 = L * 1 (Φ − α 1 ) L 2 β 2 ,(21)V 2 = L 1 Ψ L 2 β 2 ,(22)V 3 = 1 + L * 2 (Σ − β 2 ) L 2 β 2 .(23) We note that in Eq. 21 the α 1 angle appears instead of β 2 to account for the direct incidence onto a surface with slope 3α 0 . We now have for the total vignetting V tot,2 = min(1, V 1 , V 2 ) − max(1 − V, 1 − V 3 , 0),(24) which becomes, after replacing the corresponding expressions, V tot,2 = min 1, L * 1 (Φ − α 1 ) L 2 β 2 , L 1 Ψ L 2 β 2 − max L 1 β 1 L 2 β 2 , L * 2 (β 2 − Σ) L 2 β 2 , 0 .(25) Therefore, the general expression of the effective area for the secondary stray light is A SL,2 D (λ, θ) = 2R 0 π 0 [(Lβ) min,2 − (Lβ) max,2 ] ≥0 r λ (β 2 ) χ P dϕ,(26) where the superscripts, the subscript, and the χ P (ϕ) function have the same meaning as in Eq. 15, and where we have set (Lβ) min,2 = min[L 2 β 2 , L 1 Ψ, L * 1 (Φ − α 1 )],(27)(Lβ) max,2 = max[L 1 β 1 , L * 2 (β 2 − Σ), 0].(28) provided that, as usual, β 2 ≥ 0. We therefore see that the effective area for the double reflection, for primary stray light, and for the secondary one are provided all by the same formula, only differing by the definition of the terms appearing in there. We also notice that the secondary one can be effectively shaded out by reducing Ψ (denser nesting, baffle at the intersection plane) or Φ (baffle at the entrance pupil). Stray light within the detector field As indicated by the integration range [0, π], Eqs. 15 and 26 return the total stray light, i.e., reflected by the complete mirror shell, albeit limited by the obstruction, and detected over an infinitely extended focal plane. For the focused beam, this is reasonable because the detector can always be supposed to include the complete focal spot. In contrast -and fortunately -usually only a part of the stray light from an off-axis source enters the detector, assumed to be a square area of side d, located at z = −f ′ = −R 0 /(4α 0 − δ), and centered on the focal spot at x = −θf ′ . The effective area within the detector area is the quantity that matters in order to evaluate the effective amount of background generated by an X-ray source at the off-axis angle θ. The stray light pattern can be described analytically 8 using parametric equations as a function of the polar angle ϕ: x n (ϕ) = [R 0 − (2(2n − 1)α 0 + δ)f ′ ] cos ϕ + θf ′ cos 2ϕ (29) y n (ϕ) = [R 0 − (2(2n − 1)α 0 + δ)f ′ ] sin ϕ + θf ′ sin 2ϕ (30) where n = 1 for the primary (Fig. 5) and n = 2 for the secondary stray light respectively. The condition to fulfill is min [\|x n + θf ′ \|, \|y n \|] ≤ d 2 ,(31) and solving Eq. 31 for ϕ, one obtains the interval of polar angles [ϕ n,− , ϕ n,+ ] that contribute to the stray light inside the detector, with the constraints ϕ n,− ≥ 0 and ϕ n,+ ≤ π. The effective area for stray light is computed from Eqs. 15 and 26, with modified integration limits: A SL,1 D (λ, θ) = 2R 0 ϕ1,+ ϕ1,− [(Lα) min,1 − (Lα) max,1 ] ≥0 r λ (α 1 ) χ P dϕ,(32)A SL,2 D (λ, θ) = 2R 0 ϕ2,+ ϕ2,− [(Lβ) min,2 − (Lβ) max,2 ] ≥0 r λ (β 2 ) χ P dϕ,(33) whilst the integrand expressions remain unchanged, and the vignetting by the spider is still included via the χ P (ϕ) function. EXAMPLES AND RESULT VALIDATIONS VIA RAY-TRACING In this section we provide some examples of stray light effective area computation. We have implemented Eqs. 11, 32 and 33 into an IDL code, and we have checked the correctness of results by comparison with the findings of a detailed ray-tracing routine. As a first example, we have computed the stray light effective area for the JET-X module 9 (f = 3.5 m), 25 arcmin off-axis. The computation was referred to the total effective area, i.e., as detected over an infinitely extended focal plane. A 10% of the aperture was assumed to be obstructed by the spider, in 12 equally-spaced spokes. The results of the analytical calculation and of the ray-tracing are shown in Fig. 6,A and are in very good accord. The analytical formulae overestimate the ray-tracing by only some percent, in agreement with our discussion in Sect. 1. However, the application of the analytical formulae required only 15 min of computation, while more than 7 hours were needed to trace 5×10 5 rays (a number needed to reduce the statistical uncertainty) on the same IDL platform, run by a computer equipped with a commercial 2.4 GHz processor. Also, the code written to implement the analytical formulae is 8 times shorter than the code used for tracing rays. We also show in Fig. 6,B the origin points of the rays traced. Black locations are where the rays were stopped before the first reflection or absorbed at reflection; green zones are the locations where the primary stray light was originated, while blue color marks the origin locations for secondary stray light. Primary stray light can be obstructed on the rear side of the inner shell after reflection: on the primary segment (orange) or the secondary segment (purple), or be reflected twice (red), possibly obstructed after the second reflection (light blue). The origin points of secondary stray light obstructed by the secondary segment of the inner shell are marked in yellow. The distribution of colors is in agreement with the quantitative description of vignetting given in Sect. 3, and also proves that all species of obstructions are accounted for in the analytical computation, thereby validating the results. As a second example, we apply Eqs. 11, 32 and 33 to the NHXM hard X-ray optical module (f = 10 m), 10 arcmin off-axis. We have assumed a simplified coating made of 30 nm of Iridium plus 10 nm of amorphous Carbon. To furthermore simplify the visualization of the entrance pupil, we have limited the computation to the innermost 20 shells out of the 70 of the baseline design. 10 The result of the stray light effective area computation is shown in Fig. 6, C. Also in this case we have derived the total effective area without limitations concerning the detector size, and the agreement with the ray tracing is still very good. In Fig. 6, D we display the aperture pupil with colors representing the outcomes of the 5×10 5 traced rays. l y t i c a l c o m p u t a t i o n : F o c u s e d P r i m a r y s t r a y l i g h t S e c o n d a r y s t r a y l i g h t R a y -t r a c i n g : F o c u s e d P r i m a r y s t r a y l i g h t S e c o n d a r y s t r a y l i g h t J E T -X o p t i c a l m o d u l e , 2 5 a r c m i n o f f -a x i s , s o u r c e a t i n f i n i t y T o t a l e f f e c t i v e a r e Figure 6. A) Stray light simulations for the optical module of JET-X, 9 25 arcmin off-axis, total effective area. The computation using the analytical method (lines) and the ray-tracing (symbols) return the same results to within a few percent. B) Rays at the entrance pupil in colors depending on their destinations (see also Fig. 3). The meaning of colors is explained in the text. C) Stray light and focused effective area simulations for 20 innermost shells of the NHXM optical module, 10 10 arcmin off-axis, total effective area. Also in this case the analytical method (lines) and the ray-tracing (symbols) return the same results. D) The same as B, for the optical module simulated in C. Fig. 7. We have again considered the 20 innermost shells of the NHXM mirror module, but this time we have limited the computation to the effective area included in a detector of 15 mm width. The interval [ϕ n,− , ϕ n,+ ] involved varies from shell to shell. The result is -once again -in very good agreement with the ray tracing findings. We notice that now A SL,1 ≈ A SL,2 , as expected from the superposition of the primary and secondary stray light patterns from a source at infinite distance. i n n e r s h e l l s o f N H X M , 1 0 a r c m i n o f f -a x i s , s o u r c e a t i n f i n i t y E f f e c t i v e a r e a i n a 1 5 m m x 1 5 m m d e t e c t o r A n a l y t i c a l c o m p u t a t i o Figure 7. A) stray light and focused effective area simulations for 20 innermost shells of the NHXM optical module, 10 10 arcmin off-axis. Only the stray light ending into a 15 mm × 15 mm detector area was considered. The vertical scale is the same as in Fig. 6, C: the effective area for double reflection is unchanged, but the stray light is much lower. The ray-tracing predictions (symbols) are also in this case in good agreement with the analytical results (lines). B) The traced focal spot (center) and the stray light pattern (the big loop) in the detector area. The third and last example is shown in APPLICATIONS TO THE GEOMETRIC AREA FOR STRAY LIGHT We hereafter deal with some applications of the formalism exposed in Sect. 3 in the ideal case that r λ (α) = 1 (or, at least, a constant) for all α. To simplify the notation, we assume that D → ∞, that L 1 = L 2 = L * 1 = L * 2 , and that we can approximate Φ ≈ Ψ ≈ Σ (Sect. 2.1): we therefore adopt Φ as unique obstruction parameter. Moreover, we do not account for limitations by the detector size, for obstruction of the supporting structures, and, since mirror modules are usually designed as obstruction-free on-axis, we assume 8 that α 0 < Φ. Geometric area, primary stray light With the mentioned approximations, we can rewrite Eq. 15 as normalized to the on-axis geometric area for double reflection, A ∞ (0) = 2πR 0 Lα 0 : A SL,1 ∞ (θ) A ∞ (0) = 1 πα 0 π 0 [min (α 1 , Φ, Φ + 2α 0 − α 1 ) − max (2α 0 − α 1 , α 1 − Φ, 0)] ≥0 dϕ.(34) In order to explicitly solve Eq. 34, we distinguish between two main cases in the following paragraphs. 5.1.1 SL1, tight nesting: Φ/2 < α 0 In addition to α 0 < Φ, we initially assume that Φ/2 < α 0 , which implies that 0 < Φ − α 0 < Φ/2 < α 0 < Φ.(35) We can therefore distinguish 5 different angular regimes: • 0 < θ < Φ − α 0 : in this angular range, α 1 < Φ everywhere. Since θ < Φ/2, we also have that α 1 < Φ + 2α 0 − α 1 and 2α 0 − α 1 > α 1 − Φ; hence, there is no obstruction, so Eq. 34 takes the simple form: A SL,1 ∞ (θ) A ∞ (0) = 2 πα 0 π π/2 (α 1 − α 0 ) dϕ = 2θ πα 0 ,(36) where the lower integration limit is modified to keep the integrand non-negative. In this range of off-axis angles, Eq. 34 correctly returns the complement to the unity of the normalized, geometric, unobstructed area for double reflection. 1,6 • Φ−α 0 < θ < Φ/2: this time we have Φ < α 1 , i.e., obstruction of the second species for cos ϕ < −(Φ−α 0 )/θ, and Eq. 34 becomes A SL,1 ∞ (θ) A ∞ (0) = 2 πα 0 π−arccos[(Φ−α0)/θ] π 2 (α 1 − α 0 ) dϕ + 1 2 π π−arccos[(Φ−α0)/θ] (Φ − 2α 0 + α 1 ) dϕ ,(37) where the integrands are always non-negative because cos ϕ < 0. Equation 37 can be solved as A SL,1 ∞ (θ) A ∞ (0) = 2θ πα 0 1 − 1 2 S Φ − α 0 θ ,(38) where we have introduced the non-negative S function (already defined in 8 ) S(x) = 1 − x 2 − x arccos x, with 0 ≤ x ≤ 1.(39) • Φ/2 < θ < α 0 : we still have Φ < Φ + 2α 0 − α 1 everywhere. However, for cos ϕ < −Φ/2θ we also have α 1 − Φ > 2α 0 − α 1 and we begin experiencing obstruction of the first kind. Equation 34 now turns into A SL,1 ∞ (θ) A ∞ (0) = 2 πα 0 π−arccos[(Φ−α0)/θ] π 2 (α 1 − α 0 ) dϕ + 1 2 π−arccos(Φ/2θ) π−arccos[(Φ−α0)/θ] (Φ − 2α 0 + α 1 ) dϕ + 1 2 π π−arccos(Φ/2θ) (2Φ − α 1 ) dϕ ,(40) and, after a few passages, the expression can be written in an explicit form: A SL,1 ∞ (θ) A ∞ (0) = 2θ πα 0 1 − 1 2 S Φ − α 0 θ − S Φ 2θ .(41) • α 0 < θ < Φ: for cos ϕ < −α 0 /θ the first term of the integrand in Eq. 34 becomes Φ + 2α 0 − α 1 . This denotes the appearance of the obstruction of the third kind: the final expression for these values of θ is A SL,1 ∞ (θ) A ∞ (0) = 2θ πα 0 1 − 1 2 S Φ − α 0 θ − S Φ 2θ − 1 2 S α 0 θ .(42) • θ > Φ: the expression of the integrand remains unchanged, but we have to modify the upper integration limit to avoid negative contributions to the integral, and the final result is: A SL,1 ∞ (θ) A ∞ (0) = 2θ πα 0 1 − 1 2 S Φ − α 0 θ − S Φ 2θ − 1 2 S α 0 θ + S Φ θ .(43) 5.1.2 SL1, loose nesting: α 0 < Φ/2 We now admit that α 0 < Φ/2, which implies Φ/2 < Φ − α 0 < Φ. We have the following results in different ranges of the off-axis angle: • 0 < θ < Φ/2: as this implies θ < Φ/2 < Φ − α 0 , we have α 1 < Φ everywhere. Moreover, 2α 0 + Φ > 2α 1 , and then the result is equal to Eq. 36. • Φ/2 < θ < Φ − α 0 : in this case we still have α 1 < Φ everywhere, but there is some interval of ϕ <∼ π where Φ < −2θ cos ϕ; therefore, in this range of ϕ we start to have some stray light obstruction of the 1-st and the 3-rd kind. Equation 34 becomes and since θ < Φ − α 0 < Φ, we second integrand is always non-negative. Using the expression of the S function (Eq. 39), the last expression can be rewritten as A SL,1 ∞ (θ) A ∞ (0) = 2 πα 0 π−arccos(Φ/2θ) π 2 (α 1 − α 0 ) dϕ + π π−arccos(Φ/2θ) (Φ + α 0 − α 1 ) dϕ ,(44)A SL,1 ∞ (θ) A ∞ (0) = 2θ πα 0 1 − 2S Φ 2θ .(45) • Φ − α 0 < θ < Φ: in this case we potentially have an interval of polar angles affected by vignetting of the second kind. Indeed, this occurs only if Φ < Φ + 2α 0 − α 1 , i.e., only if cos ϕ > −α 0 /θ. On the other hand, this would also require Φ < α 1 , which is equivalent to cos ϕ < −(Φ − α 0 )/θ. Fulfilling the two conditions at the same time would require −α 0 < −(Φ − α 0 ), which is impossible since we initially supposed that α 0 < Φ/2. We conclude that in this interval of off-axis angles the geometric area is still described by Eq. 45. • θ > Φ: while the form of the integrand remains the same of Eq. 44, the second integrand has to be set to zero for cos ϕ < −Φ/θ to avoid negative values, i.e., A SL,1 ∞ (θ) A ∞ (0) = 2 πα 0 π−arccos(Φ/2θ) π 2 (−θ cos ϕ) dϕ + π−arccos(Φ/θ) π−arccos(Φ/2θ) (Φ + θ cos ϕ) dϕ ,(46) which can be easily written as A SL,1 ∞ (θ) A ∞ (0) = 2θ πα 0 1 − 2S Φ 2θ + S Φ θ .(47) We notice that Eq. 38 and 41 are no longer valid when α 0 = Φ/2, since their domain collapse to the single point θ = Φ/2. Moreover, when α 0 = Φ/2, Eqs. 42 and 43 correctly merge with Eqs. 45 and 47, respectively. We show in Fig. 8 some examples of normalized geometric area curves, computed using Eqs. 36 to 47 in the respective θ domains. The geometric area profiles exhibit a characteristic peak followed by a gradual decrease for increasing off-axis angles, in qualitative agreement with ray-traced data reported for the case of SIMBOL-X telescope. 3 Comparison with some ray-tracing results (symbols in Fig. 8) shows an excellent agreement. Geometric area, secondary stray light Within the same approximations, Eq. 26 yields for the normalized geometric area for secondary stray light, once normalized to the on-axis geometric area for double reflection, A SL,2 ∞ (θ) A ∞ (0) = 1 πα 0 π 0 [min (2α 0 + α 1 , Φ, Φ − α 1 ) − max (−α 1 , 2α 0 − Φ + α 1 , 0)] ≥0 dϕ,(48) where we used the relations β 2 = 2α 0 + α 1 and β 1 = −α 1 . We once more have to discriminate between the cases that Φ/2 < α 0 or Φ/2 > α 0 . 5.2.1 SL2, tight nesting: Φ/2 < α 0 We firstly consider the case of Φ/2 < α 0 , which implies the inequalities: 0 < 2α 0 − Φ < α 0 < 2α 0 − Φ/2 < 3α 0 − Φ < 2α 0 .(49) Also Eq. 48 takes different expressions in different intervals of θ. • 0 < θ < 2α 0 − Φ: in this range of off-axis angles we have 2α 0 + α 1 > Φ > Φ − α 1 , and 2α 0 − Φ + α 1 > 0 for all ϕ. Equation 48 then becomes A SL,2 ∞ (θ) A ∞ (0) = 2 πα 0 π 0 [Φ − 2α 0 + θ cos ϕ] ≥0 dϕ = 0,(50) because the expression in [ ] ≥0 is always negative; hence, the integrand has been zeroed. As expected, the secondary stray light is completely obstructed at the exit pupil for small off-axis values. • 2α 0 − Φ < θ < α 0 : the expression of the integrand is the same as in Eq. 50, but this time it becomes positive for cos ϕ > (2α 0 − Φ)/θ, and we obtain A SL,2 ∞ (θ) A ∞ (0) = 2 πα 0 arccos[(2α0−Φ)/θ] 0 (Φ − 2α 0 + θ cos ϕ) dϕ.(51) Developing the computation we obtain the result in terms of the S function (Eq. 39): A SL,2 ∞ (θ) A ∞ (0) = 2θ πα 0 S 2α 0 − Φ θ .(52) • α 0 < θ < 2α 0 − Φ/2: in this case, we have Φ < Φ − α 1 for cos ϕ > α 0 /θ, where the vignetting of the second kind takes over the first kind. However, −α 1 < 2α 0 − Φ + α 1 everywhere, and we do not need to change the second term in Eq. 48. The normalized area can now be written as (Φ − 2α 0 + θ cos ϕ) dϕ ,(53) and the first integrand is non-negative in the integration set. Hence, solving the integrals we get A SL,2 ∞ (θ) A ∞ (0) = 2θ πα 0 S 2α 0 − Φ θ − 1 2 S α 0 θ .(54) • 2α 0 − Φ/2 < θ < 3Φ − α 0 : there is now an interval of polar angles, cos ϕ > (2α 0 − Φ/2)/θ, affected by obstruction on the rear side of the reflective mirror shell (−α 1 > 2α 0 − Φ + α 1 ). In the same region, the obstruction of the second kind is still effective since Φ < 2α 0 + α 1 < Φ − α 1 : developing the passages, we remain with A SL,2 ∞ (θ) A ∞ (0) = 2θ πα 0 S 2α 0 − Φ θ − 1 2 S α 0 θ − S 2α 0 − Φ/2 θ .(55) Figure 1 . 1Origin of stray light in an unobstructed Wolter-I mirror shell. Off-axis rays reflected by the primary segment above the V (ϕ) line (Eq. 3) do not make the double reflection. Stray light also stems from a direct reflection below the z =0 plane. Figure 2 . 2A) three possible causes for obstruction of doubly-reflected rays in a Wolter-I mirror shell: (1) before the first reflection, at the entrance pupil: (2) before the second reflection, on the rear side of the primary segment of the obstructing shell: (3) after the second reflection, on the rear side of the secondary segment of the obstructing shell. The shaded regions are grayed. B) geometrical meaning of the obstruction parameters (after 8 ). Figure 3 . 3Initial positions and destinations of 40000 rays at the entrance pupil (points) for a Wolter-I mirror shell with L = L * , and the angular parameters reported in the legend (Φ ≈ Ψ ≈ Σ). The radial scale has been expanded. Only rays that would have struck the primary mirror were traced. The limits of the regions of different vignetting (dashes) are computed from the vignetting coefficients (after 8 ). Figure 4 . 4A) obstructions of primary stray light of a Wolter-I mirror shell: (0) for double reflection; (1) at the entrance pupil: (2) on the primary segment of the obstructing shell: (3) on the secondary segment of the obstructing shell. B) obstructions of secondary stray light: (0) on the rear side of the primary segment of the reflective shell; (1) by the obstructing shell, on the front side; (2) on the rear side of the obstructing shell, before reflection; (3) on the rear side of the obstructing shell, after reflection. The shaded regions are grayed. Figure 5 . 5The primary stray light pattern described by a mirror shell with f = 10 m, R0 = 105.5 mm, δ = 0, θ = 10 arcmin. The analytical curve was computed with Eqs. 29 and 30. Figure 8 . 8Total, normalized geometric area for primary stray light of a mirror shell with α0 = 0.3 deg, as a function of the off-axis angle, for variable values of the obstruction parameter Φ (assuming that Φ ≈ Ψ ≈ Σ). The expressions in Sect. 5.1 (lines) have been used in the respective intervals of validity. Some values computed by ray-tracing (symbols) are also shown for verification. ( 2Φ − 3α 0 + θ cos ϕ) dϕ + Figure 9 . 9Total, normalized geometric area for secondary stray light of a mirror shell with α0 = 0.3 deg, as a function of the off-axis angle, for variable values of the obstruction parameter Φ (assuming that Φ ≈ Ψ ≈ Σ). The expressions in Sect. 5.2 (lines) have been used in the respective intervals of validity. Some values computed by ray-tracing (symbols) are also shown for verification. In this paper we have reviewed the possible sources of obstruction for focused and stray rays in nested modules of Wolter-I mirrors. We have thereby found integral formulae to compute, in addition to the already known expression for the double-reflection (focused) intensity, the effective area for stray light off the primary (Eq. 15) and the secondary (Eq. 26) mirror segments, also accounting for the finite size of the detector (Eqs. 32 and 33). The predictions are in very good agreement with the ray-tracing findings. In the ideal case of a mirror with constant reflectivity, algebraic expressions for the geometric area could be provided in different ranges of off-axis angles.The formalism provided here can be useful in designing a mirror module maximizing the focused effective area and, at the same time, minimizing the stray light impact. In fact, the solution of designing a completely obstruction-free mirror module within the field of view 8 might leave too much spacing for the stray light to propagate throughout the mirror nesting. In contrast, the formalism provided here enables, from a given mirror module design, not only a fast assessment of the stray light impact from off-axis sources; it also returned useful relations between the tolerable stray-light magnitude and the obstruction parameters. They therefore provide a way to establish the optimal obstruction to minimize the effective area for stray light while preserving the required effective area in the field of view. Should these formulae be solved numerically for Φ, Ψ, and Σ, the complex task of mirror module design problem could be solved easily without the need to run a complex raytracing program. Finally, the same method might be applied to the problem of designing an X-ray baffle, only by a simple re-definition of the obstruction parameters.As a future development of this work, the formalism might be extended to include the case of segmented mirrors, as the ones foreseen for the ATHENA telescope. 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[ "Three-dimensional dust density structure of the Orion, Cygnus X, Taurus, and Perseus star-forming regions", "Three-dimensional dust density structure of the Orion, Cygnus X, Taurus, and Perseus star-forming regions" ]	[ "T E Dharmawardena \nMax Plank Institute for Astronomy (MPIA)\nKönigstuhl 1769117HeidelbergGermany\n", "C A L Bailer-Jones \nMax Plank Institute for Astronomy (MPIA)\nKönigstuhl 1769117HeidelbergGermany\n", "M Fouesneau \nMax Plank Institute for Astronomy (MPIA)\nKönigstuhl 1769117HeidelbergGermany\n", "D Foreman-Mackey \nCenter for Computational Astrophysics\nFlatiron Institute\n162 5th Ave10010New YorkNYUSA\n" ]	[ "Max Plank Institute for Astronomy (MPIA)\nKönigstuhl 1769117HeidelbergGermany", "Max Plank Institute for Astronomy (MPIA)\nKönigstuhl 1769117HeidelbergGermany", "Max Plank Institute for Astronomy (MPIA)\nKönigstuhl 1769117HeidelbergGermany", "Center for Computational Astrophysics\nFlatiron Institute\n162 5th Ave10010New YorkNYUSA" ]	[]	Interstellar dust affects many astronomical observations through absorption and reddening, yet this extinction is also a powerful tool for studying interstellar matter in galaxies. Three-dimensional reconstructions of dust extinction and density in the Milky Way have suffered from artefacts such as the fingers-of-god effect and negative densities, and have been limited by large computational costs.Here we aim to overcome these issues with a novel algorithm that derives the three-dimensional (3D) extinction density of dust in the Milky Way using a latent variable Gaussian Process in combination with variational inference. Our model maintains nonnegative density and hence monotonically non-decreasing extinction along all lines-of-sight, while performing the inference within a reasonable computational time.Using extinctions for hundreds of thousands of stars computed from optical and near infrared photometry, together with distances based on Gaia parallaxes, we use our algorithm to infer the structure of the Orion, Taurus, Perseus, and Cygnus X star-forming regions. A number of features that are superimposed in 2D extinction maps are clearly deblended in 3D dust extinction density maps. For example, we find a large filament on the edge of Orion that may host a number of star clusters. We also identify a coherent structure that may link the Taurus and Perseus regions, and show that Cygnus X is located at 1300-1500 pc, in line with VLBI measurements. We compute dust masses of the regions and find these to be slightly higher than previous estimates, likely a consequence of our input data recovering the highest column densities more effectively. By comparing our predicted extinctions to Planck data, we find that known relationships between density and dust processing, where high-extinction lines of sight have the most processed grains, hold up in resolved observations when density is included, and that they exist at smaller scales than previously suggested. This can be used to study the changes in size or composition of dust as they are processed in molecular clouds.	10.1051/0004-6361/202141298	[ "https://arxiv.org/pdf/2111.06672v1.pdf" ]	244,102,968	2111.06672	2f9faef9cf277e93ea7488ada5eb8864f7d68e2a	Three-dimensional dust density structure of the Orion, Cygnus X, Taurus, and Perseus star-forming regions T E Dharmawardena Max Plank Institute for Astronomy (MPIA) Königstuhl 1769117HeidelbergGermany C A L Bailer-Jones Max Plank Institute for Astronomy (MPIA) Königstuhl 1769117HeidelbergGermany M Fouesneau Max Plank Institute for Astronomy (MPIA) Königstuhl 1769117HeidelbergGermany D Foreman-Mackey Center for Computational Astrophysics Flatiron Institute 162 5th Ave10010New YorkNYUSA Three-dimensional dust density structure of the Orion, Cygnus X, Taurus, and Perseus star-forming regions Received Month Date, Year; accepted Month Date, YearAstronomy & Astrophysics manuscript no. Main ©ESO 2021 November 15, 2021Methods: numerical -ISM: clouds -dust, extinction -ISM: structure -local interstellar matter -Galaxy: structure Interstellar dust affects many astronomical observations through absorption and reddening, yet this extinction is also a powerful tool for studying interstellar matter in galaxies. Three-dimensional reconstructions of dust extinction and density in the Milky Way have suffered from artefacts such as the fingers-of-god effect and negative densities, and have been limited by large computational costs.Here we aim to overcome these issues with a novel algorithm that derives the three-dimensional (3D) extinction density of dust in the Milky Way using a latent variable Gaussian Process in combination with variational inference. Our model maintains nonnegative density and hence monotonically non-decreasing extinction along all lines-of-sight, while performing the inference within a reasonable computational time.Using extinctions for hundreds of thousands of stars computed from optical and near infrared photometry, together with distances based on Gaia parallaxes, we use our algorithm to infer the structure of the Orion, Taurus, Perseus, and Cygnus X star-forming regions. A number of features that are superimposed in 2D extinction maps are clearly deblended in 3D dust extinction density maps. For example, we find a large filament on the edge of Orion that may host a number of star clusters. We also identify a coherent structure that may link the Taurus and Perseus regions, and show that Cygnus X is located at 1300-1500 pc, in line with VLBI measurements. We compute dust masses of the regions and find these to be slightly higher than previous estimates, likely a consequence of our input data recovering the highest column densities more effectively. By comparing our predicted extinctions to Planck data, we find that known relationships between density and dust processing, where high-extinction lines of sight have the most processed grains, hold up in resolved observations when density is included, and that they exist at smaller scales than previously suggested. This can be used to study the changes in size or composition of dust as they are processed in molecular clouds. Introduction Interstellar dust affects how we view the Universe by absorbing and scattering starlight, leading to interstellar reddening and extinction. Although interstellar dust encompasses only a small fraction (∼ 1%) of the baryonic matter in our Universe, it is crucial to understand the properties of dust in order to interpret observations correctly. The absorption of starlight also plays a key role in the thermal balance of the interstellar medium, driving chemical reactions and enabling gas clouds to collapse. Knowing the spatial distribution of interstellar dust is therefore key both to understanding the interstellar medium as whole and to being able to account for the effects of interstellar extinction. Past studies have made great strides in mapping the spatial distribution of the interstellar medium in both extinction and dust density. One of the most comprehensive works was by Schlegel et al. (1998), who used IRAS and COBE far-IR emission to produce an all-sky 2D dust extinction map. This was followed by two decades of improvements fuelled by advances in machine learning techniques and the application of Bayesian statistics leading to both new 2D and 3D dust extinction and extinction density maps (e.g., Marshall et al. 2006; Our code is available at https://github.com/Thavisha/ Dustribution and the latest results are available from www.mwdust. com [email protected] 2011; Sale & Magorrian 2014;Lallement et al. 2014;Green et al. 2015;Hanson et al. 2016;Rezaei Kh. et al. 2017;Sale & Magorrian 2018;Lallement et al. 2019;Green et al. 2019;Babusiaux et al. 2020;Leike et al. 2020). Gaussian Processes (GPs) have been widely use to study the 3D structure of interstellar dust. Sale & Magorrian (2014) used them to predict extinction in simulated data on individual linesof-sight and then applied it to IPHAS photometry in a slice of the Galactic plane ). This was followed by Sale & Magorrian (2018) who improved the method to account for correlations between lines-of-sight and applied it to simulated data along individual line-of-sight. Green et al. (2019) took a similar approach and produced a 3D extinction map covering the entire sky north of −30 • up to 2 kpc. The authors used 2MASS, Gaia and Pan-STARRS data to carry out important sampling on a parameter grid assuming a GP prior on the logarithm of the dust reddening density. These maps have become a go-to resource for estimating interstellar extinction along lines-of-sight as they are easily accessible. The works by Rezaei Kh. et al. (2018a,b) used GP techniques to predict the 3D dust density of the Orion region and the Milky Way disk using Gaia DR2, 2MASS and WISE and APOGEE DR14 data for 30 000 stars respectively. Leike & Enßlin (2019) and Leike et al. (2020) modelled the dust density as a GP and simultaneously inferred the power-spectrum of dust density. Using this method Article number, page 1 of 33 arXiv:2111.06672v1 [astro-ph.GA] 12 Nov 2021 A&A proofs: manuscript no. Main they modelled the inner 400 pc molecular clouds coverage using 30 million sources from multiple surveys including Gaia. Modelling the 3D dust distribution of the Milky Way with GPs is not without its challenges and issues, however. One is the so-called fingers-of-god effect, in which the density distribution is elongated along the line-of-sight due to higher tangential than radial accuracy. It can be avoided if high-accuracy distances are available and/or if correlations between points in 3D space are incorporated explicitly rather than as individual lines-of-sight coupled together in the plane of the sky. Another key feature which must be considered is the physical requirement that densities be positive and hence extinction must be monotonically non-decreasing along any line-of-sight. One way of achieving this is to model the logarithm of the density instead of density or extinction itself. GPs are computationally intensive, naively scaling with O(N 3 ) in number of sources. Methods with improved scaling are therefore important to ensure that a wide range of problems are feasible with minimal trade-offs between, for example, resolution, map size, and number of sources. Finally, the reproducibility of the results may be hampered by the lack of code availability or non-use of public libraries. In this paper we present a technique that directly accounts for the 3D correlations between density points and enforces positive density throughout. To accelerate the processing, we use GP Latent Variable Models in combination with Variational Inference to infer the logarithm of the dust density. We describe our method in section 2, validate it using simulated data in section 3, then using input data from section 4 apply it to the four well-known star-forming regions, Orion, Cygnus X, Taurus and Perseus in sections 5 and 6 to map their dust in 3D. From this we can compute their total masses (section 7), and in section 8 we compare our results to Planck dust emission measures. 3D Extinction and Density Mapping Model Overview To ensure that density is always positive, we model the logarithm of the density, and impose a GP prior to account for correlations between the densities at points in three dimensions. While we will use the term "dust density" or simply "density" for ease throughout this work, in reality we are actually mapping dust extinction density in mag pc −1 units. For it to be a traditional density in g cm −3 units the dust extinction density needs to be converted using the dust optical depth and dust opacity. We optimise the hyperparameters for this GP along with several other parameters which approximately condition the GP as described in Sect. 2.3 and 2.4, to infer the logarithm of the density. At the end we draw sample maps from the final conditioned GP to explore the dust density distribution and its uncertainty; these are integrated to predict extinction and its uncertainty. Our approach requires measurements and uncertainties of the extinctions towards a sample of stars, as well as their positions in three dimensions. Uncertainty on distance is included following the same approach as Rezaei Kh. et al. (2020), which involves incorporating them into the observed extinctions by inflating their uncertainties (see their sec. 2 and eqns. 1 and 2). We utilise GP package GPyTorch (Gardner et al. 2018) and probabilistic programming package Pyro (Bingham et al. 2018;Phan et al. 2019) in our implementation. Both packages are built upon PyTorch (Paszke et al. 2019), an open-source machine learning framework. Latent Gaussian Process Function Mathematically, our quantity of interest, dust density (ρ), can be described as a latent quantity or function -a quantity that we do not directly observe. This latent quantity can be transformed by integrating along lines-of-sight to forward model the observed quantity, which is extinction. Dust density is expected to vary in a complex way spatially, which cannot be accurately captured by a parametric function. As a result, the model for the latent function should ideally be flexible and non-parametric, which makes GPs ideal for this purpose. We model the measured extinctions with a Gaussian likelihood. Given that A mod = d 0 ρ ds i , then if we adopted a GP prior on ρ, then the posterior would also be Gaussian in ρ, with analytic solutions for its mean and covariance. This is the approach of Rezaei Kh. et al. (2017) (their appendix A). However, to enforce non-negative densities, we define our GP prior on log 10 (ρ), with the consequence that the exact posterior no longer has a simple analytic form (see appendix A). We handle this using variational inference (see section 2.3) and approximate the posterior in log 10 (ρ) as normally distributed. The prior on log 10 (ρ)which we now denote φ for brevity -is modelled using a Gaussian Process with a constant mean (see chapter 2.2 and eqn. 2.14 in Rasmussen & Williams 2006), P (φ (x, y, z)) = GP (c, αK(x 1 − x 2 )) ,(1) and elements of the covariance matrix given by a 3D radial basis function (RBF) kernel K(x 1 , x 2 ) = exp − 1 2 (x 1 − x 2 ) T γ −2 (x 1 − x 2 ) ,(2) where x 1 and x 2 are two position vectors in 3D space, γ is a diagonal matrix of scale lengths, α is a scale factor, and c is a constant mean (see chapter 4.2 eqn. 4.9 and chapter 2.2 eqn. 2.14 in Rasmussen & Williams 2006) 1 2 . We abbreviate the right hand side of equation 1 to GP(Θ) throughout this paper, where Θ represents the hyperparameters of the GP. Our GP has five hyperparameters: three physical scale lengths in the three Heliocentric Cartesian coordinates (x, y, z) of physical space; one exponential scale factor; the mean density. These hyperparameters are used to generate the GP prior from which sets of φ are predicted. In effect, treating φ as a latent variable introduces one free parameter at each point in x, y, z, all coupled together through the GP prior. We evaluate the GP on a grid, which we take to be regular in Galactic (spherical) coordinates (l, b, d) as this eases the implementation of line-of-sight integration. We then convert the coordinates of the centres of the cells to Heliocentric Cartesian coordinates for use in eq. 1. To compare to observed extinction (A obs ) to what our model density distribution predicts, we must integrate this density distribution along the line-of-sight. We first exponentiate the distribution of φ to obtain ρ and then numerically integrate along the lines-of-sight to all stars in our observed data set. The numerical integration is described in Sect. 2.5. These integrated densities, i.e. the model extinctions (A mod ), are compared to our observed extinctions (via the likelihood) in order to optimise the model. The posterior for our model is therefore P(φ, Θ \| A obs ) ∝ P(A obs \| φ) P(φ \| Θ) P (Θ)(3) where the first term on the right does not depend on Θ because A obs is independent of Θ once conditioned on φ. As mentioned above, as we have a prior on log density rather than density, we no longer have close form solutions for the integrals over the model densities. We therefore use numerical methods to compute the integrals to achieve a Gaussian approximation of the posterior in φ. We do this via variational inference, as explained in the next section. When doing this, we are free to infer the hyperparameters of the GP prior at the same time. As shown in eq. 3, our model is implicitly hierarchical: the GP is a prior on (the logarithm of) dust density, the plausible values of which are determined by the hyperparameters, and we optimise the dust density as a function of position. Variational Inference The above approach to modelling the dust density effectively makes the problem one of inferring the joint distribution of densities at all points, and as a result the problem has a very highdimensional parameter space. To handle this effectively we use two techniques. The first is to reduce the dimensionality of the problem by conditioning the GP only on a subset of points, known as the inducing points. The locations of these inducing points are parameters which can be optimised along with the other free parameters. Although the use of inducing points lowers the dimensionality, it is still high, because there are three free parameters per inducing point. There is no "correct" choice of the number of inducing points. A rule of thumb is to incorporate as many inducing points as computationally feasible given the available resources (e.g., Wang et al. 2019;Wu et al. 2021;Yi 2020). Results in the literature show that there is minimal gain once the number of inducing points rises above several percent of the full data set (Wang et al. 2019). We typically use 500-1000 of inducing points in this work, which is as much our computational setup allowed at the time. To circumvent the still large number of free parameters, we employ variational inference to approximate the target marginalisation integrals. Variational inference replaces the target posterior with an approximate posterior that is easier to work with, and finds the parameters for this approximation that best reproduce the true posterior (Bishop 2006;Blei et al. 2017). This allows the direct computation of the approximate posterior and its gradient with respect to the free parameters, thereby enabling the use of gradient-descent optimisers, as described below in Sect. 2.4. This approximate posterior is referred to as the variational distribution. As we are using a GP, we use a multivariate Gaussian distribution for this approximate posterior (as described above in Sect. 2.2) to give the joint distribution of φ at the inducing points. The mean and covariance of this Gaussian are what we need to infer. In our case, the variational distribution approximates the true (non-Gaussian) posterior of φ at the inducing points, analogous to conditioning a GP on a function which is approximating the real (non-Gaussian) distribution, and only conditioning at a subset of the available points (Bishop 2006;Blei et al. 2017). We want to find that approximate posterior that is most similar to the true one. We achieve this by finding the parameters of the (variational) distribution that minimizes the difference between this distribution and P(φ \| A obs ). To minimise this difference, we maximise the Evidence Lower Bound (ELBO). ELBO is a variational bound, i.e. it bounds a quantity by a functional over some set of functions (Bishop 2006, chapter 10). It computes a minimum value for the evidence (the integral of the like-lihood over all possible parameter values) given a set of likelihood evaluations. The motivation for the ELBO is that it is equal to the logarithm of the evidence (which is equal to the expectation of the likelihood over the prior) plus the negative of the Kullback-Leibler (KL) divergence (Bishop 2006;Blei et al. 2017) ELBO (V) = E log P(A obs \| φ) − KL (V (φ) \|\|P (φ \| A obs )) ,(4) where the KL divergence is a measure of the dissimilarity of two distributions, and V is the variational distribution. In our case the two distributions whose KL divergence is measured are the exact posterior P (φ \| A obs ) and the approximation, the variational distribution. This is calculated as: KL (V (φ) \|\|P (A obs , φ)) = V (φ) ln V (φ) P (φ\|A obs ) dφ,(5) following Bishop (2006, chapter 10 eq. 10.2-10.4) and Blei et al. (2017, sec. 2.2 eq. 11-14). This can be efficiently minimised using stochastic gradient descent algorithms. The full set of free parameters to be inferred in our model therefore consists of the GP hyperparameters, the locations of the inducing points and the parameters of variational distribution i.e. the means and covariances. Once the inference is complete, we have our conditioned GP. From this we can draw samples of φ which can be exponentiated to then determine the 16th, 50th and 84th percentiles of ρ. We can also numerically integrate (as described in Sect. 2.5) the samples of ρ over distance to determine the 16th, 50th, and 84th percentiles of extinction. We use samples to determine confidence intervals as our distributions of ρ and extinction are not Gaussian; if they were, the resulting integration (or sum) would itself be Gaussian, however, we end up needing to sum over lognormal random variates which does not necessarily have similar convenient properties. Sampling, on the other hand, is computationally efficient. Implementation The GP is implemented using GPytorch (Gardner et al. 2018), and Pyro (Bingham et al. 2018;Phan et al. 2019) is used for variational inference. The inputs to our algorithm are the positions and extinctions (including uncertainties) of a sample of stars within a region of interest in the Galaxy. We define a training grid on which the GP can infer densities. The grid is composed of cells, and every cell is assumed to be uniform, that is, the value of the density is the same at every point within a given cell. The boundaries of this grid are defined in galactic coordinates to enclose the region of interest. The number of cells along each axis of the grid is independent, and can be spaced arbitrarily. The number of cells define the resolution of our training grid and hence the minimum size of the recovered features in the final reconstruction. We have three sets of free parameters in our model which are optimised. The GP hyperparameters, the locations of the inducing points within our training grid and the means and covariances of the variational distribution. The hyperparameters of the GP require starting values to begin the optimisation. For the mean of the RBF kernel we estimate the mean density of our region of interest based on prior knowledge and the specific values are given in sec. 5 and 6. The scale factor of the kernel is initiated from the expected scale (amplitude) of deviations from the mean density in dex (since our model is logarithmic); these values must be below 1 dex because we do not expect variations Article number, page 3 of 33 in dust density of an order of magnitude on scales of 1 pc in the ISM. Finally, we choose the starting scale length based on the typical size of clumps to be recovered. Table 1 below presents the input hyperparameters as well as their conditioned counterparts. We randomly select a subset of the positions of stars in our input sample for the initial locations of the inducing points (however, a user is free to place them in any initial configuration of their choice and can leave these positions fixed throughout the optimisation process if they wish). The variational distribution is initiated by GPyTorch with means of zero and variances of 1. As seen in Eqn. 5, estimating ELBO requires integration; this is calculated using 32 Monte Carlo samples at each iteration, and Adaptive Moment Estimation with Weight decay (AdamW; Loshchilov & Hutter 2019), implemented in Pyro, is used to maximise ELBO. AdamW has several tunable parameters, including the learning rate, which determines the step size at each iteration. We choose relatively large learning rates to avoid the optimisation becoming stuck in a local minimum or taking too long to optimise, while also avoiding moving around too much and so potentially missing the optimum values while maintaining numerical stability. AdamW is set to terminate training after a fixed number of iterations chosen based on when the variation of the ELBO is less than 1% across the last 10 iterations and the model has converged. For our training grid, we choose a set of boundaries in all three dimensions that closely encompass our region of interest with padding in l, b, d. This speeds up the run time of the model compared to using a larger region, but requires that we account for the effects of the foreground and background dust that have been omitted. The padding added in the d dimension accounts for the fore and background dust. In the case of foreground dust, which would directly affect all observed extinctions, the model would increase the density of the closer cells in order to account for the missing foreground cells (i.e. from zero to the lower boundary). To accommodate this we add a set of ghost cells to hold this additional dust, similar to an approach used in grid-based hydrodynamics models (e.g. O'Brien & Bussmann 2018). No stars are given as input to these ghost cells, giving the model the freedom to insert foreground dust to explain the extinctions of the stars closest to the lower distance boundary. This avoids biasing the final scale factor and mean density. Fewer ghost cells are required when the foreground extinction is low (for example nearby regions, or at high galactic latitude) while high-extinction lines-of-sight, particularly for distant star-forming regions in the galactic plane, will require more cells. The exact amount must be tuned on a case-by-case basis. The case of background regions is slightly more complex. Because our model is conditioned on extinction, and the density at a point is informed by the extinctions of stars at larger distances on nearby lines-of-sight, we require a sufficient number of sources in the background to the star formation region in order to infer the density at a point within the star formation region unambiguously. Restricting the input sources exclusively to the star formation region in which we want to infer the dust density would discard vital information, so sources must be included at greater distances such that they sample the density at a sufficient number of lines-of-sight inside the region of interest. Hence, we pad the model volume at greater distances. Unlike with the foreground ghost cells, these background cells contain input sources, and so the padding must be increased until a sufficient number of sources are available to encapsulate the underlying structure of the region. Once again, this was determined for each case individually by iteratively increasing the distance un-til the changes in the structure are no longer significant by eye. Beyond the range probed by the input set of stars, the predicted density will rapidly converge to the mean of the GP prior, as this is the maximum a posteriori value in the absence of data on which to condition. This does not preclude low-extinction linesof-sight from introducing low dust-density regions in this mean density if the source is sufficiently distant and if there is little dust in front of it, as there is information available here. Once the GP is conditioned, we are free to predict the density at arbitrary points. We exploit this to produce visualisations of the reconstructed density at higher resolution, making them easier to interpret. We therefore define a grid with 2-3 times higher resolution than the training grid but the same boundaries, which we refer to as the prediction grid hereafter. Although it does not provide any new information compared to the training grid, it is especially beneficial for visualising structures that are small and only a few pixels in size in the training grid. As our optimisation routine AdamW requires the gradient of the ELBO all quantities are stored as Torch tensors or distributions to exploit PyTorch's autograd module. These tools have been scaled to large datasets and give us good performance for our problem. Integration along lines-of-sight The integration of density is approximated as a summation. This is carried out in spherical coordinates with the Sun at the origin to optimise the computations by minimising the number of cell boundary crossings. We want to compute A mod = d * ,i 0 10 φ ds i(6) for each star, where φ is the logarithm of extinction density, s i is the path to the ith star and d * ,i is its distance. As φ is discretised on a grid, we approximate this integral as the sum A mod = j * j=0 10 φ j ∆d j(7) where the index j iterates over the grid cells along the line-ofsight, j * is the index of the cell that contains the ith source, φ j is the density in the jth cell and ∆d j is the length of the path crossing the jth cell. This summation is refined slightly to account for the partial crossing of the final cell on the path, i.e. A mod = j * −1 j=0 10 φ j ∆d j + 10 φ j * d * − d j * −1(8) where d j * −1 is the distance to the boundary between the j * − 1th and the j * th cell. Testing Model on Simulated Data As an initial test of our model we run it with a set of simulated data with known densities. We generate a 3D density field comprising of a constant background overlaid by three Gaussian overdensities randomly located within a cube. This superposition of clumps on a background provides a simple analogue to the real distribution of matter that is easy to analyse. The background of the density field is set to 1.6 × 10 −3 mag pc −1 , which Article number, page 4 of 33 Dharmawardena et al.,: 3D dust density Fig. 1: Synthetic extinction estimates used as input for the model, drawn from the simulated density distribution. Each point in the plot corresponds to one simulated star, with a randomlygenerated (l, b, d) and its corresponding extinction. gives a total background extinction of 0.5 mag when integrating over the model distance range (see below) providing sufficient extinction to be visible in the model. We generate extinctions toward 120 000 simulated sources within the model volume by integrating along lines-of-sight to a representative sample of coordinates spread randomly throughout the 3D density field. The coordinates are distributed in l, b and d 3 , assuming a uniform space-density of sources. The integration is performed on a grid with boundaries l = 50 • to 90 • , b = −10 • to +10 • , and d = 180 − 500 pc and consists of l × b × d = 100 × 100 × 115 cells. We add zero-mean Gaussian noise to these extinctions with a standard deviation equal to 10% of the true line-of-sight extinction. These noisy extinctions, along with their noise-free 3D positions, serve as the input to the model and are shown in Fig. 1. We only apply the distance uncertainty method of (Rezaei Kh. et al. 2020) to our real star formation data and it is not applied here. Our algorithm was trained on a grid of n l ×n b ×n d = 20×20× 35 cells 3 with 1000 inducing points and the output predictions were made on a grid of n l × n b × n d = 100 × 100 × 105 cells to be able to directly compare to the ground truth maps. The grid boundary coordinates were the same during training and for the prediction. Figure 2 shows the variation and convergence of the ELBO for our model run. The variations in ELBO over the last few iterations are below the 1% level, demonstrating that the model has converged. As we can see from our comparison of the simulated and reconstructed data, our model reproduces the overall shape of the extinction and density distributions well. In the histogram presented in Fig 3 the predicted source-by-source extinction follows the same trend as true extinction with only a small fraction (< 10%) of the sources being underestimated. We do see that the model reconstructions of extinction and density are smoothed out compared to the ground truth, as is expected from a GP method. This smoothing is more evident in the plots of the resid-uals (Figs. 4 and 5). In the total extinction we see an average underprediction of less than 1 magnitude and in density it's slightly larger, with a typical (maximal) variation of ∼ 12% (∼ 20%). The underestimation of extinction, particularly the highest extinctions, is driven by a combination of two effects; the density is smoothed, spreading the same extinction over a larger range of distances, and the vanishingly-small probability of having sources that sample the very highest extinctions makes it difficult to measure them. On the other hand, densities are both over-estimated in some regions and underestimated in others, which also arises because of smoothing -when some material is moved along the line-of-sight, the clump becomes larger and hence the core will be underestimated and the wings slightly overestimated. The smoothing performed by the GP inevitably introduces some error into the reconstructed positions of structures in density, particularly along the line-of-sight. However, from the predicted and residual plots of density shown in Fig. 5 we can see that this is typically less than one scale length (i.e. features may be offset from their true positions by up to 20 pc in the final map in this case). For our density reconstructions we also need to consider the typical uncertainty scales on the values of the density. In appendix B we present the density and extinction as a function distance for several lines of sight, along with cuts along l and b as a function distance. From these figures we infer a typical uncertainty of 20% in density. Overall, the size of the uncertainties and errors of the output of our model are small, demonstrating that our algorithm provides good performance in both localising structures and estimating their magnitude, provided that the scale length is smaller than the typical molecular cloud size. The time taken to train our model depends on several factors. (The run times of our model for our star formation region are given in the upcoming table 1). The number of cells in the training grid is the largest factor, with the run time empirically scaling slightly faster than linear (∼ O n 1.4 ) with the number of cells n. The run time relative to the number of inducing points, however, scales approximately O n 0.95 but consistent with linear scaling. Finally, as the number of sources changes, the run time varies O n 0.8 . The prediction run time makes up a negligible fraction of the run time taking only several hundred seconds compared to the hours -days required for training depending on the described factors above. Data In the rest if this work, we use data from the stellar parameter catalogue of Fouesneau et al. (submitted) 4 . Specifically we use their inferred extinction at 547 nm (A 0 ) as A obs and their distances as d. Our code is wavelength agnostic, however, and could be used with extinctions at any wavelength from any catalogue. The Fouesneau et al., parameters are inferred from photometry from Gaia DR2, 2MASS, and WISE together with Gaia parallaxes. Their model simultaneously estimates A 0 , distances, R 0 , effective temperature, luminosity, surface gravity, mass and age for each star independently. The models predict the reddened spectral energy distributions (SEDs) from the data using PARSEC isochrone models (Chen et al. 2014;Marigo et al. 2013;Rosenfield et al. 2016), the ATLAS9 atmospheric library Pyro reports ELBO as a loss function and therefore seeks to minimise −1× ELBO instead of directly maximising ELBO, and the y-axis on this plot is therefore positive instead of negative. Minimising −1× ELBO is therefore analogous to minimising the KL divergence. (Castelli & Kurucz 2004), and Fitzpatrick extinction law (Fitzpatrick 1999). Their estimates result from Markov chain Monte Carlo (MCMC) samples of the posterior parameter distribution. During this procedure, the models are interpolated using a neural network. In the regions we study in this present work, the typical (median) uncertainties in extinction are 0.34 mag, and the fractional parallax uncertainties are typically between 0.13 and 0.19. We show their Galactic A 0 map and the regions we study in this work in Fig. 6. We do not make any cuts on the set of sources based on stellar type or stellar parameters; we simply include all sources that fall within our region boundaries. Young Stellar Objects (YSOs) often have circumstellar dust that may be incorrectly attributed to interstellar extinction when that is determined in the Fouesneau et al., catalogue. We did not filter out such objects. However, we expect these objects to be rare, because they are often very red and so would lack a BP measurement, and thus be excluded from the catalogue. Using this catalogue as our input data provides us with advantages compared with previous Studies. First, one can achieve more reliable estimates of stellar parameters by combining multiple spectroscopic and photometric surveys. This catalogue relies on more than Gaia-only data instead of using GDR2 A G from Andrae et al. (2018). Leike & Enßlin (2019) used those in their analysis and were limited to the systematics of the GDR2 extinctions. Second, infrared indicators such as RJCE (Majewski et al. 2011), optimized for particular applications, are less sensitive to low column density than optical bands. These A k estimates need careful considerations to extrapolate their usage on non-giant stars (e.g. Rezaei Kh. et al. 2020). Finally, the method in Fouesneau et al., subm. jointly estimates distance with the extinction (and other properties). As a result, we do not rely on the inverse parallax as distance measurements (Bailer-Jones 2015), and we obtain a coherent set of input data. Orion The Orion star-forming region (SFR) is the nearest cluster of young, massive stars (> 8 M ) at an approximate distance of 400 pc (Menten et al. 2007). The stellar ages in the Orion SFR range from 2-12 Myr and the SFR is separated into three main components, the Orion A and Orion B molecular clouds, and the λ Orionis molecular ring (Bally 2008). Orion has been studied extensively in literature, making it an ideal real-world test case for our model. Its 3D dust density distribution has been explored by Rezaei Kh. et al. (2018b, 2020, who found a foreground cloud in front of the Orion A molecular cloud at 345 pc. The dust extinction distribution was investigated by Lombardi et al. (2011) (hereafter L11) and Schlafly et al. (2015) (hereafter S15). L11 explores the distances to several clouds within the Orion SFR using foreground stellar densities and infers masses for them, while S15 explore the 3D structure using dust reddening to study dust ring structures in detail. Attempts to understand the structure of Orion are particularly motivated by the need to study the feedback of massive stars on their environment and the impact this has on the star-forming process (Bally 2008). Orion Model setup The model set up including the total number sources from Sec. 4, chosen region size along galactic longitude l, galactic latitude b and distance d along line-of-sight (l bounds, b bounds, d bounds), spacing of cells along l, b, d (n l , n b , n d ) and initial input and final conditioned Gaussian kernel hyperparameters are presented in Table 1. We begin testing our model for the Orion SFR by setting the scale lengths of the Gaussian kernel to 10 pc, a value we arrive at based on the distance to the complex and the projected sizes of the molecular clouds. We initialised the mean density to that used by Rezaei Kh. et al. (2017). We set the initial scale factor based on the size of the perturbations expected from the mean background. We then ran the model a number of times, starting the next run from the end point of the previous one, until we found a satisfactory region of parameter space for the scale factor and mean density for the final model run. This was an intuitive process, and resulted in the initial values listed in Table 1. Inferred structure of Orion We summarise the coordinates and the sizes of the main components of Orion identified here in Table 2 We do not see evidence for a separate foreground cloud along the line-of-sight to Orion A as suggested by Rezaei Kh. et al. (2020). We do, however, see that Orion A is quite extended, stretching from 350 to 450 pc as first suggested by (Großschedl et al. 2018) using YSO data from Gaia DR2 and later by (Rezaei Kh. et al. 2020) using 3D dust densities. The Planck 850 µm and dust intensity maps (see Figs. D.1 and D.2) show a bubble at l = 215 • and b = −20 • at the edge of Orion A in the direction of the Galactic center. This bubble is also seen in the infrared extinction by L11, and S15 suggests it lies beyond 550 pc. However, our maps show this bubble is closer to us along the line-of-sight, appearing as an over-density in all three of our maps. It is densest at 405 pc while spanning 380 ≤ d ≤ 430 pc. We recover Orion B and find it to be composed of two segments and to be elongated along the line-of-sight (seen in Fig. 8). The component adjacent to Orion A (Ori B comp 1 in Fig. 8) appears at around 380 pc and disappears at around 400 pc before reappearing at larger distances with the densest region around 430 pc (consistent with Rezaei Kh. et al. 2020), while the component closer to λ Orionis (Ori B comp 2 in Fig. 8) appears around 380 pc and extends up to 420 pc. λ Orionis itself is the bubble-like structure whose nearside is around 360 pc and the far side is around 110 pc further along the line-of-sight, placing it closer than previous measurements such as L11 who infer a distance 445±50pc. Additionally, we recover a peak in density at the centre of the ring (on sky) at a distance of 360-390 pc; this was not seen by L11 or S15 but is clearly visible in the Planck dust intensity map shown in Fig. D.2. Another arc-like feature linking Orion A and Orion B can be seen at 205 • ≤ l ≤ 210 • and −19 • ≤ b ≤ −15 • between 400 and 450 pc, possibly corresponding to NGC2170 and the dust surrounding the Barnard Loop hii region. The large cloud seen at 205 • ≤ l ≤ 215 • and b = −5 • in the Planck 850 µm map and in extinction by L11 is not visible in our results. This may imply the cloud is located beyond 550 pc, possibly indicating its association with the Canis Major star-forming region instead of Orion. Finally, we recover a large filament at l 186 • visible at 270-300 pc, even though it is not clearly visible as a coherent structure in extinction. This emphasises the power of 3D recon-Article number, page 7 of 33 A&A proofs: manuscript no. Main (x, y, z; pc) 9.5, 9.3, 9.5 19.1, 19.1, 19.2 12.8, 5.8, 6.4 9.2, 6.7, 4.9 initial mean density (mag pc −1 ) 4.6 × 10 −4 3.0 × 10 −3 4.6 × 10 −4 1.5 × 10 −3 final mean density (mag pc −1 ) 1.06 × 10 −3 1.2 × 10 −3 6.0 × 10 −4 1. structions recovering structure lost in integration to 2D (however it must be noted that if differentiated, 2D extinction maps may also have the potential to show similar localisation). We refer to this filament as the λ filament in Fig. 7 as it is closest to λ Orionis. A similar structure is also visible in the Planck maps (figures D.1 and D.2). This filament may be host to several small clouds such as LDN 1558 and TGU L156 (L11) as they overlap on sky. However without confirmed distances to these condensations we cannot corroborate this. Cygnus X, Perseus, and Taurus We select three more star forming regions to demonstrate the capabilities of our algorithm. These were chosen on the grounds that their 3D dust densities have been less well studied, and because they have different properties from Orion. Once again, we summarise the coordinates and the sizes of the main components of these regions identified through this work in Table 2. Cygnus X is the most massive star forming region within 2 kpc of the Sun, with a total mass of 3 × 10 6 M (Schneider et al. 2006) and cluster ages of up to 18 Myr (Maia et al. 2016). It is home to the largest number of massive protostars and the largest OB stellar association Cygnus OB2 (Guarcello et al. 2013). While the Orion SFR is located south of the Galactic plane, Cygnus X is in the Galactic plane, so shows higher foreground extinction and crowding. Of the four regions studied here, Cygnus X is the furthest away from us, with estimated distances in the range 1300-2000 pc (Rygl et al. 2010(Rygl et al. , 2012Schneider et al. 2006). The Taurus star formation region is the nearest to us at a distance of 145 pc (Yan et al. 2019) and an age below 5 Myr. Deviating from the common embedded cluster mode star formation seen in other SFRs such as Orion and Perseus, Taurus is the prototype low-mass star formation region (0.7 − 1.0M ) where stars appear to form in relative isolation (Kraus et al. 2017). The proximity of Taurus to us also makes it one of the largest SFRs on sky. This, combined with its large population of young stars, makes it an interesting star formation region for analysis. Finally, we study the Perseus star formation region which neighbours Taurus on sky and is at a distance of 310 pc (Yan et al. 2019). Perseus is home to several regions of active or recent star formation with typical ages of 1-5 Myr (Pavlidou et al. 2021). It is the smallest of the four regions in angular size presented in this work. There are several filamentary structures spreading towards it from the other nearby star formation regions including Taurus and California. With a mass of around 100 M , this SFR is the closest region which is still actively forming low-intermediatemass stars . The model-setup for these three regions is similar to the one we used for Orion, bar a few modifications to match properties of these regions as given in table 1. 6.1. Inferred structure of the regions 6.1.1. Cygnus X The distance to Cygnus X has historically proven difficult to determine, in part because of its location in the Galactic Plane. Distances in the literature span from 1.3 kpc (Rygl et al. 2010) to 2 kpc (Schneider et al. 2006). With our 3D density mapping algorithm we are able to localise the densest regions of Cygnus X to 1.3-1.5 kpc as shown by fig 10. Our distance estimate is consistent with the maser parallax distances measured by Rygl et al. (2010) and Rygl et al. (2012). Using CO data in which adjacent regions appear to be interacting, Schneider et al. (2006) inferred that Cygnus X is a monolithic structure at 1.7 kpc. However, we detect small clumps throughout our distance range, corroborating suggestions in the literature that the Cygnus X region may contain both the large star-forming region (which we place at 1.3-1.5 kpc) and a number of smaller clouds spread along the line of sight, as first suggested by Dickel et al. (1969) Article number, page 8 of 33 based on extinction measurements. The clumpy structure and upper distance limit also matches well with Zucker et al. (2020). While Zucker et al. (2020) recover some clumps closer to us than 1200 pc, we do not because we do not have any stars closer than 1200 pc included in our model. We see a separation in the Cygnus X North and South Clouds in figure 10 and find that Cygnus X South is closer to us, with the dense region in Cygnus X South starting at 1300 pc. The south cloud is densest at 1350 pc, while the north cloud appears to be densest at around 1500 pc. We also see the presence of the North American Nebula (NAN) in the extinction figure 9 at l = 0 • , b = 85 • . At 600 pc (Schneider et al. 2006) it is much closer to us than the region encompassed by our model and therefore not visible in our 3D density plot. Further, as expected given the high extinction in the region and its placement in the Galactic plane, Cygnus X has the highest mean density of any of the regions we examine. Perseus Our density distribution (Fig. 12) shows that Perseus itself lies between 300 and 350 pc. This is larger than the mean distance of 240 pc determined by Lombardi et al. (2010) (hereafter L10) who used foreground stars to arrive at this distance. However it agrees well with the mean distance of 310 pc from Yan et al. (2019) (using Gaia DR2 extinction and parallaxes) and the range of 294-350 pc found by Zucker et al. (2018) who used photometry, Gaia DR2 astrometry and 12 CO data. We identify the two largest clusters IC348 (l = 160 • , b = −18 • ) and NGC1333 (l = 158 • , b = −20 • ) at the eastern and western edges of the cloud complex at similar distances to one another. We also detect a clump just above IC 348 at l = 160 • , b = −17.5 • located at a distance of ∼ 306 − 330 pc. It is densest at a distance of ∼ 325 pc and is clearly detected in both the Planck dust emission in Fig. 15 as well as Zucker et al. (2020) who place it at a distance of 331 ± 16 pc. We do not localise the smaller clumps associated with the Perseus SFR. Although IC348 is also visible in our total extinction map (Fig. 11) NGC1333 is not as distinct, further demonstrating the importance of 3D mapping to fully understand the structure of molecular cloud regions. Interestingly, we also recover the elongated filaments which appear to extend towards Perseus from the Taurus and California molecular clouds when viewed on sky. We also detect filamentary structure extending from the western edge of Perseus away from the Galactic plane. We localise these filaments in 3D for the first time. These filaments are visible in Planck emission (see Figs.15, D.1, D.2) as well as extinction maps presented by L10, but they have not been localised in 3D before this work. The filament associated with California may in fact be on the Article number, page 9 of 33 edge of Perseus with a distance starting at 250 pc out to nearly 350 pc. The filament between Perseus and Taurus corresponds to a distance of 300 pc extending to ∼ 330 pc. The filament furthest from the Galactic plane is at 300 pc but appears to be more diffuse than the other filaments and we cannot clearly associate it to Perseus or any other nearby star forming regions. We refer to these filaments as California filament, Taurus filament, and Perseus filament respectively in our figures. Taurus Taurus has been separated into two components, TMC 1 and TMC 2, based on their sky positions (Lombardi et al. 2010). In our 3D density distribution plot shown in Fig. 14 Fig. 13 corresponds to the main component of TMC 2. This component is elongated along the lineof-sight (Fig. 14), starting at 110 pc and continuing to 145 pc, therefore TMC 2 is closer to us than TMC 1. While the main component of TMC 2 ("TMC 2 main" in Figs. 13 and 14) is home to several smaller clumps we do not separate these clumps in our predictions. TMC 1, on the other hand, appears at 145 pc and continues to 190 pc, showing that it is distinct from TMC 2. Similar to TMC 2 main, we cannot separate the smaller components of TMC 1 in our maps. This suggests that TMC 1 is a coherent structure containing the clumps. The overdensity at l = 170 • , b = −19 • and 195 pc can be associated with the L1498 clump, part of TMC 2. Furthermore, the elongated structure at 195-220 pc which appears to connect TMC 2 main and clump L1498 on sky can be associated with the B215/L1506 extended clump, also part of TMC 2 on sky. While the B215/L1506 is clearly visible in 3D density in figure 14 it is barely visible in total extinction as seen by figure 13; this reiterates the importance of 3D density localisation compared to extinction maps where information is lost in the integration along the line-of-sight. We also see evidence for a structure at 220-250 pc at l = 165 • , b = −17 • (Taurus filament in Fig. 14). This likely corresponds to the other (closer) end of the filament identified in the Perseus data, corroborating this feature. It appears to lie at a distance intermediate to the two star-forming regions, along a trajectory linking them. Whether this is an independent feature or a bridge between the two regions remains to be seen. In our density figure, we also see two arc-like structures at ∼ b < −14 • appearing one after the other extending from 201 pc to the end of our distance range. This could be associated to the filamentary structure adjoining TMC 1, and holds the L1540, L1507 and L1503 clumps. As noted in the Sec. 3 there are uncertainties associated with the localisation of structure in all four regions. These uncertainties include shifts in position by approximately one scale length. Total Mass To calculate the dust and total masses of our regions, we take our integrated extinction (calculated up to the upper distance boundary) as described in Sect. 2.5 and convert it to a column density using the dust mass-extinction coefficient (opacity) of κ 0 = 26000 cm 2 g −1 calculated at 547 nm. To calculate this opacity we use an absorption coefficient of 8.5 × 10 3 cm 2 g −1 and albedo of 0.67 at 547 nm 5 (Draine 2003a,b). We integrate the total extinction along and l and b to derive a dust mass as follows: M d = d 2 max b max b min cos b db l max l min dl A mod,d max (l, b) 1.086κ 0 ,(9) where the factor 1.086 = 1 ln 2.512 converts from magnitudes of extinction to optical depth, A mod,d max is the total extinction map predicted by our algorithm up to upper distance boundary and D is the distance assumed for the mass calculation which in our case is assumed to be the upper distance boundary of our model setup presented in table 1, d max . To derive the total mass we use a gas:dust ratio of 124 (Draine 2003a,b) and multiply the dust mass by this ratio. We estimate the uncertainties in the masses in Table 3 by sampling over the uncertainties in our extinction maps, which in turn come from the finite spread of the Gaussian posteriors in the dust densities. Table 3 presents the calculated masses. Both L11 and S15 derived the total masses for Orion using the same l and b range as each other. This range is approximately two degrees smaller on each axis compared to our measured region and excludes the filamentary structure we detect. They also assumed smaller upper distance boundaries. In order to be able to directly compare our masses to L11 and S15 we carry out the following steps: 1. Recalculate the mass from this work using the same l, b boundaries used in the literature. 2. Scale the literature masses up to the d max used in this work. Following step 1, our total mass for Orion recalculated to the smaller l, b range used in L11 and S15 is 735 +260 −177 × 10 3 M . This estimate is now directly comparable to the literature masses from L11 and S15 scaled to our distance of 550 pc (step 2), which are 504 × 10 3 M and 444 × 10 3 M respectively. We find our recalculated mass to be 1.4 times larger than L11 and 1.6 times larger S15. Our input A 0 values (see Sec. 4) are derived using longer wavelengths than either L11 or S15, and hence are sensitive to higher total extinction. This means we are more likely to recover higher column densities and therefore greater masses in the present work. Another reason for the variations between the masses are the two different approaches used. While we use a conversion from extinction to mass using κ 0 and a dust:gas ratio, both L11 and S15 use an approach that adopts a hydrogen column density to extinction relationship. The typical (systematic) uncertainties on the conversions in each case are approximately a factor of two (e.g. Draine 2003a). In the cases of Perseus and Taurus we use the same l, b range as L10, but not the same d max boundaries. Therefore we can skip step 1 given for Orion and only carry out step 2, where we scale up the literature mass up to our d max in order to be able to directly compare our masses. Once scaled we derive total masses of 117 × 10 3 M for the Perseus region and a total mass of 110 × 10 3 M for the Taurus region for L10. Our total masses are 1.5 times and 1.3 times larger for Perseus and Taurus respectively similar to the variation observed in Orion. The causes of variations discussed above also applies to these two regions as well. A recent publication by Zucker et al. (2021) predicted total masses for Perseus, Taurus, Orion A and B as well as λ Ori and found them to be of the order of 10 4 M , which is inconsistent with our work as well as the works we compare to. We have not identified a clear reason for this and will be exploring it in a forthcoming paper. For the Cygnus X region Schneider et al. (2016) derived a total mass of 7.9 × 10 6 M using CO 2-1 and 3-2 data which we have scaled up to our d max of 2200 pc. However the l, b range employed by Schneider et al. (2016) is about a degree shifted from ours; thus recalculating our mass to match this l, b range we derive 8.7 +0.7 −0.6 × 10 6 M , making the two masses now directly comparable. The two masses are consistent with each other with our mass being 1.1 times larger compared to Schneider et al. (2016). Further the mass of Cygnus X is more than 10 times larger than the masses of all three of our other regions, as expected given that Cygnus X is the most massive SFR within 2 kpc while the other regions are low-or intermediate-mass. Comparison to the Planck legacy products In addition to its all-sky flux maps, the Planck sub-mm survey has produced maps of various physical quantities, such as dust optical depth, at a resolution of around 10 arcmin (Planck Collaboration et al. 2016). We compare these here to our dust maps. A natural product to compare to extinction is the sub-mm optical depth of dust at 353 GHz (τ 353 ). Like extinction, this measures the column density, but along the entire line-of-sight instead of just to certain stars. Both of these quantities depend only on the dust column density and the dust properties at their respective wavelengths. Extinction is proportional to the dust cross sections at optical wavelengths, and τ 353 is proportional to the dust cross sections at sub-mm wavelengths. This means that taking the ratio the two cancels out the column density, leaving only the dust properties. We show the Planck τ 353 maps of our star-formation regions in Fig. 15. It also includes ratio maps of our predicted extinction to Planck τ 353 . This ratio is proportional to the ratio of the optical to the sub-mm dust cross-section. A similar comparison of the predicted extinction to Planck sub-mm flux and predicted dust emission is shown in figs D.1, D.2. In Fig. 15 we can identify in the Planck optical depth map the same features we see in our extinction map. The global structure of the clouds is reproduced well, but the finer details e.g. of filaments in Taurus TMC 2 are not; our model is not able to resolve such small structures on account of the length scale we have adopted, but instead encloses several features that are distinct in the Planck data as a single overdensity. Similar behaviour is seen in the smallest substructure in Perseus as well. The ratio of extinction to τ 353 changes significantly between high and low extinction lines-of-sight, as seen in figure 15. Lines-of-sight that intersect with the molecular clouds (dense regions) all have rather low ratios of extinction to sub-mm (dust) emission, while the values are much higher, by up to a factor of 10, in the lines-of-sight with low density (diffuse) regions. The most likely physical explanation for the variation in the A 0 τ 353 ratio between dense and diffuse regions is a change in the ratio of the cross-sections between the visual and the sub-mm between these lines-of-sight, for example as a result of changes in grain size or composition. As grains grow, the ratio of the optical to sub-mm cross sections tends to shrink, because the optical properties change from Rayleigh to Mie scattering and eventually to the geometric-optics regime, yet the sub-mm opacities stay in the Rayleigh regime. This relationship between opacity and extinction (and hence changes in grain properties) has been described in the near-infrared by, for example, Lombardi et al. (2014) in an unresolved sense when looking only within the cores of molecular clouds. We can extend this relationship to the wider environments of our star formation regions and their surroundings. If the dust properties are forced to stay the same, the observed trends could also be explained if there is additional material at larger distances not traced in our extinction maps, but that is detected by Planck in emission; this would, however, require an unlikely preference for additional dust clouds only along lines-of-sight that already have dust clouds near to them. Following the argument above, the variation in the ratio of optical extinction to sub-mm emission allows us to map changes in grain properties in detail on large scales in the ISM and molecular clouds. This probes the dust processing (probably grain growth) that occurs as density increases. As the grains get larger, the opacities change, and the range of variation tells us something about how much they change. However as the plots are normalised they probe only the changes in size and not the absolute sizes. Models of star formation show that grains start off in the diffuse ISM (smaller grains and standard composition, higher ratios) and as the cloud collapses the density increases, allowing them to grow larger or change in composition (lower ratios) (e.g., Guillet et al. 2007;Hirashita & Kuo 2011). What we see in the plot is that the grain size changes relatively little in the outer layers of the clouds (but still enough to differentiate them from the diffuse material around it) and as you go deeper into the cloud the processing accelerates at higher densities, so that the ratio changes by a much larger factor moving from the outer layers of the cloud into their cores than it does when moving from the diffuse medium to the outer parts of the cloud. In effect, we are mapping the trend between A (λ) /τ 353 and R V identified by Zelko & Finkbeiner (2020), showing that this trend can be used to trace where and when grain processing occurs in detail. The trend persists on much smaller scales than those identified by Schlafly et al. (2016), more similar to the assumption of star-forming regions as the typical scale. Conclusions We have introduced an algorithm that reconstructs the 3D dust density distribution of our Galaxy from the extinction to stars sources along multiple lines-of-sight. Based on latent variable GPs and variational inference techniques, our algorithm enforces positive density and thus monotonically non-decreasing extinction along any given line-of-sight. Our approach scales favourably in computation time and memory up to large sam-Article number, page 11 of 33 A&A proofs: manuscript no. Main ples and distances. It is built on publicly-available software for GPs, machine learning, and Bayesian inference, thereby ensuring portability and reproducibility while maximising stability of the code. We applied our algorithm to four well known star forming regions in the Milky Way: Orion, Cygnus X, Perseus, and Taurus. Our predicted 3D dust densities improve on previous work by recovering features in 3D density, for example filaments in Orion and condensations in Taurus, that are otherwise lost in 2D extinction maps. Our maps shed light on debates over the structure of Cygnus X, suggesting that the main star-forming regions are at 1300-1500 pc and that there is a collection of more distant clumps which likely contributes to its on sky projection. We find that the filament seen in extinction between Taurus and Perseus is a unified structure that lies upon a line joining the two regions and may connect the two clouds. We suggest that several clusters in the Orion region outside the main star-forming regions may be components of a large filament at around 300 pc from us; distance measurements to these clusters are required to confirm this. Based on the predicted extinction maps, we estimate masses for the clouds. Our estimates are comparable to those in the literature, but higher by a factor of 1.1 -1.5, with a typical uncertainty of 10% but 35% in the case of Orion. This may be a result of the inclusion of WISE photometry in the determination of the input extinctions, improving the recovery of the highest column densities. Comparing our results with the Planck maps shows that lines-of-sight that intersect dense clouds are systematically different from those that do not, most likely as a result of differences in grain size or composition; dense regions have much smaller ratios between optical and sub-mm cross-sections, as expected if the average grain size is larger or the composition is different in dense regions. This shows that the previously observed trend toward a lower ratio of optical extinction to sub-mm optical depth when passing from diffuse regions to more dense regions also holds in a resolved sense. The flexibility of our method means that it can easily be extended. For example, future work could use multiple kernels to account for different structures in the interstellar medium, enabling structures of variable sizes to be recovered more effectively, something that is particularly relevant when mapping a large fraction of the Galaxy in one go. Alternatively, the model could take into account the wavelength dependence of extinction (R 0 ) to map changes in the grain-size distribution in three dimensions. Fig. 2 : 2Variation of ELBO through model iterations. Fig. 3 : 3Top: Histograms of true extinctions and predicted extinctions; Bottom: Residual, i.e. predicted extinctions -true extinctions, for the set of simulated sources. Fig. 4 : 4Top: Simulated extinction map up to 500 pc; Middle: Our model reconstructed Extinction map up to 500 pc; Bottom: Residual of Model Predicted -Simulated Extinction. inferred dust densities and the line-of-sight extinctions computed from these. These maps recover the structure of the Orion region well when compared to other literature results, such as L11, S15 and Zucker et al. (2020), as well as the Planck dust emission (shown in Fig. D.2; Planck Collaboration et al. 2016). , the Taurus cloud complex extends from 110-190 pc, with the highest density region at 150-180 pc. This is consistent with Yan et al. (2019), who found a mean distance of 145 pc. The dense cloud at l = 174 • , b = −16 • in Table 1 : 1Summary of model setup and behaviour for the selected star formation regionsOrion Cygnus X Perseus Taurus l bounds ( • ) 180 ≤ l ≤ 217 73 ≤ l ≤ 87 154 ≤ l ≤ 164 165 ≤ l ≤ 180 b bounds ( • ) −25.5 ≤ b ≤ −3.8 −4 ≤ b ≤ 6 −25 ≤ b ≤ −14 −20 ≤ b ≤ −10 d bounds (pc) 250 ≤ d ≤ 550 800 ≤ d ≤ 2200 180 ≤ d ≤ 500 40 ≤ d ≤ 350 stars incorporated from (pc) 270 1200 200 90 training n l , n b , n d (cells) 40, 40, 55 40, 40, 65 30, 30, 35 30, 30, 35 training resolution (pc) 2.3 3.5 1.0 0.23 number of sources 87283 703291 92829 107051 initial scale length (x, y, z; pc) 10 15 10 5 final scale length Table 2 : 2Locations and approximate sizes of the main components identified in the star formation regionscentral l, b Component coordinates ( o ) los extent (pc) Notes Orion: Orion complex 200,-15 280 -500 Orion A 210,-20 340-450 Orion A bubble 215,-20 380-430 Orion B comp 1 206,-16 380-400, 430-475 separated into two segments along los Orion B comp 2 204,-11 380-420 λ Ori 195,-12 360-470 λ Ori center 195,-12 360-390 AB arc 210,-15 400-450 Filament λ l =186 270-300 could hold LDN 1558 and TGU L156 clumps Cygnus X: Cygnus X complex 80,0 1300 -1500 Cyg X North 82,0 1500 densest region Cyg X South 79,-1 1350 densest region NAN 0,85 <800 located closer than Cygnus X Perseus: Perseus complex 160,-20 300 -350 IC348 160,-18 325 NGC 1333 158,-20 325 California filament 158,-16 250-350 Taurus filament 163,-17 300-330 Perseus filament 157,-24 300-320 connecting Perseus and Taurus Taurus: Taurus complex 172,-15 110-190 densest at 150-180 pc TMC 1 174, -14 145-190 TMC 2 main 174,-16 110-145 L1498 clump 170,-19 195 part of TMC 2 B215/L1506 extended clump 172,-17.5 195-220 Taurus filament 165,-17 220-250 connecting Perseus and Taurus Arcs 1+2 b < −14 • 190-250 could hold L1540, L1507 and L1503 Table 3 : 3Inferred MassesRegion Dust Mass (10 3 M ) Total Mass (10 3 M ) Orion 9.1 +3.2 −2.2 1130 +400 −270 Cygnus 88.2 +7.0 −6.2 10900 +900 −800 Perseus 1.5 +0.1 −0.1 187 +17 −14 Taurus 1.2 +0.1 −0.1 149 +17 −14 available online at http://www.gaussianprocess.org/gpml/ 2 see the Gpytorch documentation for the n-dimensional generalisation of the RBF kernel: https://docs.gpytorch.ai/en/stable/ kernels.html Article number, page 2 of 33 Dharmawardena et al.,: 3D dust density The choice of these values affect the final reconstruction; we recommend using the largest grid that fits in memory during integration on the machine used for training. available at http://dc.zah.uni-heidelberg.de/ Article number, page 5 of 33 A&A proofs: manuscript no. Main Article number, page 6 of 33 Dharmawardena et al.,: 3D dust density data obtained from https://www.astro.princeton.edu/ draine/dust/dustmix.html Article number, page 10 of 33 Dharmawardena et al.,: 3D dust density Acknowledgements. This project is partially funded by the Sonderforschungsbereich SFB 881 "The Milky Way System" of the German Research Foundation (DFG).Appendix A: The approximate posteriorWe use two approximations to compute our posterior. The first is that the posterior distribution over log 10 (ρ) = φ can be approximated as a Gaussian. The second is to compute the covariances of the GP prior only using a subset of points at which the prior is defined.The first approximation arises as follows. We adopt a Gaussian likelihood for the observed extinctions, A obs , which have uncertainties σ obs , i.e.where A mod are the corresponding extinctions predicted by the model, and the sum is taken over the measured extinctions for different lines-of-sight. Our goal is to infer the densities, ρ(s), at various distances s along the different lines-of-sight. For one line-of-sight, the integral of these dust densities gives the extinction to a star at distance d,Thus the likelihood can be writtenIn principle we could use this to solve for ρ mod , for which we require some regularisation of these model densities because they are otherwise degenerate. Specifically, if we adopted a Gaussian prior on ρ mod we have a Gaussian posterior in ρ mod and so a straight forward closed form solution (see sections 2.2 and 2.3 of (RezaeiKh. et al. 2017)). However, we also want to impose non-negativity on ρ, and so we choose to infer φ = log 10 ρ instead of ρ. The likelihood iswhich makes it explicit that the posterior will not be Gaussian in φ when we have a Gaussian prior in φ.We therefore have two options for computing the posterior: we could solve for the distribution of φ at all points in space by sampling the full posterior (e.g. using MCMC), or we could approximate the posterior with something more tractable. We choose the second option, and approximate the posterior as a Gaussian using variational inference. That is, when we compute the likelihood from equation A.4 and combine it with the prior, we assume that the resulting posterior has a Gaussian distribution. This is our first approximation.Ideally we would calculate the covariances of the GP using all the grid points, but for computational time reasons we do this instead only at a subset of points, known as the inducing points. Using an iterative procedure we then find the set of inducing points, as well as the mean and covariance of the distribution at these points that best approximates the exact posterior over all the data. This is our second approximation. This procedure is implicit to Gpytorch.Once these two approximations have been made, the Gaussian approximation of the posterior can be writtenwhere φ i is the set of log densities log 10 ρ at the inducing points whose means and covariances are those of the variational distribution (see Sect. 2.3), x i is the set of positions of the inducing points, and x is the set of other positions at which the posterior is computed (e.g. the grid points). The mean of the posterior iswhere M is the number of inducing points and K x = K(x i , x) is an M × N matrix where N = n l × n b × n d is the total number of points in the grid. The covariance of the posterior isAppendix B: Additional figures showing the behaviour of the model with simulated data introduced in Sec 3In this appendix we present the additional figures which demonstrate the behaviour of our model using the simulated data and their model predictions introduced in Sec. 3. First we show the simulated vs. model predicted extinctions and densities as a function of distance for a selected set of lines-of-sight. This is followed by simulated and model predicted densities as a function of l and b for the same set of distance points asFig 5.Appendix B.1: Extinction and density along selected lines-of-sight for the simulated and model predicted data introduced in Sec 3In this subsection we present the extinctions and densities along a set of selected lines-of-sight for our simulated data in comparison to our model predicted data of the simulations. From these figures we infer a typical uncertainty of 20% in density. 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[ "Thermodynamic behavior of the XXZ Heisenebrg s = 1/2 chain around the factorizing magnetic field", "Thermodynamic behavior of the XXZ Heisenebrg s = 1/2 chain around the factorizing magnetic field" ]	[ "J Abouie \nDepartment of physics\nShahrood University of Technology\n36199-95161ShahroodIran\n\nSchool of physics\nInstitute for Research in Fundamental Sciences (IPM)\n19395-5531TehranIran\n", "A Langari \nDepartment of Physics\nSharif University of Technology\n11155-9161TehranIran\n", "M Siahatgar \nDepartment of Physics\nSharif University of Technology\n11155-9161TehranIran\n" ]	[ "Department of physics\nShahrood University of Technology\n36199-95161ShahroodIran", "School of physics\nInstitute for Research in Fundamental Sciences (IPM)\n19395-5531TehranIran", "Department of Physics\nSharif University of Technology\n11155-9161TehranIran", "Department of Physics\nSharif University of Technology\n11155-9161TehranIran" ]	[]	We have investigated the zero and finite temperature behaviors of the anisotropic antiferromagnetic Heisenberg XXZ spin-1/2 chain in the presence of a transverse magnetic field (h). The attention is concentrated on an interval of magnetic field between the factorizing field (h f ) and the critical one (h c ). The model presents a spin-flop phase for 0 < h < h f with an energy scale which is defined by the long range antiferromagnetic order while it undergoes an entanglement phase transition at h = h f . The entanglement estimators clearly show that the entanglement is lost exactly at h = h f which justifies different quantum correlations on both sides of the factorizing field. As a consequence of zero entanglement (at h = h f ) the ground state is known exactly as a product of single particle states which is the starting point for initiating a spin wave theory. The linear spin wave theory is implemented to obtain the specific heat and thermal entanglement of the model in the interested region. A double peak structure is found in the specific heat around h = h f which manifests the existence of two energy scales in the system as a result of two competing orders before the critical point. These results are confirmed by the low temperature Lanczos data which we have computed.	10.1088/0953-8984/22/21/216008	[ "https://arxiv.org/pdf/0809.1987v2.pdf" ]	206,031,099	0809.1987	2354546f47efe0372ded556de644b3b55d236da8	Thermodynamic behavior of the XXZ Heisenebrg s = 1/2 chain around the factorizing magnetic field 23 Apr 2010 J Abouie Department of physics Shahrood University of Technology 36199-95161ShahroodIran School of physics Institute for Research in Fundamental Sciences (IPM) 19395-5531TehranIran A Langari Department of Physics Sharif University of Technology 11155-9161TehranIran M Siahatgar Department of Physics Sharif University of Technology 11155-9161TehranIran Thermodynamic behavior of the XXZ Heisenebrg s = 1/2 chain around the factorizing magnetic field 23 Apr 2010arXiv:0809.1987v2 [cond-mat.str-el]PACS numbers: 7510Jm, 7540Cx We have investigated the zero and finite temperature behaviors of the anisotropic antiferromagnetic Heisenberg XXZ spin-1/2 chain in the presence of a transverse magnetic field (h). The attention is concentrated on an interval of magnetic field between the factorizing field (h f ) and the critical one (h c ). The model presents a spin-flop phase for 0 < h < h f with an energy scale which is defined by the long range antiferromagnetic order while it undergoes an entanglement phase transition at h = h f . The entanglement estimators clearly show that the entanglement is lost exactly at h = h f which justifies different quantum correlations on both sides of the factorizing field. As a consequence of zero entanglement (at h = h f ) the ground state is known exactly as a product of single particle states which is the starting point for initiating a spin wave theory. The linear spin wave theory is implemented to obtain the specific heat and thermal entanglement of the model in the interested region. A double peak structure is found in the specific heat around h = h f which manifests the existence of two energy scales in the system as a result of two competing orders before the critical point. These results are confirmed by the low temperature Lanczos data which we have computed. Introduction The zero temperature phase diagram (i.e. quantum phase diagram) of a model gives important information on the low temperature behaviors of the system [1,2]. The anisotropic antiferromagnetic Heisenberg (XXZ) spin-1/2 model shows different quantum phases with respect to a symmetry-breaking transverse field (non-commuting magnetic field) [3,4,5]. The non-commuting field imposes quantum fluctuations into the ground state which can induce new phases. Cs 2 CoCl 4 is a quasi-one dimensional spin-1/2 XY-like antiferromagnet with weak inter-chain couplings (J ′ /J = 0.014) which can be studied in terms of XXZ chain with anisotropy parameter ∆ = 0.25 [6,7]. The scaling behavior and quantum phase diagram of the XXZ model in the presence of a transverse magnetic field (h x ) have been investigated [3,4,5,8]. Recently, the quantum spin models received much attentions from the quantum information point of views. These are prototype models to implement and examine the idea of quantum computations which requires the quantum correlations measured by entanglement. Hence, disentangled ground states have to be avoided for such implementations. On the other hand, the zero entanglement property of a ground state provides an exact form for it in terms of product of single particle states. The investigation in this direction resumed recently following the seminal work of J. Kurman and his collaboratores [9]. Several efforts have been devoted to this direction which is important for condensed matter researchers, i.e finding an exact (factorized) ground state even at particular values of the coupling constants [10,11,12,13]. The factorized (exact) ground state is an accurate starting point to investigate the quantum nature of a phase close to the factorizing point in addition to some exact knowledge which gives at the factorized point. This property is implemented in this article to initiate a spin wave theory to describe the thermodynamic properties of the XXZ model in the presence of a transverse magnetic field. The U(1) symmetry of the XXZ model is lost upon adding the transverse magnetic field. Initially, a perpendicular antiferromagnetic order is stabilized by promoting a spin-flop phase (which has a partial moment projection along the field direction). At the factorizing field (h f ) in the spin-flop phase the ground state is known exactly as a direct product of single spin states and the staggered magnetization along y-direction is close to its maximum value. In our model the factorizing field is h f = J 2(1 + ∆), where quantum fluctuations are uncorrelated and the ground state is the classical one (J is the scale of energy and ∆ is the anisotropy parameter). For slightly larger magnetic field very close to the critical one (h f < h < h c ) the antiferromagnetic order becomes unstable and the staggered magnetization falls rapidly to vanish at the critical point. For h > h c the spins become aligned in x direction and a fully polarized phase will be appeared. The region between the factorizing and the critical fields (h f < h < h c ) is the main issue of our study which is not well understood so far. The zero temperature properties of the intermediate region (h f < h < h c ) induces its signature into the thermodynamic functions of the model. We have found that this region is characterized by two energy scales and its fingerprint will appear as a double peak in the specific heat. Moreover, the existence of more than an energy scale in the model can be related to a spontaneous symmetry breaking (SSB) [10]. Each broken phase is described by an order parameter which can be zero in the disordered phase. For the aforementioned model the symmetry breaking occurs where the staggered magnetization becomes zero. The thermodynamic properties of the spin 1/2 XXZ chain in a transverse magnetic field have been studied using the Low temperature Lanczos method [15]. However, in this paper we will focus on the intermediate values of h and more precisely on the region where a double peak appears in the specific heat of the model. We implement the exact factorized ground state at h = h f (where the entanglement vanishes) to build up a spin wave theory appropriate to describe the model in the intermediate values of the magnetic field. In the linear spin wave approximation we calculate the specific heat and thermal entanglement of the model. Moreover, the spin wave theory gives two different types of quasi particles which are representing the two energy scales. We have also studied both the zero and finite temperature properties of the model on a finite chain using the low temperature Lanczos method [14]. Our numerical results are in agreement with the spin wave theory counterparts. In the next section, we will briefly review the zero temperature properties of the model from the quantum information point of view. We will provide the spin wave theory in Sec.3 where the quantum property, specific heat and thermal entanglement are obtained. Finally, the numerical Lanczos results will be presented in addition to discussions on the mentioned topics. Zero temperature phase diagram The anisotropic spin-1/2 Heisenberg model in the presence of a transverse magnetic field is described by the following Hamiltonian, H = J i (S x i S x i+1 + S y i S y i+1 + ∆S z i S z i+1 + hS x i ),(1) where S α i 's are spin-1/2 operators, ∆ is the anisotropy parameter, h is proportional to the transverse magnetic field and the exchange antiferromagnetic coupling J which defines the scale of energy is set to one. This model has several quantum phases with respect to the transverse magnetic field and anisotropy parameter (∆). The αcomponent spin structure factor at momentum p is defined by G αα (p) = N x=1 S α 1 S α 1+x e ipx ,(2) and its increasing behavior in term of system size shows the magnetic order at the specified momentum. We have plotted in Fig.(1) the magnetization along x direction (M x = (1/N) S x i ) and the y-component of spin structure factor at momentum p = π versus h. The value of ∆ = 0.25 has been fixed to fit the case of real material Cs 2 CoCl 4 . The numerical data has been obtained by zero temperature Lanczos method on a finite chain of length N = 20 with periodic boundary condition. The magnetization curve can be distinguished in three parts: (1) Magnetization for 0 < h < h f , (2) Magnetization for h f < h < h c , and (3) the paramagnetic phase for h > h c . At zero magnetic field, there is no order in the system and all order parameters are zero i.e M x,y,z = 0. The nonzero value of h starts to align the spins in x direction and induces a small ferromagnetic order in x direction. The magnetization, M x , increases monotonically by increasing the magnetic field. For h < h f the quantum effects are considerable and the magnetization changes parabolically versus h. Increasing h, suppresses the quantum correlations and they goes to zero around h f . For h f < h < h c , the magnetization is increased linearly versus h. Magnetization increases up to the critical point (h c ) and saturates at infinite field. Although the staggered magnetization along y-direction becomes zero at the finite critical field, the magnetization in x-direction will be fully saturated only for the isotropic case (∆ = 1) [13]. In other words, the full saturation in x-direction for ∆ = 1 will happen for h → ∞. For h > h c all of spins align almost completely (for ∆ = 1) in the x direction and we have a fully polarized paramagnetic phase. We would like to draw your attentions to the region h f < h < h c , where the model behaves surprisingly. Let us first study the spin chain through the entanglement (τ 2 ) of two spins [16,17] which have no classical counterpart. The entanglement is defined by the following relation, τ 2 = i =j C 2 ij .(3) Where, C ij is the concurrence [18] which is implemented instead of the pairwise zero temperature entanglement between two spins at sites i and j. For XXZ model, M z is zero thus the concurrence takes the form [17], C ij = 2 max{0, C (1) ij , C (2) ij }, where C (1) ij = S x i S x j + \| S y i S y j − S z i S z j \| − 1 4 , C (2) ij = \| S y i S y j + S z i S z j \| − ( 1 4 + S x i S x j ) 2 − (M x ) 2 .(4) Using quantum Monte Carlo simulation, T. Roscilde and collaborators [10] have shown that unlike the standard magnetic order parameters (Fig.1) the pairwise entanglement, plays an essential role at the factorizing field. At the factorizing field (also called classical field, h cl = h f = 2(1 + ∆)) [9] the ground state takes a product form [9,13] and its entanglement is zero. In Fig.(2 ), we have plotted C (1) i,i+1 and C (2) i,i+1 versus the transverse field h for XXZ spin-1/2 chain with ∆ = 0.25 by using exact diagonalization Lanczos method. At the classical field (h = h f ≃ 1.58), C i,i+1 = C (1) i,i+1 = C (2) i,i+1 = 0. At this point the ground state of the model is factorized (disentangled) where the Neel order is roughly maximized in y direction. i,i+1 = C (2) i,i+1 = 0.(1) Let us describe the behaviors of C (1) ij and C (2) ij from a spontaneous symmetry breaking (SSB) point of view. It is found in Ref. [10] that the competition between the two functions C (1) ij and C (2) ij demonstrates the appearance of a SSB. In other words, for our model, a SSB can be occurred when C (2) ij < C (1) ij . At this condition the magnetic field is greater than the factorizing value (h > h f ). Moreover, SSB is usually recognized to happen at the position where the order parameter becomes zero. In our model two standard order parameters, M x and SM y (staggered magnetization along y direction) can represent the quantum phases of the model. We have plotted in Fig.(1) the magnetization along x direction (M x ) and the y-component of spin structure factor (G yy ) at momentum π for ∆ = 0.25 XXZ spin-1/2 chain. M x is nonzero in the whole range of magnetic fields however, G yy (π) which is used to show the antiferromagnetic order of the system has a different behavior. As it is observed from our Lanczos data, G yy (π) increases by h up to h f . This behavior is also found in the SM y curve which has been obtained by density matrix renormalization group (DMRG) [8]. Thus, no SSB occurs for h < h f . By increasing the magnetic field for h > h f , G yy (π) (or equivalently SM y ) decreases rapidly and falls to zero at the critical point h c . In this respect, a symmetry breaking can occur only for h > h f . In this region, as mentioned before the slop of M x with respect to h is different from the corresponding one for h < h f . Spin waves theory As discussed in the previous section the entanglement is zero at the factorizing field (h f ). It allow s us to write the many body ground state as the direct product of the single spin states on odd and even sublattices [9] \|GS = i∈odd,j∈even \|S i \|S j(5) The spin state on each sublattice is expressed in terms of polar angles (θ, φ) which defines the rotation of the up-spin eigen-state of S z to the specific direction defined by the factorized state. Let us label the odd (even) sublattice by A(B). Thus, (θ, φ) represents A-sublattice while (β, α) is the corresponding one for B-sublattice. It has been shown [13] that one can consider φ = 0 = α without loss of generality. Moreover, the magnitude of remaining polar angles are equal \|θ\| = \|β\| in the case of homogeneous spin model (like here) and is given by cos(θ) = − 1 + ∆ 2(6) For the antiferromagnetic model the factorized ground state is defined by β = −θ. Before starting the spin wave approach we implement a unitary transformation on the Hamiltonian. All spins on the A-sublattice are rotated with angle θ counterclockwise around y-direction and clockwise for spins on B-sublattice. The rotated Hamiltonian (H) is the result of rotations on all lattice points,H =D † HD and D = i∈A,j∈B D i (−θ)D j (θ). The single spin rotation operator is D i (θ) = exp(−iθS y i / ). In the rotated spin model, the Hamiltonian will have a factorized ground state which is a ferromagnet at the factorizing field. The rotated spin Hamiltonian is written in terms of boson operators a, b with the following Holstein-Primakoff transformation, S + Ai = (2S − a † i a i ) 1/2 a i ,S x Ai = S − a † i a i , S + Bj = (2S − b † j b j ) 1/2 b j ,S x Bj = S − b † j b j , whereS A(B) = D † A(B) S A(B) D A(B) are the rotated spin operators and D A(B) is the unitary single spin rotation operator. The bosonic Hamiltonian in the linear spin wave approximation, i.e.S + Ai ≃ √ 2Sa i ;S + Bi ≃ √ 2Sb i , is given bỹ H = N[ ∆ 2 + h h f (1 + ∆)] + N l=0 [ h h f (1 + ∆) − ∆](a † l a l + b † l b l ) − ∆ 2 [a l (b † l + b † l+1 ) + h.c] + 1 2 ( h h f − 1) √ 1 − ∆ 2 (a l + b l + h.c) .(7) To diagonalize the bosonic model we first implement the Fourier transformation and then apply a rotation to the boson operators (a k , b k ) a l = 1 √ N k e −ikl a k , b l = 1 √ N k e −i(kl+ k 2 ) b k , ψ k = cos η k a k − sin η k b k , χ k = sin η k a k + cos η k b k .(8) The diagonalized Hamiltonian in terms of two sets of quasi-particle operators is given byH = E 0 + k (ω + k χ † k χ k + ω − k ψ † k ψ k ),(9) where the excitation spectrums have the following forms ω ± k = h h f (1 + ∆) − ∆ ± ∆ cos k 2 , E 0 = N 2 (∆ − hh f ) + ω + 0 (t + ) 2 + 2N(1 − ∆ 2 )( h h f − 1)t + , t + = N 2 (1 − ∆ 2 )(1 − h h f ) ω + 0 .(10) In the above calculations, a translation χ 0 → χ 0 + t + has been performed in the diagonalization procedure of the Hamiltonian (7). The Hamiltonian is now represented in terms of two quasi-particles (bosons), each defines an energy scale. The two energy scales which are the excitation energies of each bosons lead to two different dynamics for the system. The consequence of two types of dynamics will be seen in the structure of specific heat which will be studied. However, the existence of two different quasi-particle energies (ω + k = ω − k ) is the first sign of two dynamics in the model. Specific heat To get the finite temperature properties of the model, we assumẽ n ± k = n − ,n + n ± k P k (n + , n − ), for the spin wave distribution functions, where P k (n + , n − ) is the probability of parallel (n + k = χ † k χ k ) and perpendicular (n − k = ψ † k ψ k ) normal modes appearing in the k-momentum state which satisfies n + ,n − P k (n + , n − ) = 1 for all k's. The substitutions ofñ + k = χ † k χ k andñ − k = ψ † k ψ k (where · · · represents the thermal average) in the spin-wave Hamiltonian (9) gives the free energy, F = E 0 + k (ω + kñ + k + ω − kñ − k ) + T k n + ,n − P k (n − , n + ) ln P k (n − , n + ). The number of bosons are controlled by the following constraint which is the magnetization in x-direction, M x = s − 1 2N k (ñ + k +ñ − k ) − (t + ) 2 2N .(11) The free energy is minimized with respect to P k (n + , n − )s under the constraint of (11) which is applied by a Lagrange multiplier (µ) via the boson's occupation number n ± k = 1 e 1 k B T (ω ± −µ) − 1 .(12) The constraint (11) is applied by the values of M x (h, T ) which have been obtained by the numerical Lanczos method. We have plotted in Fig.(3) the specific heat of ∆ = 0.25 XXZ spin-1/2 chain versus temperature and different values of magnetic field. A double peak is observed in the specific heat which is the signature of the existence of two comparable energy scales. More precisely, the specific heat for h = 1.4 and 1.5 have a narrow peak at low temperature (T ≃ 0.2) and a broaden one at higher T . We will discuss more on this point in the next section. 3.0.2. Thermal entanglement The established spin wave theory close to the factorizing point is used to obtain the thermal behaviors of the correlation functions in Eqs. (4) and consequently to calculate the concurrence. This method can describe the thermal entanglement of two spins in linear spin wave approximation. In this approach, one can find the following expression for C (1) ij and C (2) ij C (1) ij ≃ − 1 2N k (ñ + k +ñ − k ) + (t + ) 2 , C (2) ij ≃ 1 2N k Cos(k · r + k/2)(ñ + k −ñ − k ) + (t + ) 2 . As it is seen from these functions, C ij is always less than zero thus the concurrence is 2 max{0, C (2) ij }. We have plotted in Fig.(4) uncorrelated and the thermal entanglement is very close to zero (∼ 10 −3 ). It means that the thermal entanglement is mainly originated from the ground state and the excited states have very tiny contribution to the entanglement. Summary, Discussions and Lanczos results We have studied the effects of transverse magnetic field on the zero and finite temperature properties of ∆ = 0.25 XXZ spin-1/2 chain. We have focused our attentions on the intermediate region of h f < h < h c where the model behaves more interestingly. M. Kenzelmann and his collaborators [6] have investigated experimentally the effects of a transverse magnetic field on the quasi-one dimensional spin-1/2 antiferromagnet Cs 2 CoCl 4 , using single-crystal neutron diffraction. Due to the weak inter-chain couplings in Cs 2 CoCl 4 (J ′ /J = 0.014) [7] where J is the coupling within a chain, it is proposed in Ref. [6] that the bulk material has a spin liquid phase at the interval h f < h < h c . In a spin liquid phase all order parameters should be zero and there should be no long range order in the system. The existence of very weak coupling between magnetic chains in this material makes it feasible to be described by a one dimensional spin-1/2 XXZ model with the anisotropy parameter ∆ = 0.25. However, for the 1D XXZ model in the region h f < h < h c both magnetization and staggered magnetization are nonzero. Thus the ground state of the model could not be a spin liquid phase. Although the behaviors of the magnetic orders for this region are clearly known, the lacuna of a perfect study of the properties of the system at the intermediate region is still felt. In this respect, we have devoted our attentions to survey the magnetic and thermodynamic behavior of the system at the intermediate region of the transverse field. We have implemented the low temperature Lanczos method (LTLM) [14,15] to compute the thermodynamic behaviors of the model for a chain of finite length. LTLM has been used since it is accurate for low temperatures and specially the thermodynamic averages reach the ground state expectation values as temperature approaches zero. The specific heat versus temperature for different values of the magnetic field on a chain of N = 20 and with ∆ = 0.25 has been plotted in Fig. (5). The sampling is taken over R = 100 Lanczos numerical data which have been obtained by different initial random states. The finite size effect can be ignored since the number of sampling (R) is rather high. The result of the spin wave theory (Fig.3) and LTLM (Fig.5) are in mutual agreement; moreover, both figures show the presence of double peak in the specific heat for h = 1.4, 1.5 which is an evidence for the existence of two scales of energy or equivalently two dynamics in the system. The specific heat versus T for all values of the magnetic fields has Schottky like peak at low temperatures which is a remarkable feature of the antiferromagnetic behavior. For small values of the magnetic field (h < 1.4) the Schottky anomaly is the only one which justifies the existence of a single dynamics in the model. Further increasing of the magnetic field (being close to the factorizing point) a broaden peak is emerged in the specific heat data versus T . This bump is the result of the paramagnetic order. Two different orders are in competition and its signature is specially observable for magnetic fields in the region h f h < h c where the magnitude of the two types of ordering becomes comparable. The antiferromagnetic order in y-direction is the result of exchange coupling in the broken U(1) symmetry phase. The effect of transverse magnetic field as a paramagnetic order in x-direction shows its presence when it is enough large to define a new scale of energy. Around the factorizing field the two orders manifest their influence on the model as a narrow and broaden peak of the specific heat which happens for h f h < h c . This is the region where the spontaneous symmetry breaking is started to happen. At the critical point (h c ) the antiferromagnetic order vanishes and paramagnetic order is the only representative of the model. A broaden peak in the specific heat versus T is significant for h > h c which justifies the paramagnetic order. Employing the specific heat data, one can also scan the behavior of the energy gap (E g ). The existence of energy gap in the model is clearly observed by the exponential decay of the specific heat at very low temperatures (T 0.02). At enough low temperatures (T < E g ) the specific heat and energy gap are related by [7] c v T 3/2 ∝ e − Eg T . Thus, the slop of (c v T 3/2 ) curves versus 1 T in log scale gives good information about the energy gap. In Figs.(6,7) we have plotted (c v T 3/2 ) versus 1 T for different values of transverse field. Fig.(6) shows the gap behavior close to the factorizing point, h < h f . While its behavior for h > h f is presented in Fig.(7). A turning point of different plots in Fig.(7) for different transverse magnetic field is the result of different behavior for h < h c ≃ 1.65 (decreasing gap) and h > h c (linear increasing paramagnetic gap). Our results fit very well with Figs. (5.14 and 5.15) of Ref. [7]. Motivated by neutron-scattering results [6], T. Radu [7] investigated the effects of non-commuting field on the ground state of Cs 2 CoCl 4 . To be more precise in comparison, let us refer also to Fig.13 of chapter.5 of Ref. [7] where the experimental data of the specific heat have been shown. The experimental data (specially in the inset of Fig.13 of chapter.5 of Ref. [7]) display a double peak in the specific heat versus temperature which is for the intermediate range of magnetic field (less than the critical one). It is also worth to point out the behaviors of the system from the internal energy points of view. The internal energy and specific heat are related by relation C = dU dT . The double peak structure of the specific heat presages that at the intermediate values of the transverse field, there are two temperatures where the internal energy of the system gets its maximum variation. In other words, at these temperatures the maximum amount of states contribute to the response functions of the systems. Thus, one can also conclude that the appearance of the two energy scales in the system is appropriate with the number of contributed states. Knowledge on these properties could be important in the study of the magneto-caloric effects of the XXZ model in the transverse field which is a work in progress. Figure 1 . 1(color online) Results of numerical zero temperature Lanczos method for the magnetization along x-direction (right vertical axis) and the y-component spin structure factor at momentum π (left vertical axis) versus the transverse magnetic field h for a chain with N = 20 and ∆=0.25. The magnetization in z-direction is zero. Figure 2 . 2(color online) Entanglement estimators of the XXZ spin-1/2 chain versus transverse field h with parameter ∆ = 0.25. C (1) is depicted by (black) circles and C (2) by (red) squares. At the classical field (h = h f ≃ 1.58), C Figure 3 . 3the thermal entanglement of the ∆ = 0.25 XXZ spin-1/2 chain versus transverse field at temperature T = 0.05. By increasing the magnetic field, quantum fluctuations are decreased and the thermal entanglement is declined. At the factorizing point, quantum fluctuations become approximately (color online) The spin wave results for specific heat of the XXZ model versus T for different values of the transverse magnetic field h and anisotropy parameter ∆ = 0.25. (Black) circle is for h = 1.2, (red) square: h = 1.4, (green) gradient: h = 1.5 and (blue) diamond: h = 1.6. A narrow peak at low temperature followed by a broaden one for higher temperature are observed for h = 1.4, 1.5. Figure 4 . 4Thermal entanglement of the XXZ model versus h and anisotropy parameter ∆ = 0.25 at T = 0.05 which has been obtained by spin wave approximation. Because of the tiny contribution of excited states to the entanglement, its value at h f is ∼ 10 −3 . Figure 5 . 5The numerical Lanczos results for the specific heat of XXZ model versus T for different values of transverse magnetic field and ∆ = 0.25. At the intermediate region of h, a double peak structure is observed which indicates the existence of two energy scales in the system. Figure 6 . 6c v T 3/2 of XXZ model versus 1/T for different values of h < h f and ∆ = 0.25. At low temperatures the slope of the curve shows increasing of the energy gap in the system. Figure 7 . 7c v T 3/2 of XXZ model versus 1/T for different values of h > h f and ∆ = 0.25. At low temperatures, the slop of the curve is decreased from h = 1.45 to h = 1.6 and then starts to increase by increasing of h. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 AcknowledgmentsWe would like to thank M. Rezai for his helpful comments. This work was supported in part by the Center of Excellence in Complex Systems and Condensed Matter (www.cscm.ir). S Sachdev, Quantum phase transitions. Cambridge, UKCambridge University PressSachdev S 1999 Quantum phase transitions, Cambridge University Press, Cambridge, UK . M Vojta, Rep. Prog. Phys. 66Vojta M 2003 Rep. Prog. Phys. 66 . D V Dmitriev, V Krivnov, Ya, A A Ovchinnikov, Phys. Rev. B. 65172409Dmitriev D V, Krivnov V Ya and Ovchinnikov A A 2002 Phys. Rev. B 65 172409 ; . D V Dmitriev, V Krivnov, Ya, A Ovchinnikov, A Langari, JETP. 95538Dmitriev D V, Krivnov V Ya, Ovchinnikov A A and Langari A 2002 JETP 95 538 . A Langari, Phys. Rev. B. 69100402RLangari A 2004 Phys. Rev. B 69 100402(R) . A Langari, S Mahdavifar, Phys. Rev. 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[ "Jamming-assisted Eavesdropping over Parallel Fading Channels", "Jamming-assisted Eavesdropping over Parallel Fading Channels" ]	[ "Yitao Han ", "Senior Member, IEEELingjie Duan ", "Fellow, IEEERui Zhang " ]	[]	[]	Unlike passive eavesdropping, proactive eavesdropping is recently proposed to use jamming to moderate a suspicious link's communication rate for facilitating simultaneous eavesdropping. This paper advances the proactive eavesdropping research by considering a practical half-duplex mode for the legitimate monitor (e.g., a government agency) and dealing with the challenging case that the suspicious link opportunistically communicates over parallel fading channels. To increase eavesdropping success probability, we propose cognitive jamming for the monitor to change the suspicious link's long-term belief on the parallel channels' distributions, and thereby induce it to transmit more likely over a smaller subset of unjammed channels with a lower transmission rate. As the half-duplex monitor cannot eavesdrop the channel that it is simultaneously jamming to, our jamming design should also control the probability of such "own goal" that occurs when the suspicious link chooses one of the jammed (uneavesdroppable) channels to transmit. We formulate the optimal jamming design problem as a mixed integer nonlinear programming (MINLP) and show that it is nonconvex. Nevertheless, we prove that the monitor should optimally use the maximum jamming power if it decides to jam, for maximally reducing suspicious link's communication rate and driving the suspicious link out of the jammed channels. Then we manage to simplify the MINLP to integer programming and reveal a fundamental trade-off in deciding the number of jammed channels: jamming more channels helps reduce the suspicious link's communication rate for overhearing more clearly, but increases own goal probability and thus decreases eavesdropping success probability. Finally, we extend our study to the two-way suspicious communication scenario, and show there is another interesting trade-off in deciding the common jammed channels for balancing bidirectional eavesdropping performances. Numerical results show that our optimized jamming-assisted eavesdropping schemes greatly increase eavesdropping success probability as compared with the conventional passive eavesdropping.	10.1109/tifs.2019.2901821	[ "https://arxiv.org/pdf/1902.07420v1.pdf" ]	67,769,703	1902.07420	f654826f6ec7893ae0044a1f079dd9f250ab6c91	Jamming-assisted Eavesdropping over Parallel Fading Channels Yitao Han Senior Member, IEEELingjie Duan Fellow, IEEERui Zhang Jamming-assisted Eavesdropping over Parallel Fading Channels 1Index Terms-Wireless surveillancejamming-assisted eaves- droppinghalf-duplex monitorparallel fading channelseaves- dropping success probabilitybidirectional eavesdropping Unlike passive eavesdropping, proactive eavesdropping is recently proposed to use jamming to moderate a suspicious link's communication rate for facilitating simultaneous eavesdropping. This paper advances the proactive eavesdropping research by considering a practical half-duplex mode for the legitimate monitor (e.g., a government agency) and dealing with the challenging case that the suspicious link opportunistically communicates over parallel fading channels. To increase eavesdropping success probability, we propose cognitive jamming for the monitor to change the suspicious link's long-term belief on the parallel channels' distributions, and thereby induce it to transmit more likely over a smaller subset of unjammed channels with a lower transmission rate. As the half-duplex monitor cannot eavesdrop the channel that it is simultaneously jamming to, our jamming design should also control the probability of such "own goal" that occurs when the suspicious link chooses one of the jammed (uneavesdroppable) channels to transmit. We formulate the optimal jamming design problem as a mixed integer nonlinear programming (MINLP) and show that it is nonconvex. Nevertheless, we prove that the monitor should optimally use the maximum jamming power if it decides to jam, for maximally reducing suspicious link's communication rate and driving the suspicious link out of the jammed channels. Then we manage to simplify the MINLP to integer programming and reveal a fundamental trade-off in deciding the number of jammed channels: jamming more channels helps reduce the suspicious link's communication rate for overhearing more clearly, but increases own goal probability and thus decreases eavesdropping success probability. Finally, we extend our study to the two-way suspicious communication scenario, and show there is another interesting trade-off in deciding the common jammed channels for balancing bidirectional eavesdropping performances. Numerical results show that our optimized jamming-assisted eavesdropping schemes greatly increase eavesdropping success probability as compared with the conventional passive eavesdropping. I. INTRODUCTION Security issues in wireless communication have drawn increasingly more attentions from both academia and industry. Due to the broadcast nature of wireless communication, its physical layer is vulnerable to eavesdropping (interception of confidential information) and jamming (interruption of legitimate transmission) [1], and there are many works focusing on defence schemes against eavesdropping and jamming, such as secrecy beamforming [2]- [4], channel-based secret key [5], Y. Han and L. Duan are with the Engineering Systems and Design Pillar, Singapore University of Technology and Design (e-mail: yitao [email protected], lingjie [email protected]). R. Zhang is with the Department of Electrical and Computer Engineering, National University of Singapore (e-mail: [email protected]). using cooperative networks to avoid eavesdropping [6] and using hopping to avoid jamming attacks [7]. These works view eavesdropping or jamming as malicious attacks and assume all the communication links are rightful (see [8]- [13]). However, they overlook the emerging case that wireless links or devices can be established and used by criminals or terrorists to present severe public security threats. With the fast development of wireless technologies and devices, user-controlled or infrastructure-free communications (e.g., ad hoc network and short-range communication) now become popular. For examples, mobile applications such as MeshMe and FireChat can network users in the vicinity with reliable mutual connection, and drones can take nice photos or videos and send back to their users. While providing great convenience to normal users, these new technologies and devices can be misused to commit crimes. Terrorists can use them to facilitate their plotting and acts, and spies can use them to send out commercial or military secrets. Since the data do not go through any public infrastructure under internet service providers (ISPs), it is difficult to be monitored by government surveillance program. As a result, prior methods (e.g., deploying dedicated wiretapping devices in network infrastructure) for eavesdropping infrastructurebased communications (e.g., cellular networks) no longer work. There is thus a growing need for authorized parties to develop new approaches to legitimately eavesdrop these infrastructure-free suspicious wireless communications. For example, in the USA, the National Security Agency has launched Terrorist Surveillance Program and aims to intercept all wireless devices [14] to protect public security. Traditionally, passive eavesdropping is used for such surveillance purpose but it does not provide good eavesdropping performance once the suspicious transmitter (ST) is far away from the monitor or hops to an undesirable channel. Recently, a novel approach called proactive eavesdropping via jamming is proposed in [15], [16], where the legitimate monitor, ideally operating in full-duplex mode, uses jamming to moderate the suspicious communication rate for facilitating simultaneous eavesdropping. [17] extends this work by assuming the legitimate monitor's knowledge of full channel state information (CSI) and designs adaptive jamming power in each fading block. [18] studies the case that the suspicious link adopts HARQ-based communication. [19] further considers that the monitor is equipped with multiple antennas to achieve more efficient jamming and better eavesdropping performance. [20] proposes another efficient eavesdropping method, where the monitor disguises as a fake relay to overhear the suspicious communication. These works largely assume that there is only one communication channel between the ST and suspicious receiver (SR), and they require the monitor to operate in interference-free full-duplex mode for enabling simultaneous jamming and eavesdropping. However in practice, full-duplex mode is difficult to implement, and self interference cancellation is hard to achieve as perfect [21] [22]. Rather, half-duplex mode is more widely used, and usually there is more than one channel for the suspicious link to communicate over. In this paper, we study a practical wireless surveillance scenario: a half-duplex legitimate monitor eavesdrops from a suspicious communication link over parallel independent Rayleigh fading channels. As shown in Fig. 1, at the beginning of each fading block, based on the conditions of the parallel independently fading channels, the ST hops to the best one for transmission in this fading block. 1 Here we assume a typical delay-sensitive application (e.g., video talk) on the suspicious link, i.e., the transmitter adjusts its transmission rate by maintaining a certain target outage probability at the receiver [16]. Usually, the monitor is far away from the ST to stay undetected, which makes the traditional passive eavesdropping difficult or even infeasible. Under this challenging setup, the monitor can deliberately send jamming signals to the SR to induce the ST to transmit more likely over a smaller subset of unjammed channels with a lower transmission rate, so that the monitor can still eavesdrop effectively. To avoid getting exposed, the monitor will not change its jamming power and jammed channels over time, it just disguises itself as a normal device-to-device (D2D) user in sharing the network, by sending randomly modulated messages over fixed channels with fixed power. The ST/SR is aware of its co-existence in the same network by updating the long-term belief of the parallel channels' distributions, but does not consider it as a jammer. If the monitor keeps changing its jammed channels or jamming power, then it is no longer like a normal user and will cause the ST/SR's suspicion to hop over channels to increase resilience in a game theoretic setting or directly stop transmitting any message as in [7] [23] [24]. The key novelty and main contributions of this paper are summarized as follows. • Novel jamming-assisted eavesdropping approach over parallel fading channels: To our best knowledge, this is the first paper studying wireless surveillance of parallel fading channels via a half-duplex monitor. The monitor uses jamming to change the suspicious link's long-term belief on the parallel channels' distributions, and thereby induce it to transmit more likely in a smaller subset of unjammed channels with a lower transmission rate for higher eavesdropping success probability. For practical concern, we consider a challenging case that the monitor has no instantaneous CSI of any suspicious link channels, and the monitor in half-duplex mode cannot eavesdrop a channel that it is simultaneously jamming to. • Joint optimization of jamming power and number of jammed channels: We formulate the problem for optimal jamming design over parallel fading channels as a mixed integer nonlinear programming (MINLP) and show it is non-convex. Nevertheless, we prove that the monitor should use the maximum jamming power if it decides to jam. Then we manage to simplify the MINLP to integer programming and further show that there is a fundamental trade-off in deciding the number of jammed channels: jamming more channels helps reduce the suspicious communication rate for overhearing more clearly, but at the risk that the ST is more likely to choose among the jammed channels to transmit and as a result cannot be overheard. • Jamming-assisted eavesdropping over two-way communications: We extend the model to consider the two-way communications of the suspicious link. As the monitor cannot change its jamming strategy to avoid getting exposed, it needs to jam the same subset of channels for both communication directions. To decide the optimal number of jammed channels, we show there is another trade-off to balance the eavesdropping performances of the two-way communications. • Performance evaluation: Numerical results show that our jamming-assisted eavesdropping schemes achieve great performance gain over conventional passive eavesdropping. We also show that the monitor will perform passive eavesdropping only when it is close to the ST, and will jam increasingly more channels when it is moving away from the ST to SR, due to deteriorating eavesdropping channels and improving jamming channels. The rest of this paper is organized as follows. In Section II, we present the system model and formulate the legitimate monitor's proactive eavesdropping design problem. In Section III, we solve the eavesdropping optimization problem in the special two-channel case to gain useful insights. In Section IV, we extend to multi-channel case and show an interesting trade-off in deciding the number of jammed channels, for eavesdropping success probability maximization. In Section V, we further consider jamming-assisted eavesdropping over two-way communications. In Section VI, we provide more numerical results to evaluate our jamming-assisted eavesdropping approach. Finally, we conclude this paper in Section VII. Monitor's jamming power on channel i. Q max Jamming power budget of the monitor. σ 2 a , σ 2 b , σ 2 c Noise power level at the SR, the monitor, the ST. R I , R II Suspicious transmission rate of passive eavesdropping, jamming-assisted eavesdropping. δ Target outage probability at the SR. ρ Own goal probability (probability that one of the jammed channels is chosen by the ST for transmission). ϕ I , ϕ II Eavesdropping success probability of passive eavesdropping, jamming-assisted eavesdropping. II. SYSTEM MODEL AND PROBLEM FORMULATION As shown in Fig. 1, the ST communicates with the SR over N ≥ 2 parallel channels with independent Rayleigh fading, and there is a legitimate monitor aiming to eavesdrop their communications. We consider a quasi-stationary system model, where the monitor has sufficient time (before the ST and SR's movement to another locations) to learn the global channel distribution information (CDI) and launch jamming to eavesdrop from the suspicious link's transmission. The ST and SR are both equipped with one antenna, while the legitimate monitor is purposely equipped with two antennas, one for receiving (eavesdropping) and the other for transmitting (jamming). In order to characterize the fundamental performance limit of this jamming-assisted eavesdropping approach, we assume the encryption method used by the ST/SR is known to the monitor beforehand (e.g. via eavesdropping the related encryption codebook). Note that the focus of this work is on decoding the message instead of decrypting the message itself. The monitor disguises itself as a normal user in sharing the same set of channels with the suspicious link, and operates at half-duplex mode, which means it cannot eavesdrop the channel that it is jamming to. Thus, the monitor will not jam all N channels, otherwise, it overhears nothing. There are two eavesdropping schemes to investigate and compare: • Scheme I (passive eavesdropping): The legitimate monitor performs passive eavesdropping over all N channels while jamming no channels. This is also a benchmark case for our proposed jamming-assisted eavesdropping to compare with. • Scheme II (jamming-assisted eavesdropping): The legitimate monitor performs jamming-assisted eavesdropping by jamming n channels and eavesdropping from the rest N − n channels, where 1 ≤ n ≤ N − 1. We consider a block fading model, where the channel stays unchanged in each fading block and may vary over different fading blocks. As shown in Fig. 1, we respectively denote the channel power gains of the suspicious communication link, eavesdropping link (from the ST to the monitor) and jamming link (from the monitor to the SR) on channel i ∈ {1, · · · , N } as g ai , g bi and g ci . By considering independent Rayleigh fading, g ai , g bi and g ci are modelled as independent exponentially distributed random variables with mean 1/λ a , 1/λ b and 1/λ c , respectively, with i ∈ {1, · · · , N }. This suggests that all the suspicious link channels (g ai , ∀i) are independent and identically distributed (i.i.d.), so are the eavesdropping channels (g bi , ∀i) and the jamming channels (g ci , ∀i). Hence in Scheme II, in statistical sense it does not matter which channels to jam given the jammed channel number by the monitor (who does not know the instantaneous CSI of any suspicions link channels). Thus, without loss of generality, we assume the half-duplex monitor picks the first n out of N channels to jam, and eavesdrops from the rest N −n channels. We assume that the monitor only knows the global CDI (1/λ a , 1/λ b and 1/λ c ), which can be obtained by the monitor through long-term observation as mentioned earlier. On the other hand, we consider that the ST knows the CSI of all the suspicious communication channels (i.e., g ai 's instantaneous values). The ST transmits at a fixed power P and keeps hopping to the best channel for transmission in each fading block. For ease of reading, Table I summarizes the main symbol notations used in this paper and their physical meanings. A. Monitor's expected performance of the suspicious link without or with jamming Both the signal sent by the ST and the jamming signal sent by the monitor are assumed to be circularly symmetric complex Gaussian (CSCG) random variables. This is because that CSCG message will achieve channel capacity given CSCG noise, and monitor's CSCG jamming signal will achieve the best jamming effect [8]. In Scheme I, the monitor does not jam, and the achievable rate of suspicious communication on channel i ∈ {1, · · · , N } is log 2 (1 + gaiP σ 2 a ) in bits/second/Hertz (bps/Hz), where σ 2 a denotes the noise power at the SR. The monitor expects the signal-to-noise ratio (SNR) gaiP σ 2 a at the SR on the i th channel is a random variable, with cumulative distribution function (CDF) given by P g ai P σ 2 a ≤ γ = 1 − e − λa σ 2 a P γ , γ ≥ 0.(1) In Scheme II, the monitor jams by allocating Q i ≥ 0 power to channel i ∈ {1, · · · , N }, and thus the achievable rate of suspicious communication channel i is log 2 (1 + gaiP gciQi+σ 2 a ). The monitor expects the signal-to-interference-plus-noise ratio (SINR) at the SR on the i th channel is a random variable, with CDF given by the following lemma. Lemma 2.1: In Scheme II, the CDF of SINR gaiP gciQi+σ 2 a at the SR under jamming is given by P g ai P g ci Q i + σ 2 a ≤ γ = 1 − λ c P e − λa σ 2 a P γ λ c P + λ a Q i γ , γ ≥ 0. (2) Proof: See Appendix A. Based on (1) and (2) under the two eavesdropping schemes, we are ready to formulate the monitor's design objective. In Scheme I, the legitimate monitor performs passive eavesdropping. The ST will choose the best channel, i.e., the one with the highest SNR, in each fading block. From (1), the monitor's expected CDF of the maximum SNR γ I at the SR among all the channels is P γ I = max g a1 P σ 2 a , · · · , g aN P σ 2 a ≤ γ = 1 − e − λa σ 2 a P γ N , γ ≥ 0.(3) We consider a typical delay-sensitive transmission model for the suspicious link, where the ST adjusts its transmission rate to keep a target outage probability δ at the SR. Only when the transmission rate R I is no larger than the achievable rate of the best suspicious communication channel r I a = log 2 (1+γ I ), the SR can successfully decode the delay-sensitive message. Thus we have P(r I a < R I ) = δ,(4) which yields the suspicious transmission rate R I as R I = log 2 1 + P λ a σ 2 a ln 1 1 − δ 1 N .(5) In Scheme II, the legitimate monitor performs jammingassisted eavesdropping and jams n ≥ 1 channels. Under jamming, the ST will choose the channel with the highest SINR among the n jammed channels or the channel with the highest SNR among the remaining N −n channels without jamming in each fading block. Note that ST's chosen channel may still be a jammed channel by the monitor due to independent channel fading. From (1) and (2), the monitor's expected CDF of the maximum SINR or SNR γ II at the SR among all the channels is P γ II = max g a1 P g c1 Q 1 + σ 2 a , · · · , g an P g cn Q n + σ 2 a , g a(n+1) P σ 2 a , · · · , g aN P σ 2 a ≤ γ = 1 − e − λaσ 2 a P γ N −n n i=1 1 − λ c P e − λa σ 2 a P γ λ c P + λ a Q i γ , γ ≥ 0.(6) To maintain target outage probability δ at the SR, the ST sets the transmission rate R II to ensure P(log 2 (1 + γ II ) < R II ) = δ,(7) which yields 1− e − λa σ 2 a P (2 R II −1) N −n n i=1 1− λ c P e − λa σ 2 a P (2 R II −1) λ c P +λ a Q i (2 R II −1) = δ.(8) With its jamming power allocations {Q i } n i=1 , the monitor believes the ST will use rate R II to transmit. Note that there is no closed-form solution R II to equation (8). B. Monitor's problem formulation for jamming-assisted eavesdropping The legitimate monitor aims to maximize the eavesdropping success probability on the suspicious communication, which is the percentage of fading blocks it can successfully decode. In Scheme I, assuming the suspicious communication is on the i th channel in a certain fading block, only when the achievable rate of the i th eavesdropping channel is no smaller than the suspicious transmission rate R I in (5), the monitor can successfully eavesdrop in this fading block. Thus, the eavesdropping success probability ϕ I under Scheme I (passive eavesdropping) is ϕ I = 1−P r b = log 2 1+ g bi P σ 2 b < R I = e − λ b σ 2 b P (2 R I −1) ,(9) where σ 2 b denotes the noise power at the legitimate monitor. In Scheme II, since all the suspicious communication channels have independent fading, it is possible that a jammed channel is still chosen by the ST, and in this case the halfduplex monitor cannot eavesdrop anything. We define the probability that any jammed channel is chosen by the ST (i.e., own goal probability of the monitor's self-jamming) as ρ ∈ [0, 1] and we will detail its analysis later in Sections III and IV. Now, assuming the suspicious communication is on the i th channel in a certain fading block, only when none of the jammed channels is chosen by the ST, and the achievable rate of the i th eavesdropping channel is no smaller than the suspicious transmission rate R II in (8), the monitor can successfully eavesdrop. Thus, the eavesdropping success probability ϕ II under Scheme II (jamming-assisted eavesdropping) is ϕ II (n,Q 1 ,· · ·,Q n ) = (1−ρ(n,Q 1 ,· · ·,Q n )) × 1−P r b = log 2 1+ g bi P σ 2 b < R II (n,Q 1 ,· · ·,Q n ) = (1 − ρ(n,Q 1 ,· · ·,Q n ))e − λ b σ 2 b P (2 R II (n,Q 1 ,···,Qn) −1) .(10) From (9) and (10), we can formulate the optimization problem for jamming design as a mixed integer nonlinear programming (MINLP), given by (P1) : max n,Q1,··· ,Qn max{ϕ I , ϕ II (n, Q 1 , · · · , Q n )} s.t. 0 < n i=1 Q i ≤ Q max , 1 ≤ n ≤ N − 1, n ∈ Z. Note that ϕ I does not depend on n or Q i and it is a constant. In other words, the monitor only jams if the optimized ϕ II (n, Q 1 , · · · , Q n ) is larger than ϕ I . The joint optimization of n and Q i 's in problem (P1) is difficult due to discrete n and non-concave objective ϕ II (n, Q 1 , · · · , Q n ). In the next section, we will first look into the two-channel case (N = 2) to simplify this problem and provide tractable analysis and clean insights. We will generalize the results to the multi-channel case in Section IV. III. OPTIMAL JAMMING-ASSISTED EAVESDROPPING OVER TWO CHANNELS If there are only two parallel channels (N = 2), in jammingassisted eavesdropping approach, the half-duplex legitimate monitor will only jam one channel (i.e., n = 1), otherwise, both channels are jammed and it cannot overhear anything due to own goal. Its eavesdropping success probability ϕ II under our jamming-assisted eavesdropping in (10) now only depends on jamming power Q, i.e., ϕ II (Q) = (1 − ρ(Q))e − λ b σ 2 b P (2 R II (Q) −1) ,(11) and (P1) is simplified to (P2) : max Q max{ϕ I , ϕ II (Q)} s.t. 0 < Q ≤ Q max . Proposition 3.1: In the two-channel case, the own goal probability ρ(Q) in (11) due to self-jamming at the monitor is given by ρ(Q) = − λ c σ 2 a Q e 2λc σ 2 a Q Ei(− 2λ c σ 2 a Q ),(12) where Ei(·) is the exponential integral function [25, Eq. 8.21]. As jamming power Q increases, ρ(Q) decreases. This is because a higher jamming power helps drive the suspicious link to the other unjammed (eavesdropped) channel. More specifically, we have lim Q→0 + ρ(Q) = 1/2 due to trivial jamming effect on changing the ST's belief of channel distributions, and the suspicious link is equally likely to choose both i.i.d. channels to transmit. Moreover, lim Q→∞ ρ(Q) = 0, since the jammed channel will never be chosen by the suspicious link under infinite jamming power. Proof: See Appendix C, where we choose N = 2 and n = 1. By substituting Q = 0 + and Q = ∞ into (12), we can derive the two limits for ρ(Q). Proof: Next, we analyze the non-outage probability at the monitor e −λ b σ 2 b (2 R II (Q) −1)/P , a part of (11), which is a function of R II (Q). By simplifying (8) under N = 2 and n = 1, we have the following results. Proposition 3.2: In the two-channel case, the suspicious link's transmission rate R II (Q) in (11) is the unique solution to 1−e − λa σ 2 a P (2 R II (Q) −1) 1− λ c P e − λa σ 2 a P (2 R II (Q) −1) λ c P +λ a Q(2 R II (Q) −1) = δ. (13) As jamming power Q increases, R II (Q) decreases. This is because the ST faces more noisy channel, and has to transmit at a lower rate in order to maintain target outage probability δ. More specifically, lim Q→0 + R II (Q) = log 2 1+ P ln(1−δ 1 2 ) −1 /λ a σ 2 a , which equals to R I in (5) with N = 2 under passive eavesdropping, due to trivial jamming effect on the suspicious communication. Moreover, lim Q→∞ R II (Q) = log 2 1+P ln(1−δ) −1 /λ a σ 2 a , which equals to R I in (5) with N = 1 under the passive eavesdropping, since the jammed channel will never be chosen. Proof: By substituting N = 2 and n = 1 into (8), we have (13). Denote the left-hand-side (LHS) of (13) to be g(R II (Q), Q). According to the implicit function theorem, we have dR II (Q) dQ = − ∂g(R II (Q), Q)/∂Q ∂g(R II (Q), Q)/∂R II (Q) < 0, thus R II (Q) monotonically decreases as Q increases. By substituting Q = 0 + and Q = ∞ into (13), we can derive the two limits for R II (Q). Theorem 3.1: We denote Q th as the unique solution to 1 − ρ(Q th ) e − λ b σ 2 b P (2 R II (Q th ) −1) = e − λ b σ 2 b P (2 R I −1) . (14) If Q max > Q th , the legitimate monitor will jam with the maximum power Q max , otherwise it will perform passive eavesdropping without jamming (as illustrated in Fig. 2). According to Propositions 3.1 and 3.2, ϕ II (Q) in (11) monotonically increases with Q, thus the monitor will use the maximum jamming power Q max if it decides to jam, while ϕ I in (9) is a constant regardless of Q. When jamming power Q is close to zero, lim Q→0 + ϕ II (Q) = 1 2 e − λ b σ 2 b P (2 R I −1) < ϕ I . On the other hand, when jamming power goes to infinity, lim Q→∞ R II (Q) = R I \| (N =1) < R I , hence lim Q→∞ ϕ II (Q) = e − λ b σ 2 b P (2 R I \| (N =1) −1) > ϕ I . Given that when jamming power goes to zero, the performance of jamming-assisted eavesdropping is worse than passive eavesdropping, while the performance of jammingassisted eavesdropping increases with jamming power, and eventually when jamming power goes to infinity, becomes better than passive eavesdropping, it follows that there exists a unique intersection point between ϕ I and ϕ II (Q) at point Q = Q th , which is given in (14). If Q max > Q th , the monitor will jam with the maximum jamming power Q max , otherwise it will perform passive eavesdropping to obtain a greater ϕ I than ϕ II (Q max ). As we can see from above, in jamming-assisted eavesdropping, jamming with a higher power helps reduce own goal probability ρ(Q) and transmission rate R II (Q) at the same time. Thus, by using up jamming power budget Q max , the monitor can achieve the maximum eavesdropping success probability. IV. OPTIMAL JAMMING-ASSISTED EAVESDROPPING OVER MULTIPLE CHANNELS In this section, we consider the general case with multiple i.i.d. fading channels for the optimal jamming design, and need to further decide how many channels to jam. Similar to the N = 2 case in Theorem 3.1, we also expect to jam with the maximum power Q max in the general case of N ≥ 2 channels if the monitor decides to jam. More specifically, we have the following result. Proposition 4.1: Given that n ≥ 1 of N channels are jammed, the monitor should allocate all the jamming power over n jammed channels equally, i.e., Q * i = Q max /n, ∀i ∈ {1, · · · , n}. Proof: See Appendix B. Thanks to Proposition 4.1, we know that the monitor will evenly allocate all the jamming power over the jammed channels, as a result the eavesdropping success probability ϕ II under jamming-assisted eavesdropping in (10) only depends on the number of jammed channels n, i.e., ϕ II (n) = (1 − ρ(n))e − λ b σ 2 b P (2 R II (n) −1) .(15) Thus, we manage to simplify the non-convex MINLP in (P1) to the following single-variable problem: (P3) : max n max{ϕ I , ϕ II (n)} s.t. 1 ≤ n ≤ N − 1, n ∈ Z. As n is an integer, problem (P3) is an integer programming problem and still difficult to solve analytically. Next, we will analyze the monotonic properties of own goal probability ρ(n) and suspicious link's transmission rate R II (n) with respect to n in the objective, to understand the key insights and solve (P3). Proposition 4.2: In the general multi-channel case, the own goal probability ρ(n) at the monitor in (15) is given by ρ(n) = (N −n)λ a σ 2 a P N−n−1 i=0 n j=1 N −n−1 i n j (−1) i+j+1 × λ a Q max nλ c P −j (1+i+j) λ a σ 2 a P j−1 e (1+i+j)nλc σ 2 a Qmax ×Γ 1−j, (1+i+j)nλ c σ 2 a Q max ,(16) where Γ(·, ·) is the upper incomplete gamma function [25,Eq. 8.35]. Further, ρ(n) increases as n increases. Proof: See Appendix C. Here, ρ(n) increases with n due to two reasons. First, more channels are jammed and potentially they can be selected by the suspicious link for transmission. Second, the jamming power on each jammed channel weakens as n increases given the total jamming power budget, and thus each jammed channel is more likely to be chosen by the suspicious link. Thus, the suspicious link is more likely to transmit on the jammed channels and this increases the self-jamming (own goal) probability ρ(n) for the monitor. Next, we determine the non-outage probability e −λ b σ 2 b (2 R II (n) −1)/P at the monitor in (15), which is a function of R II (n). Similar to ρ(n), here R II (n) only depends on n. (17) Further, R II (n) monotonically decreases as n increases. Proof: See Appendix D. The reason why R II (n) decreases with n is because jamming more channels increases the chance that the ST chooses the jammed channels, and the ST will transmit at a lower rate to maintain target outage probability at the SR. A lower transmission rate R II (n) leads to a higher non-outage probability e −λ b σ 2 b (2 R II (n) −1)/P at the monitor, and thus the monitor can overhear more clearly. (16), non-outage probability in (17), and their product ϕ II (n) in (15). As n increases, it is observed that 1 − ρ(n) decreases and e −λ b σ 2 b (2 R II (n) −1)/P increases. To maximize ϕ II (n), there is thus a trade-off in deciding the optimal number of jammed channels n * (here n * = 5 in this numerical example). It should be noted that it is still difficult to analytically solve n * in (P3) even by relaxing n to be continuous for tractable analysis. This is because not only R II (n) in (17) is not in closed-form, but also it is difficult to approximate ρ(n) in (16) to be a continuous function due to the combinatorial nature and the involved upper incomplete gamma function. Still, we can numerically obtain n * , by a one-dimensional exhaustive search in the set {1, 2, · · · , N − 1} with low computation complexity of O(N ). Proposition 4.4: In jamming-assisted eavesdropping scheme, as the monitor's jamming power budget Q max goes to zero, it is optimal for the monitor to jam as few channels as possible (n * = 1), and the eavesdropping success probability is lim Qmax→0 + ϕ II (n * = 1) = 1 − 1 N e λ b σ 2 b λa σ 2 a ln 1−δ 1 N .(18) On the other hand, as the monitor's jamming power budget Q max goes to infinity, it is optimal for the monitor to maximally jam n * = N − 1 channels and eavesdrop the remaining one with ideal zero own goal probability, and the eavesdropping success probability is Proof: When the monitor's jamming power budget goes to zero, the jammed channels and unjammed channels are the same in distribution, while the half-duplex monitor cannot eavesdrop the jammed channels, making the own goal proba-bility proportional to n, i.e., lim Qmax→0 + ρ(n) = n N . Meanwhile, the transmission rate R II (n) in (17) now becomes the same as constant transmission rate of passive eavesdropping in (5). Thus, the eavesdropping success probability in (15) becomes lim Qmax→0 + ϕ II (n) = 1 − n N e λ b σ 2 b λa σ 2 a ln 1−δ 1 N .(20) As we can see from (20), lim Qmax→0 + ϕ II (n) monotonically decreases as n increases, thus it is optimal to only jam n * = 1 channel when employing the jamming-assisted eavesdropping. On the other hand, as jamming power budget goes to infinity, the own goal probability becomes lim Qmax→∞ ρ(n) = 0, as the severely jammed channel will never be chosen by the ST. Meanwhile, the transmission rate R II (n) becomes As we can see from (21), lim Qmax→∞ ϕ II (n) monotonically increases as n increases, thus it is optimal to jam n * = N − 1 channels in this case. Note that under optimal n * , if ϕ II (n * ) > ϕ I , the legitimate monitor will perform jamming-assisted eavesdropping, otherwise it will perform passive eavesdropping. V. EXTENSION TO EAVESDROPPING TWO-WAY COMMUNICATIONS So far, we have considered the one-way communication from the ST to the SR for the suspicious link, while in practice, the two users may need to alternately exchange information with each other over time periods or fading blocks (see Fig. 4). Our jamming-assisted eavesdropping approach in Section IV is designed for the one-way communication, and will be extended in this section for the two-way communications. The distributions of channel power gains of the communication from user A to user B follow the same model defined in Section II, i.e., g ai ∼ exp(λ a ), g bi ∼ exp(λ b ), g ci ∼ exp(λ c ). Due to the reciprocity of wireless channel, the suspicious communication channels are the same from user B to user A, i.e., g ai ∼ exp(λ a ). However, the original jamming channels now become eavesdropping channels, i.e., g bi ∼ exp(λ c ), and the original eavesdropping channels now become jamming channels, i.e., g ci ∼ exp(λ b ). As we can see, the optimal number of jammed channels are in general different for user A to B communication (n * AB as computed in last section) and user B to A communication (n * BA ), where subscript (·) AB denotes the communication direction from user A to user B, and (·) BA denotes the communication from user B to user A. But alternately jamming n * AB = n * BA channels over time will easily arouse the suspicion of suspicious users by examining the channel statistics. Then the suspicious link can tell that the monitor is a jammer instead of a normal D2D user with time-division-duplex. To intercept the two-way communications between user A and user B, we need to balance these two communication ways for maximizing the minimum eavesdropping success probability between both communication ways, keep the same jammed channels and jamming power in the long run. Without loss of generality, if n out of N i.i.d. fading channels are jammed in two-way communications, we consider the monitor picks the first n out of N channels to jam. Similar to Proposition 4.1 in the one-way communication, the monitor should also use up all the jamming power budget and evenly partition over the jammed channels in two-way communications. Thus, the maxmin optimization problem can be reformulated as follows. (P4) compares the performance of jamming-assisted eavesdropping min{ϕ II AB (n), ϕ II BA (n)} and the performance of passive eavesdropping min{ϕ I AB , ϕ I BA } in two-way communications. Since both ϕ I AB and ϕ I BA are constants, we can only focus on optimizing the performance of jamming-assisted eavesdropping min{ϕ II AB (n), ϕ II BA (n)}. To numerically solve this integer programming, we can obtain the optimal number of jammed channels n * in two-way communications, by efficiently performing one-dimensional exhaustive search in the set {1, 2, · · · , N − 1} with low computation complexity of O(N ). Then we compare min{ϕ I AB , ϕ I BA } with min{ϕ II AB (n * ), ϕ II BA (n * )}: if the former is smaller, the monitor will perform jamming-assisted eavesdropping with n * jammed channels, otherwise the monitor will perform passive eavesdropping. We first use a numerical example to illustrate the eavesdropping success probability of jamming-assisted eavesdropping in two-way communications. Assuming there are N = 8 parallel channels, we set the mean channel power gains to be 1/λ a = 1/5, 1/λ b = 1 and 1/λ c = 1/4, additive white Gaussian noise (AWGN) power as σ 2 a = σ 2 b = σ 2 c = 1, target 2 Our problem (P4) can further include different weights for the two communication ways. For example, if the monitor values the message from user A (e.g., a leader of a criminal gang) to user B (e.g., a follower) more important, it will assign a large weight to this way's eavesdropping success probability. outage probability at user A/B as δ = 0.05, transmitting power of user A/B as P = 10 dB and jamming power budget as Q max = 20 dB. As Fig. 5 shows, the optimal number of jammed channels n * AB for suspicious user A's communication to user B is 2, while the optimal number of jammed channels n * BA for suspicious user B's communication to user A is 6; and we find the optimal number of jammed channels n * = 5 to balance between the communications from user A to B and from user B to A. It is difficult to analytically derive the optimal solution to (P4), due to the non-concave objective involving incomplete gamma function and the discrete nature of decision variable n. Despite of these, we still manage to derive some analytical results by assuming the one-way eavesdropping success probability ϕ II (n) (i.e., ϕ II AB (n) for user A to B and ϕ II BA (n) for user B to A) is unimodal (having only one peak) or monotonic in n as in Fig. 3. Actually, this is always the case in our extensive simulations though it is difficult to rigorously prove. Proposition 5.1: Assuming the objective functions ϕ II AB (n) and ϕ II BA (n) in (P4) are unimodal or monotonic in n, the optimal jamming-assisted eavesdropping scheme is given as follows, depending on the jamming power budget Q max . • If the monitor's jamming power budget Q max is low (i.e., Q max < Q), jamming more channels hurts the eavesdropping performances on both ways, where Q = min{Q max \|ϕ II AB (1) ≥ ϕ II AB (2), ϕ II BA (1) ≥ ϕ II BA (2)}. In this case, it is optimal for the monitor to minimally jam n * = 1 channel. As a special case, when Q max goes to zero, n * = 1 (as a two-way extension of Proposition 4.4). • If the monitor's jamming power budget Q max is high (i.e., Q max >Q), jamming more channels improves the eavesdropping performances on both ways, whereQ = max{Q max \|ϕ II AB (N − 2) ≤ ϕ II AB (N − 1), ϕ II BA (N − 2) ≤ ϕ II BA (N −1)}. In this case, it is optimal for the monitor to maximally jam n * = N − 1 channels. As a special case, when Q max goes to infinity, n * = N − 1 (as a two-way extension of Proposition 4.4). Passive eavesdropping with N=2 Passive eavesdropping with N=4 Passive eavesdropping with N=8 Jamming-assisted eavesdropping with N=2 Jamming-assisted eavesdropping with N=4 Jamming-assisted eavesdropping with N=8 Fig. 7. Eavesdropping success probability versus mean channel power gain of the eavesdropping and jamming links with different total numbers of channels N 's. • If the jamming power budget Q max is medium (i.e., Q ≤ Q max ≤Q), the optimal number of jammed channels n * is between n * AB and n * BA , by balancing the eavesdropping performances of the two ways. Proof: See Appendix E. VI. NUMERICAL RESULTS In this section, we provide more numerical results to validate our studies and designs. Assuming there are N = 8 parallel channels, we set the mean channel power gain of the suspicious communication link, eavesdropping link and jamming link in the one-way communication to be 1/λ a = 1, 1/λ b = 1 and 1/λ c = 1/3, respectively. We also set the AWGN power as σ 2 a = σ 2 b = σ 2 c = 1, the target outage probability at the SR as δ = 0.05 and transmitting power of the ST as P = 10 dB. Fig. 6 shows the eavesdropping success probability ϕ II as a function of n and Q max for the case of one-way communication from suspicious user A to user B. When Q max is small (e.g., Q max = 4 dB curve in Fig. 6), the monitor will perform passive eavesdropping. When Q max is sufficiently large (starting from Q max = 10 dB), the monitor will jam to assist eavesdropping for a larger eavesdropping success probability. We can see that as Q max increases, the monitor will jam more channels by optimally controlling the trade-off between own goal probability and transmission rate. When Q max is further large (e.g., Q max = 40 dB), the monitor will optimally jam N − 1 channels to overhear the lowest-rate suspicious communication in the remaining single channel, without worrying about the own goal of self-jamming. This result is consistent with Proposition 4.4. Then we consider the passive eavesdropping as a benchmark for performance comparison. We set the mean channel power gain of the suspicious communication link to be 1/λ a = 1 and jamming power budget to be Q max = 30 dB. Here we consider the monitor is far away from the ST and SR, thus the mean channel power gains of eavesdropping channels and jamming channels are nearly the same, i.e., 1/λ b = 1/λ c . Fig. 7 shows the eavesdropping success probability as a function of N and mean channel power gain of eavesdropping and jamming links. As their mean channel power gains increase, eavesdropping success probabilities of both jammingassisted and passive eavesdropping increase. But jammingassisted eavesdropping greatly outperforms passive eavesdropping. The performance of jamming-assisted eavesdropping is better when total number of channels is smaller (e.g., N = 2 in Fig. 7), as the monitor can more efficiently induce the suspicious link to use a smaller subset of unjammed channels and a lower transmission rate under the same jamming power budget. While the performance gain of jammingassisted eavesdropping comparing with passive eavesdropping is greater when total number of channels is large (e.g., N = 8 in Fig. 7), as more channels provides more degrees of freedom for jamming. Next, we study the effect of the monitor's location. To capture the effect of large-scale fading, we consider that for any two points with coordinates (x 1 , y 1 ) and (x 2 , y 2 ) in the twodimensional (2D) ground plane, the mean channel power gain between the two points is inversely proportional to the square of their distance, i.e., 1/λ = 1/ (x 1 − x 2 ) 2 + (y 1 − y 2 ) 2 . The ST is located at (3, 4.5), and the SR is located at (7, 4.5). The distances are normalized with transmit power. The legitimate monitor is placed in different locations in this plane, and its jamming power budget is Q max = 30 dB. Fig. 8(a) shows the optimal number of jammed channels n * in different monitor's locations in the 2D ground plane. When the monitor is close to the ST, passive eavesdropping can already provide good eavesdropping performance. When the monitor is moving away from the ST, passive eavesdropping can no longer provide good eavesdropping performance, and the monitor will jam more channels to lower the suspicious link's transmission rate in order to overhear more clearly. When the monitor is close to the SR, which means now the efficiency of jamming is high, the monitor will jam most channels (up to N − 1) for improving eavesdropping performance. Fig. 8(b) and Fig. 8(c) compare the eavesdropping success probability of jamming-assisted eavesdropping and passive eavesdropping with different monitor's locations in the 2D ground plane. As we can see, the eavesdropping success probability of passive eavesdropping is fully determined by the distance between the monitor and the ST, and a good eavesdropping performance can only be guaranteed when the monitor is close to the ST. While our proposed jamming-assisted eavesdropping greatly outperforms passive eavesdropping even when the monitor is not close to the ST, because it can efficiently jam the SR and drive the ST to transmit in a smaller subset of channels with a lower transmission rate so that the monitor can eavesdrop more effectively. The performance gain of jamming-assisted eavesdropping is significant when the monitor is close to the SR, since now the efficiency of jamming is high. This clearly shows that passive eavesdropping is dramatically sensitive to the ST-monitor distance, while our proposed jamming-assisted eavesdropping is no longer that sensitive even when the monitor is geometrically far away from the ST. Finally, we examine the performance of jamming-assisted eavesdropping in two-way communications. Here we still assume the mean channel power gain between any two points is inversely proportional to the square of their distance, similar to the previous simulation. Suspicious user A is located at (1, 1) and suspicious user B at (3,1). The legitimate monitor can be at any point between (0, 0) and (4, 0) to eavesdrop the two-way communications, and its jamming power budget is set to Q max = 30 dB. We provide two benchmark cases for performance comparison with our optimal solution. Benchmark n * AB tells that the monitor just focuses on the one-way communication from A to B, and jams n * AB channels according to the one-way surveillance problem in (P3). Meanwhile, benchmark n * BA focuses on the one-way communication from B to A, and jams n * BA channels. Fig. 9(a) compares the optimal number of jammed channels n * in (P4) with n * AB and n * BA when the monitor moves horizontally between (0, 0) and (4, 0). Fig. 9(b) shows the eavesdropping success probability versus the monitor's location in two-way communications. We have the following observations. • When the monitor is between (0, 0) and (0.3, 0), it is far away from user B and the overall eavesdropping performance is bottlenecked by user B as the ST, thus jamming n * BA channels will give the best eavesdropping performance as the optimal n * . Note that n * AB is smaller than n * BA (n * ) in this case, since user A is much closer to the monitor than user B, which is consistent with Fig. 8(a) in the sense that the monitor will jam more channels when it is moving away from the ST. • When the monitor is moving from (0.3, 0) to (1.6, 0), it is getting closer to user A than user B, thus the eavesdropping performance of benchmark n * AB improves more significantly than that of benchmark n * BA . Still, n * outperforms benchmarks n * AB and n * BA . Note that n * BA is greater than n * in this case, because now the monitor is very close to user A, and according to Fig. 8(a), the monitor will jam most channels when it is close to the SR. • When the monitor is between (1.6, 0) and (2, 0), the monitor's distances to users A and B are close, thus both benchmarks n * AB and n * BA are symmetric and achieve the optimal eavesdropping performance as n * . Note that in Fig. 9(b) the sharp performance increase of benchmarks n * AB and n * BA at point (1.6, 0) is caused by changing the number of jammed channels. Taking n * AB as an example, the monitor is moving closer to user B (SR in n * AB 's point of view), and at the point (1.6, 0), the monitor increases the number of jammed channels due to better jamming efficiency, which causes the obvious eavesdropping performance change. Finally, the eavesdropping success probability reaches the maximum when the monitor is at (2, 0), since now the monitor's distances to user A and user B are the same. The eavesdropping performance analysis of the monitor moving from (2, 0) to (4, 0) is similar as above by symmetry. VII. CONCLUSIONS This paper proposes a new wireless security model, which is jamming-assisted legitimate eavesdropping over parallel independently fading channels. The legitimate monitor uses jamming in order to achieve better eavesdropping performance. Assuming Rayleigh fading, we formulate the optimization problem for jamming design as a mixed integer nonlinear programming (MINLP). Despite its non-convexity, we show that the legitimate monitor should use the maximum jamming power for the best eavesdropping performance if it decides to jam. Then we simplify the MINLP to integer programming and further show that there is a trade-off in deciding the number of jammed channels in the general multi-channel case, where jamming more channels helps reduce the suspicious communication rate for overhearing more clearly, but at the risk that the ST is more likely to choose jammed channels to transmit and as a result cannot be overheard. Finally, we extend our study to two-way communications, and show another trade-off in deciding the common jammed channels for balancing the bidirectional eavesdropping performances. Numerical results show that our jamming-assisted eavesdropping schemes greatly improve eavesdropping success probability comparing with conventional passive eavesdropping. This work can be extended in various directions. For example, the suspicious link can transmit at multiple channels and perform combining at the receiver, or there can be multiple suspicious link pairs, which will bring more challenges to legitimate eavesdropping. The more general case of parallel channels with correlated (non-independent) fading is also worth investigating in future work. We aim to show the distribution of SINR Y = gaiP gciQi+σ 2 a on jammed channel i. Denote its numerator and denominator as Y 1 = g ai P and Y 2 = g ci Q i + σ 2 a , respectively. As g ai and g ci follow independent exponential distributions with mean 1/λ a and 1/λ c , respectively, we have f Y1 (y 1 ) = λ a P e − λa P y1 , y 1 ≥ 0, f Y2 (y 2 ) = λ c Q i e − λc Q i (y2−σ 2 a ) , y 2 ≥ σ 2 a . The probability density function (PDF) of Y = Y 1 /Y 2 can be calculated as follows f Y (y) = λ a λ c P Q i e λc σ 2 a Q i ∞ σ 2 a y 2 e −( λc Q i + λa y P )y2 dy 2 . With the help of [25, Eq. 3.351.2], we have f Y (y) = λ a λ c (λ c P σ 2 a + P Q i + λ a Q i σ 2 a y) (λ c P + λ a Q i y) 2 e − λa σ 2 a y P , y ≥ 0, and the CDF of Y can be calculated as y 0 f Y (y) dy = 1 − λ c P e − λa σ 2 a P y λ c P + λ a Q i y , y ≥ 0. B. Proof of Proposition 4.1 For certain jamming scheme S 1 , the own goal probability can be expressed as ρ S1 (n) =P max g a1 P g c1 Q 1 + σ 2 a , · · · , g aj P g cj Q j + σ 2 a , · · · , g an P g cn Q n + σ 2 a > max g a1 P σ 2 a , · · · , g a(N −n) P σ 2 a . There exists a jamming scheme S 2 , where the monitor reduces jamming power Q j on channel j to Q j − ∆ with ∆ > 0, for which ρ S2 (n) = P max g a1 P g c1 Q 1 +σ 2 a , · · · , g aj P g cj (Q j −∆)+σ 2 a , · · · , g an P g cn Q n +σ 2 a > max g a1 P σ 2 a , · · · , g a(N −n) P σ 2 a . Clearly, ρ S2 (n) is larger than ρ S1 (n). For certain jamming scheme S 1 , the transmission rate of the ST satisfies 1−e − λa σ 2 a P (2 R II S 1 −1) N −n n i=1 1− λ c P e − λaσ 2 a P (2 R II S 1 −1) λ c P +λ a Q i (2 R II S 1 −1) = δ. There exists a jamming scheme S 2 , where the monitor reduces jamming power Q j on channel j to Q j −∆, for which 1−e − λaσ 2 a P (2 R II S 2 −1) N −n 1− λ c P e − λa σ 2 a P (2 R II S 2 −1) λ c P +λ a (Q j −∆)(2 R II S 2 −1) × n i=1,i =j 1 − λ c P e − λa σ 2 a P (2 R II S 2 −1) λ c P + λ a Q i (2 R II S2 − 1) = δ. Similar to the proof of Proposition 3.2, R II S2 is larger than R II S1 , while maintaining the same target outage probability δ at the receiver, but the non-outage probability at the monitor will decrease due to the higher transmission rate R II S2 . From above, if the jamming power on any jammed channel decreases, the product of non-outage probability and non-owngoal probability at the monitor, i.e. the eavesdropping success probability, will degenerate. So the monitor will always use up all the jamming power. Further since all channels are i.i.d. fading, by symmetry it is optimal to allocate the same amount of jamming power over n jammed channels. Thus, if n of N channels are jammed, the monitor should evenly allocate all the jamming power over n jammed channels, i.e., Q max /n. C. Proofs of Propositions 3.1 and 4.2 We first prove that the own goal probability ρ(n) increases as the number of jammed channels n increases. Assuming k out of N channels are jammed, the own goal probability can be expressed as ρ(k) = P max g a1 P g c1 Qmax k + σ 2 a , · · · , g ak P g ck Qmax k + σ 2 a > max g a1 P σ 2 a , · · · , g a(N −k) P σ 2 a . If now k + 1 channels are jammed, then the own goal probability becomes ρ(k+1) = P max g a1 P g c1 Qmax k+1 +σ 2 a , · · · , g a(k+1) P g c(k+1) Qmax k+1 +σ 2 a > max g a1 P σ 2 a , · · · , g a(N −k−1) P σ 2 a . Since the number of jammed channels changes from k to k + 1, the jamming power on each jammed channel gets smaller, which makes them easier to be chosen by the ST for transmission. Also, there are more jammed channels (from k to k +1) and less unjammed channels (from N −k to N −k −1). Combining these two effects, clearly as the number of jammed channels n increases, the own goal probability ρ(n) will also increase. Note that Proposition 3.1 is a special case of Proposition 4.2 and it is sufficient to prove Proposition 4.2 here. Denote X i = gaiP σ 2 a and Y i = gaiP gciQi+σ 2 a , the CDFs of X i and Y i are given in (1) and (2), respectively. Then we have ρ(n) = ∞ 0 P{max{Y (1) , · · · , Y (n) ≥ x}} × f max{X (1) ,··· ,X (N −n) } (x) dx, where f max{X (1) ,··· ,X (N −n) } (x) is the PDF of the maximum SNR of the N − n unjammed channels. Thus we have = comes from the fact that the total number of channels N and the number of jammed channels n are both integers, following the binomial expansion of the two polynomial terms; and equality (b) = comes from reference [25,Eq. 3.353.2], and Γ(·, ·) is the incomplete Gamma function. Note that Γ(0, x) = −Ei(−x), which completes the proof of Proposition 3.1 with N = 2 and n = 1. ρ(n) = ∞ 0 ∞ x d[F Y (y)] n d[F X (x)] N −n = (N − n)λ a σ 2 a P ∞ 0 1 − (1 − e − λa D. Proof of Proposition 4.3 Denote u = 2 R II −1. We define the LHS of (17) as h(n, u), which is the outage probability at the SR. Then we have h(n, u) − δ = 0. By taking the first-order derivative of this implicit function over u and n, we have the relationship between u and n as follows du dn = − ∂h(n, u)/∂n ∂h(n, u)/∂u . For ∂h(n,u) ∂u , as the transmission rate R II increases, outage probability at the SR h(n, u) increases, thus ∂h(n, u) ∂u > 0. Combining (22) and (23), we have du dn = − ∂h(n, u)/∂n ∂h(n, u)/∂u < 0. By substituting R II = log 2 (1 + u) back, we have dR II dn = du dn dR II du < 0. Thus, as the number of jammed channels n increases, the transmission rate R II (n) decreases. E. Proof of Proposition 5.1 Consider the expression of eavesdropping success probability ϕ II (n) in (15), which is the product of non-own goal probability 1 − ρ(n) multiplied by non-outage probability at the monitor e − λ b σ 2 b P (2 R II (n) −1) . For any fixed n, as jamming power budget Q max increases, the first part, non-own goal probability, 1 − ρ(n) increases, since the jammed channels become less likely to be chosen by the suspicious link. The second part, non-outage probability at the monitor, e − λ b σ 2 b P (2 R II (n) −1) also increases, since the suspicious link will transmit at a lower rate to maintain target outage probability and the monitor can eavesdrop more clearly. Thus ϕ II (n) monotonically increases with Q max for any fixed n. For ϕ II AB (n), we can see from (20) that lim Qmax→0 + ϕ II AB (1) > lim Qmax→0 + ϕ II AB (2), and from (21), we can see that Thus there exists a unique solution Q max , so that ϕ II AB (1) = ϕ II AB (2), and we call this solution as Q AB . By assuming ϕ II AB (n) is unimodal or monotonic in n, if Q max < Q AB , from ϕ II AB (1) > ϕ II AB (2) we can conclude that ϕ II AB (n) now monotonically decreases with n. For ϕ II BA (n), we can derive Q BA so that when Q max < Q BA , ϕ II BA (n) decreases with n. Define Q = min{Q AB , Q BA }, when Q max < Q, both ϕ II AB (n) and ϕ II BA (n) decrease with n, thus it is optimal for the monitor to jam n * = 1 channel. Similarly for ϕ II AB (n), we can see from (20) that lim Qmax→0 + ϕ II AB (N − 2) > lim Qmax→0 + ϕ II AB (N − 1), and from (21), we can see that lim Qmax→∞ ϕ II AB (N − 2) < lim Qmax→∞ ϕ II AB (N − 1). There exists a unique solution Q max , so that ϕ II AB (N − 2) = ϕ II AB (N − 1), and we call this solution asQ AB . By assuming ϕ II AB (n) is unimodal or monotonic in n, if Q max >Q AB , from ϕ II AB (N − 2) < ϕ II AB (N − 1) we can conclude that ϕ II AB (n) now monotonically increases with n. Similarly for ϕ II BA (n), we can deriveQ BA so that when Q max >Q BA , ϕ II BA (n) increases with n. Definē Q = max{Q AB ,Q BA }, when Q max <Q, both ϕ II AB (n) and ϕ II BA (n) increase with n, thus it is optimal for the monitor to jam n * = N − 1 channels. When Q ≤ Q max ≤Q, at least one of ϕ II AB (n) and ϕ II BA (n) is unimodal. Without loss of generality, we assume n * AB ≤ n * BA , clearly when n < n * AB , both ϕ II AB (n) and ϕ II BA (n) are monotonically increasing, and when n > n * BA , both ϕ II AB (n) and ϕ II BA (n) are monotonically decreasing. Thus n * must lie between n * AB and n * BA , and can be numerically searched. Fig. 1 . 1System model of jamming-assisted eavesdropping over the suspicious link (suspicious transmitter to receiver). Fig. 2 . 2Eavesdropping success probability versus jamming power at the monitor, where N = 2, λa = 1, λ b = λc = 3, P = 10 dB, σ 2 a = σ 2 b = 1 and δ = 0.05. Fig. 3 . 3ϕ II (n), 1 − ρ(n) and non-outage probability at the monitor versus number of jammed channels n, where N = 8, λa = λ b = 1, λc = 3, P = 10 dB, Qmax = 20 dB, σ 2 a = σ 2 b = 1 and δ = 0.05. Proposition 4. 3 : 3In the general multi-channel case, the suspicious link's transmission rate R II (n) in(15)is the unique solution to 1−e − λa σ 2 a P (2 R II −1) N −n 1− λ c P e − λa σ 2 a P (2 R II −1) λ c P +λ a Qmax n (2 R II −1) n = δ. Fig. 4 . 4Jamming-assisted eavesdropping over two-way communications (user A and user B alternately communicate with each other over different time slots). Fig. 3 3numerically illustrates 1 − ρ(n) in ϕ II (n * = N − 1) = e λ b σ 2 b λa σ 2 a ln(1−δ) . AB , ϕ I BA }, min{ϕ II AB (n), ϕ II BA (n)}} s.t. 1 ≤ n ≤ N − 1, n ∈ Z. Fig. 5 . 5Eavesdropping success probability in jamming-assisted eavesdropping of two-way communications, which is the minimum of the eavesdropping success probabilities from user A to B and from user B to A. Fig. 6 . 6Eavesdropping success probability ϕ II versus number of jammed channels n under different power budget Qmax's. number of jammed channels n * versus different monitor's locations in the 2D ground plane (N = 8 channels in total). success probability of jamming-assisted eavesdropping versus different monitor's locations in the 2D ground plane. success probability of passive eavesdropping versus different monitor's locations in the 2D ground plane. Fig. 8 .Fig. 9 . 89Effect of monitor's geometric locations on jamming strategy design, jamming-assisted eavesdropping performance, and passive eavesdropping performance. Eavesdropping success probability versus monitor's location in two-way communications. 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Dogfight in spectrum: combating primary user emulation attacks in cognitive radio systems, part I: known channel statistics. H Li, Z Han, IEEE Trans. Wireless Commun. 911H. Li and Z. Han, "Dogfight in spectrum: combating primary user emulation attacks in cognitive radio systems, part I: known channel statistics," in IEEE Trans. Wireless Commun., vol. 9, no. 11, pp. 3566- 3577, Sep. 2010. Dogfight in spectrum: combating primary user emulation attacks in cognitive radio systems, part II: unknown channel statistics. H Li, Z Han, IEEE Trans. Wireless Commun. 101H. Li and Z. Han, "Dogfight in spectrum: combating primary user emulation attacks in cognitive radio systems, part II: unknown channel statistics," in IEEE Trans. Wireless Commun., vol. 10, no. 1, pp. 274- 283, Jan. 2011. A Jeffrey, D Zwillinger, Table of Integrals, Series, and Products. Academic PressA. Jeffrey and D. Zwillinger, Table of Integrals, Series, and Products. Academic Press, 2007. He is currently pursuing the Ph. D. degree in Engineering System and Design pillar. Yitao Han, S'19) received the B. Eng. degree in information and communication engineering from Zhejiang University, Hangzhou, China, in 2015 and the M. Sc. degree in. Singapore; SingaporeDepartment of Electrical and Computer Engineering from National University of Singapore ; Singapore University of Technology and Designunder SUTD-NUS joint PhYitao Han (S'19) received the B. Eng. degree in information and communication engineering from Zhejiang University, Hangzhou, China, in 2015 and the M. Sc. degree in Department of Electrical and Computer Engineering from National University of Singapore, Singapore, in 2016. He is currently pur- suing the Ph. D. degree in Engineering System and Design pillar, Singapore University of Technology and Design, Singapore, under SUTD-NUS joint Ph. D. programme with President's Graduate Fellowship. D. programme with President's Graduate Fellow- ship. His current research interests include physical layer security and UAV communications. His current research interests include physical layer security and UAV communications. Lingjie Duan (S'09-M'12-SM'16) received the. Lingjie Duan (S'09-M'12-SM'16) received the	[]
[ "Article Nr", "Article Nr" ]	[ "János Kollár \nÉ pijournal de Géométrie Algébrique\n\n" ]	[ "É pijournal de Géométrie Algébrique\n" ]	[]	Let X be a general conic bundle over P 2 with branch curve of degree at least 19. We prove that there is no normal projective variety Y that is birational to X and such that some multiple of its anticanonical divisor is effective.Titre. Fibrés en coniques qui ne sont pas birationnelsà des paires de Calabi-Yau numériques Résumé. Soit X un fibré en coniques général sur P 2 avec une courbe de branchement de degré au moins 19. Nous montrons qu'il n'existe pas de variété projective normale Y qui soit birationnelleà X et telle qu'un multiple de son diviseur anticanonique soit effectif.	10.46298/epiga.2017.volume1.1518	[ "https://arxiv.org/pdf/1605.04763v3.pdf" ]	73,521,467	1605.04763	3e3976cbbc4fd3332eb7b150c037c400fb0e14d3	Article Nr 2017 János Kollár É pijournal de Géométrie Algébrique Article Nr 112017Received by the Editors on June 24, 2016, and in final form on January 17, 2017. Accepted on January 30, 2017.Rationally connected varietyconic bundleCalabi-Yau varietybirational equivalence 2010 Mathematics Subject Classification 14M2214J4514J20 (Primary); 14J3214E05 (Secondary) [Français] Let X be a general conic bundle over P 2 with branch curve of degree at least 19. We prove that there is no normal projective variety Y that is birational to X and such that some multiple of its anticanonical divisor is effective.Titre. Fibrés en coniques qui ne sont pas birationnelsà des paires de Calabi-Yau numériques Résumé. Soit X un fibré en coniques général sur P 2 avec une courbe de branchement de degré au moins 19. Nous montrons qu'il n'existe pas de variété projective normale Y qui soit birationnelleà X et telle qu'un multiple de son diviseur anticanonique soit effectif. Understanding the difference between rationally connected varieties and Fano varieties has long been a goal of birational geometry. In any dimension, smooth Fano varieties come in finitely many families but rationally connected varieties form infinitely many families. Thus one expects that not every rationally connected variety is birational to a Fano variety, but actual proofs of this fact have been quite subtle; see [Sar80] or [Cor95]. A more general form of this problem asks if every rationally connected variety X is birational to the underlying variety of a Q-Fano pair (Y, ∆). Since Q-Fano pairs form infinitely many families up-to birational equivalence [Oka09], this is a harder variant. This form of the question was posed in [CG13] and a negative answer is established in [Kry14]. Closely related results describing Calabi-Yau fiber-space structures on certain Fano 3-folds are proved in [Che04,Ryd06,CK10]. Probably the most general question in this direction is whether every rationally connected variety is birational to the underlying variety of a numerical Calabi-Yau pair (Y, ∆). Here we allow the most general definition of Calabi-Yau pairs, that is, Y is normal, proper, ∆ is a pseudo-effective R-divisor, K Y + ∆ is R-Cartier and K Y + ∆ ≡ 0, but we impose no restrictions on the singularities of Y . (For most purposes the more restrictive definition of Calabi-Yau pairs adopted in [KX16] is the best; but the above numerical version also seems natural.) It turns out that the singularities of Y are not very important and the difference between effective and pseudo-effective divisors also may not be significant, but allowing divisors with coefficient > 1 in ∆ leads to many more cases; see Example 4 and Definition 6. If X is birational to a numerical Calabi-Yau pair (Y, ∆) then the birational transform of ∆ on X is frequently a quite interesting divisor. Understanding such divisors was a key to proving unirationality of degree 1 conic bundle surfaces [KM17]. We discuss several methods to show that certain conic bundles (see Definition 5) are not birational to any numerical Calabi-Yau pair. Typical results are the following. Theorem 1. There are conic bundles S → P 1 defined over Q that are not birational to any numerical Calabi-Yau surface. This property seems to depend very subtly on the coefficients involved in the definition of S and we give only sufficient conditions in Example 23. The following are some concrete special cases. Example 2. Let p ≥ 11 be a prime such that p ≡ −1 mod 4 and choose m ∈ Z not divisible by p. Then the conic bundle z 2 = s p−1 −1 s 2 −1 x 2 + s p−1 − 1 + mp y 2 ⊂ P 2 xyz × A 1 s is not birational-over Q-to any numerical Calabi-Yau surface. In dimension 3 we get the following. Theorem 3. Let X d,2 ⊂ P 2 × P 2 be a general hypersurface of bidegree (d, 2) over a field of characteristic = 2, 3, 5. Then X d,2 is not birational to the underlying variety of a numerical Calabi-Yau pair for d ≥ 7. I could not write down explicit examples with smooth branch curve, but it is easy to get many with reducible branch curve using Corollary 17. It is quite likely that the theorem also holds over any infinite field. The main arguments in our paper work whenever the characteristic is = 2, but a key reference [Sar80] is stated in the literature only for characteristic = 2, 3 and we also use resolution of singularities. The most important numerical invariant of a conic bundle π : X → P n is the degree of its branch divisor B X ⊂ P n . Typical results say that for smooth branch divisors the degree is the only important invariant and the higher the degree of B X , the more complicated X is. This is, however, not the case for our question. We see in Examples 20-21 that there are conic bundles π : X → P 2 with smooth branch curve of arbitrary high degree (resp. π : S → P 1 with many singular fibers) that are birational to numerical Calabi-Yau varieties. The following example shows that many varieties are birational to the underlying variety of a smooth numerical Calabi-Yau pair. Example 4. If (Y, ∆) is a numerical Calabi-Yau pair and ∆ = 0 then Y is uniruled by [MM86] but Y need not be rationally connected. In fact, for any smooth, projective variety X, the product X × P 1 is birational to a numerical Calabi-Yau pair. To see this, let H be an ample divisor on X such that H − K X is effective and note that the anticanonical class of P X O X ⊕ O X (H) is effective. It is the sum of twice the negative section and of the pull-back of H − K X . In particular, the conic bundles in Theorem 3 are not birationally ruled. However, we use even stronger non-rationality results of [Sar80] during the proof. Definition 5. A conic bundle is a flat, proper morphism π : X → Z such that every fiber of π is isomorphic to a plane conic. We are mainly interested in the cases when Z is regular, but for the basic definition it is enough to assume that Z is normal. (For many purposes one should allow non-flat morphisms and more singular fibers, but for us the restrictive version is more convenient.) A conic bundle is called extremal if for every codimension 1 point z ∈ Z the fiber X z is irreducible over k(z). Equivalently, the relative Picard number is 1. A conic bundle is called minimal if it is extremal and has no rational sections. (This is the "right" definition for conic bundles but note that a product Z × P 1 → Z is extremal but not minimal in our sense.) If X is regular then π : X → Z is minimal iff Pic(X) = π * Pic(Z) ⊕ Z[ω −1 X/Z ]; this is the key property that we are interested in. This is equivalent to saying that if L is any line bundle on X and C ⊂ X is a fiber then deg(L\| C ) is even and if C is reducible then L\| C has the same degree on both irreducible components. The branch locus, denoted by B X ⊂ Z, is the subscheme parametrizing singular fibers of π. Set theoretically it consists of points z ∈ Z such that X z := π −1 (z) is singular. If X is regular then B X is reduced. In general, the scheme structure is given by the formula (25.3) which also shows that B X has pure codimension 1 in Z, except possibly when char k(Z) = 2. In order to avoid various complications, we assume from now on that all residue characteristics are = 2. Our main interest is in conic bundles over C or Q, but we will use some examples that are defined over Z p . Assume that Z is regular and let z ∈ B X be a regular point. Then X z is a pair of lines and X is regular along X z ; see Paragraph 25. This defines a double coverB X → B X that isétale over the regular locus of B X . Then X → Z is minimal iff, for every irreducible component B i ⊂ B X , the correspondingB i is also irreducible. I do not know a good general introduction to conic bundles, but everything can be gleaned from the basic sources [Isk67,Bea77,Sar80,Isk87] or [Cor00,Chap.4], [KSC04,. The key results are discussed in Paragraphs 25-28. Definition 6. Let k be a field and X a normal, proper k-variety of dimension n. A divisor class B is called effective (resp. Q-effective) if B is linearly (resp. Q-linearly) equivalent to an effective Z-divisor (resp. Q-divisor). An R-divisor B is called pseudo-effective if its class in N n−1 (X) R is a limit of Q-effective divisors. Birational transformation of divisors does not preserve linear equivalence, so it is not very useful to ask whether a divisor is birationally effective or not. However, the birational transform of a mobile linear system is well defined and the canonical class makes sense on any birational model of X. Let \|M \| be a mobile linear system on X. We say that −K X + \|M \| is birationally effective (resp. birationally Q-effective or birationally pseudo-effective) if there is a normal, proper k-variety φ : X bir ∼ X such that −K X + φ * \|M \| is effective (resp. Q-effective or pseudo-effective). For now our main interest is in the case \|M \| = 0. We stress that we do not impose a priori restrictions on the singularities of X but it is easy to improve them, at least in characteristic 0. Assume that −K X ∼ Q D is pseudo-effective and let p : X → X be a terminal modification [Kol13, 1.33]. Write −K X ∼ Q E + D where E is p-exceptional and D is the birational transform of D . By the Negativity lemma [KM98,3.39] we see that E is an effective R-divisor, hence −K X is pseudo-effective. If K X is not numerically trivial, then it is also not pseudo-effective, hence a suitable minimal model program terminates with a Mori fiber space X m → Z. That is, −K X m is relatively ample and the relative Picard number of X m /Z is 1. Note that φ : X X m is a birational contraction, that is, φ −1 has no exceptional divisors. Thus if −K X ∼ Q D then −K X m = −φ * (K X ) ∼ Q φ * (D ) =: D m shows that −K X m is also pseudo-effective. Thus the following holds. Claim 6.1. Let k be a field of characteristic 0 and X a normal k-variety. Then −K X is birationally pseudo-effective iff (a) either X has canonical singularities and K X is numerically trivial, (b) or there is a projective variety X m with terminal singularities such that X m is birational to X, −K X m is pseudo-effective and there is a Mori fiber space structure π : X m → Z. This suggests that our question should be treated using the Noether-Fano method. (See [KSC04, Chap.5] for an introduction and [Cor00] for a more detailed treatment.) This is the approach taken in [Kry14] and the examples given there are also not birational to a numerical Calabi-Yau pair. We work with conic bundles. These have a rich birational geometry yet all of their birational models are quite well understood. The characteristic 0 assumption is only needed to guarantee that resolutions and Mori fiber space models exist. Thus Claim 6.1 holds over any field k if dim X = 2 or if dim X = 3 and char k > 5. Note also that in birational geometry one would usually like to control the singularities of the pair (X , ∆ ), not just the singularities of X . However, even if ∆ is effective, we do not assume that the coefficients in ∆ are ≤ 1, thus a similar reduction to the log canonical case is not possible. Remark 7. Our notion is close to the concept of Fano type varieties introduced in [PS09], which asks for a normal, proper k-variety X bir ∼ X such that −K X ∼ Q ∆ + H where (X , ∆ ) is klt and H is an ample Q-divisor. (That is, (X , ∆ ) is a log-Fano pair.) ¿From the technical point of view the difference between −K X being birationally pseudo-effective and X being of Fano type can be substantial, but in our examples none of the difficulties appear. 8. (Outline of the proof of Theorems 1 and 3) Start with X → P 2 over C and restrict to a generic line P 1 ⊂ P 2 . We get a 2-dimensional conic bundle S → P 1 over the function field C(s, t). A slight complication is that −K X \| S = −K S + F where F is a general fiber, but this is easy to deal with. Next we study when −K S + F is birationally pseudo-effective for 2-dimensional conic bundles S → P 1 over C(s, t), or, more generally, over any field k. We show in Corollary 11 that this holds iff S contains a double section C ⊂ S whose normalization has genus ≤ 2. The double section C ⊂ S then extends to a generically finite double section D ⊂ X whose normalization is a birational to a K3 surface. (The projection π D : D → P 2 has degree 2 and the ramification curve is a sextic, but π D need not be finite.) Then we study how the branch curve of X → P 2 and the branch curve of D → P 2 intersect. In order to prove Theorem 1, we extend a 2-dimensional conic bundle over P 1 Q to a 3-dimensional conic bundle over P 1 Z and argue as above. The answer seems to depend on subtle properties of S → P 1 Q . In particular, the branch divisor B S ⊂ P 1 Q alone is not enough to decide what happens; see Example 21. Surface conic bundles Fix an arbitrary field k. Assume for simplicity that char k = 2. Let S be a smooth surface over k and S → P 1 a minimal conic bundle. The number of singular geometric fibers, which is the degree of the branch locus B S , is denoted by δ(S). A typical fiber is denoted by F . Thus (K 2 S ) = 8 − δ(S) and (K S · F ) = −2. Lemma 9. Let k be a field and π : S → P 1 a minimal conic bundle. Fix a natural number m such that δ(S) > 12 + 6m. Assume that −K S + mF is pseudo-effective. Then there is a unique irreducible curve C ⊂ S such that (C 2 ) < 0. Furthermore, π : C → P 1 has degree 2 and \| − K S + mF \| = C + \|bF \| for some b ≥ 0. In particular, −K S + mF is effective. Proof. By assumption there is a sequence of effective Q-divisors D t converging to −K S + mF . Note that (−K S + mF ) 2 = 8 − δ(S) + 4m < 0, hence (D 2 t ) < 0 for some t. So there is a k-irreducible component C ⊂ Supp D t with C 2 < 0. Thus the cone of curves is generated by C and a fiber F . Write C ∼ Q a(−K S + m F ). Here a is an integer but m could be rational. The degree of the dualizing sheaf of C is deg ω C = 2p a (C) − 2 = C(C + K S ) = a(a − 1)(K 2 S ) + 2a(2a − 1)m . Since Ck has at most 2a irreducible components, deg ω C ≥ −4a, hence a(a − 1)(K 2 S ) + 2a(2a − 1)m ≥ −4a. If a ≥ 2 then this rearranges to δ(S) − 8 = −(K 2 S ) ≤ 4 + 2 a−1 m + 4 a−1 ≤ 6m + 4. Note that −K S + mF ∼ Q 1 a C + (m − m )F , hence m ≤ m since −K S + mF is pseudo-effective. Thus, if δ(S) > 12 + 6m then a = 1. Note that we also could have used Bend-and-break (as stated in [Kol96, II.5.5.3]) to show that a ≤ 2. Example 10. There are minimal conic bundles S → P 1 with δ(S) = 12 for which −2K S is effective but −K S is not effective. To construct such examples, let Q ⊂ P 2 be an irreducible degree 4 point. Then B Q P 2 → P 1 is a minimal conic bundle with with δ = 3. The exceptional curves give a conjugate set of 4 sections, each with self-intersection −1. Pull it back by a general degree r map P 1 → P 1 to get S → P 1 . We get a minimal conic bundle S → P 1 with δ(S) = 3r and a conjugate set of 4 sections, each with self-intersection −r. The sum of these 4 sections is in \| − 2K S + (r − 4)F \|. Thus the bound δ(S) > 12 + 6m in Lemma 9 is sharp. (It is interesting to note that for r = 4 contracting the 4 sections gives S → T where T is a singular Enriques surface. It has 4 singular points and 2K T ∼ 0.) Corollary 11. Let k be a field and S → P 1 a minimal conic bundle. Assume that δ(S) > 12 + 6m. Then −K S + mF is birationally pseudo-effective iff S contains a double section C ⊂ S whose normalization has genus ≤ m + 1. Proof. Assume that S S is birational and −K S + mF is pseudo-effective. By [Isk67] (see also Theorem 27), S is another minimal conic bundle with the same δ, S → P 1 and S → P 1 have the same generic fiber and −K S + m F is linearly equivalent to an irreducible double section C for some m ≤ m by Lemma 9. Thus C is obtained as the birational transform of C . Its normalization has genus ≤ m + 1 by Lemma 12. Conversely, let C ⊂ S be a double section. We can resolve the singularities of C by performing elementary transformations at its singular points; see Paragraph 26. (Clearly, these points are on smooth fibers.) At the end we have C ⊂ S and C is smooth. If p a (C ) ≤ m + 1 then C is a sub-curve of \| − K S + mF \| by Lemma 12. Lemma 12. Let C ⊂ S be a double section. Then C is a sub-curve of \|−K S +mF \| iff p a (C) ≤ m+1. Proof. C ∼ −K S + bF for some b ∈ Z. Thus 2p a (C) − 2 = C(C + K S ) = b(C · F ) = 2b hence b ≤ m iff p a (C) ≤ m + 1. Threefold conic bundles We use Corollary 11 to prove a similar result for 3-dimensional conic bundles. Theorem 13. Let π : X → P 2 be a smooth, minimal conic bundle over a field of characteristic = 2, 3, 5 whose branch curve B X ⊂ P 2 has degree ≥ 19. The following are equivalent. 1. −K X is birationally pseudo-effective. 2. There is a generically finite double section D ⊂ X with normalization τ :D → D such that the branch curve B D of π • τ :D → P 2 has degree ≤ 6. 3. X is birational to a smooth, minimal conic bundle π : X → P 2 (with the same branch curve) such that \| − K X \| = ∅. Proof. Assume that φ : X X shows that −K X is birationally pseudo-effective. By Complement 28, we may choose X such that π : X → P 2 is also a conic bundle with the same branch curve. Write −K X ∼ Q ∆ where ∆ is pseudo-effective. Let L ⊂ P 2 be the generic line. By restriction we get π S : S → L. Since −K S + F = −K X \| S , we see that −K S + F ∼ Q ∆ \| S is pseudo-effective. Moreover, if ∆ t is a sequence of Q-effective divisors converging to ∆ then ∆ t \| S is a sequence of Q-effective divisors converging to ∆ \| S . Thus, by Lemma 9, ∆ t \| S has an irreducible component C ⊂ S with negative self-intersection and its normalization has genus ≤ 2 by Lemma 12. Since L is the generic line, C is the restriction of an irreducible component D of ∆ t . Thus D is a double section and its birational transform on X is a double section D ⊂ X. Furthermore, we know that the preimage of L in the normalization τ :D → D has genus ≤ 2. Thus L intersects the branch curve of π • τ :D → P 2 in ≤ 6 points. This shows that (1) ⇒ (2). Assume next that π has a double section D ⊂ X as in (2). By Lemma 14, there is a sequence of elementary transformations X X such that the branch curve of π : D → P 2 has degree ≤ 6. By Lemma 15 this implies that D is a sub-divisor of \| − K X \|. Thus (2) ⇒ (3) and (3) ⇒ (1) holds by definition. Lemma 14. Let π : X → Z be a conic bundle. Assume that X and Z are smooth and dim Z = 2. Let D ⊂ X be a generically finite double section. Then there is a sequence of elementary transformations X X such that the resulting D ⊂ X is normal, except possibly along finitely many fibers. Proof. Let C ⊂ Sing D be a curve not contained in a fiber of π. Let F ⊂ F ⊂ X be an irreducible component of a fiber that is not contained in D but meets C at a point p. Since D is singular at p, the local intersection number (D · F ) p is at least 2. Since D is a double section, (D · F ) ≤ 2. Thus F is irreducible and D ∩ F = {p}, hence π C : C → π(C) is birational. As we discuss in Paragraph 26, the elementary transformation centered at C decreases the degree of the branch curve of D → P 2 . After finitely many such steps we get π : X → Z such that D ⊂ X is normal, except possibly along finitely many fibers. Lemma 15. Let π : X → P 2 be a minimal conic bundle and D ⊂ X a generically finite double section. Then D is a sub-divisor of \| − K X \| iff the branch curve B D ⊂ P 2 of π D : D → P 2 has degree ≤ 6. Proof. Since X is minimal, we know that D ∼ −K X + bπ * H for some b ∈ Z where H is the class of a line in P 2 . Thus K D ∼ bπ * D H. Let D → D → P 2 denote the Stein factorization. Then K D ∼ bπ * D H pushes forward to K D ∼ bπ * D H. By the Hurwitz formula K D ∼ π * D K P 2 + 1 2 B D . Thus b ≤ 0 iff deg B D ≤ 6. K3 surfaces on conic bundles In order to prove Theorem 3, it remains to show that (13.2) does not hold for suitable branch curves B ⊂ P 2 . Example 20 shows that there are conic bundles with high degree smooth branch curve for which \| − K X \| does contain a K3 surface. Thus we need to focus on more subtle properties of B. We present 2 approaches. The first uses branch curves with many nodes; we prove that all the nodes have to lie on a sextic curve. This leads to quite explicit examples starting with deg B = 12; see Corollary 17. Note also that, in a flat family of conic bundles, double sections are parametrized by countably many components of the Chow variety. Thus if we find one conic bundle without certain type of double sections then the very general conic bundle also has no double sections of the same type. The other approach proves that there has to be a sextic curve that is everywhere tangent to B X . This is easily seen to be a non Proof. If p / ∈ B D then both of theétale local branches ofD → Z giveétale local sections of π. As we discuss in Claim 25.5, there are noétale local sections over the nodes of B X for X smooth. Corollary 17. Let X ⊂ P 2 x × P 2 y be given by an equation of bidegree (d, 2) i g i (x 0 , x 1 , x 2 )y 2 i = 0. Assume that the curves B i := g i (x 0 , x 1 , x 2 ) = 0 ⊂ P 2 x are smooth and they intersect each other transversally in 3d 2 distinct points. Let D ⊂ X be a double section and B D ⊂ P 2 x the branch curve of π\| D • τ :D → Z. Then deg B D ≥ 2d. Proof. The assumptions imply that X is smooth. The branch curve of X is B X = B 0 + B 1 + B 2 . By Lemma 16, B D passes through all the nodes of B. In particular, B D intersects B 0 in the 2d 2 points B 0 ∩ (B 1 ∪ B 2 ). Thus deg B D ≥ 2d unless B 0 ⊂ B D . We can repeat the argument for B 1 to get that deg B D ≥ 2d unless B 0 + B 1 ⊂ B D . The letter case also implies that deg B D ≥ 2d. For conic bundles with smooth branch curves we have a less precise condition. (The 2 results are closely related since passing through a node guarantees that the intersection multiplicity is ≥ 2.) Proposition 18. Let π : X → P 2 be a smooth, minimal conic bundle with branch locus B X ⊂ P 2 . Assume that there is no reduced curve C ⊂ P 2 of degree ≤ 6 such that all points of C ∩ B X have intersection multiplicity ≥ 2. Then −K X is not effective. Proof. Assume to the contrary that −K X ∼ D where D is an effective Z-divisor. Since X is minimal, there are no rational sections. Thus there is a unique irreducible component D 0 ⊂ D that is a double section of π. Let τ :D 0 → D 0 denote the normalization and let B D ⊂ P 2 be the branch curve of τ • π\| D 0 :D 0 → P 2 . Note that D 0 ∼ −K X − mH for some m ≥ 0 and deg B D ≤ 6 by Lemma 15. We claim that all points of B D ∩ B X have intersection multiplicity ≥ 2. To see this, assume to the contrary that there is a point p ∈ B D ∩ B X where B D , B X are smooth and intersect transversally. In a neighborhood of p the surface D 0 is a smooth double cover ramified along B D . Then (π\| D 0 ) −1 (B X ) is a smooth double cover of B X ramified at p. On the other hand, (π\| D 0 ) −1 (B X ) = D 0 ∩ π −1 (B X ) hence its normalization factors through theétale coverB X → B X . This is a contradiction. (This argument in fact shows that these intersection multiplicities are even.) Next we check that the assumptions of Proposition 18 hold for a general branch curve of degree ≥ 15. Lemma 19. Let B ⊂ P 2 be a general curve of degree d. Then there is no reduced curve C ⊂ P 2 of degree < 1 2 d − 1 such that all points of C ∩ B have intersection multiplicity ≥ 2. Proof. Fix a reduced curve C and let L be a line bundle of degree m. Let W (L) ⊂ H 0 (C, L) denote the subvariety consisting of sections without simple zeros. Note that any s ∈ W (L) has at most 1 2 m zeros. The map that sends a section of L to its set of zeros has 1-dimensional fibers. Thus dim W (L) ≤ 1 2 m + 1. Fix a reduced plane curve C ⊂ P 2 of degree c. Let W (C, d) denote the set of degree d curves B such that all points of C ∩ B have intersection multiplicity ≥ 2. Applying the above estimate to (Note that the bound 1 2 d − 1 is sharp for d = 4 but one should be able to prove slightly better bounds for larger values of d.) L = O P 2 (d)\| C we get that dim W (C, d) ≤ 1 2 cd + d− Example 20. Let Z be a smooth variety and E a rank 3 vector bundle one Z. Set P := P Z (E) with projection π : P → Z and note that ω P/Z ∼ = O P (−3) ⊗ π * det E. Let L be a line bundle on Z and X ⊂ P the zero set of a section of O P (2) ⊗ π * L. Then ω X ∼ = O P (−1) ⊗ π * (L ⊗ det E) \| X and π * ω −1 X ∼ = (ω Z ⊗ L ⊗ det E) −1 ⊗ E. Note that H 0 P, O P (2) ⊗ π * L = H 0 Z, L ⊗ Sym 2 E and H 0 X, O X (−K X ) = H 0 Z, L(K Z ) ⊗ det E −1 ⊗ E . We would like X to be smooth, this suggests that L ⊗ Sym 2 E should be generated by global sections, hence L should be positive. By contrast, the condition \| − K X \| = ∅ suggests that L should be negative. It seems that both of these can be satisfied only if E is rather unstable. For example, take Z = P 2 , set E = O P 2 (c) ⊕ O P 2 (3) ⊕ O P 2 where c > 3 and L = O P 2 . Then L ⊗ Sym 2 E is generated by global sections, so a general section gives a smooth conic bundle X. Furthermore, π * ω −1 X ∼ = O P 2 (3 − 3 − c) ⊗ E ∼ = O P 2 ⊕ O P 2 (3 − c) ⊕ O P 2 (−c) has a unique section. Thus X → P 2 is a conic bundle such that \| − K X \| contains a unique K3 surface. It is obtained by intersecting X with the divisor in P corresponding to the O P 2 (3) ⊕ O P 2 summand of E. The branch curve of X is given by the equation det   g 2c g c+3 g c g c+3 g 6 g 3 g c g 3 g 0   = 0, where g i := g i (x, y, z) denotes a homogeneous polynomial of degree i. (We can thus assume that g 0 = 1.) Eliminating g c , g 3 in the last row writes g 0 times the determinant, hence the equation of B X , in the form (g 6 g 0 − g 2 3 )(g 2c g 0 − g 2 c ) − (g c+3 g 0 − g c g 3 ) 2 = 0. Thus the degree 6 curve g 6 g 0 − g 2 3 = 0 intersects B X at the points where g c+3 g 0 − g c g 3 = 0 and all intersection multiplicities are even; as needed in Lemma 18. Examples of conic bundle surfaces Working backwards from Theorem 3 we see that there are surface conic bundles S → P 1 over C(s, t) such that −K S is not birationally Q-effective. We will exhibit similar examples over Q and Q p . Example 21. Let k be a field and S → P 1 a conic bundle over k that becomes trivial after a quadratic extension k ⊃ k. Then S k has a section C , thus S has a conjugate pair of sections C. The normalization of C has p a = −1. Thus Corollary 11 implies that S → P 1 is birational to a conic bundle S → P 1 such that −K S is effective. (Typically the base locus of \| − K S \| consists of the disjoint conjugate sections and the moving part of 2 fibers.) There are such examples over Q, even with arbitrary branch locus. Let g(s) ∈ Q[s] be a polynomial of degree 2d with simple roots only. Choose a prime p such that √ p is not contained in the splitting field of g. Let S g → P 1 be the projective model of the surface g(s)z 2 = x 2 − py 2 ⊂ P 1 xyz × A 1 s . Then S g → P 1 is minimal and the singular fibers lie exactly over the roots of g(s). Thus there are surface conic bundles S → P 1 with δ(S) arbitrarily large and branch locus B S in general position for which −K S is birationally effective. Note also that the above argument implies that −K S is birationally effective for every conic bundle over R since every surface conic bundle over C has a section. The situation over finite fields is unclear to me. The examples where −K S is not birationally Q-effective rest on the following observation. Lemma 22. Let Z be a 2-dimensional regular scheme and π : X → Z a conic bundle. Let C ⊂ Z be an irreducible 1-dimensional subscheme such that X C := π −1 (C) → C has no rational sections. Let W ⊂ C be the set of points z ∈ C such that X z is a double line and X is regular along X z . Let D ⊂ X be a double section with normalization τ :D → D and B D ⊂ Z the branch locus of π\| D • τ :D → Z. Then C ⊂ B D but W ⊂ B D . Proof. If C ⊂ B D then red D ∩ X C is a rational section of X C → C. This is contrary to our assumptions and W ⊂ B D follows from Claim 25.3. Example 23. We will apply Lemma 22 with Z = P 1 Zp and C = P 1 Fp for some prime p ≥ 3. Fix a natural number d and set P : + 1) . The conic bundle X ⊂ P will be given by a section of O P (2). For simplicity we choose a section of = P Z O Z ⊕ O Z (d) ⊕ O Z (dO Z + O Z (2d) ⊕ O Z (2d + 2) ⊂ π * O P (2). Choosing an affine coordinate s on P 1 , one can give such an X by an equation X := z 2 = a(s)x 2 + b(s)y 2 where deg a(s) = 2d, deg b(s) = 2d + 2. We choose a(s) and b(s) as follows. 1.ā(s) has only simple zeros where¯denotes reduction mod p, Let α be a root of a(s). Near α the equation of X has the form z 2 = (s − α)u 1 x 2 + (s − α)u 2 + pu 3 y 2 , where the u i are units. Thus the fiber X α is a double line and X is regular along X α by (25.5). Over the point (s = 1) ∈ P 1 Fp the fiber is (z 2 =ā(1)x 2 ) ⊂ P 2 Fp . Sinceā(1) is not a square, its only F p -point is (0:1:0) whereX is smooth. HenceX → P 1 Fp has no sections. Thus Lemma 22 implies that 2g(D Qp ) + 2 = deg B D ≥ deg a(s) = 2d. The choices (23.1-23.4) can be satisfied for a(s), b(s) ∈ Q[s], hence we proved the following more precise form of Theorem 1. Corollary 24. For every g there are conic bundles S → P 1 defined over Q with δ(S) = 4g + 6 and such that every double section of S has geometric genus ≥ g. For g = 2 this gives conic bundles S → P 1 with δ(S) = 14 defined over Q such that −K S is not birationally pseudo-effective. Birational maps of conic bundles We summarize the results on birational maps of conic bundles that we used. As before, all residue characteristics are assumed to be = 2. (Extending conic bundles) We will need to understand the following Problem 25.1. Let Z be a regular surface, W ⊂ Z a finite subset and π 0 : X 0 → Z 0 := Z \ W a conic bundle with branch locus B 0 X . We would like to extend π 0 : X 0 → Z 0 to a conic bundle π : X → Z and control the singularities of X in terms of B X . (This is also interesting if dim Z > 2 but the 2-dimensional case is simpler.) We may assume that W = {p} is a single point and Z is local. The push-forward E 0 := π 0 * ω −1 x 0 /Z 0 is a locally free sheaf of rank 3. Set E := j * E 0 where j : Z 0 → Z is the natural injection. Then E is a reflexive sheaf but, since Z is regular and 2-dimensional, E is locally free, hence free. Set P := P Z (E) and let X ⊂ P be the closure of X 0 ⊂ P 0 . Choose an isomorphism P ∼ = P 2 Z , then X is given by an equation ij g ij (z 1 , z 2 )x i x j = 0, (25.2) where g ij (z 1 , z 2 ) ∈ O Z . The scheme structure of the branch locus is defined by det g ij (z 1 , z 2 ) = 0. (25.3) The worst case is when the central fiber X p equals P 2 ; thus π : X → Z is not even equidimensional. If this happens then all the g ij vanish at p so det g ij (z 1 , z 2 ) ∈ m 3 p . Thus B X has a triple (or higher) point at p. Otherwise the central fiber X p is conic. X p is smooth iff p ∈ B X . Next assume that X p is a pair if lines. Possibly after a quadratic residue field extension, in suitable formal coordinates we can diagonalize the equation of X as x 2 0 = x 2 1 + f (z 1 , z 2 )x 2 2 and B X = f (z 1 , z 2 ) = 0 where f (0, 0) = 0. We see that X is regular along X p iff f ∈ (z 1 , z 2 ) 2 ; that is, iff B X is regular at p. Furthermore, (z 1 , z 2 ) → (1:1:0; z 1 , z 2 ) is a formal section. Finally consider the case when X p is a double line. Then det g ij (z 1 , z 2 ) ∈ m 2 p and B X has a double (or higher) point at p. If the branch curve has a node then, after a quadratic residue field extension, in suitable formal coordinates we can write X as X = x 2 0 = z 1 x 2 1 + z 2 x 2 2 . The fiber of the projection π : X → A 2 z is a double line over the origin, a pair of lines over the coordinate axes (z 1 z 2 = 0) and smooth otherwise. In contrast with the previous nodal case, the equation has no solutions in the quotient field of the completion of O z,Z ; see Claim 25.5. We have thus proved the following. Claim 25.4. Let Z be a regular surface whose residue characteristics are = 2, B ⊂ Z a curve with only nodal singularities and W ∈ Z a finite subset containing the nodes of B. Let π 0 : X 0 → Z \ W be a conic bundle with (scheme theoretic) branch locus B \ W . Then X 0 extends to a unique conic bundle π : X → Z. Furthermore, X is regular iff the following holds: for every p ∈ W that is a node of B, the projection π : X → Z has no sections over the quotient field of the strict henselization O sh p,Z of the local ring O p,Z . We also used the following well known result. Claim 25.5. Let R, m = (s, t) be a regular local ring of dimension 2. Then x 2 0 = sx 2 1 + tx 2 2 has no nonzero solutions in the quotient field of R. Proof. After clearing denominators, we may assume that the x i are in R. Let c be the largest such that the x i are in m c . We can thus write x i = p i (s, t) + r i where p i is a homogeneous polynomial of degree c and r i ∈ m c+1 . Then x 2 0 ∈ m 2c but sx 2 1 + tx 2 2 ∈ m 2c+1 . Thus in fact p 0 ≡ 0 and so sp 2 1 + tp 2 2 ∈ m 2c+2 . Thus sp 2 1 + tp 2 2 ≡ 0 hence p 1 and p 2 are both identically 0. This is a contradiction. (Elementary transformations of conic bundles) Let π : X → Z be a conic bundle over a regular surface and C ⊂ Z a regular curve such that π is smooth over C and s : C → X C a section of X C → C. We first blow up s(C) ⊂ X and then contract the birational transform of X C . We get another conic bundle π (s) : X (s) → Z. The rational map X X (s) is called the elementary transformation with center s(C). Next consider the case when C ⊂ Z is a geometrically reduced curve, π is smooth over the generic points of C and s : C X C is a rational section of X C → C. Then there is a finite subset W ⊂ Z such that Z 0 := Z \ W , C 0 := C \ W and X 0 := X \ π −1 (W ) satisfy the previous assumptions. We can thus construct the elementary transformation π 0,(s) : X 0,(s) → Z 0 of π 0 : X 0 → Z 0 with center s(C 0 ). Finally, whenever possible, we use the method of Paragraph 25 to extend π 0,(s) : X 0,(s) → Z 0 to a conic bundle π (s) : X (s) → Z called the elementary transformation with center s(C). Claim 26.1. Let π : X → Z be a conic bundle over a surface, C ⊂ Z a geometrically reduced curve and s : C X C a rational section of X C → C. Assume that X and Z are regular and π is smooth over the generic points of C. Then the elementary transformation with center s(C) exists and it is a conic bundle π (s) : X (s) → Z such that X (s) is also regular and has the same branch curve as X → Z. The following result of [Sar80], whose idea goes back to [Isk67], describes birational transformations of conic bundles over the same base. Claim 26.2. Let π i : X i → Z be smooth conic bundles over a smooth surface. Let φ : X 1 X 2 be a birational equivalence over Z. That is, the following diagram commutes X 1 φ X 2 π 1 ↓ ↓ π 2 Z = Z. Then φ is a composite of elementary transformations. The key result about birational maps of conic bundles is the following. The surface case is due to [Isk67]. (The minimal model program for surfaces over any field is established in [Mor82], thus the arguments of [Isk67] extend to any field.) The much harder 3-fold case is treated in [Sar80]. See [Cor95,Cor00] for more conceptual proofs. Theorem 27. Let π : X → Z be a minimal conic bundle over a field of characteristic = 2, 3, 5 such that X, Z are smooth and B X + 4K Z is effective. Let π : X → Z be a Mori fiber space and φ : X X a birational map. Then π is also a conic bundle, and there is a birational map φ Z : Z Z such that the following diagram commutes X φ X π ↓ ↓ π Z φ Z Z. (27.1) For our applications we need the following more precise version for which I could not find an explicit reference. Complement 28. Assume in addition that B X + K Z is ample. Then the diagram (27.1) can be factored as X ρ X τ X π ↓ ↓ π ↓ π Z ρ Z → Z = Z,(28.1) where τ : X X is a composite of elementary transformations, ρ : X X is a birational contraction and −K X = ρ * −K X . -empty Zariski open condition for deg B ≥ 15, though I could not write down explicit examples. As pointed out by [Sar82, Sec.5], the results of [AM72, Sec.3] imply that every smooth plane curve of degree ≥ 3 is the branch curve of a minimal conic bundle. Lemma 16. Let π : X → Z be a minimal, smooth conic bundle with branch curve B X . Let D ⊂ X be a double section with normalization τ :D → D and B D ⊂ Z the branch curve of π\| D • τ :D → Z. Then Sing(B X ) ⊂ B D . 2. b(s) = (s 2 − 1)a(s) + pc(s) for some c(s), 3. s 2 − 1,ā(s),c(s) are pairwise relatively prime and 4.ā(1) is not a square in F p .Claim 23.5. Let π : X Qp → P 1Qp be the generic fiber of the above X → Z and D Qp ⊂ X Qp a double section with normalization τ :D Qp → D Qp . Then g(D Qp ) ≥ d − 1.Proof. The closure of D Qp gives a double section D ⊂ X. Let B D ⊂ Z be the branch locus of π\| D • τ :D → Z. We aim to apply Lemma 22. c+2 2 . 2Next let W (d) denote the set of degree d curves B such that all points of C ∩ B have intersection multiplicity ≥ 2 for some curve C of degreec. Then dim W (d) ≤ 1 2 cd + d−c+2 2 + c+2 2 − 1. After expanding the binomials we see that dim W (d) < d+2 2 − 1 for d > 2c + 2. Acknowledgements I thank A. Corti, Y. Gongyo, A. Skorobogatov, C. Xu and the referees for helpful comments, corrections and references.Proof. Let π : X → Z be a conic bundle over a field of characteristic 0 such that X, Z are smooth. Pick a point z ∈ Z and let p : Z 1 := B z Z → Z denote the blow-up. The pull-back X × Z Z 1 is birational to a minimal conic bundle π 1 : X 1 → Z 1 . We want to describe its branch locus B X 1 . It is clear that B X 1 ⊃ p −1 * B X , the only question is what happens with the exceptional curve E z . The following possibilities are listed in [Sar80, 2.4-2.5].IfIn all these cases we see that for any sequence of blow ups p r :Applying this to a common resolution Z ← Z r → Z we conclude that if K Z + B X is ample then (Z, B X ) is the unique log-canonical model of (Z r , B Xr ). In particular, the rational map φ Z : Z Z in (27.1) is a morphism. This establishes the bottom row of (28.1).In order to get the top row, let A be an ample divisor on Z and choose m such that \|H \| := \| − K Z + m(p ) * A \| is a very ample linear system on Z . Its push-forward φ * \|H \| is a mobile linear system consisting of rational double sections of π. Its base locus consists of some horizontal curves (that is curves C i ⊂ X such that C i → π(C i ) is birational) and some vertical curves (that is curves that are contracted by π). 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[ "A hybrid model for the evolution of galaxies and Active Galactic Nuclei in the infrared", "A hybrid model for the evolution of galaxies and Active Galactic Nuclei in the infrared" ]	[ "Zhen-Yi Cai [email protected] \nAstrophysics Sector\nSISSA\nVia Bonomea 265I-34136TriesteItaly\n\nDepartment of Astronomy and Institute of Theoretical Physics and Astrophysics\nXiamen University\n361005XiamenP. R. China\n", "Andrea Lapi \nAstrophysics Sector\nSISSA\nVia Bonomea 265I-34136TriesteItaly\n\nDipartimento di Fisica, Università 'Tor Vergata'\nVia della Ricerca Scientifica 1, I00133RomaItaly\n", "Jun-Qing Xia \nAstrophysics Sector\nSISSA\nVia Bonomea 265I-34136TriesteItaly\n\nInstitute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Science\nP.O.Box 918-3100049BeijingP.R.China\n", "Gianfranco De Zotti \nAstrophysics Sector\nSISSA\nVia Bonomea 265I-34136TriesteItaly\n\nINAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio 5, I35122PadovaItaly\n", "Mattia Negrello \nINAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio 5, I35122PadovaItaly\n", "Carlotta Gruppioni \nINAF -Osservatorio Astronomico di Bologna\nvia Ranzani 1, I-40127BolognaItaly\n", "Emma Rigby \nLeiden Observatory\nThe Netherlands 8 APC, 10, rue Alice Domon et Léonie Duquet, France -2P.O. Box 95132300 RA, 75205Leiden, Paris Cedex 13\n", "Guillaume Castex ", "Jacques Delabrouille ", "Luigi Danese \nAstrophysics Sector\nSISSA\nVia Bonomea 265I-34136TriesteItaly\n" ]	[ "Astrophysics Sector\nSISSA\nVia Bonomea 265I-34136TriesteItaly", "Department of Astronomy and Institute of Theoretical Physics and Astrophysics\nXiamen University\n361005XiamenP. R. China", "Astrophysics Sector\nSISSA\nVia Bonomea 265I-34136TriesteItaly", "Dipartimento di Fisica, Università 'Tor Vergata'\nVia della Ricerca Scientifica 1, I00133RomaItaly", "Astrophysics Sector\nSISSA\nVia Bonomea 265I-34136TriesteItaly", "Institute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Science\nP.O.Box 918-3100049BeijingP.R.China", "Astrophysics Sector\nSISSA\nVia Bonomea 265I-34136TriesteItaly", "INAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio 5, I35122PadovaItaly", "INAF -Osservatorio Astronomico di Padova\nVicolo dell'Osservatorio 5, I35122PadovaItaly", "INAF -Osservatorio Astronomico di Bologna\nvia Ranzani 1, I-40127BolognaItaly", "Leiden Observatory\nThe Netherlands 8 APC, 10, rue Alice Domon et Léonie Duquet, France -2P.O. Box 95132300 RA, 75205Leiden, Paris Cedex 13", "Astrophysics Sector\nSISSA\nVia Bonomea 265I-34136TriesteItaly" ]	[]	We present a comprehensive investigation of the cosmological evolution of the luminosity function of galaxies and active galactic nuclei (AGN) in the infrared (IR). Based on the observed dichotomy in the ages of stellar populations of early-type galaxies on one side and late-type galaxies on the other, the models interprets the epoch-dependent luminosity functions at z ≥ 1.5 using a physical model for the evolution of proto-spheroidal galaxies and of the associated AGNs, while IR galaxies at z < 1.5 are interpreted as being mostly late-type "cold" (normal) and "warm" (starburst) galaxies. As for proto-spheroids, in addition to the epoch-dependent luminosity functions of stellar and AGN components separately, we have worked out, for the first time, the evolving luminosity functions of these objects as a whole (stellar plus AGN component), taking into account in a self-consistent way the variation with galactic age of the global SED. The model provides a physical explanation for the observed positive evolution of both galaxies and AGNs up to z ≃ 2.5 and for the negative evolution at higher redshifts, for the sharp transition from Euclidean to extremely steep counts at (sub-)mm wavelengths, as well as the (sub-)mm counts of strongly lensed galaxies, that are hard to account for by alternative, physical or phenomenological, approaches. The evolution of late-type galaxies and of z < 1.5 AGNs is described using a parametric phenomenological approach. The modeled AGN contributions to the counts and to the cosmic infrared background (CIB) are always subdominant. They are maximal at mid-IR wavelengths: the contribution to the 15 and 24 µm counts reaches 20% above 10 and 2 mJy, respectively, while the contributions to the CIB are of 8.6% and of 8.1% at 15 µm and 24 µm, respectively. The model provides a good fit to the multi-wavelength (from the mid-IR to millimeter waves) data on luminosity functions at different redshifts and on number counts (both global and per redshift slices). A prediction of the present model, useful to test it, is a systematic variation with wavelength of the populations dominating the counts and the contributions to the CIB intensity. This implies a specific trend for cross-wavelength CIB power spectra, that is found to be in good agreement with the data.	10.1088/0004-637x/768/1/21	[ "https://arxiv.org/pdf/1303.2335v1.pdf" ]	119,295,473	1303.2335	6ff80f885b645972c8b289907bcc12652ad71d46	A hybrid model for the evolution of galaxies and Active Galactic Nuclei in the infrared 10 Mar 2013 Zhen-Yi Cai [email protected] Astrophysics Sector SISSA Via Bonomea 265I-34136TriesteItaly Department of Astronomy and Institute of Theoretical Physics and Astrophysics Xiamen University 361005XiamenP. R. China Andrea Lapi Astrophysics Sector SISSA Via Bonomea 265I-34136TriesteItaly Dipartimento di Fisica, Università 'Tor Vergata' Via della Ricerca Scientifica 1, I00133RomaItaly Jun-Qing Xia Astrophysics Sector SISSA Via Bonomea 265I-34136TriesteItaly Institute of High Energy Physics Key Laboratory of Particle Astrophysics Chinese Academy of Science P.O.Box 918-3100049BeijingP.R.China Gianfranco De Zotti Astrophysics Sector SISSA Via Bonomea 265I-34136TriesteItaly INAF -Osservatorio Astronomico di Padova Vicolo dell'Osservatorio 5, I35122PadovaItaly Mattia Negrello INAF -Osservatorio Astronomico di Padova Vicolo dell'Osservatorio 5, I35122PadovaItaly Carlotta Gruppioni INAF -Osservatorio Astronomico di Bologna via Ranzani 1, I-40127BolognaItaly Emma Rigby Leiden Observatory The Netherlands 8 APC, 10, rue Alice Domon et Léonie Duquet, France -2P.O. Box 95132300 RA, 75205Leiden, Paris Cedex 13 Guillaume Castex Jacques Delabrouille Luigi Danese Astrophysics Sector SISSA Via Bonomea 265I-34136TriesteItaly A hybrid model for the evolution of galaxies and Active Galactic Nuclei in the infrared 10 Mar 2013Subject headings: galaxies: formation -galaxies: evolution -galaxies: elliptical -galaxies: high redshift -submillimeter We present a comprehensive investigation of the cosmological evolution of the luminosity function of galaxies and active galactic nuclei (AGN) in the infrared (IR). Based on the observed dichotomy in the ages of stellar populations of early-type galaxies on one side and late-type galaxies on the other, the models interprets the epoch-dependent luminosity functions at z ≥ 1.5 using a physical model for the evolution of proto-spheroidal galaxies and of the associated AGNs, while IR galaxies at z < 1.5 are interpreted as being mostly late-type "cold" (normal) and "warm" (starburst) galaxies. As for proto-spheroids, in addition to the epoch-dependent luminosity functions of stellar and AGN components separately, we have worked out, for the first time, the evolving luminosity functions of these objects as a whole (stellar plus AGN component), taking into account in a self-consistent way the variation with galactic age of the global SED. The model provides a physical explanation for the observed positive evolution of both galaxies and AGNs up to z ≃ 2.5 and for the negative evolution at higher redshifts, for the sharp transition from Euclidean to extremely steep counts at (sub-)mm wavelengths, as well as the (sub-)mm counts of strongly lensed galaxies, that are hard to account for by alternative, physical or phenomenological, approaches. The evolution of late-type galaxies and of z < 1.5 AGNs is described using a parametric phenomenological approach. The modeled AGN contributions to the counts and to the cosmic infrared background (CIB) are always subdominant. They are maximal at mid-IR wavelengths: the contribution to the 15 and 24 µm counts reaches 20% above 10 and 2 mJy, respectively, while the contributions to the CIB are of 8.6% and of 8.1% at 15 µm and 24 µm, respectively. The model provides a good fit to the multi-wavelength (from the mid-IR to millimeter waves) data on luminosity functions at different redshifts and on number counts (both global and per redshift slices). A prediction of the present model, useful to test it, is a systematic variation with wavelength of the populations dominating the counts and the contributions to the CIB intensity. This implies a specific trend for cross-wavelength CIB power spectra, that is found to be in good agreement with the data. Subject headings: galaxies: formation -galaxies: evolution -galaxies: elliptical -galaxies: high redshift -submillimeter Introduction The huge amount of infrared (IR) to millimeter-wave data that has been accumulating in the last several years have not yet led to a fully coherent, established picture of the cosmic star-formation history, of the IR evolution of Active Galactic Nuclei (AGNs), and of the inter-relations between star formation and nuclear activity. Many, increasingly sophisticated, phenomenological models for the cosmological evolution of the galaxy and AGN luminosity functions over a broad wavelength range have been worked out (e.g. Béthermin et al. 2012aBéthermin et al. , 2011Gruppioni et al. 2011;Rahmati & van der Werf 2011;Marsden et al. 2011;Franceschini et al. 2010;Valiante et al. 2009;Le Borgne et al. 2009;Rowan-Robinson 2009). These models generally include multiple galaxy populations, with different spectral energy distributions (SEDs) and different evolutionary properties, described by simple analytic formulae. In some cases also AGNs are taken into account. All of them, however, admittedly have limitations. The complex combination of source properties (both in terms of the mixture of SEDs and of evolutionary properties), called for by the richness of data, results in a large number of parameters, implying substantial degeneracies that hamper the interpretation of the results. The lack of constraints coming from the understanding of the astrophysical processes controlling the evolution and the SEDs limits the predictive capabilities of these models. In fact, predictions of pre-Herschel phenomenological models, matching the data then available, yielded predictions for Herschel counts quite discrepant from each other and with the data. The final goal is a physical model linking the galaxy and AGN formation and evolution to primordial density perturbations. In this paper we make a step in this direction presenting a comprehensive 'hybrid' approach, combining a physical, forward model for spheroidal galaxies and the early evolution of the associated AGNs with a phenomenological backward model for late-type galaxies and for the later AGN evolution. We start from the consideration of the observed dichotomy in the ages of stellar populations of early-type galaxies on one side and late-type galaxies on the other. Early-type galaxies and massive bulges of Sa galaxies are composed of relatively old stellar populations with mass-weighted ages of 8-9 Gyr (corresponding to formation redshifts z 1-1.5), while the disc components of spiral and irregular galaxies are characterized by significantly younger stellar populations. For instance, the luminosity-weighted age for most of Sb or later-type spirals is 7 Gyr (cf. Bernardi et al. 2010, their Fig. 10), corresponding to a formation redshift z 1. Thus proto-spheroidal galaxies are the dominant star-forming population at z ≥ 1.5, while IR galaxies at z < 1.5 are mostly late-type "cold" (normal) and "warm" (starburst) galaxies. Fuller hierarchical galaxy formation models, whereby the mass assembly of galaxies is related to structure formation in the dark matter and the star formation and merger histories of galaxies of all morphological types are calculated based on physical prescriptions have been recently presented by several groups (Lacey et al. 2008;Fontanot et al. 2009;Narayanan et al. 2010;Shimizu et al. 2012). However, the predictions for the IR evolution of galaxies are limited to a small set of wavelengths and frequently highlight serious difficulties with accounting for observational data (Lacey et al. 2010;Niemi et al. 2012;Hayward et al. 2012). While the evolution of dark matter halos in the framework of the 'concordance' ΛCDM cosmology is reasonably well understood thanks to N-body simulations such as the Millennium, the Millennium-XXL and the Bolshoi simulations (Springel et al. 2005;Boylan-Kolchin et al. 2009;Angulo et al. 2012;Klypin et al. 2011), establishing a clear connection between dark matter halos and visible objects proved to be quite challenging, especially at (sub-)mm wavelengths. The early predictions of the currently favoured scenario, whereby both the star-formation and the nuclear activity are driven by mergers, were more than one order of magnitude below the observed SCUBA 850 µm counts (Kaviani et al. 2003;Baugh et al. 2005). The basic problem is that the duration of the star-formation activity triggered by mergers is too short, requiring non standard assumptions either on the Initial Mass Function (IMF) or on dust properties to account for the measured source counts. The problem is more clearly illustrated in terms of redshift-dependent far-IR/sub-mm luminosity function, estimated on the basis of Herschel data (Eales et al. 2010;Gruppioni et al. 2010;). These estimates consistently show that z ≃ 2 galaxies with Star Formation Rates SFR ≃ 300 M ⊙ yr −1 have comoving densities Φ 300 ∼ 10 −4 Mpc −3 dex −1 . The comoving density of the corresponding halos is n(M vir ) ∼ Φ 300 (t exp /τ SFR ), where M vir is the total virial mass (mostly dark matter), τ SFR is the lifetime of the star-forming phase and t exp is the expansion timescale. For the fiducial lifetime τ SFR ≃ 0.7 Gyr advocated by , log(M vir /M ⊙ ) ≃ 12.92 while for τ SFR ≃ 0.1 Gyr, typical of a merger-driven starburst, log(M vir /M ⊙ ) ≃ 12.12. Thus while the model implies a SFR/M vir ratio easily accounted for on the basis of standard IMFs and dust properties, the latter scenario requires a SFR/M vir ratio more than a factor of 6 higher. To reach the required values of SFR/M vir or, equivalently, of L IR /M vir , Baugh et al. (2005) resorted to a top-heavy IMF while Kaviani et al. (2003) assumed that the bulk of the sub-mm emission comes from a huge amount of cool dust. But even tweaking with the IMF and with dust properties, fits of the sub-mm counts obtained within the merger-driven scenario (Lacey et al. 2010;Niemi et al. 2012) are generally unsatisfactory. Further constraints on physical models come from the clustering properties of sub-mm galaxies that are determined by their effective halo masses. As shown by Xia et al. (2012) both the angular correlation function of detected sub-mm galaxies and the power spectrum of fluctuations of the cosmic infrared background indicate halo masses larger than implied by the major mergers plus top-heavy initial stellar mass function scenario (Kim et al. 2011) and smaller than implied by cold flow models but consistent with the self-regulated baryon collapse scenario Lapi et al. 2006;. As is well known, the strongly negative K-correction emphasizes high-z sources at (sub-)mm wavelengths. The data show that the steeply rising portion of the (sub-)mm counts is indeed dominated by ultra-luminous star-forming galaxies with a redshift distribution peaking at z ≃ 2.5 (Chapman et al. 2005;Aretxaga et al. 2007;Yun et al. 2012;Smolčić et al. 2012). As shown by , the selfregulated baryon collapse scenario provides a good fit of the (sub-)mm data (counts, redshift-dependent luminosity functions) as well as of the stellar mass functions at different redshifts. Moreover, the counts of strongly lensed galaxies were predicted with remarkable accuracy (Negrello et al. 2007Lapi et al. 2012;González-Nuevo et al. 2012). Further considering that this scenario accounts for the clustering properties of sub-mm galaxies (Xia et al. 2012), we conclude that it is well grounded, and we adopt it for the present analysis. However, we upgrade this model in two respects. First, while, on one side, the model envisages a co-evolution of spheroidal galaxies and active nuclei at their centers, the emissions of the two components have been, so far, treated independently of each other. This is not a problem in the wavelength ranges where one of the two components dominates, as in the (sub-)mm region where the emission is dominated by star-formation, but is no longer adequate at mid-IR wavelengths, where the AGN contribution may be substantial. In this paper we present and exploit a consistent treatment of proto-spheroidal galaxies including both components. Second, while the steeply rising portion of (sub-)mm counts is fully accounted for by proto-spheroidal galaxies, late-type (normal and starburst) galaxies dominate both at brighter and fainter flux densities and over broad flux density ranges at mid-IR wavelengths. At these wavelengths, AGNs not associated to proto-spheroidal galaxies but either to evolved early type galaxies or to late-type galaxies are also important. Since we do not have a physical evolutionary model for late-type galaxies and the associated AGNs, these source populations have been dealt with adopting a phenomenological approach. Another distinctive feature of the present model is that we have attempted to fit simultaneously the data over a broad wavelength range, from mid-IR to mm waves. As mentioned in several papers, this faces us with several challenges. First, the data come from different instruments and the relative calibration is sometimes problematic (see the discussion in Béthermin et al. 2011). Systematic calibration offsets may hinder simultaneous fits of different data sets. For example, Marsden et al. (2011) pointed out that there is considerable tension between the SCUBA 850 µm counts and the AzTEC counts at 1.1 mm, and indeed the 850 µm and mm-wave counts have been repeatedly corrected (generally downwards) as biases were discovered and better data were acquired. Also, the very complex SEDs in the mid-IR, where strong polycyclic aromatic hydrocarbon (PAH) emission features show up, make the counts exceedingly sensitive to the details of the spectral response function of the specific instrument and introduce large uncertainties in the conversion from broad-band measurements to monochromatic flux densities giving rise to strong discrepancies among data sets nominally referring to the same wavelength. In fact, large discrepancies are present among different determinations of 15 µm and 60 µm source counts. The plan of the work is the following. In Section 2 we describe the physical model for the evolution of proto-spheroidal galaxies and of the associated AGNs and the SEDs adopted for these sources. Section 3 deals with the evolutionary model for late-type galaxies and z ≤ 1.5 AGNs. In Section 4 we present the formalism to compute the source counts of unlensed and lensed sources, the cumulative flux density as a function of redshift and the contributions to the CIB. In Section 5 we report on the determination of the best fit values of the model parameters. In Section 6 the model results are compared with data on multifrequency luminosity functions at various redshifts and on source counts, both total and per redshift slices. The multi-frequency power-spectra of CIB fluctuations implied by the model are discussed in Section 7. Finally, Section 8 contains a summary of the paper and our main conclusions. Tabulations of multi-frequency model counts, redshift distributions, SEDs, redshift-dependent luminosity functions at several wavelengths, and a large set of figures comparing model predictions with the data are available in the Web site http://people.sissa.it/∼zcai/galaxy agn/. Throughout this paper we adopt a flat cosmology with present day matter and baryon density, in units of the critical density, Ω m,0 = 0.27 and Ω b,0 = 0.044; Hubble constant h = H 0 /100 = 0.71; spectrum of primordial density perturbations with slope n = 1 and normalization on a scale of 8 h −1 Mpc σ 8 = 0.81. 2. Star-forming proto-spheroidal galaxies 2.1. Overview of the model We adopt the model by Granato et al. (2004, see also Lapi et al. 2006, 2011Mao et al. 2007) that interprets powerful high-z sub-mm galaxies as massive proto-spheroidal galaxies in the process of forming most of their stellar mass. It hinges upon high resolution numerical simulations showing that dark matter halos form in two stages (Zhao et al. 2003;Wang et al. 2011;Lapi & Cavaliere 2011). An early fast collapse of the halo bulk, including a few major merger events, reshuffles the gravitational potential and causes the dark matter and stellar components to undergo (incomplete) dynamical relaxation. A slow growth of the halo outskirts in the form of many minor mergers and diffuse accretion follows; this second stage has little effect on the inner potential well where the visible galaxy resides. The star formation is triggered by the fast collapse/merger phase of the halo and is controlled by selfregulated baryonic processes. It is driven by the rapid cooling of the gas within a region with radius ≈ 30% of the halo virial radius, i.e. of ≃ 70(M vir /10 13 M ⊙ ) 1/3 [(1 + z vir )/3] −1 kpc, where M vir is the halo mass and z vir is the virialization redshift, encompassing about 40% of the total mass (dark matter plus baryons). The star formation and the growth of the central black-hole are regulated by the energy feedback from supernovae (SNe) and from the active nucleus, is very soon obscured by dust and is stopped by quasar feedback. The AGN feedback is relevant especially in the most massive galaxies and is responsible for their shorter duration (5 − 7 × 10 8 yr) of the active star-forming phase. In less massive proto-spheroidal galaxies the star formation rate is mostly regulated by SN feedback and continues for a few Gyr. Only a minor fraction of the gas initially associated to the dark matter halo is converted into stars. The rest is ejected by feedback processes. The equations governing the evolution of the baryonic matter in dark matter halos and the adopted values for the parameters are given in the Appendix where some examples of the evolution with galactic age (from the virialization time) of quantities related to the stellar and to the AGN component are also shown. For additional details and estimates of physically plausible ranges for each parameter we refer to Granato et al. (2004), Lapi et al. (2006) and Mao et al. (2007). Since spheroidal galaxies are observed to be in passive evolution at z 1 − 1.5 (e.g. Renzini 2006), they are bright at sub-mm wavelengths only at higher redshifts. Luminosity functions The bolometric luminosity function (LF) of proto-spheroids is obtained convolving the halo formation rate dN ST (M vir , z)/dt with the galaxy luminosity distribution, P (L, z; M vir ). The halo formation rate is well approximated, for z 1.5, by the positive term of the cosmic time derivative of the halo mass function N ST . For the latter, giving the average comoving number density of haloes of given mass, M vir , we adopt the Sheth & Tormen (1999) analytical expression N ST (M vir , z)dM vir =ρ m,0 M 2 vir f ST (ν) d ln ν d ln M vir dM vir ,(1) whereρ m,0 = Ω m,0 ρ c,0 is the present day mean comoving matter density of the universe and ν ≡ [δ c (z)/σ(M vir )] 2 , with δ c (z) = δ 0 (z)D(0)/D(z). The critical value of the initial overdensity that is required for spherical collapse at z, δ 0 (z), is δ c (z) = δ 0 (z)D(0)/D(z) with (Nakamura & Suto 1997) δ 0 (z) = 3(12π) 2/3 20 [1 + 0.0123 log Ω m (z)] ≃ 1.6865[1 + 0.0123 log Ω m (z)]. The linear growth factor can be approximated as (Lahav et al. 1991;Carroll et al. 1992) D(z) = 5Ω m (z) 2(1 + z) 1 70 + 209 140 Ω m (z) − 1 140 Ω 2 m (z) + Ω 4/7 m (z) . The mass variance σ(M vir ) of the primordial perturbation field smoothed on a scale containing a mass M vir with a top-hat window function was computed using the Bardeen et al. (1986) power spectrum with correction for baryons (Sugiyama 1995), for our choice of cosmological parameters (see Section 1). The results are accurately approximated (error < 1% over a broad range of M vir , 10 6 < M vir /M ⊙ < 10 16 ) by σ(M vir ) = 0.8 0.84 14.110393 − 1.1605397x − 0.0022104939x 2 + 0.0013317476x 3 − 2.1049631 × 10 −6 x 4 (2) where x ≡ log(M vir /M ⊙ ). Furthermore f ST (ν) = A[1 + (aν) −p ] aν 2 1/2 e −aν/2 π 1/2 , where A = 0.322, p = 0.3, and a = 0.707. The halo formation rate is then dN ST (M vir , z) dt = N ST (M vir , z) d ln f ST (ν) dt = −N ST (M vir , z) aδ c σ 2 + 2p δ c σ 2p σ 2p + a p δ 2p c − 1 δ c dδ c dz dz dt ≃ N ST (M vir , z) aν 2 + p 1 + (aν) p d ln ν dz dz dt ,(3)where dz/dt = −H 0 (1 + z)E(z) with E(z) ≡ Ω Λ,0 + Ω m,0 (1 + z) 3 . The comoving differential luminosity function Φ(log L, z), i.e. the number density of galaxies per unit log L interval at redshift z, is given by Φ(log L, z) = where P (log L, z; M vir , z vir ) is the luminosity distribution of galaxies at redshift z inside a halo of mass M vir virialized at redshift z vir . We set z min vir = 1.5 and z max vir = 12. As mentioned in Section 2.1 the total luminosity of a galaxy is the sum of those of the stellar component and of the active nucleus. For each component we assume a log-normal luminosity distribution P [log L\| logL]d log L = exp[− log 2 (L/L)/2σ 2 ] √ 2πσ 2 d log L,(5) with dispersion σ * = 0.10 around the mean stellar luminosityL * (z; M vir , z vir ) and σ • = 0.35 around the mean AGN luminosityL • (z; M vir , z vir ). The mean luminosities are computed solving the equations detailed in the Appendix. The higher luminosity dispersion for the AGN component reflects its less direct relationship, compared to the stellar component, with M vir and z vir . The distribution of the total luminosity, L tot = L * + L • , is then (Dufresne 2004) P [log L tot \| logL * , logL • ]d log L tot = d log L tot × log Ltot −∞ dx 2πσ * σ • L tot L tot − 10 x exp{−(x − logL * ) 2 /2σ 2 * } × exp{−[log(L tot − 10 x ) − logL • ] 2 /2σ 2 • }.(6) In the upper left panel of Fig. 1 we show, as an example, the bolometric luminosity functions at z = 1.5 of the stellar and of the AGN components, as well as the luminosity function of the objects as a whole. As shown by eq. (6) the latter is different from the sum of the first two, although in this case the difference is difficult to perceive. The bright end is dominated by QSOs shining unobstructed after having swept away the interstellar medium of the host galaxy. In this phase the QSOs reach their maximum luminosity. Around log(L bol /L ⊙ ) ≃ 13 the AGNs and the starbursts give similar contributions to the bolometric luminosity function. The inflection at log(L IR /L ⊙ ) ≃ 11.7 corresponds to the transition from the regime where the feedback is dominated by supernovae (lower halo masses) to the regime where it is dominated by AGNs. While the star formation in massive halos is abruptly stopped by the AGN feedback after 0.5-0.7 Gyr, it lasts much longer in smaller galaxies, implying a fast increase of their number density. The upper right panel of the same figure illustrates the evolution with cosmic time of the global luminosity function. The cooling and free-fall timescales shorten with increasing redshift because of the increase of the matter density and this drives a positive luminosity evolution, thwarted by the decrease in the comoving density of massive halos. The two competing factors result, for both the starburst and the AGN component (see the lower panels of the figure), in a positive evolution up to z ≃ 2.5 followed by a decline at higher z, consistent with the observational determinations by Gruppioni et al. (2010) and for the starburst component and by Assef et al. (2011) and Brown et al. (2006) for AGNs. The decrease of the luminosity function at low luminosities, more clearly visible at the higher redshifts, is an artifact due to the adopted lower limit for the considered halo masses. This part of the luminosity function however does not contribute significantly to the observed statistics and therefore is essentially irrelevant here. Below the minimum virialization redshift, z min vir = 1.5, the bolometric luminosity function of proto-spheroidal galaxies rapidly declines as they evolve towards the 'passive' phase. The decline is faster at the bright end (above log(L bol /L ⊙ ) ≃ 12) since the switching off of the star formation for the more massive halos occurs on a shorter timescale. The monochromatic luminosity functions of each component or of objects as a whole can obviously be computed using the same formalism, given the respective spectral energy distributions (SEDs). We definē L * ,ν ≡ νL * ,ν = νf * (ν)L * ,IR ,L •,ν ≡ νL •,ν = νf • (ν)L •,bol , andL ν ≡L * ,ν +L •,ν , where f (ν) is the normalized SED ( dν f (ν) = 1). Since the model cannot follow in detail the evolution of the AGN SEDs during the short phase when they shine unobstructed by the interstellar medium of the host galaxy, the distinction between obscured and unobscured AGNs in the model is made in two ways. First, following Lapi et al. (2006), we choose a fixed optical (B-band) "visibility time", ∆t vis = 5 × 10 7 yr, consistent with current estimates of the optically bright QSO phase. Alternatively, we set the beginning of the optical bright phase at the moment when the gas mass fraction is low enough to yield a low optical depth. We estimate that this corresponds to a gas fraction within the dark matter potential well f gas = M gas /M vir f gas,crit = 0.03. The two approaches give very similar results and we have chosen the criterion f gas f gas,crit to compute the luminosity functions at optical wavelengths. Spectral energy distributions Although there is evidence that the galaxy SEDs vary with luminosity (e.g. Smith et al. 2012), have shown that the sub-mm data can be accurately reproduced using a single SED for protospheroidal galaxies, i.e. the SED of the strongly lensed z ≃ 2.3 galaxy SMM J2135-0102 Ivison et al. 2010), modeled using GRASIL ). The basic reason for the higher uniformity of the SEDs of high-z active star-forming galaxies compared to galaxies at low-z is that the far-IR emission of the former objects comes almost entirely from dust in molecular clouds, heated by newly formed stars, while in low-z galaxies there are important additional contributions from colder 'cirrus' heated by older stellar populations. This SED worked very well at sub-mm wavelengths but yielded mm-wave counts in excess of the observed ones. To overcome this problem the sub-mm slope of the SED has been made somewhat steeper, preserving the consistency with the photometric data on SMM J2135-0102 (see Fig. 2). Moreover, the SED used by Lapi et al. has a ratio between the total (8-1000 µm) IR and the 8 µm luminosity (IR8 = L IR /L 8 ) of ≃ 30, far higher than the mean value for z ≃ 2 galaxies (IR8 ≃ 9, Reddy et al. 2012). We have therefore modified the near-and mid-IR portions of the SED adopting a shape similar to that of Arp 220. The contribution of the passive evolution phase of early-type galaxies is small in the frequency range of interest here and will be neglected. As mentioned in Section 2.1, the model follows the AGN evolution through two phases (a third phase, reactivation, will be considered in Section 3.2). For the first phase, when the black-hole growth is enshrouded by the abundant, dusty interstellar medium (ISM) of the host galaxy, we adopt the SED of a heavily absorbed AGN taken from the AGN SED library by Granato & Danese (1994). Note that these objects differ from the classical type-2 AGNs because they are not obscured by a circum-nuclear torus but by the more widely distributed dust in the host galaxy. They will be referred to as type-3 AGNs. In the second phase the AGN shines after having swept out the galaxy ISM. For this phase we adopted the mean QSO SED by Richards et al. (2006) extended to sub-mm wavelengths assuming a grey-body emission with dust temperature T dust = 80 K and emissivity index β = 1.8. These SEDs imply that the IR (8-1000 µm) band comprises 92% of the bolometric luminosity of obscured AGNs and 19% of that of the unobscured ones. As illustrated by Fig. 2, except in the rare cases in which the AGN bolometric luminosity is much larger than that of the starburst, the AGN contribution is small at (sub-)mm wavelengths, while it is important and may be dominant, in the mid-IR. This implies that the statistics discussed here are insensitive to the parameters describing the extrapolation of the Richards et al. SED to (sub-)mm wavelengths. Figure 3 shows the global SEDs and the contributions of the stellar and AGN components for 2 galaxy ages and three host halo masses virialized at z vir = 3. The shorter evolution timescale of the AGNs is clearly visible. It is worth noticing that the effect of feedback as a function of halo mass on the SFR is very different from that on accretion onto the supermassive black-hole. In the less massive halos the AGN feedback has only a moderate effect on the evolution of the SFR and of the accretion rate, that are mostly controlled by the SN feedback. With reference to the figure, for log(M vir /M ⊙ ) = 11.4, the star-formation continues at an almost constant rate for a few Gyrs. On the other hand the accretion rate onto the central black-hole is at the Eddington limit only up to an age of ≃ 0.3 Gyr and afterwards drops to a strongly sub-Eddington regime. This is because the growth rate of the reservoir is approximately proportional to the SFR (and therefore slowly varying for few Gyrs) while the accretion rate grows exponentially until the mass contained in the reservoir is exhausted. From this moment on the accretion rate is essentially equal to the (strongly sub-Eddington) inflow rate. For more massive halos the quenching of both the SFR and of the accretion occurs more or less simultaneously at ages of ≃ 0.5-0.6 Gyr, but while the SFR stops very rapidly, the AGN activity continues until the flow of the matter accumulated in the reservoir runs out. At ages > ∼ 0.6 Gyr the more massive galaxies are in passive evolution and therefore very weak in the far-IR while star-formation and the dust emission are still present in lower-mass galaxies. 3. Low redshift (z 1.5) populations 3.1. Late-type and starburst galaxies We consider two z 1.5 galaxy populations: "warm" starburst galaxies and "cold" (normal) late-type galaxies. For the IR luminosity function of both populations we adopt the functional form advocated by Saunders et al. (1990): Φ(log L IR , z)d log L IR = Φ * L IR L * 1−α × exp − log 2 (1 + L IR /L * ) 2σ 2 d log L IR(7) where the characteristic density Φ * and luminosity L * , the low-luminosity slope α and the dispersion σ of each population are, in principle, free parameters. However, the low-luminosity portion of the luminosity function is dominated by "cold" late-type galaxies and, as a consequence, the value of α of the warm population is largely unconstrained; we have fixed it at α warm = 0.01. In turn, the "warm" population dominates at high luminosities so that the data only imply an upper limit to σ cold . We have set σ cold = 0.3. For the "warm" population we have assumed power law density and luminosity evolution [Φ * (z) = Φ * 0 (1 + z) αΦ ; L * (z) = L * 0 (1 + z) αL ] up to z break = 1, α Φ and α L being free parameters. The "cold" population comprises normal disc galaxies for which chemo/spectrophotometric evolution models (Mazzei et al. 1992;Colavitti et al. 2008) indicate a mild (a factor ≃ 2 from z = 0 to z = 1) increase in the star formation rate, hence of IR luminosity, with look-back time. Based on these results we take, for this population, α L = 1 and no density evolution. At z > z break both Φ * (z) and L * (z) are kept to the values at z break multiplied by the smooth cut-off function {1 − erf[(z − z cutoff )/∆z)]}/2, with z cutoff = 2 and ∆z = 0.5. The choice of the redshift cutoff for both populations of late-type galaxies is motivated by the fact that the disc component of spirals and the irregular galaxies are characterized by relatively young stellar populations (formation redshift z 1-1.5). Above z = 1.5 proto-spheroidal galaxies (including bulges of disk galaxies) dominate the contribution to the luminosity function, at least in the observationally constrained luminosity range. The other parameters are determined by minimum χ 2 fits to selected data sets, as described in Sect. 5. Their best fit values and the associated uncertainties are listed in Table 1. Although there is clear evidence of systematic variations of the IR SEDs of low-z galaxies with luminosity (e.g., Smith et al. 2012), we tried to fit the data with just 2 SEDs, one for the "warm" and one for the "cold" population. These SEDs were generated by combining those of Dale & Helou (2002), that are best determined at mid-IR wavelengths, with those of Smith et al. (2012), primarily based on Herschel data in the range 100-500 µm. Dale & Helou (2002) give SED templates for several values of the 60 to 100 µm flux density ratio, log[f ν (60µm)/f ν (100µm)]. Using the relation between this ratio and the 3 to 1100 µm luminosity, L TIR , given by Chapman et al. (2003) we established a correspondence between their SEDs and those by Smith et al., labeled by the values of log(L IR /L ⊙ ). The combined SEDs are based on Smith et al. above 100 µm and on Dale & Helou at shorter wavelengths. By trial and error we found that the best fit to the data is obtained using for the "cold" population the SED corresponding to log(L IR /L ⊙ ) = 9.75 (actually the SEDs change very slightly for log(L IR /L ⊙ ) 9.75) and for the "warm" population the SED corresponding to log(L IR /L ⊙ ) = 11.25. These 2 SEDs are displayed in Fig. 4. Reactivated AGNs In the framework of our reference galaxy and AGN evolutionary scenario, most of the growth of supermassive black holes is associated to the star forming phase of spheroidal components of galaxies at z 1.5 when the great abundance of interstellar medium favours high accretion rates, at, or even slightly above, the Eddington limit. At later cosmic times the nuclei can be reactivated by, e.g., interactions, mergers or dynamical instabilities. The accretion rates are generally strongly sub-Eddington. Our evolutionary scenario cannot predict their amplitudes and duty cycles. We therefore adopted, also for these objects, a phenomenological backward evolution model analogous to that used for the "warm" galaxy population, i.e. luminosity functions of the same form of eq. (7) and power-law density and luminosity evolution with the same break and cutoff redshifts. However the parameters of the luminosity functions refer to 12 µm (see Sect. 3.1 and Table 1). The data do not allow a determination of the slopes, α, of the faint portions of the luminosity functions. We have set α = 1.1 for type-1 AGNs and α = 1.5 for type-2. The steeper slope for type-2 was chosen on account of the fact that these dominate over type-1 at low luminosities. As in the case of normal late-type and of starburst galaxies, the other parameters are obtained by minimum χ 2 fits, as detailed in Sect. 5, and the best fit values are listed, with their uncertainties, in Table 1. For type-2 AGNs pure density evolution was found to be sufficient to account for the data. For type-1 AGNs we adopted the mean QSO SED by Richards et al. (2006), extended to mm wavelengths as described in Sect. 2.3, while for type-2 AGNs we adopted the SED of the local AGN dominated ULIRG Mrk 231, taken from the SWIRE library (Polletta et al. 2007). These SEDs are shown in Fig. 5 where the SED of type-3 AGNs associated to dusty star-forming proto-spheroidal galaxies is also plotted for comparison. The SED of type-3 AGNs is the most obscured at optical/near-IR wavelengths due to the effect of the dense, dusty interstellar medium of the high-z host galaxies. This means that the counts at optical/near-IR wavelengths are dominated by type-1 AGNs with type-2 AGNs becoming increasingly important in the mid-IR. The 3 AGN populations have approximately the same ratio between the rest-frame 12 µm and the bolometric luminosity, as first pointed out by Spinoglio & Malkan (1989). The type-1/type-2 space density ratio yielded by the model increases with luminosity, consistent with observations (e.g, Burlon et al. 2011) and with the receding torus model (Lawrence 1991). In the framework of the standard unified model of AGNs type-1 and type-2 AGNs differ only in terms of the angle which the observers line of sight makes with the axis of a dusty torus. If the line of sight to the central region is blocked by the torus, the AGN is seen as a type-2. According to the receding torus model the opening angle of the torus (measured from the torus axis to the equatorial plane) is larger in more luminous objects, implying that obscuration is less common in more luminous AGNs. Since our model implies that type-1 AGNs (but not type-2's) are evolving in luminosity, they become increasingly dominant with increasing redshift. Source counts and contributions to the background The surface density of sources per unit flux density and redshift interval is d 3 N (S ν , z) dS ν dzdΩ = Φ(log L ν ′ , z) L ν ′ ln 10 dL ν ′ dS ν d 2 V dzdΩ ,(8) where ν ′ = ν(1 + z), S ν = (1 + z)L ν ′ 4πD 2 L (z) ,(9) the comoving volume per unit solid angle is d 2 V dzdΩ = c H 0 (1 + z) 2 D 2 A (z) E(z) ,(10) and the luminosity distance D L and the angular diameter distance D A are related, in a flat universe, by D L 1 + z = (1 + z)D A = c H 0 z 0 dz ′ E(z ′ ) .(11) The differential number counts, i.e., the number of galaxies with flux density in the interval S ν ± dS ν /2 at an observed frequency ν per unit solid angle, are then d 2 N dS ν dΩ (S ν ) = zmax zmin dz Φ(log L ν ′ , z) L ν ′ ln 10 dL ν ′ dS ν d 2 V dzdΩ .(12) The integral number counts, i.e., the number of galaxies with flux density S ν > S ν,inf at frequency ν per unit solid angle, are given by dN dΩ (S ν > S ν,inf ) = zmax zmin dz d 2 V dzdΩ ∞ log L ν ′ ,inf Φ(log L ν ′ , z)d log L ν ′ ,(13) where ν ′ = (1 + z)ν and L ν ′ ,inf is the monochromatic luminosity of a source at the redshift z observed to have a flux density S ν,inf . Counts (per steradian) dominated by local objects (z ≪ 1) can be approximated as S 2.5 ν d 2 N dS ν dΩ ≃ 1 4π 1 4 √ π ∞ 0 Φ(log L ν , z ≃ 0)L 3/2 ν d log L ν .(14) The redshift distribution, i.e. the surface density of sources with observed flux densities greater than a chosen limit S ν,inf per unit redshift interval, is d 2 N dzdΩ (z, S ν > S ν,inf ) = ∞ S ν,inf d 3 N dS ′ ν dzdΩ dS ′ ν .(15) The steepness of the (sub-)mm counts of proto-spheroidal galaxies and their substantial redshifts imply that their counts are strongly affected by the magnification bias due to gravitational lensing (Blain 1996;Perrotta et al. 2002Perrotta et al. , 2003Negrello et al. 2007): d 3 N lensed (S ν , z) d log S ν dzdΩ = µ dµ d 3 N (S ν /µ, z) d log S ν dzdΩ dP (µ\|z) dµ ,(16) where dP/dµ is the amplification distribution that describes the probability for a source at redshift z to be amplified by factor µ. Here we have approximated to unity the factor 1/ µ that would have appeared on the right-hand side, as appropriate for large-area surveys (see Jain & Lima 2011). We have computed dP/dµ using the SISSA model worked out by Lapi et al. (2012). The differential counts including the effect of lensing can be computed integrating eq. (16) over z. The effect of lensing on counts of other source populations and on proto-spheroidal counts at shorter wavelengths is small and will be neglected in the following. Interesting constraints on the halo masses of proto-spheroidal galaxies come from the auto-and crosscorrelation functions of intensity fluctuations. A key quantity in this respect is the flux function, d 2 S ν /dzdΩ, i.e. the redshift distribution of the cumulative flux density of sources below the detection limit S ν,lim d 2 S ν dzdΩ = S ν,lim 0 d 3 N dS ν dzdΩ S ν dS ν .(17) The contribution of a source population to the extragalactic background at the frequency ν is I ν = ∞ 0 S ν d 2 N (S ν ) dS ν dΩ dS ν .(18) Determination of the best fit values of the parameters A minimum χ 2 approach for estimating the optimum values of the parameters of the physical model for proto-spheroidal galaxies and associated AGNs is unfeasible because of the lengthy calculations required. Some small adjustments compared to earlier versions Lapi et al. 2006;Mao et al. 2007) were made, by trial and error, to improve the agreement with observational estimates of luminosity functions at z > 1.5. An outline of the model, including the definition of the relevant parameters, is presented in Appendix A. The chosen values are listed in Table 2. Discussions of physically plausible ranges can be found in Granato et al. (2004), Cirasuolo et al. (2005), Lapi et al. (2006), Shankar et al. (2006), Cook, Lapi, &Granato (2009) andFan et al. (2010). On the contrary, the minimum χ 2 approach was applied to late-type/starburst galaxies and to reactivated AGNs. The χ 2 minimization was performed using the routine MPFIT 1 exploiting the Levenberg-Marquardt least-squares method (Moré 1978;Markwardt 2009). The huge amount of observational data in the frequency range of interest here and the large number of parameters coming into play forced us to deal with subsets of parameters at a time using specific data for each subset. The parameters of the evolving AGN luminosity functions were obtained using: • the B -band local QSO luminosity function of Hartwick & Schade (1990), The B -and g-band luminosity functions were used to constrain the parameters of type-1 AGNs (type-2 being important only at the low luminosity end) while the 1.24 µm luminosity functions were regarded as made by a combination of type-1 and type-2 AGNs, the latter being dominant at low luminosities. As for the evolving luminosity functions of "warm" and "cold" galaxy populations we used the following data sets: • the IRAS 60 µm local luminosity function of Soifer & Neugebauer (1991), • the Herschel SPIRE counts at 250, 350, and 500 µm (Béthermin et al. 2012b). The fits of the counts were made after having subtracted the contributions of proto-spheroidal galaxies, which are only important at wavelengths ≥ 160 µm. The best-fit values of the parameters are listed in Table 1, where values without errors denote parameters that were kept fixed, as mentioned in Sect. 3. In comparing model results with observational data the instrumental spectral responses were taken into account. This is especially important in the mid-IR because of the complexity of the SEDs due to PAH emission lines. The monochromatic luminosity at the effective frequency ν eff in the observer's frame is given by: L(ν eff ) ≡ T (ν ′ )L ν ′ (1+z) dν ′ T (ν ′ )dν ′(19) where T (ν) is spectral response function and the integration is carried out over the instrumental band-pass. When the model is compared with luminosity function data at frequency ν i (in the source frame) coming from different instruments for sources at redshift z we use the response function of the instrument for which ν eff is closest to ν i /(1 + z). In the case of source counts we use the response function appropriate for the most accurate data. 6. Results Model versus observed luminosity functions and redshift distributions The most direct predictions of the physical model for proto-spheroidal galaxies are the redshift-dependent SFRs and accretion rates onto the super-massive black-holes as a function of halo mass. During the dust enshrouded evolutionary phase the SFRs can be immediately translated into the IR (8-1000 µm) luminosity functions of galaxies. As mentioned above, according to our model, the transition from the dust-obscured to the passive evolution phase is almost instantaneous and we neglect the contribution of passive galaxies to the IR luminosity functions. In turn, the accretion rates translate into bolometric luminosities of AGNs given the mass-to-light conversion efficiency for which we adopt the standard value ǫ = 0.1. The SEDs then allow us to compute the galaxy and AGN luminosity functions at any wavelength. In contrast, the phenomenological model for late-type/starburst galaxies yields directly the redshiftdependent IR luminosity functions and that for reactivated AGNs yields the 12 µm luminosity functions. Again these can be translated to any wavelength using the SEDs described in the previous sections. In Fig. 6 the model IR luminosity functions are compared with observation-based determinations at different redshifts. At z > 1.5 the dominant contributions come from the stellar and AGN components of proto-spheroidal galaxies. These contributions fade at lower redshifts and essentially disappear at z < 1. The model implies that AGNs associated to proto-spheroidal galaxies are important only at luminosities higher than those covered by the luminosity functions which therefore have been converted to bolometric luminosity functions using their galaxy SED, i.e. neglecting the AGN contribution, so that log(L IR /L ⊙ ) = log(L 100 /L ⊙ ) + 0.21 and log(L IR /L ⊙ ) = log(L 250 /L ⊙ ) + 1.24. At z ≤ 1.5 "warm" and "cold" star forming galaxies take over, "cold" galaxies being important only at low luminosities. Type-2 AGNs (long-dashed pink lines) may dominate at the highest IR luminosities while type-1 AGNs (long-dashed light-blue lines) are always sub-dominant (in the IR). The scale on the top x-axis in Fig. 6 gives the star formation rates corresponding to the IR luminosities log L IR L ⊙ = log SFR M ⊙ yr −1 + 9.892,(20) and is therefore meaningful only to the extent that the AGN contribution is negligible. Moreover, the normalization constant applies to high-z proto-spheroidal galaxies whose IR luminosity comes almost entirely from star-forming regions. For more evolved galaxies older stellar populations can contribute significantly to the dust heating ; therefore L IR is no longer a direct measure of the star formation rate and therefore the upper scale has to be taken as purely indicative. Observational determinations of luminosity functions are available in many wave bands and for many cosmic epochs. The comparison between the model and the observed g-band (0.467 µm) AGN luminosity functions at several redshifts is presented in Fig. 7, while the comparison in the J -band (1.24 µm) is shown in Fig. 8. The conversion from monochromatic absolute AB magnitude M λ,AB to the corresponding monochromatic luminosity νL ν (λ) is given by log(νL ν /[L ⊙ ]) = −0.4M λ,AB − log(λ/[Å]) + 5.530. The contribution of type-2 AGNs at z < 1.5 strongly increases from the g-to the J -band. Apart from the low-luminosity portion of the J -band luminosity function, very likely affected by incompleteness, the agreement between the model and the data is remarkably good. The comparisons between the global (stellar plus AGN components) luminosity functions yielded by the model and those observationally determined at several redshifts and wavelengths are shown in Figs. 9-11. In Fig. 12 we compare model and observed redshift distributions at various wavelengths and flux density limits. The comparisons for all the other wavelengths for which estimates of the luminosity function are available can be found in the Web site http://people.sissa.it/∼zcai/galaxy agn/. Note that a substantial fraction of sources have only photometric redshifts. For example, the fraction of photometric redshifts is 91% for the VVDS-SWIRE survey with S 24 µm > 0.4 mJy , 67.5% for the GOODS-N and 36% for the GOODS-S samples with S 24 µm > 0.08 mJy ). Only few sources at z > 2 have spectroscopic redshifts (Berta et al. 2011). Note that photometric redshift errors tend to moderate the decline of the distributions at high-z. The effect is analogous to the Eddington bias on source counts: errors move more objects from the more populated lower z bins to the less populated higher z bins than in the other way. Thus the observed distributions may be overestimated at the highest redshifts. In addition, optical identifications are not always complete. On the whole, observational estimates of luminosity functions and of redshift distributions may be affected by systematic effects difficult to quantify and the true uncertainties may be larger than the nominal values. Model versus observed source counts and contributions to the CIB Model and observed source counts at wavelengths from 15 µm to 1.38 mm are compared in Fig. 13. At wavelengths ≥ 350 µm, where, in the present framework, proto-spheroidal galaxies are most important, the model provides a simple physical explanation of the steeply rising portion of the counts, that proved to be very hard to account for by other both physical (Hayward et al. 2012;Niemi et al. 2012;Lacey et al. 2010) and phenomenological (e.g. Béthermin et al. 2012a;Gruppioni et al. 2011) models. In our model the sudden steepening of the (sub-)mm counts is due to the appearance of proto-spheroidal galaxies that show up primarily at z > ∼ 1.5, being mostly in passive evolution at lower redshifts. Their counts are extremely steep because, due to the strongly negative K-correction, the sub-mm flux densities corresponding to a given luminosity are only weakly dependent on the source redshift. Then, since the far-IR luminosity is roughly proportional to the halo mass, the counts reflect the high-z luminosity function whose bright end reflects, to some extent, the exponential decline of halo mass function at high masses. This situation results in a very strong magnification bias due to gravitational lensing (Blain 1996;Perrotta et al. 2002Perrotta et al. , 2003Negrello et al. 2007). The counts of strongly lensed galaxies depend on the redshift distribution of the unlensed ones. Thus, the good agreement between the model and the observed counts of strongly lensed galaxies (see the 350 µm, 500 µm and 1380 µm panels of Fig. 13) indicates that the model passes this test on the redshift distribution. Low-z "warm" and "cold" star-forming galaxy populations become increasingly important with decreasing wavelength. At λ ≥ 160 µm proto-spheroidal galaxies yield only a minor contribution to the counts. The AGN (mostly type-2) contribution implied by the model is always sub-dominant. We find a maximum contribution in the mid-IR. At 15 µm it is ≃ 8% up to 1 mJy and then rapidly increases up to ≃ 20% above 10 mJy while at 24 µm it is ≃ 7-8% up to 0.5 mJy and increases up to ≃ 20% above 2 mJy, in fair agreement with the observational estimates (Treister et al. 2006;Teplitz et al. 2011). Another test on the redshift distribution is provided by the estimated counts in different redshift slices (Fig. 14), although we caution that the true uncertainties may be larger than the nominal ones since the observational estimates are partly based on photometric redshifts and on stacking. The consistency between the model and the data is reasonably good. Figure 15 shows the contributions of the different populations to the cosmic infrared background (CIB). The model accounts for the full CIB intensity over the whole wavelength range. Only at λ ≤ 10 µm other galaxy populations, such as passively evolving galaxies, become important. According to the model, for λ ≥ 350 µm the main contribution to the CIB comes from proto-spheroidal galaxies and the fraction contributed by these objects increases with increasing wavelengths. Below λ = 350 µm lower z "warm" galaxies take over, with "cold" galaxies adding a minor contribution. AGNs are always sub-dominant. The model gives a total (type-1 + type-2 + type-3) AGN contribution of 8.6% at 16 µm and of 8.1% at 24 µm. (2007) find a contribution of ∼ 10% at 24 µm. It must be noted that these observational estimates are endowed with substantial uncertainties: on one side they may be too low because strongly obscured AGNs may be missed, on the other side they may be too high because a significant fraction of the observed emission may come from the host galaxy. Clustering properties of dusty galaxies and power spectra of the cosmic infrared background An important test of our physical model for the evolution of dusty proto-spheroidal galaxies is provided by their clustering properties that are informative on their halo masses. A specific prediction of our model is that proto-spheroidal galaxies are the main contributors to the CIB at (sub-)mm wavelengths with "warm" starburst galaxies becoming increasingly important with decreasing wavelength. Since, in our model, protospheroidal galaxies are much more strongly clustered than starburst galaxies, the variation in the mixture with wavelength translates in quantitative predictions on the frequency dependence of the amplitude of the CIB power spectra and on the level of correlations among the maps at different frequencies. We have updated the analysis by Xia et al. (2012) taking into account the new auto-and cross-frequency power spectra obtained by Viero et al. (2012) from Herschel/SPIRE measurements and the power spectrum at 100 µm derived by Pénin et al. (2012). The latter authors actually give also an estimate of the power spectrum at 160 µm. However the amplitude of the latter is anomalously large. As an example, for the wave-number k θ = 0.03 arcmin −1 we find that the amplitude normalized to the CIB intensity δI I = [2πk 2 θ P (k θ )] 1/2 I CIB ,(21) [eq. (13) of Viero et al. (2012)] is ≃ 0.08-0.09 at 100, 250, 350 and 500 µm but jumps to ≃ 0.2 at 160 µm. Since such a jump over a small wavelength range looks odd we decided not to use the 160 µm power spectrum. All the relevant details on the formalism used are given by Xia et al. (2012). Briefly, the power spectrum of the galaxy distribution is parameterized as the sum of the 1-halo term, that dominates on small scales and depends on the distribution of galaxies within the same halo, and the 2-halo term, that dominates on large scales and is related to correlations among different halos. The Halo Occupation Distribution (HOD), which is a statistical description of how dark matter halos are populated with galaxies, is modeled using a central-satellite formalism (see, e.g., Zheng et al. 2005). This assumes that the first galaxy to be hosted by a halo lies at its center, while any remaining galaxies are classified as satellites and are distributed in proportion to the halo mass profile. The mean halo occupation function of satellite galaxies is parameterized as: N sat ∝ (M vir /M sat ) αsat , where M vir is the halo mass and the power-law index α sat is a free parameter. The key parameter in the 2-halo term is the minimum halo mass, M vir,min , that determines the amplitude of the effective bias function b eff (z). In the Xia et al. (2012) paper the only free parameters are the minimum halo mass, M min,protosph , and the power-law index of the mean occupation function of satellites, α sat,protosph , of proto-spheroidal galaxies. This is because the contribution of late-type galaxies to the power spectra at λ ≥ 250 µm is always subdominant and therefore the parameters characterizing their clustering properties were poorly constrained. This is no longer true if we add the 100 µm power spectrum, which, however, still provides only weak constraints on α sat,late−type . We therefore fixed that parameter to α sat,late−type = 1. The fits to the Herschel/SPIRE power spectra determined by Viero et al. (2012) give log(M min,protosph /M ⊙ ) = 12.15±0.04 and α sat,protosph = 1.55 ± 0.05 (1 σ errors), close to the values found by Xia et al. (2012). The 100 µm data do not constrain these parameters further but yield log(M min,late−type /M ⊙ ) = 11.0 ± 0.06. The nominal errors on each parameter have been computed marginalizing on the other and correspond to ∆χ 2 = 1. We caution that the true uncertainties are likely substantially higher than the nominal values, both because the model relies on simplifying assumptions that may make it too rigid and because of possible systematics affecting the data. Our value of M min,protosph implies an effective halo mass [eq. (17) of Xia et al. (2012)] at z ≃ 2 of proto-spheroidal galaxies, making up most of the CIB, M eff ≃ 4.5 × 10 12 M ⊙ . This value is close to the estimated halo mass of the most effective star formers in the universe. Tacconi et al. (2008) estimated their mean comoving density at z ∼ 2 to be ∼ 2 × 10 −4 Mpc −3 . For the standard ΛCDM cosmology this implies that they are hosted by dark matter halos of ∼ 3.5 × 10 12 M ⊙ . The best fit model power spectra are plotted in Fig. 16 where the 1-and 2-halo contributions of protospheroidal and late-type galaxies are also shown. The relative contribution of the latter galaxy population increases with decreasing wavelength and becomes dominant at 100 µm. This trend implies a decrease of the level of correlations among the maps with increasing separation in wavelength. As illustrated by Fig. 17 the model is in very good agreement with the cross-wavelength correlations measured by Viero et al. (2012) and defined by [eq. (14) of Viero et al. (2012)] C A×B = P A×B k θ P A k θ · P B k θ .(22) Summary and conclusions Studies of galaxy properties as a function of morphological type (e.g. Bernardi et al. 2010) have highlighted a dichotomy between the luminosity-weighted ages of early-and late-type galaxies. The former are mostly older than 8 Gyr while most of Sb or later-type spirals are younger than 7 Gyr, corresponding to a formation redshift z 1-1.5. Building on this datum we have worked out a model whereby the protospheroidal galaxies, in the process of forming the bulk of their stars, are the dominant population in the IR at z > ∼ 1.5 while late-type galaxies dominate at lower redshifts. The model is 'hybrid' in the sense that it combines a physical, forward model for spheroidal galaxies and the early evolution of the associated AGNs with a phenomenological backward model for late-type galaxies and for the later AGN evolution. To describe the cosmological evolution of proto-spheroidal galaxies and of the associated AGNs we adopted the physical model by Granato et al. (2004), upgraded working out, for the first time, the epochdependent luminosity functions of sources as a whole (stellar plus AGN component), taking into account in a self-consistent way the variation with galactic age of the global SED. With only minor adjustments of the parameters the model accurately reproduces the observed luminosity functions at all redshifts (z > ∼ 1.5) and IR wavelengths at which they have been determined. The model naturally accounts for the observed positive evolution of both galaxies and AGNs up to z ≃ 2.5 and for the negative evolution at higher redshifts. This is the result of the combination of two competing effects. On one side cooling and free-fall timescales shorten with increasing redshift because of the increase of the matter density and this yields higher star formation rates, i.e. higher galaxy luminosities at given halo mass. The higher gas densities are also responsible for a delay of the AGN switch-off time by feedback implying positive luminosity and density evolution of these objects. These effects are thwarted by the decrease in the comoving density of massive halos that prevails above z ≃ 2.5 causing a decline of the bolometric luminosity functions of both galaxies and AGNs. The model also provides a simple physical explanation of the steeply rising portion of the (sub-)mm counts, that proved to be very hard to account for by other physical and phenomenological models. The sharp steepening is due to the sudden appearance of proto-spheroidal galaxies that do not have, in this spectral band, an evolutionary connection with nearby galaxies because their descendants are in passive evolution at z < ∼ 1.5. Their (sub-)mm counts are extremely steep because, due to the strongly negative K-correction, the flux densities corresponding to a given luminosity are only weakly dependent on the source redshift. Then, since the far-IR luminosity is roughly proportional to the halo mass, the counts reflect, to some extent, the exponential decline of halo mass function at high masses. The steepness of the counts imply a strong magnification bias due to gravitational lensing. The counts of strongly lensed sources depend on the redshift distribution that determines the distribution of lensing optical depths. In fact, this model was the only one that correctly predicted (Negrello et al. 2007) the strongly lensed counts at 500 µm and the correct redshift distribution of bright (S 500 µm ≥ 100 mJy) sub-mm sources González-Nuevo et al. 2012). The epoch-dependent luminosity function of late-type galaxies has been modeled in terms of two populations, "warm" and "cold" galaxies with different SEDs and different evolution properties. Simple truncated power law models have been adopted for the evolution of these populations. "Cold" (normal) late-type galaxies evolve (weakly) only in luminosity, while "warm" (starburst) galaxies evolve both in luminosity and in density. Below z = 1.5 the far-IR emission of proto-spheroidal galaxies and the associated AGNs fade out rather rapidly. The AGNs, however, can be reactivated e.g. by interactions. This later phase of AGN emission has been described by a phenomenological model analogous to that used for late-type galaxies, distinguishing between type-1 and type-2 AGNs. In this framework, there is a systematic variation with wavelength of the populations dominating the counts and the contributions to the extragalactic background intensity. Above 350 µm the main contributors to the CIB are proto-spheroidal galaxies. In this wavelength range late-type galaxies dominate the counts only at the brightest (where normal "cold" star-forming galaxies prevail) and at the faintest flux densities (where "warm" starburst galaxies outnumber the proto-spheroids). But these galaxies become increasingly important with decreasing wavelength. Proto-spheroids are always subdominant below 250 µm. This strong variation with wavelength in the composition of IR sources implies specific predictions for the auto-and crosspower spectra of the source distribution, that may help discriminating between different models. Essentially all the alternative models have all source populations present over the full relevant redshift range. This implies a high correlation between the CIB intensity fluctuations at different frequencies. On the contrary, the present model predicts a high (close to unity) cross-correlation only at the longest wavelengths (≥ 500 µm). At shorter wavelengths the cross correlation progressively weakens and we expect little cross-correlation between CIB fluctuations at, say, 100 and 500 µm. No observational determination is available for correlations among these wavelengths, but in the Herschel/SPIRE wavelength range, where cross correlations have been measured, the model results are in good agreement with observations. According to our model, the AGN contribution to the CIB is always sub-dominant. It is maximal in the mid-IR where it reaches 8.6% at 16 µm and 8.1% at 24 µm. These contributions are close to, but somewhat lower than most observation-based estimates which however are complicated by the difficulty of separating the AGN emission from that of the host galaxy. The AGN contribution to the counts is also always subdominant. We find a maximum contribution in the mid-IR where the model gives AGN fractions in fair agreement with the observational estimates (Treister et al. 2006;Teplitz et al. 2011). We are indebted to Matthieu Béthermin for several useful clarifications on flux calibration and color correction issues, to Roberto Assef for having sent his IR luminosity functions of AGNs in tabular form and to Aurelie Pénin for having provided a tabulation of CIB power spectra at 100 and 160 µm and clarifications on error estimates. We also benefited from useful comments from an anonymous referee. Z.Y.C. acknowledges support from the joint PhD project between XMU and SISSA. A.L. thanks SISSA for warm hospitality. J.Q.X. is supported by the National Youth Thousand Talents Program and the grant No. Y25155E0U1 from IHEP. The work has been supported in part by ASI/INAF agreement n. I/072/09/0 and by INAF through the PRIN 2009 "New light on the early Universe with sub-mm spectroscopy". A. Self-regulated evolution of high-z proto-spheroidal galaxies The gas initially associated to a galactic halo of mass M vir , with a cosmological mass fraction f b = M gas /M vir = 0.165 is heated to the virial temperature at the virialization redshift, z vir . Its subsequent evolution partitions it in three phases: a hot diffuse medium with mass M inf infalling and/or cooling toward the center; cold gas with mass M cold condensing into stars; low-angular momentum gas with mass M res stored in a reservoir around the central super-massive black hole, and eventually viscously accreting onto it. In addition, two condensed phases appear and grow, namely, stars with a total mass M ⋆ and the black hole with mass M • . As mentioned in Section 2.1 we restrict ourselves to the ranges 11.3 log(M vir /M ⊙ ) 13.3 and z vir 1.5. The evolution of the three gas phases is governed by the following equations: M inf = −Ṁ cond −Ṁ QSO inf , M cold =Ṁ cond − [1 − R(t)]Ṁ ⋆ −Ṁ SN cold −Ṁ QSO cold ,(A1) M res =Ṁ inflow −Ṁ BH , that link the mass infall rate,Ṁ inf , the variation of the cold gas mass,Ṁ cold , and the variation of the reservoir mass,Ṁ res , to the condensation rate of the cold gas,Ṁ cond , to the star formation rateṀ ⋆ , to the cold gas removal by supernova and AGN feedback,Ṁ SN cold andṀ QSO cold respectively, to the fraction of gas restituted to the cold component by the evolved stars, R(t), to the inflow rate of cold gas into the reservoir around the central super-massive black hole,Ṁ inflow , and to the back hole accretion rate,Ṁ BH . The hot gas cools and flows toward the central region at a ratė M cond ≃ M inf t cond ,(A2)with M 0 inf = f b M vir and t cond ≃ 8 × 10 8 1 + z 4 −1.5 M vir 10 12 M ⊙ 0.2 yr,(A3) where the coefficient is 10% smaller than the value used by Fan et al. (2010). Note that the cooling and inflowing gas we are dealing with is the one already present within the halo at virialization. In this respect it is useful to keep in mind that the virial radius of halo (R vir ≃ 220(M vir /10 13 M ⊙ ) 1/3 [3/(1 + z vir )] kpc) is more than 30 times larger than the size of the luminous galaxy, and that only a minor fraction of the gas within the halo condenses into stars. Indeed, we need strong feedback processes, capable of removing most of the halo gas, to avoid an over-production of stars. This implies that any gas infalling from outside the halo must also be swept out by feedback; it could however become important for the formation of a disc-like structure surrounding the preformed spheroid once it enters the passive evolution phase, with little feedback (Cook, Lapi, & Granato 2009). As mentioned in Sect. 2.1, the additional material (stars, gas, dark matter) infalling after the fast collapse phase that creates the potential well, i.e. during the slow-accretion phase, mostly produces a growth of the halo outskirts, and has little effect on the inner part where the visible galaxy resides. The star formation rate is given byṀ ⋆ ≃ M cold t ⋆ ,(A4) where the star formation timescale is t ⋆ ≃ t cond /s with s ≃ 5. For a Chabrier (2003) IMF of the form φ(m) = m −x with x = 1.4 for 0.1 m 1M ⊙ and x = 2.35 for m > 1M ⊙ we find R ≃ 0.54 under the instantaneous recycling approximation. The gas mass loss due to the supernova feedback iṡ M SN cold = β SNṀ⋆ ,(A5)with β SN = N SN ǫ SN E SN E bind ≃ 0.6 N SN 8 × 10 −3 /M ⊙ ǫ SN 0.05 × E SN 10 51 erg M vir 10 12 M ⊙ −2/3 1 + z 4 −1 . (A6) We adopt the following values: number of SNe per unit solar mass of condensed stars N SN ≃ 1.4 × 10 −2 /M ⊙ ; fraction of the released energy used to heat the gas ǫ SN = 0.05; kinetic energy released per SN E SN ≃ 10 51 ergs; halo binding energy E bind ≃ 3.2 × 10 14 (M vir /10 12 M ⊙ ) 2/3 ([(1 + z)/4] cm 2 s −2 (Mo & Mao 2004). The infrared luminosity (8-1000 µm) associated to dust enshrouded star formation is L ⋆,IR (t) = k ⋆,IR × 10 43 Ṁ ⋆ M ⊙ yr −1 erg s −1 ,(A7) where the coefficient k ⋆,IR depends on the SED. We adopt k ⋆,IR ∼ 3 Kennicutt 1998). The cold gas inflow rate into the reservoir around the super-massive black hole, driven by radiation drag, is given byṀ inflow ≃ L ⋆ c 2 (1 − e −τRD ) ≃ α RD × 10 −3Ṁ ⋆ (1 − e −τRD ),(A8) with τ RD (t) = τ 0 RD Z cold (t) Z ⊙ M cold 10 12 M ⊙ M vir 10 13 M ⊙ −2/3 .(A9) For the strength of the radiation drag we adopt α RD = 2.5 and set τ 0 RD = 3.0. The model also follows the evolution of the cold gas metallicity, Z cold (t). An approximate solution of the equations governing the chemical evolution is (Lapi et al., in preparation) Z cold (t) = Z 0 inf + s sγ − 1 E Z (t) − st/t cond e (sγ−1)t/t cond − 1 · · E Z (t) + B Z ∞ i=2 1 i · i! (sγ − 1) min(t, t Z ) t cond i−1 ,(A10) where γ = 1 − R − β SN , the metallicity of the primordial infalling gas is Z 0 inf = 10 −5 , and the mass fraction of newly formed metals ejected from stars, E Z (t) is given by ≃ A Z + B Z ln min(t, t saturation ) t Z(A11) with A Z = 0.03, B Z = 0.02, t Z = 20 Myr, and t saturation = 40 Myr for the Chabrier's IMF (Z ⊙ ≃ 0.02). Equation (A11) accounts for the fact that, soon after the onset of star formation, the metal yield, mainly contributed by stars with large masses (≥ 20 M ⊙ ) and short lifetimes (t Z ≤ 20 Myr), is a relatively large fraction of the initial stellar mass (f metal ≥ 0.12) while, as the star formation proceeds, it progressively lowers to f metal ∼ 0.06 as the main contribution shifts to stars with intermediate masses ∼ 9 − 20 M ⊙ and lifetimes t Z ∼ 20 − 40 Myr, and finally saturates to values f metal ∼ 0.013 as stars with masses ≤ 9 M ⊙ and long lifetimes (t saturation ≥ 40 Myr) take over ). The two parameters A Z and B Z depends mainly on the IMF. The accretion rate into the central black hole obeys the equatioṅ M BH = min(Ṁ visc BH , λ EddṀEdd ),(A12) whereṀ visc BH is the accretion rate allowed by the viscous dissipation of the angular momentum of the gas in the reservoirṀ visc BH = M res τ visc = κ accr 5 × 10 3 V vir 500 km s −1 3 × M res M • 3/2 1 + M • M res 1/2 ,(A13) with κ accr ≃ 10 −2 and V 2 vir = GM 2/3 vir [4π∆ vir (z)ρ m (z)/3] 1/3 , ∆ vir being the overdensity of a virialized halo at redshift z vir within its virial radius r vir .Ṁ Edd ≡ M • /ǫ t Edd is the accretion rate corresponding to the Eddington luminosity given the mass to light conversion efficiency ǫ (we set ǫ = 0.1 so that the Salpeter time ǫ t Edd = 4.5 × 10 7 yr) and λ Edd (z) is the Eddington ratio that we assume to slightly increase with redshift for z 1.5 λ Edd (z) ≃ 0.1(z − 1.5) 2 + 1.0 (A14) up to a maximum value λ Edd,max = 4. The growth rate of the black hole mass iṡ M • (t) = (1 − ǫ)Ṁ BH (A15) starting from a seed mass M seed • = 10 2 M ⊙ . The bolometric AGN luminosity is L • = ǫṀ BH c 2 = 5.67 × 10 45 ǫ 0.1 Ṁ BH M ⊙ yr −1 erg s −1 .(A16) A minor fraction of it couples with the interstellar medium of the host galaxy giving rise to an outflow at a rateṀ QSO inf,cold =Ṁ wind M inf,cold M inf + M cold ,(A17)withṀ wind = L ISM QSO E bind ,(A18) and L ISM QSO ≃ 2 × 10 44 ǫ QSO Ṁ BH M ⊙ yr −1 3/2 erg s −1 . (A19) L ISM QSO is the mechanical AGN luminosity, used to unbind the gas. The coefficient quantifying the strength of the QSO feedback is chosen to be ǫ QSO = 3. The ratio of the mechanical to the total AGN luminosity L ISM QSO /L • ≃ 3.5 × 10 −3 ǫ QSO ǫ Ṁ BH M ⊙ yr −1 1/2 ,(A20) is constrained to be in the range 0.006-0.15. Examples of the resulting evolution with galactic age of properties of the stellar and of the AGN component are shown in Fig. 18 for three values of the virial mass and z vir = 3. As mentioned in Sect. 5, to improve the fits of the data we have modified, by trial and error, the values of some model parameters used in previous papers, still within their plausible ranges (see Table 2). The impact of these parameters on the derived luminosity functions can be more easily understood with reference to the time lag between the halo virialization and the peak in black hole accretion rate, ∆t peak . The duration of star formation is ∆t SF ∆t peak (see Fig. 18) due to the drastic effect of QSO feedback in massive halos which dominate the bright end of the luminosity functions. Note that longer ∆t peak (or ∆t SF ) imply higher bright tails of the luminosity functions. The final black hole mass increases with increasing the coefficient, τ 0 RD , of the optical depth of gas clouds [eq. (A9)] because it implies a higher efficiency of the radiation drag driving the gas into the reservoir. There is a degeneracy, to some extent, between τ 0 RD and the gas metallicity Z cold , implying that τ 0 RD cannot be tightly constrained (see Granato et al. 2004). The value of ∆t peak grows substantially in response to a small increase of the radiative efficiency ǫ that yields a slower growth of the black hole mass and a weaker QSO feedback. Higher values of the Eddington ratio, λ Edd , result in lower values of both ∆t peak and of the final black hole mass. A rise of λ Edd at high-z is required to account for the observed space density of very luminous QSOs (see the high-z data in Fig. 7 and Fig. 8; Lapi et al. 2006). A higher QSO feedback efficiency (higher ǫ QSO ) shortens the duration of star formation, ∆t SF , but has a minor effect on ∆t peak and on the final black hole mass. Finally, the coefficient relating the SFR to the IR luminosity, k ⋆,IR , varies with age mix of stellar populations, chemical composition and IMF. Increasing it we shift the luminosity functions towards higher luminosities. Note that, as discussed in Section 2.2, the latter is not the sum of the two components although, in this case, is very close to it. The upper right panel illustrates the evolution of the global luminosity function from z = 1.3 to z = 4.5, while the lower panels show the evolution of each component separately. The decline at low luminosities is an artifact due to the adopted lower limit to the proto-spheroid halo masses. The figure highlights the different shapes of the stellar and AGN bolometric luminosity function, with the latter having a more extended high luminosity tail, while the former sinks down exponentially above ∼ 10 13 L ⊙ . The evolutionary behaviour of the two components is qualitatively similar and cannot be described as simple luminosity or density evolution; down-sizing effects are visible in both cases. On the other hand there are also clear differences. Fig. 2.-SEDs of stellar and AGN components of proto-spheroidal galaxies. The solid black line shows the adopted SED for the stellar component, obtained modifying that of the z ≃ 2.3 galaxy SMM J2135-0102, also shown for comparison [solid orange line; the photometric data data are from Swinbank et al. (2010) and Ivison et al. (2010)]. The dotted magenta line represents the SED adopted for the dust obscured phase of the AGN evolution and is taken from the AGN SED library by Granato & Danese (1994). For unobscured AGNs we have adopted the mean QSO SED of Richards et al. (2006; solid magenta line). The original SMM J2135-0102 SED and the 2 AGN SEDs are normalized to log(L IR /L ⊙ ) = 13.85 while the modified SED is normalized to log(L IR /L ⊙ ) = 13.92 to facilitate the comparison with the original SED. Except in the rare cases in which the AGN bolometric luminosity is much larger than that of the starburst, the AGN contribution is small at (sub-)mm wavelengths, while it is important and may be dominant, in the mid-IR. 4.-Adopted SEDs for the "warm" (dashed blue line) and "cold" (dotted red line) low-z star-forming galaxies. They were generated combining SEDs of Dale & Helou (2002) and Smith et al. (2012), as described in the text. The solid orange line shows, for comparison, the SED of proto-spheroidal galaxies. The 3 SEDs are normalized to the same total IR luminosity log(L IR /L ⊙ ) = 1. . The dotted magenta line shows, for comparison, the adopted SED of AGNs associated to dusty proto-spheroidal galaxies (type-3 AGNs). The SEDs are normalized to the same, arbitrary, bolometric luminosity. Fig. 6.-Comparison between model and observational determinations of the IR (8-1000 µm) luminosity functions at several redshifts. At z > 1.0 we have contributions from proto-spheroidal galaxies (dot-dashed orange lines) and from the associated AGNs (both obscured and unobscured; triple-dot-dashed magenta lines). The thin solid black lines (that are generally superimposed to the dot-dashed orange lines) are the combination of the two components. These contributions fade at lower redshifts and essentially disappear at z < 1. At z ≤ 1.5 the dominant contributions come from "warm" (short-dashed blue lines) and "cold" (dotted red lines) star forming galaxies. Type-2 AGNs (long-dashed pink lines) or type-3 AGNs associated to dusty proto-spheroids (triple-dot-dashed magenta lines) dominate at the highest IR luminosities while type-1 AGNs Note that a substantial fraction of sources have only photometric redshifts and only few z > 2 redshifts are spectroscopic. Photometric redshift errors tend to moderate the decline of the distributions at high-z; thus the observed distributions may be overestimated at the highest redshifts (see Section 6.1). The dip at z ≃ 1.5 in the observed redshift distribution of sources with S 850 µm > 5 mJy is due to the 'redshift desert', i.e. to the lack of strong spectral features within the observational window and the fast decline at z > 2.5 is due to the lack of radio identifications (Chapman et al. 2005). The dip around z ≃ 1.5 in the redshift distributions yielded by the model signals the transition from the phenomenological approach adopted for low-z sources to the physical approach for high-z proto-spheroidal galaxies and associated AGNs. Such artificial discontinuity is a weakness of the model that needs to be cleared by further work. Fig. 13.-Euclidean normalized differential number counts at wavelengths from 15 µm to 1380 µm. The thick solid lines are the sum of contributions from: "cold" late-type galaxies (dotted red lines), "warm" (starburst) late-type galaxies (dashed blue lines), type-1 AGNs (long-dashed light-blue lines), type-2 AGNs (long-dashed pink lines), stellar component of proto-spheroids (dot-dashed orange lines), AGN component of proto-spheroids (triple-dot-dashed magenta lines), strongly lensed (µ ≥ 2) proto-spheroids (solid green lines; only significant at λ ≥ 250 µm). The thin solid black lines show the counts of unlensed proto-spheroids, including both the stellar and the AGN components; at λ ≥ 250 µm these counts essentially coincide with the counts of the stellar component only. The filled red circles in the 15 µm panel refer to AGNs only. The filled blue circles and the open red circles in the 24 µm panel refer to AGNs only and come from Treister et al. (2006) and from Brown et al. (2006), respectively. The purple filled squares in the 500 µm panel show the estimated counts of strongly lensed galaxies ). The bright counts at 1.38 mm are also generally interpreted as due to strongly lensed galaxies (Vieira et al. 2010;Greve et al. 2012). References for all the data points are given in Table 3. The model provides a physical explanation of the sudden steepening of the (sub-)mm counts: it is due to the appearance of proto-spheroidal galaxies that show up primarily at z > ∼ 1.5, being mostly in passive evolution at lower redshifts. Pénin et al. (2012), those at longer wavelengths are from Viero et al. (2012). The lines show the contributions of the 1-halo and 2-halo terms for the two populations considered here: late-type (LT) "warm" plus "cold" galaxies and proto-spheroidal (PS) galaxies. The horizonal dotted magenta lines denote the shot noise level. At λ ≥ 250 µm the signal is dominated by proto-spheroidal galaxies while late-type galaxies take over at shorter wavelengths. dt vir dz vir dN ST dt vir (M vir , z vir ) · · P (log L, z; M vir , z vir ), • the g-band QSO luminosity functions at z = 0.55 and 0.85 of Croom et al. (2009), • the z 0.75, 1.24 µm AGN luminosity functions of Assef et al. (2011),• the bright end [log(L 60 /L ⊙ ) ≥ 12] of the local 60 µm luminosity function ofTakeuchi et al. (2003),• the Spitzer AGN counts at 8 and 24 µm ofTreister et al. (2006). • the Planck 350, 550, and 850 µm local luminosity functions of Negrello et al. (2012), • the Spitzer MIPS counts at 24, 70, and 160 µm of Béthermin et al. (2010), • the Herschel PACS counts at 160 µm of Berta et al. (2011), For comparison, Teplitz et al. (2011) estimate a contribution of ∼ 15% at 16 µm; Treister et al. (2006) and Ballantyne & Papovich Fig. 1 . 1-Bolometric luminosity functions of proto-spheroidal galaxies. The upper left panel shows the luminosity functions at z = 1.5 of the stellar (dot-dashed orange line) and of the AGN component luminosity (triple-dot-dashed magenta line), as well as the global luminosity function (solid black line). Fig. 3 . 3-Global SEDs (solid black lines) for two galactic ages (0.3 and 0.48 Gyr) and three host halo masses (log(M H /M ⊙ ) = 11.4, 12.2 and 13.2, from left to right), virialized at z vir = 3. The dot-dashed orange line (overlaid by the solid black line in some panels) and the triple-dot-dashed magenta line show the stellar and the AGN component, respectively. The shorter evolution timescale of the AGNs is clearly visible. The effect of feedback as a function of halo mass on the SFR is very different from that on accretion onto the super-massive black-hole (see text). Fig. Fig. 4.-Adopted SEDs for the "warm" (dashed blue line) and "cold" (dotted red line) low-z star-forming galaxies. They were generated combining SEDs of Dale & Helou (2002) and Smith et al. (2012), as described in the text. The solid orange line shows, for comparison, the SED of proto-spheroidal galaxies. The 3 SEDs are normalized to the same total IR luminosity log(L IR /L ⊙ ) = 1. Fig. 5 . 5-SEDs of low-z type-1 AGNs (solid light-blue line) and type-2 AGNs (solid pink line) (long-dashed light-blue lines) are always sub-dominant (in the IR). The thick solid black lines show the sum of all contributions. The upper horizontal scale gives an estimate of the SFRs corresponding to IR luminosities. These estimates are only indicative (see Sect. 6.1). Data points are from Le Floc'h et al. (2005, black open squares), Caputi et al. (2007, black stars), Magnelli et al. (2009, green downward triangles), Rodighiero et al. (2010, blue open asterisks), Magnelli et al. (2011, black triangles), and Lapi et al. (2011, black open circles). Fig. 9 . 9-Comparison between model and observed 15 µm global (galaxies plus AGNs) luminosity function at several redshifts. Data are from Pozzi et al. (2004, magenta open asterisks), Le Floc'h et al. (2005, black open squares), Matute et al. (2006, black filled squares), Mazzei et al. (2007, black open circles), Magnelli et al. (2009, green triangles), Rodighiero et al. (2010, blue stars), Fu et al. (2010, red open circles for star-formation and red filled circles for AGNs), Wu et al. (2011, black downward triangles), and Magnelli et al. (2011, black triangles). The black filled squares in the panels at z = 0.05 and 0.35 show observational estimates of the luminosity function of type-2 AGNs only while the red filled circles at z = 0.7 refer to AGN of both types and at z = 1.2 refer to type-1 only. The lines have the same definition as in Fig. 6. Fig. 10 . 10-Comparison between model and observed 90 µm global (galaxies plus AGNs) luminosity function at several redshifts. The lines have the same definition as inFig. 6. Data are fromSoifer & Neugebauer (1991, asterisks, 100 µm),Gruppioni et al. (2010, triangles),Sedgwick et al. (2011, diamonds), andLapi et al. (2011, open circles, 100 µm). Fig. 11 . 11-Local luminosity functions at (sub-)mm wavelengths. As in the other figures the short-dashed blue lines refer to "warm" galaxies, the dotted red lines to "cold" galaxies, the long-dashed pink lines to type-2 AGNs and the long-dashed light-blue lines to type-1 AGNs. Data are fromDunne et al. (2000, orange open squares),Vaccari et al. (2010, light-blue stars), andNegrello et al. (2012, red open circles). Fig. 12 . 12-Comparison between model and observed redshift distributions at several wavelengths and for several flux density limits. The lines have the same definition as in Fig. 6. Data are from Le Floc'h et al. (2009, red open squares, 24 µm), Rodighiero et al. (2010, blue stars, 24 µm), Berta et al. (2011, magenta open asterisks, 70, 100, and 160 µm), Béthermin et al. (2012b, red filled circles, 250, 350, and 500 µm), Chapman et al. (2005, red stars, 850 µm), and Yun et al. (2012, blue open asterisks based on the optical photo-z and blue filled circles based on millimetric photo-z). Fig. 14 . 14-Euclidean normalized differential number counts per redshift slices. Lines have the same meaning as in Fig. 13. Data are from Le Floc'h et al. (2009, red open circles, 24 µm), Berta et al. (2011, magenta open asterisks, 70 and 100 µm), and Béthermin et al. (2012b, red filled circles, 250 and 500 µm). Fig. 15 .Fig. 16 . 1516-Contributions of the different populations to the cosmic infrared background. The lines have the same meaning as in Fig. 13. Proto-spheroidal galaxies are the main contributors to the CIB above ≃ 500 µm. Data points are from Renault et al. (2001), Stecker & de Jager (1997), Lagache et al. (1999), Elbaz et al. (2002), Miville-Deschênes et al. (2002), Smail et al. (2002), Papovich et al. (2004), Dole et al. (2006), Marsden et al. (2009), Hopwood et al. (2010), Greve et al. (2010), Scott et al. (2010), Altieri et al. (2010), and Berta et al. (2011). -CIB angular power spectra at far-IR/sub-mm wavelengths. The 100 µm data are from ) Fig. 17.-CIB cross-frequency power spectra at sub-mm wavelengths normalized according to eq. (14) ofViero et al. (2012). The solid line is the result from the model. The data are fromViero et al. (2012). Fig. 18 . 18-Evolution with galactic age of properties of the stellar and of the AGN component of protospheroidal galaxies virialized at z vir = 3 for three choices of the virial (mostly dark matter) mass: log(M vir /M ⊙ ) = 11.4 (left-hand column), 12.2 (central column) and 13.2 (right-hand column). In the first row, the left y-axis scale refers to masses related to the stellar component [infalling hot gas mass (dotdashed line), cold gas mass (dotted line), stellar mass (M ⋆ , solid orange line)] while the right-hand scale refers to quantities related to the AGN component [reservoir mass (triple-dot-dashed line) and black-hole mass (M • , solid magenta line)]. In the second row the left-hand scale refers to the SFR (dotted line) and to the BH accretion rate (dashed black line), while the right-hand scale refer to the IR (8-1000 µm) luminosity of the stellar (solid orange line) and of the AGN (solid magenta line) component. In the third row the left-hand scale refers to the gas metallicity (solid line) while the right-hand scale refers to the optical depth of individual gas clouds (dotted line). a Granato et al. (2004); b Lapi et al. (2006); c Lapi et al. (2011); d Shankar et al. (2006); e Lapi et al. (in preparation). Table 1 : 1Parameters for low-z AGNs and for "warm" and "cold" galaxy populations. The parameters of the AGN luminosity functions refer to 12 µm (νL ν ) while those for galaxies refer to IR (8-1000 µm) luminosities. Values without error were kept fixed.AGN 1 (12 µm) AGN 2 (12 µm) Warm (IR) Cold (IR) log(Φ * 0 /Mpc −3 ) -5.409 ± 0.098 -4.770 ± 0.122 -2.538 ± 0.051 -1.929 ± 0.112 log(L * 0 /L ⊙ ) 9.561 ± 0.084 10.013 ± 0.093 10.002 ± 0.076 9.825 ± 0.087 α 1.1 1.5 0.01 1.372 ± 0.121 σ 0.627 ± 0.017 0.568 ± 0.021 0.328 ± 0.014 0.3 α Φ 2.014 ± 0.400 4.499 ± 0.317 0.060 ± 0.200 0.0 α L 2.829 ± 0.297 0.0 3.625 ± 0.097 1.0 z break 1.0 1.0 1.0 1.0 z cutoff 2.0 2.0 2.0 2.0 Table 2 : 2Parameters of the physical model for the evolution of proto-spheroidal galaxies and associated AGNs. The values of the first eight parameters used here are somewhat different from those used in previous papers, but still well within the plausible ranges listed in column 3 and discussed in the references given in the footnotes. Normalization of optical depth of gas cloud [eq. (A9)] Strength of radiation drag [eq. (A8)]Parameter Value Plausible range Description τ 0 RD 3.0 1 -10 a ǫ 0.10 0.06 -0.42 Black hole accretion radiative efficiency [eq. (A16)] λ Edd 1 -4 a few b Redshift dependent Eddington ratio [eq. (A14)] ǫ QSO 3.0 1 -10 a Strength of QSO feedback [eq. (A19)] k ⋆,IR 3.1 2 -4 c Conversion factor from of SFR to IR luminosity [eq. (A7)] σ * 0.10 0.5 Dispersion of mean stellar luminosity [eq. (6)] σ • 0.35 0.5 b Dispersion of mean AGN luminosity [eq. (6)] f gas,crit 0.03 0.165 Gas mass fraction at transition from obscured to unobscured AGNs [ § 2.2)] ǫ SN 0.05 0.01 -0.1 d Strength of SN feedback [eq. (A6)] α RD 2.5 1 -10 e http://purl.com/net/mpfit Vieira et al. (2010) . 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(1996) and were further corrected for the the different cosmology. The UV magnitudes of Warren et al. (1994) were first converted to B magnitudes (M B = M C,1216Å + 1.39α ν + 0.09, with α ν = −0.5) following Pei (1995) and then to M g as before. The data by. Warren , Comparison between model and observed g-band (0.467 µm) AGN luminosity function at several redshifts. As in Fig. 6 the long-dashed light-blue and pink lines refer to type-1 and type-2 AGNs, respectively. Ross et al.At higher z only proto-spheroids are considered. Data points are from Hartwick & Schade (1990, black open circles). with spectral index α ν = −0.5. The correction for the different cosmology was also applied. Finally, the conversion from M g to νL ν (0.467 µm) is given in Section 6.1Fig. 7.-Comparison between model and observed g-band (0.467 µm) AGN luminosity function at several redshifts. As in Fig. 6 the long-dashed light-blue and pink lines refer to type-1 and type-2 AGNs, respectively, while the triple-dot-dashed magenta lines refer to AGNs associated with proto-spheroidal galaxies. At z < 2 the solid black line shows the sum of all the contributions. At higher z only proto-spheroids are considered. Data points are from Hartwick & Schade (1990, black open circles), Warren et al. (1994, black crosses), Croom et al. (2004, blue stars), Richards et al. (2006, red triangles), Croom et al. (2009, black open squares), Palanque-Delabrouille et al. (2012, black downward triangles), and Ross et al. (2012, black diamonds). The data by Hartwick & Schade (1990), given in terms of M B in the Vega system, were converted to M g adopting the B − g ≃ 0.14 colour estimated by Fukugita et al. (1996) and were further corrected for the the different cosmology. The UV magnitudes of Warren et al. (1994) were first converted to B magnitudes (M B = M C,1216Å + 1.39α ν + 0.09, with α ν = −0.5) following Pei (1995) and then to M g as before. The data by Ross et al. (2012) were converted from M i (z = 2) to M g following Richards et al. (2006) with spectral index α ν = −0.5. The correction for the different cosmology was also applied. Finally, the conversion from M g to νL ν (0.467 µm) is given in Section 6.1. 24 µm) AGN luminosity function at several redshifts. The lines have the same meaning as in Fig. 7. There are clear signs of substantial incompleteness at the lowest luminosities. Data are from. Assef, Comparison between model and observed J -band. Richards et al.The i-band data byFig. 8.-Comparison between model and observed J -band (1.24 µm) AGN luminosity function at several redshifts. The lines have the same meaning as in Fig. 7. There are clear signs of substantial incompleteness at the lowest luminosities. Data are from Richards et al. (2006, red triangles), Assef et al. (2011, black filled circles), and Ross et al. (2012, black diamonds). The data by Assef et al. (2011), given in terms of M J in the Vega system, were converted to νL ν (1.241 µm) using the relation (L ν (1.241 µm) = 1623 × 10 −0.4MJ Jy) by Rieke et al. (2008). The i-band data by Richards et al. (2006) and Ross et al. (2012) were converted	[]
[ "arXiv:quant-ph/0608023v1 2 Aug 2006 Impossibility of Probabilistic splitting of quantum information", "arXiv:quant-ph/0608023v1 2 Aug 2006 Impossibility of Probabilistic splitting of quantum information" ]	[ "I Chakrabarty \nHeritage Institute of Technology\nKolkataIndia\n\nBengal Engineering and Science University\nHowrahIndia\n", "B S Choudhury \nBengal Engineering and Science University\nHowrahIndia\n" ]	[ "Heritage Institute of Technology\nKolkataIndia", "Bengal Engineering and Science University\nHowrahIndia", "Bengal Engineering and Science University\nHowrahIndia" ]	[]	We know that we cannot split the information encoded in two non-orthogonal qubits into complementary parts deterministically. Here we show that each of the copies of the state randomly selected from a set of non orthogonal linearly independent states, splitting of quantum information can not be done even probabilistically.Here in this work we also show that under certain restricted conditions, we can probabilistically split the quantum information encoded in a qubit. *	null	[ "https://arxiv.org/pdf/quant-ph/0608023v1.pdf" ]	14,798,881	quant-ph/0608023	34da1b0b3541c52513acf79096aaed3abd630d9d	arXiv:quant-ph/0608023v1 2 Aug 2006 Impossibility of Probabilistic splitting of quantum information I Chakrabarty Heritage Institute of Technology KolkataIndia Bengal Engineering and Science University HowrahIndia B S Choudhury Bengal Engineering and Science University HowrahIndia arXiv:quant-ph/0608023v1 2 Aug 2006 Impossibility of Probabilistic splitting of quantum information We know that we cannot split the information encoded in two non-orthogonal qubits into complementary parts deterministically. Here we show that each of the copies of the state randomly selected from a set of non orthogonal linearly independent states, splitting of quantum information can not be done even probabilistically.Here in this work we also show that under certain restricted conditions, we can probabilistically split the quantum information encoded in a qubit. * Introduction : In quantum information theory understanding the limits of fidelity of different operations has become an important area of research. Noticing these kind of operations which are feasible in classical world but have a much restricted domain in quantum information theory started with the famous 'no-cloning' theorem [1]. The theorem states that one cannot make a perfect replica of a single quantum state. Later it was proved that one cannot clone two non-orthogonal quantum states [2]. However this does not rule out the possibility of producing approximate cloning machines [3][4][5]. Even though deterministic cloning is not possible, a probabilistic cloning machine can be designed , which will clone the input states with certain probabilities of success [6]. Though the unitarity of quantum mechanics prohibits accurate cloning of non orthogonal quantum states , but such a class of states secretly chosen from a set containing them can be faithfully cloned with certain probabilities if and only if they are linearly independent. Basically Quantum copying machine can be divided into two classes (i)deterministic quantum copying machine (ii) probabilistic quantum copying machine. The first type of quantum copying machine can be divided into two further subclasses: (i) State dependent quantum cloning machine , for example, Wooters-Zurek (W-Z) quantum cloning machine [1], (ii) Universal quantum copying machine, for example, Buzek-Hillery (B-H) quantum cloning machine [2]. Pati and Braunstein introduced a new concept of deletion of an arbitrary quantum state and shown that an arbitrary quantum states cannot be deleted. This is due to the linearity property of quantum mechanics. Quantum deletion [7,8] is more like reversible 'uncopying' of an unknown quantum state. The corresponding no-deleting principle does not prohibit us from constructing the approximate deleting machine [16]. J. Feng et.al. [9] showed that each of two copies of non-orthogonal and linearly independent quantum states can be probabilistically deleted by a general unitary-reduction operation. Like universal quantum cloning machine, D'Qiu [15] also constructed a universal deletion machine but unfortunately the machine was found to be non-optimal in the sense of fidelity. A universal deterministic quantum deletion machine is designed in an unconventional way that improves the fidelity of deletion from 0.5 and takes it to 0.75 in the limiting sense [19]. Many other impossible operations generally referred as 'General Impossible operations' [10] can not be achieved successfully with certainty , but one can carry out these operations at least probabilistically with certain probability of success [11]. Recently 'no-splitting' theorem [12] is an addition to these set of no-go theorems. The theorem states that the quantum information of a qubit cannot be split into complementary parts. The no-splitting theorem, can be mathematically stated as whether the two real parameters (θ, φ) can be split into two complementary qubits as follows: L(\|A(θ, φ) \|B = \|A(θ) \|B(φ) )? The answer is no. The linearity of quantum mechanics [12] as well as the unitarity of quantum mechanics doesn't allow the splitting of the information contained inside a qubit. Similarly 'partial erasure of quantum information' [13] is another operation which is not possible in quantum world. The 'no-splitting' theorem can also be obtained as a special case of 'no-partial erasure' of quantum information theorem. It remains interesting to see that whether we can split quantum information at least probabilistically. In this work we try to find out whether we can split the information in a qubit with a certain probability of success. We show that we cannot even probabilistically split the quantum information inside a qubit. Here we will find that unlike cloning and deletion if non orthogonal quantum states are secretly chosen from a set then there exists no such transformation that will split the quantum information of a qubit with certain probability of success. However we also show that under certain restricted conditions the splitting of quantum information will be possible. Probabilistic Quantum information splitting: The quantum 'no-splitting' theorem [12] says that exact splitting of quantum information encoded in two non orthogonal states cannot be done. Nevertheless, it does not get rid of the possibility of splitting the quantum states with certain probabilities or in other words one may ask that is there any unitary reduction process which will split the information encoded in two non-orthogonal states \|ψ i (θ i , φ i ) and \|ψ j (θ j , φ j ) secretly chosen from a set S = {\|ψ 1 (θ 1 , φ 1 ) , \|ψ 2 (θ 2 , φ 2 ) , ....\|ψ n (θ n , φ n ) }. Here \|ψ i (θ i , φ i ) = cos( θ i 2 )\|0 + sin( θ i 2 )e ıφ i \|1\|ψ i (θ i , φ i ) \|Σ −→ [U +M ] −→ \|ψ i (θ i ) \|Σ(φ i )(1) where \|Σ is the input state of the ancillary system B. Both the systems A and B are described by a N dimensional Hilbert space with N ≥ n . To continue with the argument of the above statement, a probe P with n p ( n p ≥ n + 1 ) dimensional Hilbert space is introduced , where {\|P 0 , \|P 0 , ....., \|P n } are n+1 orthonormal states of the probe. Now let us introduce a unitary operator U whose action on the tensor products of the Hilbert spaces associated with the system A,B and probe P is given by U(\|ψ i (θ i , φ i ) \|Σ \|P 0 ) = √ γ i \|ψ i (θ i ) \|Σ(φ i ) \|P 0 + n j=1 c ij \|Φ (j) AB \|P j , (i = 1, 2, .., n) (2) where \|ψ i (θ i ) = cos( θ i 2 )\|0 + sin( θ i 2 )\|1 and \|Σ(φ i ) = 1 √ 2 [\|0 + e ıφ i \|1 ]. Here, {\|Φ (j) AB } (j=1,...,n) are normalized states of the composite system AB, and these states are not necessarily orthogonal. After the unitary evolution , the measurement is made on the probe P . The attempt made for splitting the information into constituent parts will succeed with γ i probability of success if the measurement outcome of the probe is P 0 . We start here by showing that unlike probabilistic cloning and deletion , probabilistic splitting of quantum information will not be possible for both linearly dependent and independent states secretly chosen from the set S. Let us introduce a theorem. Proof : Consider an arbitrary state which can be expressed as the linear combination of the states in the set S. \|ψ(θ, φ) = n i=1 d i \|ψ i (θ i , φ i )(3) The unitary transformations of the arbitrary linearly dependent state vector is given by, U(\|ψ(θ, φ) \|Σ \|P 0 ) = √ p\|ψ(θ) \|Σ(φ) \|P 0 + c\|Φ AB \|P ⊥0(4) But, if we consider the action of the unitary transformation defined in (2) on the linear combination of the state vectors belonging to the set S, then the resultant is given by, U( n i=1 \|ψ i (θ i , φ i ) \|Σ \|P 0 ) = n i=1 √ p i d i \|ψ i (θ i ) \|Σ(φ i ) \|P 0 + n i=1 n j=1 d i c ij \|Φ (j) AB \|P j(5) Now it is clearly evident that the final states (4) and (5) are different quantum states . Since the state \|ψ(θ, φ) is a linear combination of the state vectors \|ψ i (θ i , φ i ) belonging to the set S, the linearity of quantum mechanics is prohibiting the existence of probabilistic quantum information splitting machine. Therefore the unitary evolution given by (2) exists for any set secretly chosen from the set S only if the states belonging to the set S are linearly independent. The interesting part is that the converse of the theorem is not true. The converse statement of the Theorem1 is given as follows : If the quantum states \|ψ i (θ i , φ i ) (i = 1, ..n) in the set S are linearly independent then the unitary evolution (2) exists. However we will find that this will not hold in general. Interestingly we will also see that there are few particular cases for which the converse is true, and consequently the information splitting is possible. In other words we can say that if the states chosen secretly are linearly independent, the unitary evolution (2) will not hold with positive definite matrices √ Γ, consequently the physical process described by (1) is not going to be realized. In order to verify the existence of the unitary evolution (2) we must introduce the following lemma. Lemma1 : If two sets of states {\|X 1 , \|X 2 , ....., \|X n } and {\|X 1 , \|X 2 , ....., \|X n } satisfy the condition X i \|X j = X i \|X j (i = 1, ...n; j = 1, ...n)(6) there exists a unitary operator U to make U\|X i = \|X i (i = 1, ..., n) The n × n inter-inner products of equation (2) yield the matrix equation D = √ ΓGH √ Γ + + CC + (7) where D = [ ψ i (θ i , φ i )\|ψ j (θ j , φ j ) ], G = [ ψ i (θ i )\|ψ j (θ j ) ], H = [ Σ(φ i )\|Σ(φ j ) ] and C = [c ij ]. The diagonal efficiency matrix Γ is defined by Γ = diag(γ 1 , γ 2 , ..., γ n ), hence √ Γ = √ Γ + = diag( √ γ 1 , √ γ 2 , ..., √ γ n ) . Now if lemma1 clearly shows that the equation (7) is satisfied with a diagonal positive-definite matrix Γ , then the unitary evolution (2)will hold, consequently the physical process (1) can be realized in physics. To show that there is a diagonal positive definite matrix Γ to satisfy equation (7), first we need to show that the matrix D is positive-definite. We introduce a Lemma to show that D is positive-definite. Lemma2 : If n states [\|ψ 1 (θ 1 , φ 1 ) , \|ψ 2 (θ 2 , φ 2 ) , ....\|ψ n (θ n , φ n ) ] are linearly independent, then the matrix D = [ ψ i (θ i , φ i )\|ψ j (θ j , φ j ) ] is positive definite. Proof : For any arbitrary n-vector B = (b 1 , b 2 , ....b n ) T , the quadratic form B + DB can be expressed as B + DB = Ψ\|Ψ = \|Ψ 2(8) where \|Ψ = b 1 \|ψ 1 (θ 1 , φ 1 ) + b 2 \|ψ 2 (θ 2 , φ 2 ) + ... + b n \|ψ n (θ n , φ n )(9) Since we know that states [\|ψ 1 (θ 1 , φ 1 ) , \|ψ 2 (θ 2 , φ 2 ) , ....\|ψ n (θ n , φ n ) ] are linearly independent, the state \|Ψ does not reduce to zero for any n-vector B and its norm will always remain positive. Hence from definition D is positive-definite. But the matrix L = D − GH is not a Hermitian matrix in general.This is because the matrix G is a real symmetric matrix while the matrix H is a Hermitian matrix, and we know that the product of the real symmetric matrix and the hermitian matrix is not going to give a resultant hermitian matrix all the times. If we observe the matrix G we see that the (i,j) th element of the matrix is given by the inner product ψ i (θ i )\|ψ j (θ j ) = cos( θ i 2 ) cos( θ j 2 ) + sin( θ i 2 ) sin( θ j 2 ) , which is a real quantity. Here we clearly see that in the matrix G (i,j)th element is equal to (j,i) th element , with one as the principal diagonal entries .Hence the matrix G is a real symmetric matrix In the matrix H the (i,j) th entry is given by, Σ(φ i )\|Σ(φ j ) = 1 2 [1 + e ı(φ j −φ i ) ] which is a complex quantity and here we see that the (j,i) th entry of the matrix H is conjugate of the (i,j) th entry, with one as principal diagonal entries. From here we conclude that the matrix H is a Hermitian matrix. In general, the principal diagonal elements of the matrix L = D − GH is given by, L ii = ψ i (θ i , φ i )\|ψ i (θ i , φ i ) − γ i ( ψ i (θ i )\|ψ i (θ i ) + n j=1,j =i ψ i (θ i )\|ψ j (θ j ) Σ j (φ j )\|Σ i (φ i ) ) . The off diagonal elements of the matrix L is given by, L ij = ψ i (θ i , φ i )\|ψ j (θ j , φ j ) − √ γ i γ j ( n k=1 ψ i (θ i )\|ψ k (θ k ) Σ k (φ k )\|Σ j (φ j ) ). Unless the matrix L is hermitian , we cannot have the corresponding quadratic form. For the matrix L to be Hermitian , we must have To show the matrix L to be hermitian we must have to show only the principal diagonal elements L ii are real, as L ij = L * ji . Now the diagonal elements L ii are real iff n j=1,j =i ψ i (θ i )\|ψ j (θ j ) Im[ Σ j (φ j ))\|Σ i (φ i ) ]) = 0 and hence the principal diagonal elements reduces to (i) L ij = L * ji , (where L * ji is the conjugate of L ji ) (ii) L iiL ii = ψ i (θ i , φ i )\|ψ j (θ j , φ j ) − γ i ( ψ i (θ i )\|ψ j (θ j ) + n j=1,j =i ψ i (θ i )\|ψ j (θ j ) Re[ Σ j (φ j )\|Σ i (φ i ) ] ). Now if we consider the corresponding quadratic form of the matrix L, XLX t = (x 1 , x 2 , ...., x n )[L ij ](x 1 , x 2 , ...., x n ) t = n i=1 L ii x 2 i + i j L ij x i x j(10) Now the matrix L is positive definite only when the above expression (10) is positive. This is possible only under the following conditions: (i)L ii = ψ i (θ i , φ i )\|ψ i (θ i , φ i ) −γ i ( ψ i (θ i )\|ψ i (θ i ) + n j=1,j =i ψ i (θ i )\|ψ j (θ j ) Σ j (φ j )\|Σ i (φ i ) )) > 0 (ii) n i=1 L ii x 2 i > i j L ij x i x j =⇒ i [1−γ i −γ i n j=1,j =i ψ i (θ i )\|ψ j (θ j ) Σ j (φ j )\|Σ i (φ i ) ]x 2 i > i j [ ψ i (θ i , φ i )\|ψ j (θ j , φ j ) − √ γ i γ j ( n k=1 ψ i (θ i )\|ψ k (θ k ) Σ k (φ k )\|Σ j (φ j ) )]x i x j ∀x i , x j . If the above conditions are satisfied, then the matrix L will be a positive definite matrix. As a consequence of which we can say that the equation (7) will be satisfied by positive definite matrix Γ. Hence under these conditions the converse of the Theorem1 will be satisfied, henceforth unitary evolution (2) will hold and equation (1) can be realized in physics. This clearly indicates that there are certain class of states on Bloch sphere satisfying the above conditions for which the information splitting will be possible. Conclusion In summary we can say that there is no possibility of splitting the quantum information either deterministically or probabilistically. The result obtained here is interesting in the sense that it will help us to understand and to classify the impossible operations in quantum information theory more specifically. As a consequence of which one can make a comment that splitting of quantum information are different from cloning and deletion in the sense that these operations unlike cloning and deletion cannot be achieved even probabilistically. However this doesn't rule out the probabilistic quantum information splitting of certain class of states under certain restricted conditions. This also doesn't rule out the possibility of approximate splitting of quantum information . Acknowledgement I.C acknowledge Prof C.G.Chakraborti for being the source of inspiration in carrying out research. Authors acknowledge S.Adhikari for having various useful discussions. Reference are the quantum states represented as a point on the Bloch sphere (where ı = √ −1 ). Let us consider two systems A and B . Now each of the states of the set S can be taken as a input state of the system A. Let us consider a unitary evolution U and measurement M, which together yield the following evolution Theorem1 : Theorem1The states which are secretly chosen from the set S = {\|ψ 1 (θ 1 , φ 1 ) , \|ψ 2 (θ 2 , φ 2 ) , ...\|ψ n (θ n , φ n ) } can be probabilistically split (realization of the unitary evolution) if the states are linearly independent . will be a real quantity.In general the matrix L is not Hermitian as the elements L ii are in general complex quantities, as a consequence of which we don't have the corresponding quadratic form, and henceforth there arise no question for showing L as a positive definite matrix. However under certain conditions we can show the matrix L to be positive definite. . 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[ "Classical field theories from Hamiltonian constraint: Local symmetries and static gauge fields", "Classical field theories from Hamiltonian constraint: Local symmetries and static gauge fields" ]	[ "Václav Zatloukal \nFaculty of Nuclear Sciences and Physical Engineering\nCzech Technical University\nBřehová 7115 19Prague, Praha 1\n\nCzech Republic and Max Planck Institute for the History of Science\nBoltzmannstrasse 2214195BerlinGermany\n" ]	[ "Faculty of Nuclear Sciences and Physical Engineering\nCzech Technical University\nBřehová 7115 19Prague, Praha 1", "Czech Republic and Max Planck Institute for the History of Science\nBoltzmannstrasse 2214195BerlinGermany" ]	[]	We consider the Hamiltonian constraint formulation of classical field theories, which treats spacetime and the space of fields symmetrically, and utilizes the concept of momentum multivector. The gauge field is introduced to compensate for non-invariance of the Hamiltonian under local transformations. It is a position-dependent linear mapping, which couples to the Hamiltonian by acting on the momentum multivector. We investigate symmetries of the ensuing gauged Hamiltonian, and propose a generic form of the gauge field strength. In examples we show how a generic gauge field can be specialized in order to realize gravitational and/or Yang-Mills interaction. Gauge field dynamics is not discussed in this article.Throughout, we employ the mathematical language of geometric algebra and calculus. * Electronic address: [email protected]; URL: http://www.zatlovac.eu	10.1007/s00006-018-0865-8	[ "https://arxiv.org/pdf/1611.02906v1.pdf" ]	119,331,037	1611.02906	f947e289e96ad2027dc55ca4b64869977f43160e	Classical field theories from Hamiltonian constraint: Local symmetries and static gauge fields Václav Zatloukal Faculty of Nuclear Sciences and Physical Engineering Czech Technical University Břehová 7115 19Prague, Praha 1 Czech Republic and Max Planck Institute for the History of Science Boltzmannstrasse 2214195BerlinGermany Classical field theories from Hamiltonian constraint: Local symmetries and static gauge fields We consider the Hamiltonian constraint formulation of classical field theories, which treats spacetime and the space of fields symmetrically, and utilizes the concept of momentum multivector. The gauge field is introduced to compensate for non-invariance of the Hamiltonian under local transformations. It is a position-dependent linear mapping, which couples to the Hamiltonian by acting on the momentum multivector. We investigate symmetries of the ensuing gauged Hamiltonian, and propose a generic form of the gauge field strength. In examples we show how a generic gauge field can be specialized in order to realize gravitational and/or Yang-Mills interaction. Gauge field dynamics is not discussed in this article.Throughout, we employ the mathematical language of geometric algebra and calculus. * Electronic address: [email protected]; URL: http://www.zatlovac.eu I. INTRODUCTION The Hamiltonian constraint is a concept useful for Hamiltonian formulation not only of general relativity [1], but, in fact, of a generic field theory, as pointed out in Ref. [2,Ch. 3], and exploited further in [3] and [4]. Characteristic features of this formulation are: finite-dimensional configuration space, and multivector-valued momentum variable. In this respect it is congruent with the covariant (or De Donder-Weyl ) Hamiltonian formalism [5][6][7][8][9][10][11][12][13] and should be contrasted with the canonical (or instantaneous) Hamiltonian formalism [14], which utilizes an infinite-dimensional space of field configurations defined at a given instant of time. Nonetheless, unlike the traditional covariant approaches, the Hamiltonian constraint formalism does not a priori distinguish between spacetime and field variables, leading to a simple, but, at the same time, rather general theory expressed in terms of symmetric and compact equations. Eventually, if the need arises, the full set of variables can be split into the spacetime and field-space component, and the equations can be expressed in terms of functions that depend on spacetime location, which is the usual point of view in field theory. Following Chapter 3 in Ref. [2], let us introduce some basic concepts and terminology of the Hamiltonian constraint formulation of classical field theories, and summarize the results of [3] and [4] relevant for the present article. (In fact, analogous results had been obtained previously in [15] with an approach based on the pataplectic differential form.) The configuration space C is a D + N -dimensional space of points q, which represent possible joint outcomes of partial observables, i.e., all physical variables of the theory. (In a typical example of scalar field theory, the partial observables are D spacetime coordinates x µ and N real field components φ a .) Motions are D-dimensional surfaces γ embedded in C, which generalize one-dimensional trajectories of classical particle mechanics. Often, they can be regarded as graphs of functions φ a (x µ ). We shall assume, for simplicity, that C is a flat (pseudo)Euclidean space. (Note that although in articles [3,4] C was assumed to be Euclidean, the general results hold in pseudoEuclidean spaces too.) The physical (or classical ) motions are denoted by γ cl . For a fixed boundary ∂γ, they extremize the action functional A[γ, P, λ] = γ [P (q) · dΓ(q) − λ(q)H(q, P (q))] , leading to the canonical equations of motion λ ∂ P H(q, P ) = dΓ, (−1) D λ∂ q H(q, P ) = dΓ · ∂ q P for D = 1 (dΓ · ∂ q ) · P for D > 1,(2a) H(q, P ) = 0. Here, dΓ is the infinitesimal oriented element of the surface γ, P is the D-vector-valued momentum field, and λ is a (scalar) Lagrange multiplier corresponding to the Hamiltonian constraint (2c). H is a generic function of q and P , the so-called relativistic Hamiltonian (or simply Hamiltonian), which characterizes the dynamics of the physical system in question. Let q = f (q) be a diffeomorphism on the configuration space C, which transforms the relevant quantities as follows: dΓ = f (dΓ) , P = f −1 (P ) , λ (q ) = λ(q) , H (q , P ) = H(q, P ). ( (The definition of induced mappings f and f −1 is recalled in Appendix A 3). A physical motion γ cl is mapped by f to γ cl = {f (q) \| q ∈ γ cl } ≡ f (γ cl ),(4) which is a physical motion of the transformed Hamiltonian H . This is a consequence of the action (1) being equal for primed and unprimed quantities. If H and H coincide, i.e, if H(f (q), f −1 (P ; q)) = H(q, P ) (∀q, P ), then physical motions are mapped to physical motions of the same system, and f is said to be a symmetry. For infinitesimal transformations f (q) = q + ε v(q), ε 1, determined by a vector field v, Eq. (5) takes the form v ·∂ q H(q, P ) − ∂ q ∧ (v · P ) · ∂ P H(q, P ) = 0. (6) Now, the canonical equations (2a) and (2b) can be combined to yield (−1) D λ v ·∂ q H(q, P ) − ∂ q ∧ (v · P ) · ∂ P H(q, P ) = dΓ · ∂ q (P · v) for D = 1 (dΓ · ∂ q ) · (P · v) for D > 1, and so we observe that if the infinitesimal symmetry condition (6) is fulfilled, the left-hand side vanishes, and Eq. (7) expresses a conservation law for the quantity P · v. So much for the results of our previous articles [3] and [4]. In this article, we will be concerned with the situation when the Hamiltonian of the system is invariant under some class of global (rigid) transformations, but fails to be invariant under more generic local transformations that possess additional position dependence. To compensate for this non-invariance, we introduce in Sec. II a gauge field with appropriate transformation properties, which acts as a q-dependent linear map on the momentum multivector P . The ensuing gauged Hamiltonian acquires thereupon an additional q-dependency, and the canonical equations (2) are modified to Eqs. (15). The structure of the latter canonical equations suggests to define, by Eq. (19) in Sec. III, the field strength corresponding to the gauge field, which is a linear q-dependent function that maps grade-r multivectors to grade-r + 1 multivectors. In Appendix B, it is interpreted as torsion corresponding to the Weitzenböck connection on the configuration space C . In the present article, the gauge field and the field strength are static background quantities in the sense that they do not obey their own dynamical equations of motion, but rather are prescribed by some external body. This is also why we do not attempt to include them in the set of partial observables, but treat them separately. We relegate the study of gauge field dynamics (presumably implemented by means of a suitable kinetic term) to the future. Conservation laws maintain the form of Eq. (7), where the left-hand side features the gauged Hamiltonian. Whether a vector field v is a symmetry generator thus depends on the concrete form of the gauge field. The issue is discussed in Sec. IV. The general theory of Sections II, III and IV is much illuminated through the examples of Sec. V. In Examples V A and V B, without any reference to a concrete form of the Hamiltonian, the partial observables are divided into D spacetime coordinates and N field components, and two subgroups of the group of all configuration-space diffeomorphisms are considered. In the first example, we "gauge" spacetime diffeomorphisms by a gauge field equivalent to the tetrad (or vierbein) in the tetrad formulation of gravity. It acts nontrivially only on the spacetime, and therefore has fewer degrees of freedom than a generic gauge field. In the second example, we consider rotations in the field space that depend on spacetime location. This leads to the Yang-Mills gauge field characterized by a bivector-valued gauge potential A µ . In Example V C, we choose the Hamiltonian of a scalar field theory coupled to a generic gauge field, which is subsequently specialized, respectively, to the gravitational field of Example V A, and the Yang-Mills field (in particular, electromagnetic field) of Example V B. In this sense, the generic gauge field unifies gravitational and Yang-Mills fields. In the last example, Sec. V D, the configuration space C is identified with a D + N -dimensional spacetime, in which point particles, strings, or higher-dimensional membranes (depending on the value of D) propagate. Complete group of spacetime diffeomorphisms is gauged by a gauge field that thus embodies the gravitational field. Relation to standard metric formulation of gravity is discussed in detail. Namely, we recover the traditional form of the geodesic equation, and notice that the symmetry condition, Eq. (26), is equivalent with the celebrated Killing equation, where the symmetry generator v is the Killing vector. The mathematical formalism we use is somewhat uncommon, but proves to be very convenient when it comes to handling higher-dimensional geometric objects such as the motions γ with their surface elements dΓ, the momentum multivector P , etc. It is more explicit and versatile than the language of differential forms, and, at the same time, maintains coordinate freedom, so that expressions can be written in a succinct form without the need to introduce multiple indices (what is the case in tensor calculus). It goes under the name geometric (or Clifford ) algebra and calculus, and we follow its exposition as provided in Refs. [16] and [17]. Although we recall some important definitions and results in Appendix A, we do not attempt to supply a complete self-contained presentation of geometric algebra and calculus in this article. In order to fully understand all manipulations that follow, the reader is encouraged to consult the above monographs, or, for concise introduction, appendices of our previous articles [3,4]. II. STATIC GAUGE FIELD We have seen that a transformation f : q → q of the configuration space C is a symmetry of Hamiltonian H if Eq. (5) is satisfied. It often happens that H(q , P ) = H(q, P ). (8) (For example, the physical system does not depend on certain partial observable or a combination of those.) But this equation by itself does not imply Eq. (5) as one has to take into account also the transformation of momentum, P = f −1 (P ; q), which depends on derivatives of f , and so can become rather complicated for local transformations f , i.e., those that vary from point to point. Therefore, roughly speaking, the more generic transformations we consider, the less likely it is that they will be symmetries of the Hamiltonian. Nevertheless, we can impose that a certain transformation, or a class of transformations, f be a symmetry of our system as follows. Consider a modified gauged Hamiltonian H h (q, P ) = H(q, h(P ; q)),(9) where h is a q-dependent linear outermorphism (see Appendix A 4), the so-called gauge field, that maps momentum multivectors P to h(P H h (q , P ) = H h (q, P ) = H(q, h(P )) = H(q , h f (P )) = H(q , h (P )) = H h (q , P ),(10) where we have defined the transformation rule for the gauge field h (b; q ) = h f (b; q); q , ∀ vectors b.(11) To summarize, we obtain the following proposition. Transformation of gauged Hamiltonian. Let f : C → C be an arbitrary diffeomorphism, and suppose that the Hamiltonian H is such that Eq. This means that f may well be called a symmetry as long as we admit that gauged Hamiltonians whose gauge fields are related by Eq. (11) describe the same physical system. The transformation f is then referred to as the gauge transformation. The gauge field associates to each point of C a linear map h, which uniquely determines its adjoint h, and the respective inverses h −1 and h −1 (we shall assume that h is invertible). One may encounter the gauge field in any of the four equivalent forms. Their transformation rules can be derived easily from Eq. (11): h = h f , h −1 = h −1 f −1 , h −1 = f −1 h −1 , h = f h.(12) Being a linear function, the gauge field has, in general, (D+N ) 2 degrees of freedom. This number can be reduced if we consider only a subgroup of the group of all diffeomorphisms of C, in which case it is sufficient (but not "obligatory") to assume that h has a certain more restricted form (see Examples V A and V B, where h is reduced, respectively, to the gravitational and Yang-Mills field). In view of Eq. (9), the gauged Hamiltonian can be regarded as an ordinary Hamiltonian with extra position dependence due to the gauge field, so the canonical equations of motion, Eqs. (2), apply to H h without change. Nevertheless, it is beneficial to express them in terms of the original ungauged Hamiltonian H. For this purpose we calculate ∂ P H h (q, P ) = ∂ P H(q, h(P )) = ∂ P h(P ) · ∂P H(q,P ) = h(∂P )H(q,P ), (13) where we have denotedP ≡ h(P ), anḋ ∂ q H h (q, P ) =∂ q H(q,ḣ(P )) =∂ q H(q,P ) +∂ qḣ (P ) · ∂P H(q,P ). With these relations in hand, we readily obtain λ ∂P H(q,P ) = h −1 (dΓ),(15a)(−1) D λ∂ q H(q,P ) + (−1) D∂ qḣ (P ) · h −1 (dΓ) = dΓ · ∂ q P for D = 1 (dΓ · ∂ q ) · P for D > 1,(15b)H(q,P ) = 0,(15c) Alternatively, the second canonical equation (15b) can be cast in terms ofP ≡ h(P ) as (−1) D λ h(∂ q )H(q,P ) − h −1 (dΓ) · h ∂ q ∧ḣ −1 (P ) = dΓ · ∂ qP for D = 1 h −1 (dΓ · ∂ q ) ·P for D > 1. (15d) To derive Eq. (15d), we employed identities (A21) and (A31) to find (dΓ · ∂ q ) · P = (dΓ · ∂ q ) · h −1 (P ) = (dΓ ·∂ q ) ·ḣ −1 (P ) + h −1 h −1 (dΓ · ∂ q ) · (P )(16) and∂ qḣ (P ) · h −1 (dΓ) =∂ q h −1ḣ (P ) · dΓ = −∂ qḣ −1 (P ) · dΓ,(17) and finally rearranged the terms using Eq. (A3). Canonical equations in the form with Eq. (15d) exhibit more clearly their invariance with respect to gauge transformation f . That is, the functions (dΓ, P, λ, h) and their counterparts (dΓ , P , λ , h ), transformed according to Eqs. (3) and (12), follow the same differential equations with the same Hamiltonian function H (which is assumed to have the property expressed by Eq. (8)). Indeed, observe that individual constituents are gauge-invariant, i.e., they transform trivially: h (P ) = h(P ) , h −1 (dΓ ) = h −1 (dΓ) , h (∂ q ) = h(∂ q ),(18) where the vector derivative ∂ q transforms according to Eq. (A23). Only the second term on the left-hand side of Eq. (15d) needs closer inspection (see Sec. III below). The differential operator h(∂ q ) is invariant under f , and hence it deserves the name gaugeinvariant derivative. Its significance has been emphasized in the context of Gauge Theory Gravity [17][18][19], where the so-called displacement gauge field h ensures invariance of the theory under spacetime diffeomorphisms. Here we do not restrict our attention only to spacetime transformations, but allow also for transformations in the field space (or even for those that mix the two spaces), keeping the same generic form of the gauge field h. It is then instructive to observe, in Example V C below, how h corresponding to local rotations in the field space gives rise to a coupling of a scalar field theory to the Yang-Mills background field, expressed in terms of the traditional Yang-Mills covariant derivative, Eq. (70). III. GAUGE FIELD STRENGTH Let us draw our attention to the second canonical equation (15d), specifically, to the second term on its left-hand side, and define the field strength corresponding to gauge field h, F (P ) ≡ −h ∂ q ∧ḣ −1 (P ) = h(∂ q ) ∧ḣ(P ).(19) (C.f. the definition of displacement-gauge field strength in [17,Ch. 13.5.2].) F is a q-dependent linear mapping that raises the grade of its argument by one (e.g., it maps vectors to bivectors, etc. -see Fig. 1). It satisfies F (A r ∧ B s ) = F (A r ) ∧ B s + (−1) r A r ∧ F (B s )(20) for any r-vector A r and s-vector B s . Iterating this identity (see Eq. (A34) in Appendix A 4), together with the fact that F (α) = 0 for scalars α, we find that F is completely determined by its action on vectors. We call h a pure gauge, if h = f for some diffeomorphism f . In this case F (b) = −f (∂ q ∧ ∂ q )f −1 · b = 0(21) for all constant vectors b, i.e., the corresponding field strength vanishes. Vice versa, if F (b) = 0, i.e., ∂ q ∧ h −1 (b) = 0,(22) then vector field h −1 (b; q) has scalar potential φ(b; q), and hence can be expressed as h −1 (b; q) = ∂ q φ(b; q) = ∂ q f −1 (q) · b,(23) where f −1 is a vector representing the linear functional φ(b). For a generic gauge field it is easy to show (see Eq. (A35) in Appendix A 4) that expression F (P ) is invariant under gauge transformations f , F (P ) = F (P ).(24) This is indeed a crucial property expected of a gauge field strength that legitimizes its definition in Eq. (19), and also confirms gauge invariance of the canonical equations (15). The here-defined gauge field strength F aims to be a universal concept unifying gauge field strengths of Yang-Mills and gravity theories. Its relation to the traditional Yang-Mills field strength is provided by Eq. (73), and its relevance for gravity stressed in Appendix B, where it is interpreted as the torsion of the teleparallel theory of gravity. IV. SYMMETRIES OF GAUGED HAMILTONIAN The symmetry condition (5) for a gauged Hamiltonian H h , expressed in terms of the original Hamiltonian H, reads H f (q), h(f −1 (P ; q); f (q)) = H(q, h(P ; q)).(25) Here, f may be any transformation of the configuration space, in particular, we do not assume, at this stage, that it obeys Eq. (8). The infinitesimal version of the symmetry condition, Eq. (6), for Hamiltonian H h can be expanded in terms of H using Eqs. (13) and (14), v ·∂ q H(q,P ) + v ·∂ qḣ (P ) − h ∂ q ∧ (v · P ) · ∂P H(q,P ) = 0.(26) If we now assume that f (q) = q + ε v(q) is such that Eq. (8) holds, the first term vanishes, and we observe that for f to be a symmetry of The conservation law corresponding to a symmetry generated by a vector field v maintains its usual form (recall Eq. (7)) H h it is sufficient that v ·∂ qḣ (P ) = h ∂ q ∧ (v · P ) ,(27)dΓ · ∂ q (P · v) = 0 for D = 1 (dΓ · ∂ q ) · (P · v) = 0 for D > 1.(28) However, note that the relation between P and dΓ, which is deduced from the canonical equations (15a) and (15c), is altered due to the presence of the gauge field h in the gauged Hamiltonian H h . V. EXAMPLES In Examples V A, V B and V C, the configurations space C is understood as a Cartesian product of D-dimensional spacetime (the "x-space" with pseudoscalar I x ), and N -dimensional space of fields (the "y-space" with pseudoscalar I y ). The points in C are decomposed accordingly as q = x + y, and vectors as a = a x + a y , where a x ≡ a · I x I −1 x and a y ≡ a · I y I −1 y are respective spacetime and field-space projections. The vector derivative operator is likewise decomposed as ∂ q = ∂ x + ∂ y . Whenever convenient, we will introduce a spacetime basis {γ µ } D µ=1 and its reciprocal {γ µ } D µ=1 (see Appendix A 1), as well as an orthonormal basis {e a } N a=1 of the field space, and assume Einstein summation convention over repeated indices. We leave the signature of spacetime arbitrary, however, the signature of the field space is, for simplicity, assumed to be Euclidean. The first two examples do not presume any particular form of the Hamiltonian. There, we are only concerned with certain classes of transformations of the configuration space, and introduce complementary gauge fields with correct transformation properties, dictated by Eq. (11). In Example V A, spacetime diffeomorphisms give rise to a gravitational gauge field, while in Example V B we investigate local (i.e., spacetime-variable) rotations in the field space, and introduce a Yang-Mills gauge field. Results of these two examples are applied in Example V C where we choose the Hamiltonian of a scalar field theory. The last Example V D is independent of the previous ones. It studies relativistic particles or strings in a spacetime endowed with a generic gauge field. A. Spacetime diffeomorphisms and gravitational gauge field Gravitational field arises from the requirement of invariance under arbitrary spacetime diffeomorphisms f x (x). Let us therefore consider transformations of the configuration space C of the form q = f Gr (q) = f x (x) + y,(29) whose adjoint mapping f Gr (b; q) = f x (b x ; x) + b y(30) is determined by an x-dependent linear function f x (b x ) = ∂ x f x (x) · b x , which maps spacetime vectors to spacetime vectors. To satisfy Eq. (11) it is sufficient to assume that the corresponding gauge field is of the form h Gr (b; q) = h x (b x ; x) + b y ,(31a) where h x is the restriction of h Gr to the x-space, whose transformation rule is h x (b x ; x ) = h x f x (b x ; x); x . (31b) Corresponding derived forms of the gauge field are h −1 Gr (a) = h −1 x (a x ) + a y , h −1 Gr (b) = h −1 x (b x ) + b y , h Gr (a) = h x (a x ) + a y .(31c) The field strength of the gravitational gauge field, as obtained from Eq. (19), reads F Gr (b) = −h Gr ∂ q ∧ḣ −1 Gr (b) = −h x ∂ x ∧ḣ −1 x (b x ) .(32) We may perceive that it annihilates field-space vectors, F Gr (b y ) = 0,(33) and maps spacetime vectors to spacetime bivectors. Hence, F Gr (I x ) = 0,(34) since a D+1-vector in a D-dimensional spacetime necessarily vanishes; and, by virtue of Eq. (20), we find F Gr e a ∧ (γ µ · I −1 x ) = −e a ∧ F Gr (γ µ · I −1 x ),(35) where F Gr (γ µ · I −1 x ) = −h x ∂ x ∧ ḣ x (γ µ ) ·ḣ −1 x (I −1 x ) = −γ µ ·ḣ x (∂ x ) h xḣ −1 x (I −1 x )(36) is a scalar multiple of I x . B. Local field-space rotations and Yang-Mills gauge field In Yang-Mills theory with the rotation gauge group SO(N ), the Yang-Mills gauge field is introduced to impose invariance under local field-space rotations of the form q = f YM (q) = R(x)q R(x), R(x) = e −By(x)/2 ,(37) where B y is an x-dependent field-space bivector, i.e., B y · I y = B y I y . (See Appendix A 4 b for details on geometric algebra representation of rotations). For an arbitrary constant vector b, the induced adjoint mapping is calculated f YM (b; q) =∂ q (Rq R) · b +∂ q (Ṙq˙ R) · b = RbR +∂ x y · (2R˙ R) · b,(38) where we have used Eq. (A45). From the structure of f YM (namely, we know from Eq. (A44) that R˙ R is a bivector) we infer a possible form of the corresponding Yang-Mills gauge field: h YM (b; q) = S(x)bS(x) − γ µ y · A µ (x) · b,(39) where S is a field-space rotor (just like R), and {A µ } D µ=1 a set of field-space bivectors. (For simplicity, we have not introduced the gauge coupling constant.) In order to find the transformation rules for A µ and S, we compose, according to Eq. (11), h YM f YM (b; q); q = S RbRS +∂ x y · (2R˙ R) · b − γ µ y · A µ (x) · ( RbR) = RSbRS + γ µ y · (2R∂ µ R) − y · (RA µ R) · b.(40) Comparison with h YM (b; q ) = S (x)bS (x) − γ µ y · A µ (x) · b(41) then yields S = RS , A µ = RA µ R − 2R∂ µ R,(42) where we have denoted ∂ µ ≡ γ µ · ∂ x . Since Sγ µ S = γ µ , we may cast Eq. (39) as h YM (b) = S h YM * (b) S , h YM * (b) ≡ b − γ µ y · A µ · b.(43a) Note that h YM * has the structure of a shear linear mapping discussed in Appendix A 4 c. This observation allows us to easily find other derived forms of the gauge field: h −1 YM (a) = S h −1 YM * (a) S , h −1 YM * (a) = a + a · γ µ y · A µ h −1 YM (b) = h −1 YM * (Sb S) , h −1 YM * (b) = b + γ µ (y · A µ ) · b h YM (a) = h YM * (Sa S) , h YM * (a) = a − a · γ µ y · A µ .(43b) The Yang-Mills field strength F YM is obtained from the defining Eq. (19) as follows. First, for a constant vector b we calculate ∂ q ∧ h −1 YM (b) =∂ q ∧ḣ −1 YM * (Sb S) +∂ q ∧ h −1 YM * (Ṡb˙ S) =∂ x ∧ γ µ (y ·Ȧ µ ) · (Sb S) + A µ · (Sb S) ∧ γ µ +∂ x ∧ h −1 YM * (2Ṡ S) · (Sb S) ,(44) where we have used Eq. (A45). Now, since h −1 YM (b x ) = b x for any spacetime vector b x , we find F YM (b) = −h YM ∂ q ∧ h −1 YM (b) = −∂ x ∧ γ µ (y ·Ȧ µ ) · (Sb S) − h YM A µ · (Sb S) ∧ γ µ −∂ x ∧ (2 SṠ) · b = γ µ ∧ γ ν [y · (∂ ν A µ ) − (y · A ν ) · A µ ] · (Sb S) + γ µ ∧ ( SA µ S − 2 S∂ µ S) · b .(45) Notice that F YM (b x ) = 0,(46) and that the pure spacetime component of the bivector F YM (b), to which only the first term contributes, reads (γ ρ ∧ γ σ ) · F YM (b) = y · (∂ ρ A σ − ∂ σ A ρ − A ρ × A σ ) · (Sb S)(47) where we have used the identity (γ ρ ∧ γ σ ) · (γ µ ∧ γ ν ) = δ µ σ δ ν ρ − δ ν σ δ µ ρ , and Formula (A7). Action of the field strength on arbitrary multivectors can be inferred from Eq. (20). In particular, due to the property (46), F YM (I x ) = 0,(48) and F YM e a ∧ (γ µ · I −1 x ) = F YM e a ) ∧ (γ µ · I −1 x ) = −( SA µ S − 2 S∂ µ S) · e a I −1 x ,(49) where we have used the fact that γ ρ ∧ γ ν ∧ (γ µ · I −1 x ) = 0, and γ ν ∧ (γ µ · I −1 x ) = δ ν µ I −1 x . C. Scalar field coupled to a gauge field It has been discussed in our previous article [4] that an N -component scalar field can be described in the Hamiltonian constraint formalism by the Hamiltonian H SF (q, P ) = P · I x + 1 2 N a=1 I x · (P · e a ) 2 + V (y),(50) where {e a } N a=1 is an arbitrary orthonormal basis of the space of fields. As shown in Sec. II, coupling to a gauge field h is achieved simply by the replacement P → h(P ) ≡P . The ensuing gauged scalar field Hamiltonian reads H SF,h (q, P ) =P · I x + 1 2 N a=1 I x · (P · e a ) 2 + V (y).(51) The first canonical equation (15b) takes the form A · h −1 (dΓ) = λA · ∂P H SF,h = λA · I x + λ N a=1 I x · (A · e a ) · I x · (P · e a ) ,(52) where A is an arbitrary D-vector. It can be used, together with the relation P · dΓ =P · h −1 (dΓ), and the Hamiltonian constraint H SF,h (q, P ) = 0, to cast the extended action (1) corresponding to the Hamiltonian H SF,h as A SF,h [γ, P, λ] = γ λ P · ∂P H SF,h − H SF,h = γ λ 1 2 N a=1 I x · (P · e a ) 2 − V (y) .(53) The choice A = I −1 x in Eq. (52) identifies the Lagrange multiplier λ as λ = I −1 x · h −1 (dΓ).(54) From the same equation we also infer that I −1 x · h −1 (dΓ) · e a = λI x · (P · e a ).(55) These relations allow us to write the action as a functional of the surface γ alone, A SF,h [γ] = γ 1 2λ N a=1 I −1 x · (h −1 (dΓ) · e a ) 2 − λV (y) .(56) The equation of motion for γ can be found either by varying this action, or from the canonical equations of motion (15) with a help of relations (34) and (35) for the gravitational field, or (48) and (49) for the Yang-Mills field. Nevertheless, in the remaining part of this example we shall stay with our considerations on the level of action, where calculations are significantly less involved. Gravity The scalar field Hamiltonian (50) is independent of x and therefore satisfies condition (8) for spacetime diffeomorphisms (29). Representing the motions as γ = {x + y(x) \| x ∈ Ω}, where Ω is a spacetime region, we find for the surface element (see Ref. [4]) dΓ = dX + (dX · ∂ x ) ∧ y + . . . ,(57) where ". . ." gathers terms with two and more y-space components. Here, dX = \|dX\|I x is the oriented infinitesimal element of the spacetime with magnitude \|dX\| (this is the usual scalar Riemann measure, traditionally denoted by d D x), and the orientation defined by the spacetime pseudoscalar I x . Let us consider the gravitational gauge field h Gr characterized by Eqs. (31). Since it acts on y-space vectors as an identity, h Gr (a y ) = a y , we find for the Lagrange multiplier given by Eq. (54) λ Gr = I −1 x · h −1 Gr (dX) = \|dX\| det(h −1 Gr ),(58) where the determinant is defined in geometric algebra terms in Appendix A 4. Moreover, we have I −1 x · (h −1 Gr (dΓ) · e a ) = I −1 x · h −1 Gr (dX) · h Gr (∂ x φ a ) = \|dX\| det(h −1 Gr ) h Gr (∂ x φ a ),(59) where the scalars φ a ≡ e a · y are the components of y. The action (56) now reads A SF,Gr [y(x)] = Ω det(h −1 Gr ) 1 2 N a=1 h Gr (∂ x φ a ) 2 − V (y) \|dX\|,(60) which can be further elucidated by writing N a=1 h Gr (∂ x φ a ) 2 = (∂ x φ a ) · h Gr h Gr (∂ x φ a ) = g µν ∂ µ φ a ∂ ν φ a ,(61) where g µν = γ µ · h −1 Gr h −1 Gr (γ ν ) , g µν = γ µ · h Gr h Gr (γ ν )(62) are the components of the metric tensor of general relativity, and its inverse, respectively. The metric, regarded as a linear mapping g = h −1 Gr h −1 Gr , has determinant det(g) = det(h −1 Gr ) 2 .(63) At the same time, by Formula (A27), det(g) = I −1 x · g(I x ) = det(γ µ · g(γ ν )) = det(γ λ · γ κ ) det(g µν ) = − det(g µν ),(64) where in the last equality we assumed, for definiteness, that the x-space has Lorentzian signature η = (1, −1, −1, −1), and γ µ · γ ν = η µν . These observations should be enough to conclude that Eq. (60) depicts the action of a scalar field coupled to a gravitational field (c.f. [20, Ch. 6.4]), with h Gr playing the role of vierbein of tetrad gravity formulations. We shall have more to say about the relation between the present gauge-field approach and the standard metric formulation of general relativity in Example V D. Yang-Mills field To ensure that the scalar field Hamiltonian (50) satisfies condition (8) for field-space rotations (37), we will now assume that the potential is of the form V (y) = U (y 2 ).(65) For the Yang-Mills gauge field h YM characterized by Eqs. (43), and the surface element of γ given by Eq. (57), Eq. (54) reads λ YM = I −1 x · h −1 YM (dΓ) = I −1 x · dX = \|dX\|.(66) Here we have used relations SI −1 x S = I −1 x , and h −1 YM * (a y ) = a y . Furthermore, since h −1 YM * (dX) = dX + (dX · γ µ ) ∧ (y · A µ ) + . . . ,(67) where ". . ." gathers terms with two and more y-space components, I −1 x · h −1 YM (dΓ) · e a = I −1 x · h −1 YM * (dΓ) · (Se a S) = I −1 x · h −1 YM * (dX) · (Se a S) + h −1 YM * (dX · ∂ x ) ∧ y · (Se a S) = \|dX\| γ µ (y · A µ ) · (Se a S) +∂ xẏ · (Se a S) = \|dX\|γ µ (∂ µ y + y · A µ ) · (Se a S).(68) Substituting Eqs. (66) and (68) into Eq. (56), we find the action of a scalar field coupled to a Yang-Mills gauge field A SF,YM [y(x)] = Ω 1 2 (∂ µ y + y · A µ ) · (∂ µ y + y · A µ ) − U (y 2 ) \|dX\|.(69) This action is independent of S due to the invariance of the scalar field Hamiltonian (50) under global field-space rotations. Let us note that although we consider only real-valued fields, the present formalism allows us to discuss also (special) unitary groups of transformations, which play the most prominent role in physics. Indeed, any Lie algebra u(n) can be represented as a bivector subalgebra of the geometric algebra of a 2n-dimensional Euclidean space, while the Lie group U (n) is represented by the corresponding group of rotors (see Refs. [17,Ch. 11.4] or [21]). In the kinetic part of action (69) we identify the traditional Yang-Mills covariant derivative D µ y ≡ ∂ µ y + y · A µ = ∂ µ y + A µ (y),(70) where A µ is a linear antisymmetric mapping characterized by the bivector A µ (see Appendix A 4 a). Commutator of two covariant derivatives yields the traditional Yang-Mills field strength F µν (y) ≡ (D µ D ν − D ν D µ )y = y · (∂ µ A ν − ∂ ν A µ − A µ × A ν ).(71) The last term on the right-hand side has been obtained by the Jacobi identity, Eq. (A7). For every µ and ν, F µν is an antisymmetric linear function acting on the y-space, with characteristic bivector F µν = ∂ µ A ν − ∂ ν A µ − A µ × A ν .(72) Eq. (47) now establishes an explicit relation between the traditional field strength of an SO(N ) Yang-Mills theory and the Yang-Mills field strength defined according to Formula (19): (γ µ ∧ γ ν ) · F YM (b) = F µν (y) · (Sb S).(73) Electromagnetic field Let us now specialize the Yang-Mills field to the electromagnetic field, i.e., take a twodimensional field space (N = 2) where all bivectors are scalar multiples of the pseudoscalar I y . Due to this simplification, the rotor R from Eq. (37) can be parametrized by a single rotation angle θ, R EM (x) = e −Iyθ(x)/2 ,(74) and the bivectors A µ can be written as A µ = α µ I y ,(75) where α µ are components of the electromagnetic vector four-potential (which is a spacetime vector). The electromagnetic gauge field has the form (c.f. Eq. (39)) h EM (b; q) = e Iyφ(x)/2 b e −Iyφ(x)/2 − γ µ α µ (y · I y ) · b,(76) and the transformation rules (42) reduce to φ = φ + θ , α µ = α µ − ∂ µ θ,(77) as expected of the electromagnetic potential. The action (69) now reads A SF,EM [y(x)] = Ω 1 2 (∂ µ y + α µ y · I y ) · (∂ µ y + α µ y · I y ) − U (y 2 ) \|dX\|.(78) Scalar field that couples to electromagnetism is usually regarded as a complex field Φ(x) = φ R + iφ I with the action [22] A SF,EM [Φ(x)] = Ω 1 2 (D µ Φ) * D µ Φ − U (Φ * Φ) \|dX\|,(79) where D µ = ∂ µ + iα µ is the traditional covariant derivative (the coupling constant has been omitted). Our use of geometric algebra offers an equivalent formulation (78) in terms of a twocomponent vector y = φ R e R + φ I e I = e R Φ,(80) where the imaginary unit of complex numbers i has been identified with the unit pseudoscalar I y = e R e I of the two-dimensional field space (see Appendix A 2). The correspondences Φ * Φ = (φ R − φ I e R e I )(φ R + φ I e R e I ) = ye R e R y = y 2 , D µ Φ = (∂ µ + α µ e R e I )(φ R + φ I e R e I ) = e R (∂ µ y + α µ yI y ), (D µ Φ) * = (∂ µ − α µ e R e I )(φ R − φ I e R e I ) = (∂ µ y + α µ yI y )e R(81) then show that the complex action (79) is indeed equivalent with the real action (78). D. String coupled to gravity The Hamiltonian describing free relativistic particles, strings or high-dimensional membranes has been introduced in Example 5.3 of Ref. [3]. For spacetimes with arbitrary signature it reads H Str (q, P ) = 1 2 ( P · P − Λ 2 ),(82) where Λ is a positive constant, and " " is the reversion operation defined in Eq. (A2). (Positive P · P can be replaced by \|P \| 2 , where \| . \| is the magnitude defined in Eq. (A1).) Motions γ are identified with world-sheets, and the configuration space C with the target space of string theory, i.e., with the spacetime. The corresponding gauged Hamiltonian H Str,h (q, P ) = 1 2 h( P ) · h(P ) − Λ 2(83) implements coupling to a static gravitational field described by the gauge field h. (For the case D = 1 see also Ref. [18,Ch. 4(d)].) The first canonical equation (15a) now reads h −1 (dΓ) = λh( P ) ⇔ P = 1 λ h −1 h −1 ( dΓ).(84) Taking the magnitude, and using the Hamiltonian constraint, Eq. (15c), we find \|λ\|Λ = \|h −1 (dΓ)\|.(85) The latter two equations allow us to eliminate P and λ, and cast the action (1) as A Str,h [γ] = γ P · dΓ = γP · h −1 (dΓ) = γ 1 λ h −1 ( dΓ) · h −1 (dΓ) = ±Λ γ \|h −1 (dΓ)\|,(86) where "±" is the sign of λ. This action can be written solely in terms of the metric g ≡ h −1 h −1 , A Str,h [γ] = ±Λ γ dΓ · g(dΓ).(87) As regards the remaining second canonical equation (15b), one can show, using the identity (A31) and∂ qḣ −1 h −1 (U ) · U = 1 2∂ qḣ −1ḣ−1 (U ) · U,(88) that it reads (−1) D−1 1 2∂ qġ (U ) · U = U · ∂ q g(U ) for D = 1 (U · ∂ q ) · g(U ) for D > 1. Here we have introduced a generalization of the four-velocity U ≡ dΓ/\|h −1 (dΓ)\|, which is normalized so that U · g(U ) = 1. Component form of the geodesic equation It is instructive to compare our results with the traditional "component" (or tensor) approach to general relativity. For this purpose we choose a basis {γ µ } D+N µ=1 , and its reciprocal {γ µ } D+N µ=1 , and write the components of the metric and its inverse g µν = γ µ · g(γ ν ) , g = h −1 h −1 , g µν = γ µ · g −1 (γ ν ) , g −1 = hh.(90) For simplicity, let us concentrate on the case D = 1, i.e., relativistic particle. With a parametrization of the trajectory γ = {q(τ ) \| τ ∈ [τ i , τ f ]} the action (87) takes the familiar form A Str,h [q(τ )] = ±Λ τ f τi g µν dq µ dτ dq ν dτ dτ.(91) Furthermore, the equation of motion (89) is an equivalent of the geodesic equation [23]. This can be seen by rearranging U · ∂ q U = g −1 1 2∂ qġ (U ) · U − U ·∂ qġ (U ) ,(92) and introducing the components U µ = γ µ · U , ∂ µ = γ µ · ∂ q , U ν ∂ ν U µ = g µν 1 2 ∂ ν g λκ − ∂ κ g νλ U λ U κ .(93) Recall the definition of the Christoffel symbols Γ µ λκ = 1 2 g µν [∂ κ g νλ + ∂ λ g νκ − ∂ ν g λκ ] ,(94) which have the symmetry property Γ µ λκ = Γ µ κλ , and take the parametrization of γ such that U = dq/dτ . Then, we finally arrive at the standard component form of the geodesic equation d 2 q µ dτ 2 + Γ µ λκ dq λ dτ dq κ dτ = 0.(95) Symmetries and Killing equation Infinitesimal symmetries of a gauged Hamiltonian are found from Eq. (26), which for the present Hamiltonian (82) reads v ·∂ qḣ (P ) − h ∂ q ∧ (v · P ) · h( P ) = 0,(96) and has to be satisfied for all P . With a help of Eq. (88), this can be cast purely in terms of the metric, 1 2 v ·∂ qġ −1 (P ) − g −1 ∂ q ∧ (v · P ) · P = 0.(97) Let us consider again the case D = 1, and put P = g(a). Using the symmetricity of g, g = g, and the relationġg −1 = −gġ −1 , we find 1 2 v ·∂ qġ (a) +∂ qv · g(a) · a = 0.(98) According to Eq. (A37), this, in turn, is equivalent with the equation v ·∂ q a ·ġ(b) + a ·∂ q b · g(v) + b ·∂ q a · g(v) = 0(99) being satisfied for all vectors a and b. In components, i.e., for v = v λ γ λ , a = γ µ and b = γ ν , this condition reads v λ ∂ λ g µν + g νλ ∂ µ v λ + g µλ ∂ ν v λ = 0,(100) which is an equivalent form of the Killing equation [24,Ch. 25.2] v µ;ν + v ν;µ = 0, with the covariant derivative v κ ;ν = ∂ ν v κ + Γ κ νλ v λ . Indeed, by expanding v µ;ν + v ν;µ = g µκ v κ ;ν + g νκ v κ ;µ = g µκ (∂ ν v κ + Γ κ νλ v λ ) + g νκ (∂ µ v κ + Γ κ µλ v λ ),(102) and substituting from Eq. (94), we obtain Eq. (100). The symmetry generator v is traditionally referred to as the Killing vector. To each Killing vector corresponds a conserved quantity P · v = 1 λ g(dΓ) · v = ±ΛU · g(v) = ±Λg µν U µ v ν(103) (c.f. Eq. (25.5) in [24]), which satisfies the conservation law (28). VI. CONCLUSION We have enriched the Hamiltonian constraint formulation of classical field theories by introducing a gauge field -a position-dependent linear mapping that ensures invariance of the theory under an extended group of local transformations of the configuration space via coupling to the momentum multivector. Canonical equations of motion for the ensuing gauged Hamiltonian have been derived (Eqs. (15)), and symmetry conditions for a fixed gauge field have been discussed. We have proposed a generic form of the gauge field strength, which is a q-dependent linear mapping given by Eq. (19) that raises the grade of its argument by one. It can be interpreted as torsion of the Weitzenböck connection used in the teleparallel theory of gravity. In principle, all diffeomorphisms of the space of partial observables are gauged if the gauge field is allowed to have the most generic form. However, in examples with an N -component scalar field, we restricted the group of transformations under which the theory has to be invariant, respectively, to spacetime diffeomorphism, and local field-space rotations in order to restrict the form of the gauge field. In the first case we generated gravitational, and in the second case Yang-Mills interaction. The generic gauge field can be therefore viewed, at least for this "toy model" of scalar field theory, as a unified classical field [25,26]. In our attempt to reconcile gravity and Yang-Mills theory we view gravitational field as a field on flat spacetime rather than as a metric of a curved Riemannian manifold. The viability of such approach to gravity has been shown by the Gauge Theory Gravity [18,19], which indeed offered significant inspiration for this article. Dynamics of the gauge field has not been addressed. However, the definition of the field strength and the brief investigation of symmetries and conservation laws for the gauged Hamiltonian are important prerequisites for future development of dynamical equations for the unified gauge field. Finally, let us remark that although this article has been concerned only with classical field theory, one of the main motivations for introducing Hamiltonian methods in field theory is the desire to develop new quantization schemes (see, e.g., Ref. [27]) that could compete with established Lagrangian formalism. Projection of a vector a onto a blade (a decomposable multivector) A is a · AA −1 . It vanishes for vectors perpendicular to A, and equals a if a ∧ A = 0, i.e., if the vector is parallel with A. If a is a vector, and A r and B s multivectors of grades r and s, respectively, then the following identity holds (the proof can be found in [16,): (−1) r a ∧ (A r · B s ) + (A r · a) · B s = A r · (a ∧ B s ) for s ≥ r > 1. (A3) The commutator product is defined between any two multivectors by A × B := 1 2 (AB − BA).(A4) For any three multivectors, it satisfies the Jacobi identity A × (B × C) + B × (C × A) + C × (A × B) = 0. (A5) For a vector y and a bivector A, the commutator product reduces to the inner product, y × A = y · A = −A · y.(A6) The Jacobi identity for one vector y and two bivectors A µ and A ν then reads y · (A µ × A ν ) = (y · A µ ) · A ν − (y · A ν ) · A µ ,(A7) where A µ × A ν is again a bivector, as can be easily proven (see Basis and reciprocal basis Basis of the vector space V is a set of n vectors {e 1 , . . . , e n } that are linearly independent, i.e., satisfy e 1 ∧ . . . ∧ e n = 0. From these, a basis for the entire geometric algebra G(V ) can be built by repeated use of the outer product [16,. We do not assume the basis {e j } n j=1 to be orthonormal, i.e., in general, e j ·e k = δ jk . (In mixedsignature spaces such basis does not even exist.) Therefore, in order to write the expansion of a vector a ∈ V a = a · e j e j = a · e j e j , we need to define the reciprocal basis {e 1 , . . . , e n } by the requirement e j · e k = δ j k (∀j, k = 1, . . . , n). The reciprocal basis can be explicitly constructed [17,Ch. 4.3]: e j = (−1) j−1 e 1 ∧ . . . ∧ě j ∧ . . . ∧ e n E −1 n ,(A10) where E n ≡ e 1 ∧ . . . ∧ e n ,(A11) and the check indicates, as usual, that the term is missing from the expression. Consider, for example, the Minkowski space with a basis {γ 0 , γ 1 , γ 2 , γ 3 }, such that γ µ ·γ ν = η µν , where η = diag(1, −1, −1, −1). The geometric algebra of this vector space coincides with the Dirac algebra of γ matrices. The quantity (A11), usually denoted γ 5 , has the inverse γ −1 5 = −γ 3 γ 2 γ 1 γ 0 . It can be verified that Eq. (A10) yields the reciprocal basis γ 0 = γ 0 , γ 1 = −γ 1 , γ 2 = −γ 2 , γ 3 = −γ 3 .(A12) Complex numbers Consider a two-dimensional real vector space V 2 with positive signature, and its orthonormal basis {e R , e I }. In real geometric algebra, complex numbers are naturally identified with G + (V 2 ), the even subalgebra of G(V 2 ) (see [16, or [17,Ch. 2.3.3]). The latter contains multivectors of the form Φ = φ R + φ I I y ,(A13) where φ R , φ I are scalars (the real and the imaginary part of the corresponding complex number), and I y = e R e I is the unit pseudoscalar of G(V 2 ). I y serves the same purpose as the imaginary unit i. One can easily check that I 2 y = −1, and that acting from the right, it rotates any vector y = φ R e R + φ I e I clockwise by 90 • , yI y = −φ I e R + φ R e I .(A14) Moreover, vectors y ∈ V 2 are in one-to-one correspondence with elements of G + (V 2 ), y = φ R e R + φ I e I = e R Φ ⇔ Φ = e R y.(A15) Complex units e iθ implement clockwise rotation of a complex number Φ through an angle θ simply via multiplication, Φ = e iθ Φ. The corresponding vectors are related analogously by y = e R Φ = e R e Iyθ Φ = e R e Iyθ e R y = e −Iyθ y = e −Iyθ/2 ye −Iyθ/2 .(A16) The final expression coincides with the geometric algebra prescription for the rotation through an angle θ in a plane defined by the bivector I y , and immediately generalizes to vector spaces with dimension greater than two using rotors [17,Ch. 4.2]. Indeed, rotors can be thought of as higher-dimensional unit complex numbers. Transformations and induced mappings Let f : q → q be an invertible smooth mapping (a diffeomorphism) on the space of partial observables C. For a vector a in the tangent space of C we define the derivative of f in direction a, f (a; q) ≡ a · ∂ q f (q) := lim ε→0 f (q + εa) − f (q) ε .(A17) This gives rise to a q-dependent linear function, the differential of f , that maps vectors a at point q to vectors f (a) at q . The adjoint of f , denoted f , is defined by f (b; q) := ∂ q f (q) · b,(A18) so that for any two vectors a and b it observes the identity b · f (a) = f (b) · a.(A19) The adjoint maps vectors b at a point q to vectors f (b) at q. It is natural to extend the domain of f and f so that they may act on generic multivectors by demanding linearity and the outermorphism property [16, Ch. 3-1] f (A ∧ B) = f (A) ∧ f (B) , f (A ∧ B) = f (A) ∧ f (B).(A20)A r · f (B s ) = f [f (A r ) · B s ] for r ≤ s, f (A r ) · B s = f [A r · f (B s )] for r ≥ s.(A21) We will refer to the differential f and the adjoint f collectively as induced mappings, since they are induced by the diffeomorphism f . Let us consider an arbitrary multivector-valued function F on C. The chain rule for differentiation, a · ∂ q F (f (q)) = f (a) · ∂ q F (q ),(A22) shows that the vector derivative operator transforms under f as ∂ q = f −1 (∂ q )(A23) (c.f. the transformation of momentum, Eq. (3)). We also see that f −1 = f −1 and f −1 = f −1 . (A24) The induced mappings f and f are functions of q which can be further differentiated. Commutativity of directional derivatives then leads to the identity ∂ q ∧ f (A) = 0,(A25) valid for any constant multivector A. Linear functions Let h : V → V be a linear map. We can extend h to an outermorphism h : G(V ) → G(V ), and introduce its adjoint h along the same lines as in the previous section for the induced mapping f . Outermorphism h acting on a pseudoscalar I produces a new pseudoscalar h(I). The proportionality constant between the two pseudoscalars is the determinant of h, h(I) = I det(h) ⇔ det(h) = I −1 h(I) = I −1 · h(I). (A26) Utilizing an arbitrary basis {e j } n j=1 of V , and the reciprocal basis {e j } n j=1 , it can be expressed as the determinant of the matrix h j k = e j · h(e k ) (see Formula (4.12) in [16,), det(h) = (e n ∧ . . . ∧ e 1 ) · h(e 1 ) ∧ . . . ∧ h(e n ) = det(h j k ).(A27) From definition (A26), two popular properties of determinants, det(h 1 h 2 ) = det(h 1 ) det(h 2 ) , det(h) = det(h),(A28) follow particularly easily. It is also known that h has an inverse if and only if det(h) = 0, in which case we can use the first of Formulas (A21), where we set f = h, A r = h −1 (A) and B s = I, to obtain an explicit expression h −1 (A) = 1 det(h) h(AI)I −1 ,(A29) where A is a generic multivector. The gauge field is an example of a linear function a → h(a; q) that varies from point to point in the configuration space. Often, it is needed to form the derivative b ·∂ qḣ (a) = b ·∂ qḣ (ȧ) − b ·∂ q h(ȧ),(A30) which differentiates only the q-dependency of h, and not an eventual change of the vector field a(q). When shall omit the scalar object "b ·∂ q " when writing differential identities, if it is clear that the remaining dots are understood as directional derivatives in a generic direction. For example, the latter equation then acquires the neat formḣ(a) =ḣ(ȧ) − h(ȧ). Since hh −1 is the identity for each point q, hh −1 = −hḣ −1 .(A31) Moreover, the outermorphism h satisfies a Leibniz rulė h(A ∧ B) =ḣ(A) ∧ḣ(B) =ḣ(A) ∧ h(B) + h(A) +ḣ(B),(A32) which can be used to prove the property (20) of the gauge field strength F : F (A r ∧ B s ) = −h ∂ q ∧ḣ −1 (A r ∧ B s ) = −h ∂ q ∧ḣ −1 (A r ) ∧ B s − h ∂ q ∧ h −1 (A r ) ∧ḣ −1 (B s ) = F (A r ) ∧ B s + (−1) r A r ∧ F (B s ).(A33) This can be iterated to yield F (b 1 ∧ . . . ∧ b r ) = F (b 1 ) ∧ b 2 ∧ . . . ∧ b r − b 1 ∧ F (b 2 ∧ . . . ∧ b r ) = r j=1 (−1) j−1 F (b j ) ∧ b 1 ∧ . . . ∧b j ∧ . . . ∧ b r .(A34) (The "checked" vectors are missing from the expression.) Finally, let us recall the transformation rules (3), (12) and (A23) for P , h and ∂ q , respectively, and check that F (P ) (withP ≡ h(P )) is invariant under gauge transformations. We have F (P ) = −h f f −1 (∂ q ) ∧ḟ −1ḣ−1 (P ) = F (P ) − h ∂ q ∧ fḟ −1 (P ) . (A35) But the second term on the right-hand side vanishes on account of Eq. (A25), sincė ∂ q ∧ fḟ −1 (P ) = −∂ q ∧ḟ f −1 (P ) = 0. (A36) a. Bivectors and antisymmetric transformations A linear transformation h is said to be antisymmetric if it satisfies any of the equivalent conditions h = −h ⇔ b · h(a) = −a · h(b) (∀a, b ∈ V ) ⇔ a · h(a) = 0 (∀a ∈ V ). (A37) (Note that the first two equations can be obtained from the third one promptly by differentiation with respect to a.) Any bivector A gives rise to an antisymmetric mapping A(a) = a · A = −A · a,(A38) and conversely, any antisymmetric mapping h can be characterized by a bivector, namely, 1 2 e j ∧ h(e j ) (summation over j), since a · e j ∧ h(e j ) = a · e j h(e j ) − e j a · h(e j ) = 2h(a). (A39) Commutator of two antisymmetric mappings is again antisymmetric, and its characteristic bivector is generated by the commutator product (A4), [A 1 , A 2 ](a) = (A 1 A 2 − A 2 A 1 )(a) = (a · A 2 ) · A 1 − (a · A 1 ) · A 2 = a · (A 2 × A 1 ).(A40) The algebra of antisymmetric matrices with the product [., .] can therefore be represented as the algebra of corresponding bivectors with the commutator product "×". b. Rotations and rotors In geometric algebra, a rotation R is conveniently represented using a rotor R as (see Refs. [4], [16] or [17]) R(a) = Ra R , R = ±e −B/2 ,(A41) where B is a bivector. This exponential form of rotors is ensured only in spaces with Euclidean or Lorentz signature (see [16, or [28]), which we shall thus assume in this subsection. (The ±-sign reflects the double-covering nature of the rotor representation, and has no effect on the expression for rotation, Ra R.) Since bivectors change sign under reversion, B = −B, and henceforth R = ±e B/2 , rotors enjoy the property R R = RR = 1. (A42) For completeness we indicate also inverse and adjoint rotations: R −1 (a) = RaR, R(b) = RbR, R −1 (b) = Rb R.(A43) When rotor-valued functions R(q) on the configuration space C are considered, differentiation of relation (A42) yieldsṘ R = −R˙ R,(A44) which is, importantly, a bivector. This fact allows to represent the Lie algebra of antisymmetric matrices so(n), corresponding to the Lie group SO(n) of rotations R, as the algebra of bivectors (see Eq. (A40), and for more details Ref. [17,Ch. 11.3]). Relation (A44) can be used to obtaiṅ Rb˙ R =Ṙ R(Rb R) − (Rb R)Ṙ R = (2Ṙ R) · (Rb R) = (Rb R) · (2R˙ R),(A45) a formula employed in our study of the Yang-Mills gauge field in Sec. V B. c. Shear mappings Elementary shear transformation is a linear mapping defined by two orthogonal vectors u and v, S uv (a) := a + u v · a , u · v = 0. (A46) During our treatment of the Yang-Mills gauge field in Sec. V B we note that h YM * has slightly more general structure S(a; u 1 , . . . , u r ; v 1 , . . . , v r ) = a + r j=1 u j v j · a , u j · v k = 0 ∀j, k = 1, . . . , r, which can, nevertheless, be written as a composition of elementary shears, S(a) = S urvr . . . S u1v1 (a) . Due to orthogonality of vectors u j and v k , it is easy to form the inverse and the adjoint of the generic shear transformation (A47), S −1 (a) = a − r j=1 u j v j · a, S(b) = b + r j=1 b · u j v j , S −1 (b) = b − r j=1 b · u j v j . (A49) ( 8 ) 8holds. If γ cl is a physical motion of the gauged Hamiltonian H h , then f (γ cl ) is a physical motion of H h , where the gauge field transforms as h = h f . FIG. 1 : 1Momentum P , gauge-invariant momentumP , and field strength F (P ) for one-dimensional motions γ. regardless of a concrete form of H. This condition, however, may be too restrictive -a vector field v may well be a symmetry generator according to Eq. (26) for a given Hamiltonian H, even though it fails to satisfy Eq.(27). In Example V D, we will show how Eq. (26) reduces to the Killing equation of general relativity, and identify v with the Killing vector. (For scalar arguments one defines f (α) = f (α) = α). For an r-vector A r and an s-vector B s , the following useful generalizations of Eq. (A19) hold[16, AcknowledgementThe author received support from Czech Science Foundation (GAČR), Grant GA14-07983S, and Deutsche Forschungsgemeinschaft (DFG), Grant KL 256/54-1. In addition, he would like to thank the organizers and participants of the workshop "Rethinking Foundations of Physics 2016", held in Dorfgastein, Austria, for creating a stimulating environment, in which many of the ideas of this article have been cultivated.Appendix A: Elements of geometric algebra and calculusLet V be an n-dimensional real vector space of possibly mixed (but non-degenerate) signature, and let G(V ) denote its geometric algebra[16,17]. (For concise introduction see also appendices of[3,4].) In this appendix we quote some key definitions and results employed in the main text.Magnitude of a grade-r multivector A is definedwhere \| A · A\| is the absolute value of the scalar A · A, andis the reversion of A, which satisfies, for any two multivectors, AB = B A. If \|A\| = 0 then A/\|A\| is normalized to unity. A normalized highest-grade element of G(V ) is referred to as the unit pseudoscalar I. For any vector a ∈ V , a · I = aI.Appendix B: The field strength as torsionLet {γ j } D+N j=1 denote a basis (or frame) in the tangent space of C, and {γ j } D+N j=1 its reciprocal. Let us assume that this frame is constant, i.e., q-independent, and introduce an additional, possibly q-dependent, coordinate frame {e µ } D+N µ=1 , for which the Lie bracket between any pair of vectors vanishes, [e µ , e ν ] = e µ · ∂ q e ν − e ν · ∂ q e µ = 0.(B1)Components of the gauge field with respect to these frames form the vielbein and its inverse (c.f. Appendix C in Ref.[18])For the sake of correspondence, we shall identify the space of partial observables C with the spacetime of the teleparallel theory of gravity[29,30], where the vielbeins are used to define the Weitzenböck (or affine) connection with coefficients h ρ j ∂ ν h j µ . The gravitational field strength of teleparallel gravity is determined by the torsion tensor, which is the antisymmetric part of the connection. 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[ "Mean field approximation for noisy delay coupled excitable neurons", "Mean field approximation for noisy delay coupled excitable neurons" ]	[ "Nikola Burić [email protected] \nInstitute of Physics\nUniversity of Beograd\nPO Box 6811080Beograd-ZemunSerbia\n", "Dragana Ranković \nDepartment of Physics and Mathematics\nFaculty of Pharmacy\nUniversity of Belgrade\nVojvode Stepe 450BelgradeSerbia\n", "Kristina Todorović \nDepartment of Physics and Mathematics\nFaculty of Pharmacy\nUniversity of Belgrade\nVojvode Stepe 450BelgradeSerbia\n", "Nebojša Vasović \nDepartment of Applied Mathematics\nFaculty of Mining and Geology\nUniversity of Belgrade\nP.O.Box 162BelgradeSerbia\n" ]	[ "Institute of Physics\nUniversity of Beograd\nPO Box 6811080Beograd-ZemunSerbia", "Department of Physics and Mathematics\nFaculty of Pharmacy\nUniversity of Belgrade\nVojvode Stepe 450BelgradeSerbia", "Department of Physics and Mathematics\nFaculty of Pharmacy\nUniversity of Belgrade\nVojvode Stepe 450BelgradeSerbia", "Department of Applied Mathematics\nFaculty of Mining and Geology\nUniversity of Belgrade\nP.O.Box 162BelgradeSerbia" ]	[]	Mean field approximation of a large collection of FitzHugh-Nagumo excitable neurons with noise and all-to-all coupling with explicit timedelays, modelled by N ≫ 1 stochastic delay-differential equations is derived. The resulting approximation contains only two deterministic delay-differential equations but provides excellent predictions concerning the stability and bifurcations of the averaged global variables of the exact large system.	10.1016/j.physa.2010.05.048	[ "https://arxiv.org/pdf/1003.5187v1.pdf" ]	119,111,004	1003.5187	9d10170fbb2c0953d542eaf550f6327a7301b59f	Mean field approximation for noisy delay coupled excitable neurons 26 Mar 2010 March 29, 2010 Nikola Burić [email protected] Institute of Physics University of Beograd PO Box 6811080Beograd-ZemunSerbia Dragana Ranković Department of Physics and Mathematics Faculty of Pharmacy University of Belgrade Vojvode Stepe 450BelgradeSerbia Kristina Todorović Department of Physics and Mathematics Faculty of Pharmacy University of Belgrade Vojvode Stepe 450BelgradeSerbia Nebojša Vasović Department of Applied Mathematics Faculty of Mining and Geology University of Belgrade P.O.Box 162BelgradeSerbia Mean field approximation for noisy delay coupled excitable neurons 26 Mar 2010 March 29, 20101PACS 0545Xt; 0230Ks * Mean field approximation of a large collection of FitzHugh-Nagumo excitable neurons with noise and all-to-all coupling with explicit timedelays, modelled by N ≫ 1 stochastic delay-differential equations is derived. The resulting approximation contains only two deterministic delay-differential equations but provides excellent predictions concerning the stability and bifurcations of the averaged global variables of the exact large system. Introduction Small parts of brain cortex may contain thousands of morphologically and functionally similar interconnected neurons. Realistic models of an individual neuron, like Hodgkin-Huxley, FitzHugh-Nagumo (FN) or Hindmarsh-Rose to mention only a few popular examples [1], are given by few-dimensional nonlinear differential equations. Transport of information between neurons can be phenomenologically described by time-delayed inter-neuronal interaction (please see [2] and the references therein). It is also well known that neurons in vivo function under influences of many sources of noise [3]. Considering all mentioned factors it is clear that a basic, relatively detailed mathematical model of a small part of realistic cortex should involve an extremely large system of nonlinear stochastic delay-differential equations (SDDE). Analyzes of such complex models is impossible without more or less severe approximations, which should be adopted to different purposes. It is our goal to study some aspects of an approximation by only two deterministic delay-differential equations (DDDE) of an example of a complex neuronal system described by many-component SDDE. We shall see that, although the approximate model is very simple, the predicted critical parameter values for the bifurcations and stability of the stationary states are in excellent quantitative agrement with those of the exact complex model within a relevant domain of parameters. Neuronal dynamics with all three factors (large number of units, delayed interaction and noisy environment ) included has been studied much less than the influence of each of the factors separately [4]. Important influence of noise alone on a single, small number or large clusters of neurons has been studied a lot in recent years [5]. It is also well known that time-delay can have important qualitative effects on the stability of stationary states (please see for example [2], [6]) and synchronization of neuronal dynamics [7]. Studies of combined effects of noise and time-delay have mostly, but not entirely ( [17]) been restricted to artificial networks [8] [9] or small number of neurons (usually two) [10], [11]. An example of a study of a large collection of noisy realistic neurons with delayed coupling can be found in [17] (see also the references therein). The mean field approach (MFA) is based on a set of approximations that replace many component system by a simpler system described by a small number of (averaged) collective or macroscopic properties. The mean field approximation has been applied on systems of excitable neurons with noise but with no time-delay for example in [12], [13], [14], [5]. On the other hand a type of MFA was devised in [15] and [16] and applied on large clusters of noisy neurons with time-delayed interaction in [17]. However, the approximations made in these papers resulted in a system of equations that is still to large to be analyzed analytically, so that the approximate system must be studied numerically. We shall derive an approximate system of only two DDDE for the dynamics of the mean fields. Such a simple system allows analytical treatment of bifurcations and the parameter domains of stability of the stationary states which turn out to be in a quite good agrement with the exact complex system. 2 The model and its mean field approximation We shall study a system of excitable neurons modelled by the following set of SDDE: ǫdx i = f (x i , y i )dt + c N N j=1 (x j (t − τ ) − x i )dt dy i = g(x i , y i ) + √ 2DdW i (1) with f (x, y) = x − x 3 /3 − y + I g(x, y) = x + b,(2) where b, I, c, D and ǫ ≪ 1 are parameters. The formulas (2) represent one of the common ways of writing the famous FitzHugh-Nagumo model [1] of the excitable behavior. For certain parameter values, like b = 1.05, I = 0 to be used throughout this paper, the ODE given by (2) have stable stationary solution (x 0 , y 0 ) such that small departures from (x 0 , y 0 ) might lead to large and long lasting excursions away from (x 0 , y 0 ) which nevertheless end up on the stable state (x 0 , y 0 ). The type of excitable behavior epitomized by the FN model is called type II [1] and is characterized by destabilization of the stationary state via the Hopf bifurcation. The variable x is called the fast variable (due to ǫ ≪ 1) and corresponds to the membrane electrical potential. The variable y is the slow recovery variable and has no direct interpretation. Each of i = 1, 2 . . . N units in (1) is coupled with each other unit and with itself. There are two major types of inter-neuronal couplings: the chemical and the electrical synapses. Time-delay τ is important especially in the first type of synapses but plays also an important role in the electrical junctions and in the transmission of an impulse through the dendrite. In (1) we use the electrical coupling with the time-lag and the strength that is equal for all pairs of neurons. The terms √ 2DdW i represent stochastic increments of independent Wiener processes, i.e. dW i satisfy E(dW i ) = 0, E(dW i dW j ) = δ i,j dt,(3) where E() denotes the expectation over many realizations of the stochastic process. Mean field approximation In order to derive the approximate dynamical equations for the mean fields X(t) = 1 N N i x i (t) ≡< x i (t) >, Y (t) = 1 N N i y i (t) ≡< y i (t) >(4) of the system (1) to be used in this paper we shall first suppose that: a) The dynamics is such that the distributions of x i and y i are Gaussian and b) for large N the average over N of local random variables is given by the expectation with respect to the corresponding distribution, i.e. for example 1 N N i x i ≈ E(x i ), where E(x i ) is the expectation with respect to the distribution of x i (t). In the limit N → ∞ the last assumption is expected to become an equality, implied by the strong low of large numbers [18]. In the mean field approach it is commonly assumed that b) is approximately true even for finite but large N despite the nonzero interaction between the local random variables. The first assumption should be expected to be true when the noise intensity is small, i.e. D ≪ 1 (see for example [13], [14]). With these assumptions the system (1) of 2N SDDE can be reduced to five DDDE for the macroscopic variables X(t), Y (t) and the second order cumulants. Further assumption concerning the time scales of first and second order cumulants enables us to derive the final approximate system of only two DDDE. Mean field assumption guaranties that global averages, like (1/N) N i x i of local quantities are equal to the expectations with respect to distribution of the corresponding variable E(x i ). Besides the mean values X(t), Y (t) we introduce deviations from the expectations: n x i (t) = X(t) − x i (t), n y i (t) = y i (t) − Y (t). Because of the assumed Gauss distribution of each variable the first and the second order cumulants of these deviations are equal to the first and second order moments ( i.e. to the first and second order centered moments of the variables x i , etc. . . ). Furthermore, due to the same Gaussian assumption higher order cumulants are equal to zero, and this enables us to terminate the cumulant expansion of the dynamical equations. Details of the derivation are given in the appendix. The result is a system of five deterministic delay-differential equations for the global variables and global centered moments: s x =< n 2 x i (t) >, s y =< n 2 y i (t) >, u =< n x n y > .(5) The equations are ǫ dX(t) dt = X(t) − X(t) 3 /3 − s x (t)X(t) − Y (t) + c(X(t − τ ) − X(t)), dY (t) dt = X(t) + b, ǫ 2 ds x (t) dt = s x (t)(1 − X(t) 2 − s x (t) − c) − u(t) 1 2 ds y (t) dt = u(t) + D, du(t) dt = u(t) ǫ (1 − X(t) 2 − s x (t) − c) − 1 ǫ s y (t) + s x (t).(6) The analogous set of ordinary differential equations was used to study the mean field approximation of the stochastic system of N neurons without delay in [5]. The equations (6) are delay-differential equations because the original system of stochastic eq. (1) contains time-delay. In order to further simplify the approximate system we shall suppose that relaxation time-scale of the second order moments is much faster then those of the first order moments. Thus we can replace in eq. (6) the stationary values of s x , s y and u obtained by setting the right hand sides of the last three equations in (6) equal to zero. As the results we obtain the following two DDDE: ǫ dX(t) dt = X(t) − X(t) 3 /3 − X(t) 2 1 − c − X(t) 2 + ((c − 1 + X(t) 2 ) 2 + 4D) 1/2 − Y (t) + c(X(t − τ ) − X(t)), dY (t) dt = X(t) + b.(7) 3 Stability and bifurcations of the stationary state Stationary states, their stability and local bifurcations of the approximate system of DDDE (7) are determined by the standard procedure. It is remarkable that such a crude approximation provides relevant information about the exact system. There is only one stationary state of (7) given by: (8) is determined from the roots of the characteristic equation. Due to the time-delay the system (7) has an infinite-dimensional state space, and the characteristic equation is transcendental with an infinite number of roots. The characteristic equation is X(t) ≡ X 0 = −b, Y (t) ≡ Y 0 = −b 2 1 + b 2 /3 + c − (4D + (c + b 2 − 1) 2 ) 1/2 . (8) Local stability ofλ 2 − 1 2ǫ 1 − c + b 2 − (m 2 + 4D) 1/2 − 2b 2 m (m 2 + 4D) 1/2 λ + 1 ǫ − c ǫ λ exp(−λτ ) = 0.(9) where m = c − 1 + b 2 . Bifurcations of the stationary state occur for those values of the parameters such that any of the infinite number of roots of (9) has the real part equal to zero [19]. This is possible only if λ = iω, where ω can be taken to be positive. Substitution of λ = iω in (9) gives √ 2ω ± = (−k 2 + c 2 /ǫ 2 + 2/ǫ ± (k 2 − c 2 /ǫ 2 − 2/ǫ) 2 − 4/ǫ 2 ) 1/2 1/2 ,(10) where k = 1 2ǫ 1 − c + b 2 − (m 2 + 4D) 1/2 − 2b 2 m (m 2 + 4D) 1/2 . The critical values of the time-lag τ are related to the other parameters c, D an b by τ j c,± = cos −1 (−kǫ/c) + 2jπ) /ω ± , j = 0, 1, 2 . . . if −ω 2 ± + 1/ǫ cω ± /ǫ ≥ 0 If (12) is not satisfied then τ j c,± = − cos −1 (−kǫ/c) + (2j + 2)π /ω ± , j = 0, 1, 2 . . . It can be shown by direct substitution that dℜλ dτ τ =τ c,+ > 0, dℜλ dτ τ =τ c,+ < 0,(14) so that on the bifurcation curves τ j c,+ or τ j c,+ one unstable direction is created or destroyed. Together with the stability properties for τ = 0 the bifurcation curves (11), (13) and (10) completely solve the problem of stability of the stationary state. It can be shown by rather lengthy calculations that the bifurcations for τ j c,± are the Hopf supercritical or subcritical bifurcations of the DDDE (7). Bifurcation curves τ j c,± (c) for fixed D, b = 1.05 and τ j c,± (D) for fixed c, b = 1.05 are illustrated in figure 1 for different values of D ( fig. 1a,b,c) and c ( fig. 1d,e,f). The value b = 1.05 renders the stationary state X 0 , Y 0 stable and excitable when τ = 0 and D = 0. The predictions given by the bifurcation values (11) and (13) of the system (7) are check versus the numerical solutions of the exact system (1). To check the approximate predictions of the bifurcations of stability for the noisy system, i.e. when D = 0, a proper notion of stochastic bifurcations would be necessary [18]. Instead we use the sample paths of the SDDE (1) with large N and for D = 0 to illustrate that these paths remain in the vicinity of the stationary solution if the approximate system's stationary state is stable, or near a periodic solution when the state of the approximate system is unstable. Figure 2 presents enlarged parts of bifurcation diagrams in figure 1, where particular values of the parameters that correspond either to stable or to unstable stationary state of (7) are indicated. These parameter values are replaced in the original system (1) with large N and particular sample paths of (1) with these parameter values are computed numerically. Time series of the global variable X(t) along such sample paths are shown in figure 3, for the system (1) with N = 95. There is nothing special with N = 95 and the same qualitative behavior of X(t), Y (t) is obtained for any moderately large N. It is clear that when the bifurcation diagrams of the approximate system (7) predict that the stationary solution is stable ( like in the cases: b,d,f,h) the sample paths of the exact system display small stochastic fluctuations around the stationary state. On the other hand, when the stationary state of (7) is unstable, as shown in the bifurcation diagrams, the sample paths of the exact system displays coherent oscillations with large amplitude, indicating that the exact system has stochastically stable periodic solution. The quantitative agrement between the domains of stability in the parameters (D, c, τ ) space of the approximate system (7) and the exact system (1) is indeed quite remarkable. It should be expected that such an agrement should be observed for small values of D since this is one of the conditions which guaranties that the local random variables have Gaussian distribution which is one of the asumptions in the derivation of the mean field equations. It is interesting to observe the domains of the time-lag where the bifurcation diagrams in fig. 1 and fig. 2 predict that the non-zero time-lag induces stabilization of the stationary state. This secession of oscillations due to the specific non-zero interval of the time-lag values is correctly predicted for the global variables of the exact system. It should be stressed that the agrement in the predictions of the approximate system and the large exact system goes only as far as the parameter domains of stability are considered. It should be expected that the predictions of the parameters stability domains based on (7) should well approximate the parameters stability domains of the exact system for small values of D since this is one of the basic assumption in the derivation of the mean field equations. Also, interaction strength c should be relatively small in order for the mean field assumption to be valid for moderately large (but finite) N. However, this domain of small values of c includes interesting bifurcations predicted by (7) and occurring in (1). On the other hand, large values of τ induce unstable stationary state and stable oscillatory behavior in both systems (7) and (1) so formally there is no restriction on the time-lag τ . We should make clear that it should not be expected that the values of X and Y for the deterministic approximate system (7) should reproduce stochastic orbits X(t), Y (t) for the large exact system or their ensemble averages. The correspondence between the orbits of the two systems for the same values of the parameters in different domains is only qualitative in the sense that they share the same types of attractors. Summary We have studied validity of the mean field approximation for the treatment of stability and bifurcations of the stationary state of a large collection of FitzHugh-Nagumo excitable neurons with noise and all-to-all coupling with delays, modelled by N ≫ 1 stochastic delay-differential equations. Standard assumptions of the mean field approach are used to derive the system of only two deterministic delay-differential equations. The stability and bifurcations of the stationary state of the approximate system can be studied analytically. The bifurcation curves of the approximate system give relevant information about the global variables of the exact large system. For zero and sufficiently small noise there is remarkable quantitative agreement of the parameters bifurcation values. On the other hand, it should not be expected that the approximation gives applicable results when the noise is to large, primarily because the assumption about the Gaussian distribution of values of the dynamical variables is not valid for large noise. Using the approximate system it is predicted, and confirmed by direct numerical simulations on the large exact system, that the time-lag in a nonzero interval can stabilize the global variables onto the stationary values even when for zero time-lag the global variables perform large oscillations. This is reminiscent of the phenomenon of the oscillation's death due to the timedelay, although in this case the relevant dynamics is that of the averaged global variables and not that of the individual neurons. We have derived the mean field approximation for the delayed coupled noisy system using the example of FitzHugh-Nagumo neurons in the excitable regime. It is expected that the approximations are equally valid for noisy delayed coupled type I excitable systems like the Terman-Wang neurons, or for bursting neurons like the Hidmarch-Rose model. Also the approximation should be applicable under the same assumptions for neurons interacting by delayed chemical rather then electrical coupling. Acknowledgements This work is partly supported by the Serbian Ministry of Science contract No. 141003. We should like to acknowledge useful comments of the two referees. Appendix The system of equations (1) and (2) can be written in the form: ǫdx i = (x − x 3 /3 − y + I)dt + c(< x(t − τ ) > −x i )dt dy i = (x + b)dt + √ 2DdW i where: < x(t − τ ) >= 1 N i (x i (t − τ ) The bracket < x > is always used to denote the average over the N units of the local variable x i , which is, by the mean field assumption, for large N approximately equal to the average over the assumed Gauss distribution of the corresponding local variable x i . Next we introduce deviations from the mean field: n x i (t) =< x(t) > −x i (t), n y i (t) =< y(t) > −y i (t). Deviations will always appear averaged over N i.e. in the form of < n x > and < n y > so that the index i is in fact redundant. Correlations between centered moments are defined as s x (t) =< n 2 x >, s y (t) =< n 2 y >, u(t) =< n x n y > Our goal is to derive the equations governing the evolution of the averages: X =< x >, Y =< y >, s x , s y , u. Due to the mean field assumption, these averages can be computed as averages over the stochastic distributions of the local quantities, which are by assumption Gaussian. The equation for the derivatives of X =< x >, Y =< y >, s x , s y , u will contain averages of monomials in local variables of various orders. In order to handel these we shall need to use the formulas for the cumulant expansions up the fourth order the local quantities. The general formulas for the cumulant expansion can be found for example in [20]. Due to the assumed Gaussian distribution the third and the fourth order cumulants (and all of the higher order) are equal to zero, which will be used to express averages of monomial in local variables that appear in the evolution equations. Using the cumulant formulas one computes the following expressions which will be used to obtain the evolution equations: From the cumulant << x 2 y >>= 0 follows < x 2 i y i >= Y s x + Y X 2 + 2Xu. From the cumulant << x 3 y >>= 0 follows < x 3 i y i >= 3s x u + 3X 2 u + Y X 3 + 3XY s x . Similarly one obtains: < x 2 i > = s x + X 2 , < x 3 i > = X 3 + 3Xs x , < x 4 i > = X 4 + 6X 2 s x + 3s 2 x , < x i y i > = U + XY. These expressions provide the necessary ingredients to obtain the equations (6). Takeing the average of the equations forẋ andẏ gives the first two equations of the system (6). Next consider the equation forṡ x . s x = 2 < X(t)Ẋ(t) − X(t)ẋ i (t) − x i (t)Ẋ(t) + x i (t)ẋ i (t) > = −2 X(t) ǫ [X(t) − X(t) 3 /3 − X(t)s x (t) − Y (t) + c(X(t − τ ) − X(t)] + 2 ǫ < x i (t) 2 − x i (t) 4 /3 − x i (t)y i (t) + cx i (t)X(t − τ ) − cx i (t) 2 > = 2 ǫ [−X 2 (t)s x (t) + s x (t) − s 2 x (t) − u(t) − cs x (y)](16) which is the third equation (6). In the last equality we used the expressions obtained from the cumulant formulas. Equation forṡ y is obtained as follows: s y = d < Y (t) 2 − 2Y (t)y i (t) + y i (t) 2 > /dt = −2Y (t)Ẏ (t) − d < y i (t) > /dt. Using the Ito chain rule this becomes: − 2Y (t)[X(t) + b]+ < 2y ( t)dy i (t)/dt + 2D > = −2Y (t)X(t) − 2Y (t)b+ < 2y i (t)x i (t) + 2y i (t)b + 2y i (t) √ 2DdW i + 2D > = −2Y (t)X(t) − 2Y (t)b + 2u(t) + 2X(t)Y (t) + 2Y (t)b + 2D = 2u(t) + 2D,(17) which is the forth equation (6). Similar calculations result in theu equation (6). u(t) = d < X(t)Y (t) − X(t)y i (t) − Y (t)x i (t) + x i (t)y i (t) > /dt = −X(t)Ẋ(t) − Y (t)Ẋ(t)+ < y i (t)ẋ i (t) > + <ẏ i (t)x i (t) >= . . . = 1 ǫ u(t)[1 − X 2 (t) − s x (t) − c] − 1 ǫ s y (t) + s x (t).(18) Figure 1 : 1Bifurcation curves (τ j c,± , c) (fig. a,b,c) for fixed D = 0 (a), D = 0.001 (b), D = 0.003 (c) and (τ j c,± , D) (fig. d,e,f) for fixed c = −0.05 (a) c = 0.05 (b) c = 0.1 . In all figures b = 1.05. Gray curves correspond to τ j c,− and black curves to τ j c,+ , for j = 0, 1, 2, 3, 4, 5. Figure 2 :Figure 3 : 23Enlarged parts of bifurcation diagrams presented in fig. 1a,c,e with parameter values, indicated by letters: a,b,c,d (fig. a),e,f (fig. c),g,h (fig. e), that are used for comparison with the exact system presented in fig. 3. Illustrates dynamics of the global variable X(t) for the exact system of N=95 units for parameter values corresponding to the stable or unstable state of the approximate system (7). 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[ "Characterization of Closed Vector Fields in Finsler Geometry", "Characterization of Closed Vector Fields in Finsler Geometry" ]	[ "Nabil L Youssef [email protected] \nDepartment of Mathematics\nFaculty of Science\nCairo University\nGizaEgypt\n" ]	[ "Department of Mathematics\nFaculty of Science\nCairo University\nGizaEgypt" ]	[]	The π-exterior derivative d, which is the Finslerian generalization of the (usual) exterior derivative d of Riemannian geometry, is defined. The notion of a d-closed vector field is introduced and investigated. Various characterizations of d-closed vector fields are established. Some results concerning d-closed vector fields in relation to certain special Finsler spaces are obtained. 1(κ being the horizontal scalar curvature).	null	[ "https://arxiv.org/pdf/0704.1986v1.pdf" ]	17,453,332	0704.1986	d7193ccaa3281695a0e3d2825b3e99a0d4ea7e88	Characterization of Closed Vector Fields in Finsler Geometry 16 Apr 2007 Nabil L Youssef [email protected] Department of Mathematics Faculty of Science Cairo University GizaEgypt Characterization of Closed Vector Fields in Finsler Geometry 16 Apr 2007Dedicated to the memory of Prof. Dr. A. Tamimπ-vector fieldπ-exterior derivatived-closed formd-closed vector fieldgradient π-vector fieldπ-distribution 2000 AMS Subject Classification 53C60 The π-exterior derivative d, which is the Finslerian generalization of the (usual) exterior derivative d of Riemannian geometry, is defined. The notion of a d-closed vector field is introduced and investigated. Various characterizations of d-closed vector fields are established. Some results concerning d-closed vector fields in relation to certain special Finsler spaces are obtained. 1(κ being the horizontal scalar curvature). Introduction In the present work, we adopt the pullback approach to Finsler geometry. In Finsler geometry, there is a canonical linear connection (corresponding to the Levi-Civita connection of Riemannian geometry), due to E. Cartan, which is not a connection on the manifold M but is a connection on π −1 (T M), the pullback of the tangent bundle T M by π : T M −→ M. The Cartan connection plays a key role in this work. We define the notion of π-exterior derivative d, which is a natural generalization to Finsler geometry of the (usual) exterior derivative d of Riemannian geometry. We then introduce and investigate an important class of π-vector fields on a Finslr manifold, which we refer to as d-closed vector fields. Various characterizations of such π-vector fields are established. Some results concerning d-closed vector fields in relation to certain special Finsler spaces are obtained. The notion of a π-distribution is also introduced and is related to d-closed vector fields. It should finally be noted that our investigation is entirely global or intrinsic (free from local coordinates). The idea of this work is due to Prof. A. Tamim, whom we miss profoundly. Notation and Preliminaries In this section, we give a brief account of the basic concepts of the pullback formalism of Finsler geometry necessary for this work. For more details refer to [1], [2] and [8]. We make the general assumption that all geometric objects we consider are of class C ∞ . The following notations will be used throughout the present paper: M: a real differentiable manifold of finite dimension n and of class C ∞ , Elements of X(π(M)) will be called π-vector fields and will be denoted by barred letters X. Tensor fields on π −1 (T M) will be called π-tensor fields. The fundamental π-vector field is the π-vector field η defined by η(u) = (u, u) for all u ∈ T M. The lift to π −1 (T M) of a vector field X on M is the π-vector field X defined by X(u) = (u, X(π(u))). The lift to π −1 (T M) of a 1-form ω on M is the π-form ω defined by ω(u) = (u, ω(π(u))). We have the following short exact sequence of vector bundles, relating the tangent bundle T (T M) and the pullback bundle π −1 (T M): 0 −→ π −1 (T M) γ −→ T (T M) ρ −→ π −1 (T M) −→ 0, where the bundle morphisms ρ and γ are defined respectively by ρ = (π T M , dπ) and γ(u, v) = j u (v), where j u is the natural isomorphism j u : T π M (v) M −→ T u (T π M (v) M). The vector 1-form J on T M defined by J = γoρ is called the natural almost tangent structure of T M and the vertical vector field C on T M defined by C := γoη is called the fundamental or the canonical (Liouville) vector field [4]. Let ∇ be a linear connection (or simply a connection) in the pullback bundle π −1 (T M). We associate to ∇ the map K : T T M −→ π −1 (T M) : X −→ ∇ X η,T u (T M) = V u (T M) ⊕ H u (T M) ∀u ∈ T M. If M is endowed with a regular connection, then the vector bundle maps γ : π −1 (T M) −→ V (T M), ρ\| H(T M ) : H(T M) −→ π −1 (T M), K\| V (T M ) : V (T M) −→ π −1 (T M) are vector bundle isomorphisms. Let us denote β = (ρ\| H(T M ) ) −1 , then ρoβ = id π −1 (T M ) , βoρ = id H(T M ) on H(TM) 0 on V(TM) (1.1) The classical torsion tensor T of a regular connection ∇ is defined by T(X, Y ) = ∇ X ρY − ∇ Y ρX − ρ[X, Y ] ∀ X, Y ∈ X(T M). The horizontal ((h)h-) and mixed ((h)hv-) torsion tensors, denoted by Q and T respectively, are defined by Q(X, Y ) = T(βXβY ), T (X, Y ) = T(γX, βY ) ∀ X, Y ∈ X(π(M)). The classical curvature tensor K of the connection ∇ is defined by K(X, Y )ρZ = −∇ X ∇ Y ρZ + ∇ Y ∇ X ρZ + ∇ [X,Y ] ρZ ∀ X, Y, Z ∈ X(T M). The horizontal (h-), mixed (hv-) and vertical (v-) curvature tensors, denoted by R, P and S respectively, are defined by R(X, Y )Z = K(βXβY )Z, P (X, Y )Z = K(βX, γY )Z, S(X, Y )Z = K(γX, γY )Z. The contracted curvature tensors, denoted by R, P and S respectively, are also known as the (v)h-, (v)hv-and (v)v-torsion tensors and are defined by R(X, Y ) = R(X, Y )η, P (X, Y ) = P (X, Y )η, S(X, Y ) = S(X, Y )η. Now, let (M, L) be a Finsler manifold, where L is the Lagrangian defining the Finsler structure on M. Let g be the Finsler metric in π −1 (T M) defined by L. Theorem 1.1. [8] Let (M, L) be a Finsler manifold. There exists a unique regular connection ∇ in π −1 (T M) such that (a) ∇ is metric : ∇g = 0, (b) The horizontal torsion of ∇ vanishes : Q = 0, (c) The mixed torsion T of ∇ satisfies g(T (X, Y ), Z) = g(T (X, Z), Y ). Such a connection is called the Cartan connection associated to the Finsler manifold (M, L). One can show that the torsion of the Cartan connection has the property that T (X, η) = 0 for all X ∈ X(π(M)). For the Cartan connection, we have Koγ = id π −1 (T M ) , γoK = id V (T M ) on V(TM) 0 on H(TM) (1.2) Then, from (1.1) and (1.2), we get βoρ + γoK = id T (T M ) (1.3) Hence, if we set h:= βoρ and v:= γoK, then every vector field X ∈X(T M) can be represented uniquely in the form X = hX + vX = βρX + γKX (1.4) The maps h and v are the horizontal and vertical projectors associated with the Cartan connection ∇: h 2 = h, v 2 = v, h + v = id X(T M ) , voh = hov = 0.Ric h (X, Y ) := T r{Z −→ R(X, Z)Y }, for all X, Y ∈ X(T M). -The horizontal Ricci map Ric h 0 is defined by g(Ric h 0 (X), Y ) := Ric h (X, Y ), for all X, Y ∈ X(T M). -The horizontal scalar curvature Sc h is defined by Sc h := T r(Ric h 0 ). We terminate this section by some concepts and results concerning the notion of a nonlinear connection in the sense of Klein-Grifone [3], [4]: It should be noted that the horizontal and vertical projectors of the Cartan connection and the barthel connection coincide. Also, the canonical spray G = βoη, and G is thus horizontal. π-Exterior derivative and d-closed vector field In this section, we introduce the notion of π-exterior differentiation operator d, which is the Finslerian version of the (usual) exterior differentiation operator d of Riemannian geometry. We then investigate the d-closed π-vector fields in Finsler geometry. Let (M, L) be a Finsler manifold. Let g be the Finsler metric defined by the Lagrangian L and ∇ be the Cartan connection associated with (M, L). We start with the following lemma which is useful for subsequent use. Lemma 2.1. For all X, Y ∈ X(π(M)) and X, Y ∈ X(T M), we have : (a) R(X, Y ) = K[βX, βY ], (b) R(X, Y ) = −γ R(ρX, ρY ), R being the curvature tensor of the Barthel connection. Proof. (a) One can easily show that [βX, βY ] = γ(R(X, Y )η) + β(∇ βX Y − ∇ βY X). (2.1) Then, (a) follows directly from (2.1) by applying the operator K on both sides, noting that K • γ = id π −1 (T M ) and K • β = 0. Let ω be a π-form of order p ≥ 0. For p > 0, we define (dω)(X 1 , ..., X p+1 ) := p+1 i=1 (−1) i+1 βX i · ω(X 1 , ..., X i , ..., X p+1 ) + i<j (−1) i+j ω(ρ[βX i , βX j ], X 1 , ..., X i , ..., X j , ..., X p+1 ) (2.3) For p = 0, we set (dω)(X) := βX · ω, that is, dω = dω o β. (2.4) (Here, the symbol " " means that the corresponding argument is omitted.) In particular, for a (1)π-form ω, we have (dω)(X, Y ) = βX · ω(Y ) − βY · ω(X) − ω(ρ[βX, βY ]).(2. 5) The operator d will be called the π-exterior derivative. Definition 2.4. A (p)π-form ω is said to be: -d-closed if dω = 0. -d-exact if ω = dα, for some (p − 1)π-form α. It should be noted that a d-exact π-form is not necessarily d-closed, contrary to the case of the (ordinary) exterior derivative d ( [5] and [7]). This is due to the fact that the property d 2 = 0 dose not hold for the π-exterior derivative d. Definition 2.5. Let X ∈ X(π(M)) be a π-vector field and let ω ∈ X * (π(M)) be the associated (1)π-form under the duality defined by the Finsler metric g : ω = i X g. The π-vector field X is said to be d-closed (resp. d-exact) if its associated π-form ω is d-closed (resp. d-exact). The following result gives a simple characterization of d-closed π-vector fields. Theorem 2.6. A π-vector field X is d-closed if and only if the operator A X is self-adjoint. Proof. Let ω be the π-form associated to the π-vector field X under the duality defined by the Finsler metric g. By (2.5), we have (dω)(Y , Z) = βY · ω(Z) − βZ · ω(Y ) − ω(ρ[βY , βZ]) = βY · g(X, Z) − βZ · g(X, Y ) − g(X, ρ[βY , βZ]) = g(A X Y , Z) + g(X, ∇ βY Z) − g(A X Z, Y ) − g(X, ∇ βZ Y ) −g(X, ρ[βY , βZ]) = g(A X Y , Z) − g(A X Z, Y ) + g(X, Q(Y , Z)). As the horizontal torsion tensor Q of the Cartan connection ∇ vanishes, the result follows. Definition 2.7. The gradient of a function f ∈ F(T M) is the π-vector field X defined by : i X g = df = df oβ. (2.6) The gradient of the function f is denoted by grad f . A π-vector field X is said to be a gradient π-vector field if it is the gradient of some function f ∈ F(T M) : X = grad f . In Riemannian geometry, it is well known that ( [5], [7], [12]) the gradient of any function f ∈ F(M) is a closed vector field or, equivalently, A X is self-adjoint. This result is not in general true in Finsler geometry. This is again due to the fact that d 2 = 0. Nevertheless, we have (c) The horizontal distribution is completely integrable. (d) The π-exterior derivative d has the property that d 2 = 0. Proof. (a) ⇐⇒ (b) : Let f ∈ F(T M) be any arbitrary function and let X := grad f . For all Y , Z ∈ X(π(M)) and for all X = grad f , we have g(A X Y , Z) − g(A X Z, Y ) = g(∇ βY X, Z) − g(∇ βZ X, Y ) = βY · g(X, Z) − g(X, ∇ βY Z) −βZ · g(X, Y ) + g(X, ∇ βZ Y ) = βY · ((df oβ)Z) − βZ · ((df oβ)Y ) −(df oβ)(∇ βY Z − ∇ βZ Y ) = βY · (βZ · f ) − βZ · (βY · f ) − β(∇ βY Z − ∇ βZ Y ) · f = ([βY , βZ] − β(∇ βY Z − ∇ βZ Y )) · f In view of (2.1), the last equation takes the form g(A X Y , Z) − g(A X Z, Y ) = γ(R(Y , Z)η) · f = γ( R(Y , Z)) · f ∀f ∈ F(T M) (2.7) Now, the required equivalence follows from (2.7), taking into account Theorem 2.6, Lemma 2.1 and the fact that γ is a monomorphism. Proof. Let X := grad f . As X is d-closed, then by Theorem 2.6 and (2.7), we have γ (R(Y , Z)η) · f = 0. But since (M, L) is of scaler curvature, then γ ((ω ∧ φ)(Y , Z)) · f = 0. Setting Z = η in the above equation, noting that φ(η) = 0, we get ω(η)(γ φ(Y )) · f = 0. As ω(η) = 1 3 L 2 (C · κ + 3κ) = 0, it follows that (γ φ(Y )) · f = 0. Consequently, by the definition of φ, we get γY · f − L −1 ℓ(Y ) C · f = 0, (2.9) from which, since f is homogeneous of degree r, df o γ − rf L −1 ℓ = 0, or df f o γ − r dL L o γ = 0. This equation is equivalent to d(log(f L −r )) o γ = 0. Hence, f L −r is independent of the directional arguments and, consequently, there exists a function h ∈ F(M) so that f = h(x)L r . In fact, this follows from (2.9 ), since in this case C · f = 0. Let U be an open subset of T M. An assignment D : u ∈ U −→ D u ⊂ P −1 (u) = {u} × T π(u) M, such that every D u is an m-dimensional vector subspace of P −1 (u), is called an m-dimensional π-distribution on U. If X is a π-vector field on U with X(u) ∈ D u for every u ∈ U, we say that X belongs to D and we write X ∈ D. For a given regular connection on π −1 (T M), an m-dimensional π-distribution D on U is said to be hinvolutive if for every π-vector fields X and Y , ρ[βX, βY ] belongs to D whenever X and Y belong to D. is called the π-distribution generated by (or associated with) X. Theorem 2.14. If X ∈ X(π(M)) is d-closed, then the π-distribution D generated by X is h-involutive. Proof. Suppose that Y and Z belong to D. As the h-torsion tensor of the Cartan connection ∇ vanishes, then g(ρ[βY , βZ], X) = g(∇ βY Z − ∇ βZ Y , X) = βY · g(Z, X) − g(Z, ∇ βY X) − βZ · g(Y , X) + g(Y , ∇ βZ X). Since g(X, Y ) = 0 = g(X, Z), it follows that g(ρ[βY , βZ], X) = g(A X Z, Y ) − g(Z, A X Y ). Then, by Theorem 2.6, D is h-involutive. The following result characterizes certain d-closed π-vector fields. Proposition 2.15. Let X be a π vector field such that i X dg = 0. Then, X is d-closed if and only if βX is an isometry. The result follows directly from the identity L βX g = i X dg + di X g, where L X is the Lie derivative with respect to X ∈ X(T M) [9]. A generalized Randers manifold [11] is a Finsler manifold (M, L * ) whose Finsler structure L * is given by L * = L + α, where L is a Finsler structure on M and α = g(b, η); b being the π-vector field defined in terms of a given 1-form δ on M by δ(X) = g(b, X) ∀ X ∈ X(M), where X is the π-life of X. The change L −→ L * = L + α is called a generalized Randers change. The next result gives a characterization of d-closeness of a remarkable π-vector field m associated with Randers changes. Proof. The relation between the Finsler metrics g and g * associated with the Finsler structures L and L * is given by [11] : g * = τ (g − ℓ ⊗ ℓ) + ℓ * ⊗ ℓ * , where τ = L * L −1 , ℓ = dL o γ and ℓ * = ℓ + dα o γ. Now, if dα = 0, then ℓ * = ℓ and consequently g * = τ g + (1 − τ )ℓ ⊗ ℓ. Then i m g * = τ i m g + (1 − τ )ℓ(m)ℓ. As ℓ(m) = 0, as one can easily show, then the above relation reduces to i m g * = i τ m g, from which the result follows. Two Finsler structure L andL on a manifold M are said to be conformal [14] if L = e σ(x) L, for some function σ(x) on M. The transformation L −→L = e σ(x) L is called a conformal transformation (or a conformal change). The relation between the Finsler metrics g andg associated with L andL respectively is given by :g = e 2σ(x) g. Let X be a π-vector field, then i Xg = e 2σ(x) i X g, and so di Xg = e 2σ(x) di X g + 2e 2σ(x) (d o β)σ ⊗ i X g = e 2σ(x) di X g + 2e 2σ(x) ∂σ ∂x k dx k ⊗ i X g. Consequently, di Xg = e 2σ(x) di X g if and only if ∂σ ∂x k = 0, or equivalently, if and only if σ(x) is a constant function (provided that M is connected). This proves the following result, which gives a characterization of homotheties in terms of d-closed π-vector fields. Theorem 2.17. A d-closed π-vector field remain d-closed under a conformal transformation if and only if this transformation is a homothety. F(M): the R-algebra of differentiable functions on M, X(M): the F(M)-module of vector fields on M, π M : T M −→ M: the tangent bundle of M, π : T M −→ M: the subbundle of nonzero vectors tangent to M, V (T M): the vertical subbundle of the bundle T T M, P : π −1 (T M) −→ T M : the pullback of the tangent bundle T M by π, P * : π −1 (T * M) −→ T M : the pullback of the cotangent bundle T * M by π, X(π(M)): the F(T M)-module of differentiable sections of π −1 (T M), X * (π(M)): the F(T M)-module of differentiable sections of π −1 (T * M), i X : interior product with respect to X ∈ X(M), df : the exterior derivative of f ∈ F(M), d L := [i L , d], i L being the interior derivative with respect to the vector form L. called the connection (or the deflection) map of ∇. A tangent vector X ∈ T u (T M) is said to be horizontal if K(X) = 0 . The vector space H u (T M) = {X ∈ T u (T M) : K(X) = 0} of the horizontal vectors at u ∈ T M is called the horizontal space to M at u . The connection ∇ is said to be regular if Definition 1.3. A nonlinear connection on M is a vector 1-form Γ on T M, C ∞ on T M, C o on T M, such that JΓ = J, ΓJ = −J. The horizontal and vertical projectors h and v associated with Γ are defined by h := 1 2 (I + Γ), v := 1 2 (I − Γ). Thus Γ gives rise to the decomposition T T M = H(T M) ⊕ V (T M), where H(T M) := Im h = Ker v, V (T M) := Im v = Ker h. The torsion T of a nonlinear connection Γ is the vector 2-form on T M defined by T := 1 2 [J, Γ]. The curvature of a nonlinear connection Γ is the vector 2-form ℜ on T M defined by ℜ := − 1 2 [h, h]. Proposition 1.4. Let (M, L) be a Finsler manifold. The vector field G on T M determined by i G Ω = −dE is a spray, called the canonical spray associated with the energy E, where E := 1 2 L 2 and Ω := dd J E. Theorem 1.5. On a Finsler manifold (M, L), there exists a unique conservative nonlinear connection (d h E = 0) with zero torsion. It is given by : Γ = [J, G], where G is the canonical spray. Such a connection is called the Barthel connection or the Cartan nonlinear connection associated with (M, L). ( b ) bSetting X = ρX and Y = ρY in (a), we get γR(ρX, ρY ) = γK[βρX, βρY ] = v[hX, hY ]. Then, (b) follows directly from the identity [13]: R(X, Y ) = −v[hX, hY ].Definition 2.2. For any given π-vector field X, the F(T M)-linear operator A X on X(π(M)) is defined by :A X (Y ) := ∇ βY X. Theorem 2. 8 . 8The following assertions are equivalent :(a) The gradient of any function f ∈ F(T M) is d-closed, or, equivalently, gradient π-vector fields are d-closed. (b) The curvature tensor R of the Barthel connection vanishes. (b) ⇐⇒ (c) : This equivalence follows immediately from the identity[13]:R(X, Y ) = −v[hX, hY ]. ( c ) c⇐⇒ (d) : For all f ∈ F(T M) and X, Y ∈ X(π(M)), we have(d 2 f )(X, Y ) = βX ·(βY ·f )−βY ·(βX ·f )−β(ρ[βX, βY ])·f = (I −β o ρ)·[βX, βY ]·f. Hence, we have (d 2 f )(X, Y ) = v[βX, βY ] · f ∀ f ∈ F(T M).The above equation, shows that the horizontal distribution is completely integrable if and only if d 2 = 0 on F(T M). This proves the implication (d) =⇒ (c). For the proof of the converse implication, refer to [9]. Definition 2.9. A Finsler manifold (M, L) is said to be of scaler curvature κ if the (v)h-torsion tensor R is written in the form : Corollary 2 . 12 . 212Under the hypothesis of Theorem 2.11, if the function f ∈ F(T M) is homogeneous of degree zero in the directional arguments, then f is a function of positional arguments only, that is f ∈ F(M). Definition 2 . 13 . 213Let X be a given π-vector field on an open subset U of T M. The (n − 1)-dimensional π-distribution D : u ∈ U −→ D u := {Y ∈ P −1 (u) : g(X, Y ) = 0} Proposition 2 . 16 . 216Let L * = L + α be a generalized Randers change with closed α. The π-vector field m := b − L −2 α η is d-closed in the Finsler manifold (M, L * ) if and only if τ m is d-closed in the Finsler manifold (M, L), where τ = L * L −1 . Definition 1.2. [10] With respect to the Cartan connection ∇, we have: -The horizontal Ricci tensor Ric h is defined by This paper was presented in " The International Conference on Finsler Extensions of Relativity Theory " held at Cairo, Egypt, November 4-10, 2006. Les espaces de Finsler et certaines de leurs généralisations. H Akbar-Zadeh, Ann. Ec. Norm. Sup., Série. 3H. AKbar-Zadeh, Les espaces de Finsler et certaines de leurs généralisations, Ann. Ec. Norm. Sup., Série 3, 80 (1963), 1-79. P Dazord, Propriétés globales des géodésiques des espaces de Finsler. 575) Publ. Dept. Math. LyonThèse d'EtatP. Dazord, Propriétés globales des géodésiques des espaces de Finsler, Thèse d'Etat, (575) Publ. Dept. Math. Lyon, 1969. Structure présque-tangente et connexions, I. J Grifone, Ann. Inst. Fourier. 221J. Grifone, Structure présque-tangente et connexions, I, Ann. Inst. Fourier, Grenoble, 22,(1) (1972), 287-334. Formes extérieures génératrices de sprays. J Klein, A Voutier, Ann. Inst. Fourier, Grenoble. 181J. Klein and A. Voutier, Formes extérieures génératrices de sprays, Ann. Inst. Fourier, Grenoble, 18(1) (1968), 241-260. Foundations of differential geometry, I, Interscience. S Kobayashi, K Nomizu, New YorkS. Kobayashi and K. Nomizu, Foundations of differential geometry, I, Inter- science, New York, 1963. Scaler and gradient vector fields of Finsler spaces and holonomy groups of nonlinear connections. M Matsumoto, L Tamassy, Demonstratio Math. 132M. Matsumoto and L. Tamassy, Scaler and gradient vector fields of Finsler spaces and holonomy groups of nonlinear connections , Demonstratio Math., 13(2) (1980), 551-564. . Y Matsushima, Differentiable Manifolds, Marcel Dekker IncNew YorkY. Matsushima, Differentiable Manifolds, Marcel Dekker Inc., New York, 1972. General theory of Finsler spaces with applications to Randers spaces. A A Tamim, Cairo UniversityPh. D. ThesisA. A. Tamim, General theory of Finsler spaces with applications to Randers spaces, Ph. D. Thesis, Cairo University, 1991. Fundamental differential operators in Finsler geomtry. Proc. Math. Phys. Soc. Egypt. 73, Fundamental differential operators in Finsler geomtry, Proc. Math. Phys. Soc. Egypt, 73 (1998), 67-93. Special Finsler manifolds. J. Egypt. Math. Soc. 102, Special Finsler manifolds, J. Egypt. Math. Soc., 10(2) (2002), 149-177. On generalised Randers manifolds, Algebras Groups and Geometries. A A Tamim, N L Youssef, 16A. A. Tamim and N. L. Youssef, On generalised Randers manifolds, Algebras Groups and Geometries, 16 (1999), 115-126. Integral formulas in Riemannian geometry. K Yano, Marcel Dekker IncNew YorkK. Yano, Integral formulas in Riemannian geometry, Marcel Dekker Inc., New York, 1970. Sur les tenseurs de coubure de la connexion de Berwald et ses distributions de nullité. N L Youssef, Tensor, N. S.36N. L. Youssef, Sur les tenseurs de coubure de la connexion de Berwald et ses distributions de nullité , Tensor, N. S., 36 (1982), 275-280. A global theory of conformal Finsler geometry. N L Youssef, S H Abed, A Soleiman, math. DG/0610052N. L. Youssef, S. H. Abed, and A. Soleiman, A global theory of conformal Finsler geometry, Submitted. ArXiv No.: math. DG/0610052.	[]
[ "Computing Volumes of Adjacency Polytopes via Draconian Sequences", "Computing Volumes of Adjacency Polytopes via Draconian Sequences" ]	[ "Robert Davis [email protected] \nDepartment of Mathematics\nDepartment of Mathematics Auburn University -Montgomery Montgomery\nColgate University\nHamilton, New York, AlabamaU.S.A., U.S.A\n", "Tianran Chen [email protected] \nDepartment of Mathematics\nDepartment of Mathematics Auburn University -Montgomery Montgomery\nColgate University\nHamilton, New York, AlabamaU.S.A., U.S.A\n" ]	[ "Department of Mathematics\nDepartment of Mathematics Auburn University -Montgomery Montgomery\nColgate University\nHamilton, New York, AlabamaU.S.A., U.S.A", "Department of Mathematics\nDepartment of Mathematics Auburn University -Montgomery Montgomery\nColgate University\nHamilton, New York, AlabamaU.S.A., U.S.A" ]	[ "the electronic journal of combinatorics" ]	Adjacency polytopes appear naturally in the study of nonlinear emergent phenomena in complex networks. The "PQ-type" adjacency polytope, denoted ∇ PQ G and which is the focus of this work, encodes rich combinatorial information about power-flow solutions in sparse power networks that are studied in electric engineering. Of particular importance is the normalized volume of such an adjacency polytope, which provides an upper bound on the number of distinct power-flow solutions.In this article we show that the problem of computing normalized volumes for ∇ PQ G can be rephrased as counting D(G)-draconian sequences where D(G) is a certain bipartite graph associated to the network. We prove recurrences for all networks with connectivity at most 1 and, for 2-connected graphs under certain restrictions, we give recurrences for subdividing an edge and taking the join of an edge with a new vertex. Together, these recurrences imply a simple, non-recursive formula for the normalized volume of ∇ PQ G when G is part of a large class of outerplanar graphs; we conjecture that the formula holds for all outerplanar graphs. Explicit formulas for several other (non-outerplanar) classes are given. Further, we identify several important classes of graphs G which are planar but not outerplanar that are worth additional study.	10.37236/9768	[ "https://arxiv.org/pdf/2007.11051v4.pdf" ]	220,686,910	2007.11051	65d468b2a679407ef079d14da88fc999c8f7b1fc	Computing Volumes of Adjacency Polytopes via Draconian Sequences 2020 Robert Davis [email protected] Department of Mathematics Department of Mathematics Auburn University -Montgomery Montgomery Colgate University Hamilton, New York, AlabamaU.S.A., U.S.A Tianran Chen [email protected] Department of Mathematics Department of Mathematics Auburn University -Montgomery Montgomery Colgate University Hamilton, New York, AlabamaU.S.A., U.S.A Computing Volumes of Adjacency Polytopes via Draconian Sequences the electronic journal of combinatorics 270202010.37236/abcdSubmitted: August 7, 2020; Accepted: March 11, 2022; Published: TBDarXiv:2007.11051v4 [math.CO]. Released under the CC BY-ND license (International 4.0). Adjacency polytopes appear naturally in the study of nonlinear emergent phenomena in complex networks. The "PQ-type" adjacency polytope, denoted ∇ PQ G and which is the focus of this work, encodes rich combinatorial information about power-flow solutions in sparse power networks that are studied in electric engineering. Of particular importance is the normalized volume of such an adjacency polytope, which provides an upper bound on the number of distinct power-flow solutions.In this article we show that the problem of computing normalized volumes for ∇ PQ G can be rephrased as counting D(G)-draconian sequences where D(G) is a certain bipartite graph associated to the network. We prove recurrences for all networks with connectivity at most 1 and, for 2-connected graphs under certain restrictions, we give recurrences for subdividing an edge and taking the join of an edge with a new vertex. Together, these recurrences imply a simple, non-recursive formula for the normalized volume of ∇ PQ G when G is part of a large class of outerplanar graphs; we conjecture that the formula holds for all outerplanar graphs. Explicit formulas for several other (non-outerplanar) classes are given. Further, we identify several important classes of graphs G which are planar but not outerplanar that are worth additional study. Introduction and background Let G = (V (G), E(G)) be a simple graph on [N] = {1, . . . , N}. We use e 1 , . . . , e N to denote the standard basis vectors of R N . The PQ-type adjacency polytope of G is defined to be ∇ PQ G = conv{(e i , e j ) ∈ R 2N \| ij ∈ E(G) or i = j} where conv(S) denotes the convex hull of elements of S. Its normalized volume, defined by NVol(∇ PQ G ) = dim(∇ PQ G )! vol(∇ PQ G ) where vol(P ) is the relative volume of P , is always a positive integer. The study of PQ-type adjacency polytopes was introduced in [4], motivated by the engineering problem known as power-flow study (or load-flow study). This study models the balance of electric power on a network of power generation or delivery "buses". Of particular importance are the alternating current (AC) variations, which produce nonlinear equations that are notoriously difficult to analyze. In the AC model for a power network with buses labeled as 1, . . . , N, the voltage on each bus is expressed as a complex variable v i = x i + iy i whose absolute value represents the voltage magnitude and whose argument encodes the phase of the AC experienced on the bus. The interaction among buses is modeled by a graph G whose nodes represent the buses and whose edges represent the junctions. Kirchhoff's circuit laws give rise to an idealized balancing condition for the power injected, power generated, and power consumed on each bus, which can be expressed as the system of nonlinear equations S i = N j=1 Y ij v i v j for i = 2, . . . , N,(1) where S i = P i + iQ i is a complex representation of the real and reactive power, Y ij , known as nodal admittance, describes the connection between the i and j buses, and Y ij and v j denote the complex conjugate of Y ij and v j respectively. By dropping the conjugate constraints between v i and v i , we obtained the algebraic version of this system, known as the algebraic power-flow equations. It was shown that the maximum number of nontrivial complex solutions this system has is bounded by the normalized volume of ∇ PQ G . We take care to call the adjacency polytopes within this paper PQ-type, since a related construction is sometimes called an adjacency polytope; see, for example, [2,6,7]. This alternate construction, motivated by counting equilibrium solutions to a network of interconnected oscillators, relies on a particular change of variables that is not available here. In engineering terms, this alternate construction arises from PV-type buses. In this article we show that the normalized volume of ∇ PQ G can be described in terms of sequences of nonnegative integers related to the Dragon Marriage Problem: a variant of Hall's Matching Theorem that has far-reaching applications and spawned the study of generalized permutohedra [13,14]. We establish this relationship in Section 2 and show how it can be immediately exploited to compute normalized volumes of some PQ-type adjacency polytopes when G is nontrivial. We explore this connection more deeply in Section 3 where we establish several recurrences. Namely, we provide recurrences for all graphs with connectivity at most 1, that is, any graph that is disconnected or has a cut-vertex. These directly imply a simple formula for NVol(∇ PQ G ) whenever G is a forest. Sections 3.1 and 3.2 consider two operations on a graph: subdivision of an edge e and replacing e with the join of e and a new vertex. Under certain conditions, these operations lead to the following two recurrences that are stated simply but nontrivial to prove. NVol(∇ PQ G△e ) = 3 NVol(∇ PQ G ). Section 3 concludes by applying the recurrences to establish a closed, non-recursive formula for NVol(∇ PQ G ) for a large class of outerplanar graphs; we conjecture that this formula holds for all outerplanar graphs. The final section addresses several classes of graphs which are planar but not outerplanar. First, we give results for a complete bipartite graph where one partite set has just two elements. Then we consider the classes of wheel graphs and series-parallel graphs, which are natural points of further study and will likely require a refinement of the techniques within this article or alternate techniques altogether. an edge of G with endpoints u and v, we will write e = uv or e = vu whenever possible. When additional clarity is helpful we may alternately write e = {u, v} or e = {v, u}. If X ⊆ V (G), then we use G − X to denote the graph obtained from deleting the vertices of X as well as any edge that is incident to some vertex in X. If X = {v}, then we will just write G − v. Similarly, if S is a set of edges, then we use G \ S to denote the graph with the edges in S deleted; if S = {e}, then we just write G − e. If X ⊆ V (G), then we use G[X] to denote the subgraph of G induced by X. Lastly, if H is a graph, then we use G ∨ H to denote the join of G and H, that is, the graph with vertex set V (G) ∪ V (H) and edge set In [14], Postnikov investigated the Dragon Marriage Problem, providing a generalization of Hall's Matching Theorem for bipartite graphs. In the Dragon Marriage Problem, a small medieval village is home to n grooms and n + 1 brides, some pairs of whom would form compatible marriages. Suppose we know all pairs of compatible grooms and brides. One day, a dragon arrives in the village and kidnaps a bride. What compatibility conditions among the original set of grooms and brides will guarantee that those who remain can still be entirely paired by compatible marriages? In graph-theoretic terms, and more generally, consider an X, Y -bigraph G such that \|Y \| = \|X\| + 1. What are necessary and sufficient conditions on G so that G − y has a perfect matching regardless of choice of y ∈ Y ? The answer relies on the following. Definition 1. Let G ⊆ K N,N . Call (a 1 , . . . , a N ) ∈ Z N 0 a G-draconian sequence if a i = N − 1 and, for any 1 i 1 < i 2 < · · · < i k N, It turns out that we can describe ∇ PQ G as a root polytope for an appropriate choice of graph. As an example, let G be the graph on [4] with edges 12, 23, 34, 24. Then D(G) is the bipartite graph with vertices {1, 2, 3, 4, 1, 2, 3, 4} and edges 11,12,21,22,23,24,32,33,34,42,43, and 44. See Figure 1 for an illustration. E(G) ∪ E(H) ∪ {uv \| u ∈ V (G), v ∈ V (H)}.a i 1 + · · · + a i k < k j=1 N G (i j ) .(2) Identifying e i in R N with −e N +i in R 2N is a unimodular equivalence; thus, we have the following simple but important result. Lemma 4. For all G, ∇ PQ G is unimodularly equivalent to Q D(G) . We now list two more theorems from [14]. In the first, denotes the Minkowski sum of polytopes and, given S ⊆ [N], ∆ S = conv{e i \| i ∈ S}. Also, for a graph G on [N], set P D(G) = N i=1 ∆ N G (i)∪{i} ⊆ R N . It is also written to reflect our particular context and does not quite capture the full strength of the original statement. These two theorems are the last pieces needed to prove the main result of this section: Theorem 8. P − D(G) = x ∈ R N \| x + ∆ [N ] ⊆ P D(G) . Then NVol(Q D(G) ) = \|P − D(G) ∩ Z N \|. As written, Theorem 5 relies on D(G) being connected. Fortunately, the connectedness of G is equivalent to the connectedness of D(G). We will use this fact occasionally so we present it as a lemma, although its proof is straightforward enough that we omit it. Since we are primarily working with D(G) rather than G directly, we let D(G) denote the set of D(G)-draconian sequences. Proof. Lemma 6 assures us that D(G) is connected. By Lemma 4, we know NVol(∇ PQ G ) = NVol(Q D(G) ). Applying Theorem 5 and Theorem 7 completes the proof. To illustrate, let G be the graph on [4] with edges 12, 23, and 24. Here, we have It will be very helpful for us to explicitly state when a sequence is D(G)-draconian. The main difference is recognizing that for every vertex i of G, deg D(G) (i) = 1 + deg G (i). N D(G) (1) = {1, 2}, N D(G) (2) = {1, 2, 3, 4}, N D(G) (3) = {2,Definition (Definition 1, rephrased). Let G be a graph on [N]. Call (a 1 , . . . , a N ) ∈ Z N 0 a D(G)-draconian sequence if a i = N − 1 and, for any 1 i 1 < · · · < i k N, a i 1 + · · · + a i k < k j=1 N D(G) (i j ) = {i 1 , . . . , i k } ∪ k j=1 N G (i j ) This translates our computation of normalized volume to a purely combinatorial computation. The following simple observation will also be helpful at several points when proving the results in Section 3. Remark 9. The normalized volume of ∇ PQ G is invariant under permutation of vertices. We now give a first nontrivial application of Theorem 8 to an infinite class of graphs. Proposition 10. Let N > 2 and let M be any matching of size k in K N . Then NVol(∇ PQ K N \M ) = 2(N − 1) N − 1 − 2k. Proof. Note that since N > 2, K N \ M is connected. First consider k = 0. The D(K N )draconian sequences are the weak compositions of N − 1 into N parts, of which there are 2(N −1) N −1 . When k > 0, the deletion of each edge uv in M prohibits two compositions: those whose entries are all 0 except for one, which is N − 1 and located at position u or v. Proposition 10 refers to a very specific class of graphs. The next section proves results that allow for much more flexibility. Draconian recurrences One of the main purposes of this article is to establish several recurrences for NVol(∇ PQ G ), using what we collectively call draconian recurrences. Certain specific recurrences will be given their own names as we encounter them. For a simple first situation we consider the disjoint union of two graphs G and H, which we denote G + H. Since Theorem 8 only applies to connected graphs, we study their adjacency polytopes directly. If P ⊆ R n and Q ⊆ R m are polytopes, each containing the origins 0 n , 0 m respectively, then their free sum is P ⊕ Q = conv{(P × 0 m ) ∪ (0 n × Q)} ⊆ R n+m . When P and Q are lattice polytopes, there is a convenient product formula we may invoke. Theorem 11 ([5, Theorem 2]). Given full-dimensional convex polytopes P ⊆ R n and Q ⊆ R m , if both P and Q contain the origin of their respective ambient spaces, then NVol(P ⊕ Q) = NVol(P ) NVol(Q). While Theorem 11 insists that P and Q are full-dimensional, we may replace them with unimodularly equivalent polytopes P ′ ⊆ R n ′ ∼ = aff(P ) and Q ′ ⊆ R m ′ ∼ = aff(P ). Since unimodular equivalence preserves normalized volume, the conclusion of Theorem 11 remains true. This gives us the last piece we need to prove the following. Proposition 12. If G and H are any two graphs, then NVol(∇ PQ G+H ) = NVol(∇ PQ G ) NVol(∇ PQ H ). Proof. Let \|V (G)\| = M and \|V (H)\| = N. First consider when M = 1 or N = 1. Without loss of generality we may assume N = 1 and that the vertices of G + H are labeled so that the isolated vertex from H is labeled M + 1. This means, in particular, that ∇ PQ H consists of a single point, hence NVol(∇ PQ H ) = 1. Let A be the matrix whose columns are the vertices of ∇ PQ G . Partition A as A = A 1 A 2 where A 1 consists of the first M rows of A and A 2 consists of the last M rows of A. The matrix of vertices of ∇ PQ G+H can then be written as B =     A 1 0 M ×1 0 1×ℓ 1 A 2 0 M ×1 0 1×ℓ 1     Since f only permutes coordinates it is a unimodular transformation. Moreover, the projection of f (∇ PQ G+H ) obtained from dropping the first, (M + 1)th, (2M + 1)th, and (2M + N)th coordinates is a lattice-preserving transformation sending ∇ PQ G+H onto P ⊕ Q. Therefore, NVol(∇ PQ G+H ) = NVol(f (∇ PQ G+H )) = NVol(P ⊕ Q) = NVol(P ) NVol(Q) = NVol(∇ PQ G ) NVol(∇ PQ H ), proving the result. In light of Proposition 12, we will focus for the rest of this section on graphs that are connected unless explicitly stated otherwise. Restricting to when G is connected allows us to use Theorem 8 and therefore we study the sets D(G) directly rather than relying on properties of their polytopes. Recall that a graph G is k-connected if for any set X of vertices, \|X\| < k, the subgraph G − X is connected. A cut-vertex (respectively, cut-edge) of G is a vertex (respectively, edge) whose deletion from G increases the number of components. A block of a graph G is an inclusion-maximal connected subgraph of G with no cut-vertex. Note that a block of a simple graph G may be an isolated vertex, a cut-edge, or an inclusion-maximal 2-connected subgraph of G. Theorem 13. Suppose G is a connected graph with cut-vertex v and B is a block con- taining v. Setting B ′ = G[(V (G) \ V (B)) ∪ {v}] we have NVol(∇ PQ G ) = NVol(∇ PQ B ) NVol(∇ PQ B ′ ). Proof. By Remark 9 we may assume without loss of generality that the cut-vertex is 1, c ′ i = N − M. Thus, the sum of entries in f (c, c ′ ) is N − 1, one of the requirements for being D(G)-draconian. Now pick any sequence 1 i 1 < · · · < i k N. If i k < M or M < i 1 , then the corresponding D(G)draconian inequality automatically holds. So, suppose there is some positive 1 ℓ < k for which i 1 < · · · < i ℓ M < i ℓ+1 < · · · < i k . If 1 < i 1 , then d i 1 + · · · + d i j = c i 1 + · · · + c i ℓ + c ′ i ℓ +1 + · · · + c ′ k < ℓ j=1 N D(B) (i j ) + k j=ℓ+1 N D(B ′ ) (i j ) − 1. Since B and B ′ share just a single vertex, we have that ℓ j=1 N D(B) (i j ) + k j=ℓ+1 N D(B ′ ) (i j ) − 1 k j=1 N D(G) (i j ) . Chaining these inequalities together, the D(G)-draconian inequality holds. A similar argument holds if 1 = i 1 , only here we explicitly write d i 1 = c 1 + c ′ 1 and proceed as before. In both cases the D(G)-draconian inequality holds, therefore f (c, c ′ ) ∈ D(G). Showing that f is injective is brief and straightforward, so we omit the details. What requires slightly more work is showing that f is surjective. Let d = (d 1 , . . . , d N ) ∈ D(G). We claim that d = f (c, c ′ ) where c = M − 1 − M i=2 d i , d 2 , . . . , d M and c ′ = N − M − N j=M +1 d j , d M +1 , . . . , d N and c ∈ D(B), c ′ ∈ D(B ′ ) . For notational convenience, we set c 1 = M − 1 − M i=2 d i and c ′ 1 = N − M − N j=M +1 d j . Since it is clear that d = f (c, c ′ ), the majority of the work will be in showing that c ∈ D(B) and c ′ ∈ D(B ′ ). The procedure is analogous for both, so we will only give the details for showing c ∈ D(B). By construction, the sum of entries in c is M − 1. Every inequality of the form d i 1 + · · · + d i k < k j=1 N D(B) (i j )(3) with 1 < i 1 < · · · < i k M instantly holds since the neighbors of 2, . . . , M are the same in D(G) and D(B). It is also clear that 0 c 1 since, otherwise, d 2 + · · · + d M > M − 1, which directly contradicts (3). Now consider a sum of a subsequence of c of the form c 1 + d i 1 + · · · + d i k . By way of contradiction, suppose that this does not satisfy the corresponding D(B)draconian inequality, that is, c 1 + d i 1 + · · · + d i k N D(B) (1) ∪ k j=1 N D(B) (i j ) . the electronic journal of combinatorics 27 (2020), #P00 Since 1 < i 1 < i k M, this inequality may be rewritten c 1 + d i 1 + · · · + d i k N D(B) (1) ∪ k j=1 N D(G) (i j ) .(4) We also now know that c ′ 1 + d M +1 + d M +2 + · · · + d N = N − M.(5) Adding the corresponding sides of (4) and (5) and remembering that c 1 + c ′ 1 = d 1 results in d 1 + k j=1 d i j + N r=M +1 d r N D(B) (1) ∪ k j=1 N D(G) (i j ) + N − M. Using the fact that B ′ contains N − M + 1 vertices, d 1 + k j=1 d i j + N r=M +1 d r N D(B) (1) ∪ k j=1 N D(G) (i j ) + N r=M +1 N D(G) (r) − 1. Combining the first two summands on the right side counts the vertex 1 twice, resulting in The next result follows quickly from induction and the recurrences proven thus far. d 1 + k j=1 d i j + N r=M +1 d r N D(B) (1) ∪ k j=1 N D(G) (i j ) ∪ N r=M +1 N D(G) (r) . Corollary 14. If F is a forest on N vertices with k connected components, then we have NVol(∇ PQ F ) = 2 N −k . Interestingly, Corollary 14 implies that any two trees with the same number of edges will produce adjacency polytopes with the same normalized volume. This does not happen for connected graphs in general: as we will show in Example 21, NVol(∇ PQ C 3 ) = 6, which is not the volume obtained from a path with three edges. Moreover, even though two trees with the same number of vertices produce adjacency polytopes with equal normalized volumes, the polytopes themselves are not combinatorially equivalent. Recall that the f -vector of a polytope P is the vector (f −1 , f 0 , . . . , f dim P ) where f i is the number of i-dimensional faces of P , using the convention f −1 = 1. Example 15. Let G 1 and G 2 be graphs on [4]. Let E(G 1 ) = {12, 23, 34} and E(G 2 ) = {12, 13, 14}. One may verify the that the f -vector of ∇ PQ G 1 is (1, 10, 39, 77, 82, 46, 12, 1) and the f -vector of ∇ PQ G 2 is (1, 10, 39, 78, 86, 51, 14, 1). Thus the two polytopes are not combinatorially equivalent even though Theorem 8 guarantees that their normalized volumes are both 8. Through the recurrences established so far, we may reduce our work to considering only 2-connected graphs. The subdivision recurrence Given e ∈ E(G) let G : e denote the graph obtained by subdividing e. Since we are using the convention V (G) = [N], we will always assume that V (G : e) = [N + 1]. The main result of this subsection is Theorem 20, which gives a recurrence for NVol(∇ PQ G:e ) under certain conditions. Establishing the recurrence requires multiple lemmas that have similar flavors but are distinct enough to warrant presenting their proofs. The next three lemmas describe how to produce D(G : e)-draconian sequences from D(G)-draconian sequences and D(G \ e)-draconian sequences. We use the notation A ⊎ B to denote the disjoint union of the sets A and B. N D(G:e) (N − 1) = N D(G) (N − 1) \ {N } ⊎ {N + 1} and N D(G:e) (N) = N D(G) (N) \ {N − 1} ⊎ {N + 1}. Pick a sequence 1 i 1 < · · · < i k N + 1. There are two cases to consider: 1. {N − 1, N, N + 1} ⊆ k j=1 N D(G:e) (i j ) and 2. {N − 1, N, N + 1} ⊆ k j=1 N D(G:e) (i j ). In the first case, we can deduce two things: that i k = N + 1 and that if N − 1 is one of the indices i 1 , . . . , i k , then no other neighbor of N in G is one of the indices i 1 , . . . , i k (and vice versa). Therefore, k j=1 N D(G) (i j ) = k j=1 N D(G:e) (i j ) the electronic journal of combinatorics 27 (2020), #P00 and c i 1 + · · · + c i k < k j=1 N D(G) (i j ) = k j=1 N D(G:e) (i j ) . In the second case, if i k < N + 1, we immediately get c i 1 + · · · + c i k < k j=1 N D(G) (i j ) < k j=1 N D(G:e) (i j ) . Otherwise, i k = N + 1 and k j=1 N D(G:e) (i j ) {N + 1} ⊎ k−1 j=1 N D(G) (i j ) = k−1 j=1 N D(G) (i j ) + 1. This time, we get c i 1 + · · · + c i k = c i 1 + · · · + c i k−1 + 1 < k−1 j=1 N D(G) (i j ) + 1 k j=1 N D(G:e) (i j ) . Since each case results in satisfying the D(G : e)-draconian inequalities, we have shown that α(c) ∈ D(G : e). Lemma 17. Let G be a 2-connected graph and let e = uv be any edge. If c ∈ D(G \ e), then β(c) ∈ D(G : e) where β(c) = α(c) + e u − e N +1 . Moreover, β is an injection. Proof. Arguing that β is injective is routine, so its details are omitted. For what remains, by Remark 9 we may assume that e = {N − 1, N}. We then want to show that, if c = (c 1 , . . . , c N ) ∈ D(G \ e), then β(c) = (c 1 , . . . , c N −2 , c N −1 + 1, c N , 0) ∈ D(G : e). Note that, by Menger's theorem and the fact that G is 2-connected, c exists since G \ e is connected. Set β(c) = (β 1 , . . . , β N +1 ). Let 1 i 1 < · · · < i k N + 1 and set ℓ = k if i k < N + 1 and ℓ = k − 1 if i k = N + 1. If N − 1 = i j for any j, then β i 1 + · · · + β i k = c i 1 + · · · + c i ℓ < ℓ j=1 N D(G\e) (i j ) k j=1 N D(G:e) (i j ) . Otherwise, N − 1 = i j for some j. In this case, ℓ j=1 N D(G\e) (i j ) = ℓ j=1 N D(G:e) (i j ) − 1 k j=1 N D(G:e) (i j ) − 1. the electronic journal of combinatorics 27 (2020), #P00 Together we have β i 1 + · · · + β i k = c i 1 + · · · + c i ℓ + 1 < ℓ j=1 N D(G\e) (i j ) + 1 k j=1 N D(G:e) (i j ) , and the D(G : e)-draconian inequality holds. Therefore β(c) ∈ D(G : e). Lemma 18. Let G be a 2-connected graph with an edge e = uv such that deg G (u) = 2 and the neighbors of u are neighbors of each other. If c ∈ D(G), then γ(c) ∈ D(G : e) where γ(c) is formed by the following rule. Set γ ′ (c) = α(c) − e u + e N +1 . 1. If c / ∈ D(G \ e), then (a) if γ ′ (c) ∈ D(G : e), then set γ(c) = γ ′ (c). (b) If γ ′ (c) / ∈ D(G : e), then set γ(c) = α(c) + e u − e N +1 . 2. If c ∈ D(G \ e), then (a) if γ ′ (c) ∈ D(G : e), then set γ(c) = γ ′ (c). (b) If γ ′ (c) / ∈ D(G : e), then set γ(c) = α(c) + e v − e N +1 . Additionally, γ is an injection. Proof. As usual, Remark 9 allows us to assume e = {N −1, N} and deg G (N −1) = 2. This allows us to more specifically rewrite γ as follows: set γ ′ (c) = (c 1 , . . . , c N −2 , c N −1 −1, c N , 2). 1. If c / ∈ D(G \ e), then (a) if γ ′ (c) ∈ D(G : e), then set γ(c) = γ ′ (c). (b) If γ ′ (c) / ∈ D(G : e), then set γ(c) = (c 1 , . . . , c N −2 , c N −1 + 1, c N , 0). 2. If c ∈ D(G \ e), then (a) if γ ′ (c) ∈ D(G : e), then set γ(c) = γ ′ (c). (b) If γ ′ (c) / ∈ D(G : e), then set γ(c) = (c 1 , . . . , c N −1 , c N + 1, 0). Throughout the proof we will use the notation γ ′ (c) = (γ ′ 1 , . . . , γ ′ N +1 ) and γ(c) = (γ 1 , . . . , γ N +1 ). First suppose c / ∈ D(G \ e) and γ ′ (c) / ∈ D(G : e), so that γ(c) = (c 1 , . . . , c N −2 , c N −1 + 1, c N , 0) . This places us in case 1(b). Let S be the set of all c ∈ D(G) satisfying these conditions. To show that γ(c) is D(G : e)-draconian we first show that c N −1 1. Partition S into disjoint subsets S 0 = {c ∈ S \| c N −1 = 0} and S > = {c ∈ S \| c N −1 > 0}. If c ∈ S 0 , then clearly c satisfies c N −1 1. For c ∈ S > , since c N −1 > 0 we know γ ′ i 0 for all i, hence there must be some sequence 1 i 1 < · · · < i k N + 1 for which γ ′ (c) violates the corresponding D(G : e)-draconian inequality. In fact, such a sequence cannot contain both N − 1 and N + 1, since, otherwise, the proof of Lemma 16 implies γ ′ i 1 + · · · + γ ′ i k = c i 1 + · · · + c i k−1 + 1 < k j=1 N D(G:e) (i j ) and therefore the D(G : e)-draconian inequality corresponding to i 1 < · · · < i k is satisfied. Similarly, if i k < N + 1, then γ ′ i 1 + · · · + γ ′ i k c i 1 + · · · + c i k , and the corresponding D(G : e)-draconian inequality is satisfied again due to the proof of Lemma 16. Hence, any violation of a D(G : e)-draconian inequality by γ ′ (c) with c ∈ S > requires i k = N + 1 and i j = N − 1 for any j < k. Since deg G (N − 1) = 2, and since we are assuming c N −1 > 0, we only need to show that c N −1 = 2. If it were possible that c N −1 = 2, then for any sequence 1 i 1 < · · · < i k−1 < i k = N + 1 not containing N − 1, we would have γ ′ i 1 + · · · + γ ′ i k = c i 1 + · · · + c i k−1 + 2 = c i 1 + · · · + c i k−1 + c N −1 < k−1 j=1 N D(G) (i j ) ∪ N D(G) (N − 1) . Notice that the set T = k−1 j=1 N D(G) (i j ) ∪ N D(G) (N − 1) contains both N and N − 1. On the other hand, γ ′ i 1 + · · · + γ ′ i k < \|T \| k j=1 N D(G:e) (i j ) . Since we have verified that all other D(G : e)-draconian inequalities hold for γ ′ (c), this would imply γ ′ (c) is a D(G : e)-draconian sequence, which is a contradiction. Therefore c N −1 = 1. The remainder of our argument in establishing 1(b) applies to all elements of S. Consider a sum of the form γ i 1 + · · · + γ i k with 1 i 1 < · · · < i k N + 1. If i k < N − 1, then γ i j = c i j and N D(G) (i j ) = N D(G:e) (i j ) for each j = 1, . . . , k, so that γ i 1 + · · · + γ i k = c i 1 + · · · + c i k < k j=1 N D(G) (i j ) = k j=1 N D(G:e) (i j ) . If i k = N − 1, then there are two subcases. First, if N ∈ ∪ k−1 j=1 N D(G) (i j ), then k j=1 N D(G:e) (i j ) = k j=1 N D(G) (i j ) ⊎ {N + 1}, hence γ i 1 + · · · + γ i k = c i 1 + · · · + c i k + 1 < k j=1 N D(G) (i j ) + {N + 1} = k j=1 N D(G:e) (i j ) . On the other hand, suppose N / ∈ ∪ k−1 j=1 N D(G) (i j ). Without loss of generality, we assume that the other neighbor of N − 1 in G is N − 2. Since the neighbors of N − 1 in G are neighbors of each other, i k−1 < N − 2, hence N − 1 / ∈ ∪ k−1 j=1 N D(G) (i j ) as well. Since we now know c N −1 1, we have γ i 1 + · · · + γ i k = c i 1 + · · · + c i k−1 + c N −1 + 1 c i 1 + · · · + c i k−1 + 2 < k−1 j=1 N D(G) (i j ) + {N − 1, N + 1} k j=1 N D(G:e) (i j ) . This completes the case for i k = N − 1. If i k = N and i k−1 < N − 1, then γ i 1 + · · · + γ i k = c i 1 + · · · + c i k < k j=1 N D(G) (i j ) k j=1 N D(G:e) (i j ) , where the second inequality follows from recognizing that while N − 1 appears in the first union it may not appear in the second, and that N + 1 appears in the second union but does not appear in the first. If i k = N and i k−1 = N − 1, then γ i 1 + · · · + γ i k = c i 1 + · · · + c i k + 1 < k j=1 N D(G) (i j ) + {N + 1} = k j=1 N D(G:e) (i j ) . the electronic journal of combinatorics 27 (2020), #P00 Finally, if i k = N + 1, then by the proof of Lemma 16, γ i 1 + · · · + γ i k c i 1 + · · · + c i k−1 + 1 < k j=1 N D(G:e) (i j ) . Therefore, γ(c) is D(G : e)-draconian when in case 1(b). Now consider case 2(b), so that c ∈ D(G \ e) and γ ′ (c) / ∈ D(G : e). Proving that γ(c) = (c 1 , . . . , c N −1 , c N +1, 0) ∈ D(G:e) follows the proof of Lemma 17 almost identically, replacing N − 1 with N. For this reason, we omit the details. To show that γ is an injection, we can restrict to comparing the sequences of 1(a) with those of 2(a) and the sequences of 1(b) with those of 2(b). Fortunately, it is straightforward to see that no sequence can arise simultaneously as γ(c) under the conditions of 1(a) and γ(c ′ ) under the conditions of 2(a). If this were possible, we would obtain c = c ′ , but c cannot simultaneously be a member of and absent from D(G \ e). (b). Let γ(d) = (γ ′′ 1 , . . . , γ ′′ N +1 ). Since γ(d) falls under the conditions of 2(b), we know d ∈ D(G \ e). In G \ e, the vertex N − 1 has degree 1, hence γ ′′ N −1 ∈ {0, 1}. We cannot have γ ′′ N −1 = 0 since this would imply 0 = γ ′′ N −1 = γ N −1 = c N −1 + 1, that is, we would have c N −1 = −1, which contradicts c ∈ D(G). If γ ′′ N −1 = 1, then we claim γ ′ (d) ∈ D(G : e) meaning γ(d) would not be obtained from case 2(b). For ease of reference, we collect notation in terms of γ ′′ 1 , . . . , γ ′′ N +1 needed to complete the argument: d = (γ ′′ 1 , . . . , γ ′′ N −2 , 1, γ ′′ N − 1), c = (γ ′′ 1 , . . . , γ ′′ N −2 , 0, γ ′′ N ), γ(d) = (γ ′′ 1 , . . . , γ ′′ N −2 , 1, γ ′′ N , 0), γ ′ (d) = (γ ′′ 1 , . . . , γ ′′ N −2 , 0, γ ′′ N − 1, 2). Since d ∈ D(G \ e), all of the D(G : e)-draconian inequalities involving the first N coordinates of γ ′ (d) are immediately satisfied. Consider, then, a sum involving the last coordinate of γ ′ (d). Note that d ∈ D(G \ e) ⊆ D(G). If the sum involves the Nth and the (N − 1)th coordinates of γ ′ (d) as well, then, since we know that α(d) ∈ D(G : e), we may write γ ′′ i 1 + · · · + γ ′′ i k−3 + 0 + γ ′′ N − 1 + 2 = γ ′′ i 1 + · · · + γ ′′ i k−3 + 1 + (γ ′′ N − 1) + 1 < k−3 j=1 N D(G:e) (i j ) ∪ N +1 j=N −1 N D(G:e) (j) . If the sum involves the Nth coordinate of γ ′ (d) but not the (N − 1)th coordinate, then, the electronic journal of combinatorics 27 (2020), #P00 this time, α(c) ∈ D(G : e) gives us γ ′′ i 1 + · · · + γ ′′ i k−2 + γ ′′ N − 1 + 2 = γ ′′ i 1 + · · · + γ ′′ i k−2 + γ ′′ N + 1 < k−2 j=1 N D(G:e) (i j ) ∪ N D(G:e) (N) ∪ N D(G:e) (N + 1) . A similar argument holds for when the sum involves the (N − 1)th coordinate of γ ′ (d) but not the Nth coordinate. Next, suppose the sum is of the form γ ′′ i 1 + · · · + γ ′′ i k−2 + γ ′′ N −2 + 2 = γ ′′ i 1 + · · · + γ ′′ i k−2 + γ ′′ N −2 + 1 + 1. Since α(d) ∈ D(G : e), we can say γ ′′ i 1 + · · · + γ ′′ i k−2 + γ ′′ N −2 + 1 + 1 < k−2 j=1 N D(G:e) (i j ) ∪   j∈{N −2,N −1,N +1} N D(G:e) (j)   . Note that N D(G:e) (N − 1) can be dropped from this union because N D(G:e) (N − 1) ⊆ N D(G:e) (N − 2) ∪ N D(G:e) (N + 1). As a result, the desired inequality holds. Finally, suppose the sum is of the form γ ′′ i 1 + · · · + γ ′′ i k−1 + 2 where i k−1 < N − 2. Since d ∈ D(G), we know γ ′′ i 1 + · · · + γ ′′ i k−1 + 2 < k−1 j=1 N D(G) (i j ) + 2. Since i k−1 < N − 2, we know neither N − 1 nor N + 1, which are elements of N D(G:e) (N + 1), are in the above union. Therefore, γ ′′ i 1 + · · · + γ ′′ i k−1 + 2 < k−1 j=1 N D(G) (i j ) + 2 k−1 j=1 N D(G:e) (i j ) ∪ N D(G:e) (N + 1) . The above completes the argument that all D(G:e)-draconian inequalities are satisfied by γ ′ (d), i.e., γ ′ (d) ∈ D(G : e). This contradicts the assumption that γ(d) was obtained from case 2(b), so γ ′′ N −1 = 1. Both possible values of γ ′′ N −1 lead to a contradiction, meaning no sequence obtained from case 1(b) can be obtained from case 2(b) and vice versa. Therefore, γ is injective. (c) = γ(c ′ ) implies α(c) + e N −1 − e N +1 = α(c ′ ) + e N −1 − e N +1 and α is injective. However, this causes a contradiction, as Lemma 17 requires c ∈ D(G\e) while condition 1 of Lemma 18 requires c ′ = c / ∈ D(G \ e). Hence we may assume c ∈ D(G \ e) and γ(c ′ ) is constructed via condition 2(b) of Lemma 18. Since both c, c ′ ∈ D(G \ e), we know c N −1 , c ′ N −1 1. By the definitions of β and γ, we make several observations: • c i = c ′ i for all i N − 2; • γ(c ′ ) N −1 = c N −1 + 1, hence c ′ N −1 = 1; and • β(c) N = c ′ N + 1. We claim that, in fact, γ ′ (c ′ ) = (γ ′ 1 , . . . , γ ′ N +1 ) ∈ D(G : e), contradicting that γ(c ′ ) was constructed via condition 2(b) of Lemma 18. Note that, based on our observations, γ ′ (c ′ ) = (c ′ 1 , . . . , c ′ N −2 , c ′ N −1 − 1, c ′ N , 2) = (c 1 , . . . , c N −2 , 0, c N − 1, 2). Consider a sum of the form γ ′ i 1 +· · ·+γ ′ i k with 1 i 1 < · · · < i k N +1. If i k N −2, then the neighbors of i j are the same in D(G) and D(G : e), hence γ ′ i 1 + · · · + γ ′ i k = c i 1 + · · · + c i k < k j=1 N D(G) (i j ) = k j=1 N D(G:e) (i j ) . If i k = N − 1, then γ ′ i 1 + · · · + γ ′ i k = c i 1 + · · · + c i k−1 + 0. As in the case when i k N − 2, the neighbors of i 1 , . . . , i k−1 are the same in D(G) and D(G : e), so that γ ′ i 1 + · · · + γ ′ i k < k−1 j=1 N D(G) (i j ) = k−1 j=1 N D(G:e) (i j ) k j=1 N D(G:e) (i j ) . If i k = N, then we have γ ′ i 1 + · · · + γ ′ i k = c i 1 + · · · + c i k−1 + c N − 1 < k j=1 N D(G) (i j ) − 1 < k j=1 N D(G:e) (i j ) . In the case i k = N + 1, we consider several subcases. If i k−1 = N, then γ ′ i 1 + · · · + γ ′ i k = c i 1 + · · · + c i k−1 − 1 + 2 < k j=1 N D(G:e) (i j ) , where the inequality holds by Lemma 16. If i k−1 = N − 1, then γ ′ i 1 + · · · + γ ′ i k = c ′ i 1 + · · · + c ′ i k−1 − 1 + 2 < k j=1 N D(G:e) (i j ) , where the inequality again holds by Lemma 16. If i k−1 = N − 2, then γ ′ i 1 + · · · + γ ′ i k = c ′ i 1 + · · · + c ′ i k−1 + c ′ N −1 − 1 + 2 < kγ ′ i 1 +· · ·+γ ′ i k c i 1 +· · ·+c i k−1 +2 < k j=1 N D(G) (i j ) + {N − 1, N + 1} k j=1 N D(G:e) (i j ) . With this, we have verified that all of the D(G : e)-draconian inequalities hold for γ ′ (c ′ ), giving us the desired contradiction. Therefore, B G (e) ∩ C G (e) = ∅, completing the proof. This result, together with the three lemmas preceding it, give A G (e) ⊎B G (e) ⊎C G (e) ⊆ D(G : e). It turns out that the reverse inclusion holds, establishing what we call the subdivision recurrence. Theorem 20 (Subdivision recurrence). Let G be a 2-connected graph with an edge e = uv such that deg G (u) = 2 and the neighbors of u are neighbors of each other. Then D(G:e) = A G (e) ⊎ B G (e) ⊎ C G (e) and, consequently, NVol(∇ PQ G:e ) = 2 NVol(∇ PQ G ) + NVol(∇ PQ G\e ). Proof. Again without loss of generality we may assume e = {N −1, N} and deg G (N −1) = 2. By Lemmas 16, 17, and 18, A G (e) ∪ B G (e) ∪ C G (e) ⊆ D(G : e). For the reverse inclusion, we will show that, given d = (d 1 , . . . , d N +1 ) ∈ D(G : e), one of the following conditions holds: 1. If d N +1 = 2, then (d 1 , . . . , d N −2 , d N −1 + 1, d N ) ∈ D(G). 2. If d N +1 = 1, then (d 1 , . . . , d N ) ∈ D(G). 3. If d N +1 = 0, then one of the following is true: If the second condition holds, then d ∈ A G (e); if condition 3(a) holds, then d ∈ B G (e); if any of the remaining conditions hold, then d ∈ C G (e). (a) (d 1 , . . . , d N −2 , d N −1 − 1, d N ) ∈ D(G \ e); (b) both (d 1 , . . . , d N −2 , d N −1 −2, d N , 2) / ∈ D(G:e) and (d 1 , . . . , d N −2 , d N −1 −1, d N ) ∈ D(G) \ D(G \ e); or First suppose d N +1 = 2 and let 1 i 1 < · · · < i k N. Set (c 1 , . . . , c N ) = (d 1 , . . . , d N −2 , d N −1 + 1, d N ). If i k < N − 1, then N D(G:e) (i j ) = N D(G) (i j ) for each j, so the corresponding draconian inequality c i 1 + · · · + c i k = d i 1 + · · · + d i k < k j=1 N D(G:e) (i j ) = k j=1 N D(G) (i j ) the electronic journal of combinatorics 27 (2020), #P00 holds. Otherwise, since d ∈ D(G : e), c i 1 + · · · + c i k d i 1 + · · · + d i k + 2 − 1 < k j=1 N D(G:e) (i j ) ∪ N D(G:e) (N + 1) − 1 = k j=1 N D(G) (i j ) ⊎ {N + 1} − 1 = k j=1 N D(G) (i j ) . Therefore, each D(G)-draconian inequality holds for (c 1 , . . . , c N ), establishing the first condition. Next suppose d N +1 = 1 and let 1 i 1 < · · · < i k N. If i k < N − 1, then the corresponding draconian inequality holds as in the case of d N +1 = 2. If i k N − 1, then we know from d ∈ D(G : e) that d i 1 + · · · + d i k + 1 < k j=1 N D(G:e) (i j ) ∪ N D(G:e) (N + 1) = k j=1 N D(G) (i j ) ⊎ {N + 1} = k j=1 N D(G) (i j ) + 1. Subtracting 1 from both sides establishes the corresponding D(G)-draconian inequality for (d 1 , . . . , d N ). Thus the second condition holds. Establishing the last condition, where d N +1 = 0, requires the most care. Since deg G (N − 1) = 2, we know that d N −1 ∈ {0, 1, 2} and we will treat each case separately. Suppose d N −1 = 0. Our aim will be to show that condition 3(c) holds. It is clear that (d 1 , . . . , d N −2 , d N −1 − 1, d N − 1, 2) / ∈ D(G : e) since d N −1 − 1 < 0. Now, if d N = 0 , then there is a contradiction, since this and the 2-connectivity of G imply N = d 1 + · · · + d N −2 < N −2 j=1 N D(G:e) (j) = N −2 j=1 N D(G) (j) = N. Thus, d N > 0. Set (c 1 , . . . , c N ) = (d 1 , . . . , d N −1 , d N − 1) and consider the sum c i 1 + · · · + c i k . If i k < N − 1, then the desired D(G)-draconian inequality holds using the same argument as for the previous conditions. If i k = N − 1, then c i 1 + · · · + c i k = d i 1 + · · · + d i k−1 < k−1 j=1 N D(G:e) (i j ) = k−1 j=1 N D(G\e) (i j ) k j=1 N D(G\e) (i j ) . the electronic journal of combinatorics 27 (2020), #P00 Lastly, if i k = N, then c i 1 + · · · + c i k = d i 1 + · · · + d i k − 1 < k j=1 N D(G:e) (i j ) − 1 = k j=1 N D(G\e) (i j ) ⊎ {N + 1} − 1 = k j=1 N D(G\e) (i j ) .d i 1 + · · · + d i k k j=1 N D(G\e) (i j )(6) with i k = N and i k−1 < N − 1. If N − 1 / ∈ k j=1 N D(G\e) (i j ), then add 2 to both sides of (6) to get Computational evidence suggests that the subdivision recurrence holds even without the condition that the neighbors of u are neighbors of each other. Indeed, Kohl [11] has verified that the recurrence holds for all graphs under this more relaxed condition having at most nine vertices. However, the subdivision recurrence does not necessarily hold if we allow both endpoints of e to have degree larger than 2 in G. For example, if G = K 1 ∨ P 3 , where P 3 is the path on three vertices, and e is the edge of G whose endpoints each have degree 3 in G, then one may show that NVol(∇ PQ G:e ) = 50 whereas 2 NVol(∇ PQ G ) + NVol(∇ PQ G\e ) = 2(18) + 16 = 52. d i 1 + · · · + d i k + 2 k j=1 N D(G\e) (i j ) + 2 = k j=1 N D(G\e) (i j ) + {N − 1, N + 1} = k j=1 N D(G:e) (i j ) , which would imply (d 1 , . . . , d N −2 , 0, d N , 2) / ∈ D(G : e). If N − 1 ∈ k j=1 N D(G\e) (i j ), One important class of 2-connected graphs that the subdivision recurrence does not directly cover is the class of cycles. The conclusion of the subdivision recurrence holds, but to prove so requires a modified proof. Corollary 22. For a cycle C N on N 3 vertices and any edge e of C N , NVol(∇ PQ C N+1 ) = 2 NVol(∇ PQ C N :e ) + NVol(∇ PQ C N \e ). Consequently, NVol(∇ PQ C N ) = N2 N −2 . Proof. We will prove this result by showing that the conclusion of the subdivision recurrence holds for C N using the same functions α, β, and γ as in Lemmas 16, 17, and 18, respectively. Notice that none of the proofs of Lemmas 16, 17, 19, or Theorem 20 rely on the neighbors of u being neighbors of each other. In Lemma 18, the only time in which this condition is invoked is in establishing γ(c) ∈ D(G : e) under case 1(b). Thus, by adapting the proof of case 1(b) to C N we will have immediately established NVol(∇ PQ C N+1 ) = 2 NVol(∇ PQ C N ) + NVol(∇ PQ C N \e ). The formula NVol(∇ PQ C N ) = N2 N −2 then follows from Corollary 14 and induction, whose details we omit. Without loss of generality we may assume that C N has vertex set 1, c N , 0). Consider a sum of the form γ i 1 +· · ·+γ i k . If i k < N −1, then N D(C N ) (i j ) = N D(C N :e) (i j ) for each j, hence γ i 1 + · · · + γ i k = c i 1 + · · · + c i k < k j=1 N D(C N ) (i j ) = k j=1 N D(C N :e) (i j ) . When i k = N − 1 is when we have the most work to do. To begin, we may assume that c i j > 0 for all j < k since, once the desired inequalities for these cases hold, if we were to check the inequality involving some i ℓ < N − 1 for which c i ℓ = 0, then we would instantly obtain γ i 1 + · · · + γ i k + γ i ℓ = γ i 1 + · · · + γ i k < k j=1 N D(C N :e) (i j ) k j=1 N D(C N :e) (i j ) ∪ N D(C N :e) (i ℓ ) . This allows us to assume for the rest of this case that c i j > 0 for all j = 1, . . . , k − 1. If i 1 = 1, then γ i 1 + · · · + γ i k = c i 1 + · · · + c i k + 1 < k j=1 N D(C N ) (i j ) + \|{N + 1}\| = k j=1 N D(C N :e) (i j ) . If i 1 > 1, then partition the set I = {i 1 , . . . , i k } into nonempty subsets S 1 , . . . , S r such that each S i consists of consecutive integers and is maximal with that property with respect to containment. Additionally, label the subsets so that min(S 1 ) < min(S 2 ) < · · · < min(S r ). Define t to be the largest index satisfying min(S t ) − max(S t−1 ) > 2, the electronic journal of combinatorics 27 (2020), #P00 or t = 1 if no such index exists. Note that we always have N − 1 ∈ S t ∪ S t+1 ∪ · · · ∪ S r . If t > 1, then let I <t = S 1 ∪ · · · ∪ S t−1 and I t = S t ∪ · · · ∪ S r . Because min(I t ) − max(I <t ) > 2, we know   i j ∈I<t N D(G) (i j )   ∩   i j ∈I t N D(G) (i j )   = ∅. Since c ∈ D(C N ), c i 1 + · · · + c i k < i j ∈I<t N D(C N ) (i j ) + c min(St) + · · · + c i k < i j ∈I<t N D(C N ) (i j ) + i j ∈I t N D(C N ) (i j ) = k j=1 N D(C N ) (i j ) . Because of the two strict inequalities here, we may therefore say γ i 1 + · · · + γ i k = c i 1 + · · · + c i k + 1 i j ∈I<t N D(C N ) (i j ) + c min(St) + · · · + c i k < i j ∈I<t N D(C N ) (i j ) + i j ∈I t N D(C N ) (i j ) = k j=1 N D(C N ) (i j ) . If t = 1, then there are two additional subcases to consider: when one of c 1 , c N > 0 and when c 1 = c N = 0. In both subcases, we will use the fact that for each ℓ = 1, . . . , r, i j ∈S ℓ c i j ∈ {\|S ℓ \|, \|S ℓ \| + 1}, that is, for a fixed ℓ, the values of c i j such that i j ∈ S ℓ are all 1 or 2 with at most one of them being 2. Indeed, if there were two 2s, then i j ∈S ℓ c i j \|S ℓ \| + 2 = i j ∈S ℓ N D(C N ) (i j ) , the electronic journal of combinatorics 27 (2020), #P00 contradicting c ∈ D(C N ). We will moreover use fact that the case t = 1 means k j=1 N D(C N ) (i j ) = k j=1 N D(C N :e) (i j ) = N − (i 1 − 2). Suppose that at least one of c 1 , c N is positive. This implies there is at least one ℓ ∈ [r] for which i j ∈S ℓ c i j = \|S ℓ \|. Therefore, γ i 1 + · · · + γ i k = c i 1 + · · · + c i k + 1 r i=1 \|S i \| + 1 − 1 + 1 < N − (i 1 − 2) = k j=1 N D(C N :e) (i j ) . If c 1 = c N = 0, then we claim that γ ′ (c ′ ) ∈ D(C N : e), contradicting that we are in case 1(b). Here, c N −1 > 0 since, otherwise, c 2 + · · · + c N −2 = N − 1 = N −2 i=2 N D(C N ) (i) , in which case c / ∈ D(C N ), a contradiction. Thus, the entries of γ ′ (c) are all nonnegative. Let γ ′ (c) = (γ ′ 1 , . . . , γ ′ N +1 ) and consider a sum γ ′ i 1 + · · · + γ ′ i k . If i k N, then γ ′ i 1 + · · · + γ ′ i k c i 1 + · · · + c i k < k j=1 N D(C N ) (i j ) k j=1 N D(C N :e) (i j ) . Otherwise, i k = N + 1. Since we are in the case c 1 = c N = 0, we may assume i 1 > 1 and i k−1 < N. if i k−1 = N − 1, then α(c) ∈ D(C N : e) gives us γ ′ i 1 + · · · + γ ′ i k = c i 1 + · · · + c i k−1 − 1 + 2 <γ ′ i 1 + · · · + γ ′ i k = c i 1 + · · · + c i k−1 + 2 < k j=1 N D(C N ) (i j ) + \|{N, N + 1}\| k j=1 N D(C N :e) (i j ) . Therefore, γ ′ (c) ∈ D(C N : e), contradicting the fact that γ(c) arises from case 1(b). This completes the case i k = N − 1. If i k = N, then consider i k−1 . If i k−1 < N − 1, then we obtain the desired inequality again from knowing α(c) ∈ D(C N : e). Otherwise i k−1 = N − 1, in which case γ i 1 + · · · + γ i k = c i 1 + · · · + c i k + 1 < k j=1 N D(C N ) (i j ) + \|{N + 1}\| = k j=1 N D(C N :e) (i j ) . Lastly, if i k = N + 1, then γ i k = 0, and by the previous cases we have γ i 1 + · · · + γ i k = γ i 1 + · · · + γ i k−1 < k−1 j=1 N D(C N :e) (i j ) k j=1 N D(C N :e) (i j ) . This final inequality establishes γ(c) ∈ D(C N : e) for the case 1(b), which completes our proof. We close this subsection with an invitation to the reader. Question 23. Under what conditions for a graph G and an edge e is there a "nice" recurrence for NVol(∇ PQ G:e )? The triangle recurrence The framework which establishes the subdivision recurrence can be adapted to a different operation. Given an edge e = uv of a graph G, let G△e denote the graph with edge set E(G) ∪ {uw, vw} where w is a new vertex. We will continue to assume V (G) = [N] and V (G△e) = [N + 1]. As in Section 3.1, establishing a recurrence formula for D(G△e) will require establishing several smaller results first. The first two of these have proofs analogous enough to the proofs of Lemma 16 and Lemma 17, respectively, that we omit their details. then β △ (c) ∈ D(G△e) where β △ (c) = α △ (c) + e u − e N +1 . Additionally, β △ is injective. The next lemma is analogous to Lemmas 18, but this time its proof is different enough for us to justify providing it. It will be helpful to introduce the analogues of A G (e) and B G (e) here: let A △ G (e) and B △ G (e) be the D(G△e)-draconian sequences constructed with α △ and β △ in Lemmas 24 and 25, respectively. Lemma 26. Let G be a connected graph on [N] and let e = uv be any edge for which deg G (u) = 2. If c ∈ D(G), then γ △ (c) ∈ D(G△e) where γ △ (c) = α △ (c) + e v − e N +1 if not in B △ G (e) α △ (c) − e u + e N +1 otherwise. Additionally, γ △ is injective. Proof. That γ △ is injective is clear. For what remains, by Remark 9 we again assume without loss of generality that e = {N − 1, N}. We also assume that the other neighbor of N − 1 in G is N − 2. So, if c = (c 1 , . . . , c N ) ∈ D(G), then we must prove γ △ (c) ∈ D(G△e) where γ △ (c) = (γ △ 1 , . . . , γ △ N +1 ) = (c 1 , . . . , c N −2 , c N −1 , c N + 1, 0) if not in B △ G (e) (c 1 , . . . , c N −2 , c N −1 − 1, c N , 2) otherwise. Note that, in both cases, the entries sum to N. If c ′ = (c 1 , . . . , c N −2 , c N −1 − 1, c N + 1) ∈ D(G).(7) Consider a sum γ △ i 1 + · · · + γ △ i k with 1 i 1 < · · · < i k N + 1. If i k N, then γ △ i 1 + · · · + γ △ i k c i 1 + · · · + c i k < k j=1 N D(G) (i j ) k j=1 N D(G△e) (i j ) . If i k = N + 1 then there are four subcases to consider depending on the value of i k−1 . If i k−1 = N, then we may write the sum as γ △ i 1 + · · · + γ △ i k = c i 1 + · · · + c i k−2 + (c N + 1) + 1. By (7), we know that α △ (c ′ ) ∈ D(G△e). The above sum appears when verifying this fact, so we know that γ △ i 1 + · · · + γ △ i k < k j=1 N D(G△e) (i j ) . If i k−1 = N − 1, then γ △ i 1 + · · · + γ △ i k = c i 1 + · · · + c i k−2 + c N −1 + 1 < k j=1 N D(G△e) (i j ) , the electronic journal of combinatorics 27 (2020), #P00 where the inequality again comes from knowing α △ (c) ∈ D(G△e). If i k−1 = N − 2, then again by applying α △ , we may say γ △ i 1 + · · · + γ △ i k c i 1 + · · · + c i k−1 + c N −1 − 1 + 2 < k j=1 N D(G△e) (i j ) ∪ N D(G△e) (N − 1) . Noticing that N D(G△e) (N − 1) ⊆ N D(G△e) (N − 2) ∪ N D(G△e) (N + 1) ⊆ k j=1 N D(G△e) (i j ) we may drop N D(G△e) (N − 1) from the union, which establishes the desired inequality. Lastly, if i k−1 < N − 2, then neither N − 1 nor N + 1 is a neighbor of i j in D(G△e) for j k − 1, so we may say γ △ i 1 + · · · + γ △ i k < k−1 j=1 N D(G△e) (i j ) ⊎ {N − 1, N + 1} k j=1 N D(G△e) (i j ) . In all cases, the required D(G△e)-draconian inequality holds. Therefore, we have shown γ △ (c) ∈ D(G△e) for all c ∈ D(G). Let C △ G (e) be the D(G△e)-draconian sequences constructed from γ △ in Lemma 26. The proof of the following is completely analogous to the proof of Lemma 19. Let d = (d 1 , . . . , d N +1 ) ∈ D(G△e). As with the subdivision recurrence, there are three statements we must establish: • If d N +1 = 0, then (d 1 , . . . , d N −2 , d N −1 − 1, d N ) ∈ D(G) or, if this is not the case, then (d 1 , . . . , d N −2 , d N −1 , d N − 1) ∈ D(G); • If d N +1 = 1, then (d 1 , . . . , d N ) ∈ D(G); and • If d N +1 = 2, then both (d 1 , . . . , d N −2 , d N −1 + 1, d N + 1, 0) ∈ B △ G (e) as well as (d 1 , . . . , d N −2 , d N −1 + 1, d N ) ∈ D(G). For ease of readability, the case d N +1 = 0 is deferred to Lemmas 31 and 32, where the two different conditions on the vertices N − 1 and N are treated individually. Suppose, then, that d N +1 = 1. Pick any D(G△e)-draconian sequence of the form (d 1 , . . . , d N , 1). Let 1 i 1 < · · · < i k N. If i j = N − 1, N for all j, then the neighbors of i j are the same in D(G△e) and D(G), so the corresponding D(G)-draconian inequality instantly holds. Otherwise, d i 1 + · · · + d i k = d i 1 + · · · + d i k + 1 − 1 < k j=1 N D(G△e) (i j ) ∪ N D(G△e) (N + 1) − 1 = k j=1 N D(G△e) (i j ) ∪ N + 1 − 1 = k j=1 N D(G) (i j ) Thus (d 1 , . . . , d N ) ∈ D(G). Lastly, suppose d N +1 = 2. For this case we first show that (d 1 , . . . , d N −1 +1, d N +1, 0) ∈ B △ G (e). This can be rephrased as wanting to show (d 1 , . . . , d N −1 + 1, d N + 1, 0) = β △ (c) for some c, or, in yet other words, that (d 1 , . . . , d N −1 , d N + 1) ∈ D(G). Set d ′ = (d ′ 1 , . . . , d ′ N ) = (d 1 , . . . , d N −1 , d N + 1) and consider 1 i 1 < · · · < i k N. If i k < N − 1, then the D(G)-draconian inequality holds as usual. If i k = N − 1, then observe d ′ i 1 + · · · + d ′ i k < d i 1 + · · · + d i k + 2 − 1 < k j=1 N D(G) (i j ) ∪ {N + 1} − 1 = k j=1 N D(G) (i j ) . If i k = N, then repeat this argument but by inserting "+1 − 1" instead of "+2 − 1". In all cases, the D(G)-draconian inequality holds, so d ′ ∈ D(G), as needed. In fact, showing that (d 1 , . . . , d N −2 , d N −1 + 1, d N ) is D(G)-draconian has an entirely analogous argument. Therefore, this completes the case for d N +1 = 2. By Lemma 27, As in the case of the subdivision recurrence, by relaxing the requirement that e has an endpoint of degree 2 in G, the result may no longer hold. The same example as before, where G = K 1 ∨ P 3 and e is the edge whose endpoints each have degree 3 in G, demonstrates this. The normalized volume of ∇ PQ G△e is 52 whereas a naive attempt to apply the triangle recurrence would predict 54. D(G△e) = A △ G (e) ⊎ B △ G (e) ⊎ C △ G (e). Although the conclusion of the triangle recurrence may not hold when the endpoints of e do not have degree 2, there are cases when the conclusion still does hold. For example, the graph G 2 in Figure 2 cannot be constructed from G 1 in a way that allows us to combine the subdivision and triangle recurrences, yet we still have \|D(G 2 )\| = 3\|D(G 1 )\|. Note that the conditions under discussion are local conditions; this will contrast with global conditions that we examine in Section 3.3. This leads us to ask the following. Question 30. Under what local conditions for a graph G and an edge e is there a "nice" recurrence for NVol(∇ PQ G△e )? To close this section, we state and prove the lemmas needed to complete the proof of Theorem 28. N − 1 in G is N − 2. We first show that if d N −1 1, then (d 1 , . . . , d N −2 , d N −1 − 1, d N ) ∈ D(G) . For notational convenience, we will write d ′ = (d ′ 1 , . . . , d ′ N ) = (d 1 , . . . , d N −2 , d N −1 − 1, d N ). Consider a sum d ′ i 1 + · · · + d ′ i k with 1 i 1 < · · · < i k N. If i k < N − 1, then the neighbors of each i j is the same in D(G) and D(G△e), so d ′ i 1 + · · · + d ′ i k = d i 1 + · · · + d i k < k j=1 N D(G△e) (i j ) = k j=1 N D(G) (i j ) . If i k = N − 1 or if both i k = N and i k−1 = N − 1 then we have d ′ i 1 + · · · + d ′ i k = d i 1 + · · · + d i k − 1 < k j=1 N D(G△e) (i j ) − 1 = k j=1 N D(G) (i j ) since N + 1 is a neighbor of N − 1 in D(G△e) but not D(G). Lastly, if i k = N but i k−1 < N − 1, recall that we have required N to be a neighbor of both N − 1 and N − 2. Thus, the neighbors of N − 1 in D(G△e) are necessarily also neighbors of N in D(G△e). Therefore, d ′ i 1 + · · · + d ′ i k d i 1 + · · · + d i k + d N −1 − 1 < k j=1 N D(G△e) (i j ) ∪ N D(G△e) (N − 1) − 1 = k j=1 N D(G△e) (i j ) − 1 = k j=1 N D(G) (i j ) . Thus, d ′ ∈ D(G) when d N −1 1. If d N −1 = 0, then d N 1 since we may not have d N −1 = d N = d N +1 = 0. We will show that, in this case, (d 1 , . . . , d N −1 , d N − 1) ∈ D(G). Again for notational convenience, we will write d ′′ = (d ′′ 1 , . . . , d ′′ N ) = (d 1 , . . . , d N −2 , d N −1 , d N − 1). Consider a sum d ′′ i 1 + · · · + d ′′ i k with 1 i 1 < · · · < i k N. If i k < N − 1, then, as for d ′ , d ′′ i 1 + · · · + d ′′ i k = d i 1 + · · · + d i k < k j=1 N D(G△e) (i j ) = k j=1 N D(G) (i j ) . If i k = N − 1, then since we know d ′′ N −1 = d N −1 = 0, we may say d ′′ i 1 + · · · + d ′′ i k = d i 1 + · · · + d i k−1 < k−1 j=1 N D(G△e) (i j ) = k−1 j=1 N D(G) (i j ) k j=1 N D(G) (i j ) . Lastly, if i k = N, then d ′′ i 1 + · · · + d ′′ i k = d i 1 + · · · + d i k − 1 < k j=1 N D(G△e) (i j ) − 1 = k j=1 N D(G) (i j ) . Thus, d ′′ ∈ D(G). This completes the proof. d ′ ∈ D(G), where d ′ = (d ′ 1 , . . . , d ′ N ) = (d 1 , . . . , d N −1 , d N − 1) Consider a sum d ′ i 1 + · · · + d ′ i k with 1 i 1 < · · · < i k N. First suppose d N −1 = 0. Since we cannot have d N −1 = d N = d N +1 = 0 in a D(G△e)-draconian sequence, it must be true that d N 1, so that d ′ consists of nonnegative integers. If i k < N − 1, then d ′ i 1 + · · · + d ′ i k = d i 1 + · · · + d i k < k j=1 N D(G△e) (i j ) = k j=1 N D(G) (i j ) since the neighbors of each i j are the same in G and G△e. If i k = N − 1, then d ′ i k = 0, so that d ′ i 1 + · · · + d ′ i k = d i 1 + · · · + d i k−1 < k−1 j=1 N D(G△e) (i j ) = k−1 j=1 N D(G) (i j ) k j=1 N D(G) (i j ) . the electronic journal of combinatorics 27 (2020), #P00 Lastly, if i k = N, then d ′ i 1 + · · · + d ′ i k = d i 1 + · · · + d i k − 1 < k j=1 N D(G△e) (i j ) − 1 = k j=1 N D(G) (i j ) . Thus, d ′ ∈ D(G) when d N −1 = 0. Now suppose d N −1 > 0. Our assumption that (d 1 , . . . , d N −2 , d N −1 − 1, d N ) / ∈ D(G) implies d N > 0 as well. If d N −1 = 3, then d N −1 + d N < N D(G△e) (N − 1) ∪ N D(G△e) (N) = 5 implies d N 1. We claim that this means (d 1 , . . . , d N −2 , d N −1 − 1, d N ) ∈ D(G), which is a contradiction. Note that this means (d N −1 − 1, d N ) = (2, 0) or (d N −1 − 1, d N ) = (2, 1). In either situation, set d u = (d u 1 , . . . , d u N ) = (d 1 , . . . , d N −2 , d N −1 − 1, d N ) and consider a sum of the form d u i 1 + · · · + d u i k . If i k < N − 1, then N + 1 is not a neighbor of any i j in D(G△e), so d u i 1 + · · · + d u i k = d i 1 + · · · + d i k < k j=1 N D(G△e) (i j ) = k j=1 N D(G) (i j ) . If i k = N − 1, then N + 1 is a neighbor of N − 1 in D(G△e), so d u i 1 + · · · + d u i k = d i 1 + · · · + d i k − 1 < k j=1 N D(G△e) (i j ) − 1 = k j=1 N D(G) (i j ) . For the same reason, this inequality holds when i k = N and i k−1 = N − 1. If i k = N and i k−1 = N − 2, notice that the neighbors of N − 1 are in the union of the neighbors of N − 2 and N in D(G△e). Therefore, it follows from the case in which i k−1 = N − 1 that d u i 1 + · · · + d u i k d i 1 + · · · + d i k + d N −1 − 1 < k j=1 N D(G△e) (i j ) ∪ N D(G△e) (N − 1) − 1 = k j=1 N D(G) (i j ) . Lastly, if i k−1 < N − 2, then N − 1 is not a neighbor of any i j < N − 2 in D(G△e). Moreover, for each i j < N − 2, its neighbors in D(G) are the same as its neighbors in D(G△e). Putting this together with the fact that d N 1, we see d u i 1 + · · · + d u i k d i 1 + · · · + d i k−1 + 1 < k−1 j=1 N D(G△e) (i j ) + {N − 1} k j=1 N D(G) (i j ) . Therefore, if d N −1 = 3, then d u ∈ D(G), which is a contraction. Now suppose that d N −1 = 2. Analogous to before, this implies d N 2, leading us to the three cases (d N −1 − 1, d N ) = (1, 0) or (d N −1 − 1, d N ) = (1, 1) or (d N −1 − 1, d N ) = (1, 2). If d N = 2, then an argument symmetric to the one in the previous paragraph draws the same contradiction. If d N = 1, then an argument identical to that of the previous paragraph holds. Finally, if d N = 0, then the desired inequalities hold since those not involving the index N hold for the case of d N = 1, and each inequality involving an index N can be obtained from adding d N = 0 to the left hand side and including N D(G) (N) in the union on the right hand side. Therefore, d u ∈ D(G) whenever d N −1 > 1, which is a contradiction. Knowing now that d N −1 = 1, set d ′′ = (d ′′ 1 , . . . , d ′′ N ) = (d 1 , . . . , d N −2 , 1, d N − 1) and consider a sum d ′′ i 1 +· · ·+d ′′ i k with 1 i 1 < · · · < i k N. If i k = N −2 or i k = N, then the corresponding D(G)-inequalities hold via now-standard arguments. If i k = N − 1, then there are two subcases to consider. First suppose at least one of N , N − 1, or N − 2 does not appear in k−1 j=1 N D(G) (i j ). Without loss of generality, assume that N does not appear. We can therefore say d ′′ i 1 + · · · + d ′′ k−1 + 1 < k−1 j=1 N D(G) (i j ) + {N} k j=1 N D(G) (i j ) . Otherwise suppose {N − 2, N − 1, N} ⊆ k−1 j=1 N D(G) (i j ). the electronic journal of combinatorics 27 (2020), #P00 Since deg G (N − 1) = deg G (N) = 2, we know that this can only happen if i k−1 = N − 2 and i k−2 = N − 3. In particular, i j = N for all j and N G (N) ⊆ k−1 j=1 N D(G) (i j ), which implies from the case i k = N that d ′′ i 1 + · · · + d ′′ k−1 + 1 d i 1 + · · · + d k−1 + 1 + d N − 1 < k−1 j=1 N D(G) (i j ) ∪ N D(G) (N) = k j=1 N D(G) (i j ) . This completes the proof. Application: outerplanar graphs Recall that a plane graph is a planar graph G together with a particular embedding of G into the plane. Also recall that the weak dual of a plane graph G, denoted G ( * ) , is the subgraph of the dual G * induced by the vertices corresponding to bounded faces of G. We denote by E k the empty graph on k vertices, that is, the disjoint union of k distinct vertices. Further, given a bounded face F , let o G (F ) denote the number of edges of G bounding both F and the outer face and let v F denote the vertex of G ( * ) corresponding to F . Let F(G) be the set of bounded faces of G. Definition 33. Let G be a plane graph. The extended weak dual of G, denoted G ( * * ) , is G ( * * ) = G ( * ) ∪   F ∈F(G) v F ∨ E o(F )   Informally, G ( * * ) extends the weak dual of G by including an additional edge for each edge of G that bounds the outer face. See Figure 3 for illustrations of a plane graph G and its duals G ( * ) , G ( * * ) . Recall that a graph is outerplanar if it has a planar embedding such that every vertex is incident to the outer face. It is known [9] that a graph is outerplanar if and only if its weak dual is a forest. Putting together the results of Section 3 we can produce a simple formula for NVol(∇ PQ G ) whenever G can be constructed inductively by using the subdivision and triangle operations. The formula follows quickly from the following theorem. Theorem 34. Suppose G is a 2-connected outerplane graph obtained from C N by a sequence of applications of the subdivision recurrence and the triangle recurrence. Then NVol(∇ PQ G ) = 2 1+S(G) F ∈F(G) deg G ( * * ) (v F ), where S(G) = F ∈F (deg G ( * * ) (v F ) − 3). Proof. We will induct on the number of edges of G, which we will denote by \|E\|. If G has 3 or 4 edges, then since G is 2-connected, G = C \|E\| , hence N = \|E\|. By Corollary 22, NVol(∇ PQ G ) = 2 1+(deg G ( * * ) (v F )−3) deg G ( * * ) (v F ) as claimed. One may verify that this holds for C 4 directly as well. Now suppose \|E\| > 4. If G is of the form G = G ′ △e for some edge e of G ′ , then let F 0 be the unique face of F(G) \ F(G ′ ). We can then say that deg G ′( * * ) (v F ) = deg G ( * * ) (v F ) for all internal vertices v F of G ′ ( * * ) , from which it follows that S(G) = S(G ′ ). By the triangle recurrence and the inductive assumption, NVol(∇ PQ G ) = 3 NVol(∇ PQ G ′ ) = (deg G ( * * ) (v F 0 ))2 1+S(G ′ ) F ∈F(G ′ ) deg G ′( * * ) (v F ) = 2 1+S(G) F ∈F(G) deg G ( * * ) (v F ) as desired. Suppose instead that G is of the form G = G ′ : e for some edge e of G ′ . Since G is outerplanar, so is G ′ , and e is incident to a unique bounded face. Let B be the set of cut-edges of G \ e and again let F 0 be the unique face of F(G) \ F(G ′ ). The graph H = G \ (B ∪ {e}) is the disjoint union of k = deg G ( * * ) (v F 0 ) − 1 − \|B\| graphs,deg H ( * * ) (v F ) = deg G ( * * ) (v F ) for all F ∈ F(H)∩F(G). By the subdivision recurrence, Corollary 14, and Proposition 12, we obtain NVol(∇ PQ G ) = NVol(∇ PQ G ′ :e ) = 2 NVol(∇ PQ G ′ ) + NVol(∇ PQ G ′ \e ) = 2   2 1+S(G ′ ) F ∈F(G ′ ) deg G ′( * * ) (v F )   + 2 \|B\| NVol(∇ PQ H ) =   2 1+S(G) (deg G ( * * ) (v F 0 ) − 1) F ∈F(G)\{F 0 } deg G ( * * ) (v F )   + 2 \|B\| 2 ω F ∈F(G)\{F 0 } deg G ( * * ) (v F )(8) where ω = k + S(G) − (deg G ( * * ) (v F 0 ) − 3) = deg G ′( * * ) (v F 0 ) − 1 − \|B\| + S(G) − (deg G ( * * ) (v F 0 ) − 3) = 1 − \|B\| + S(G). Simplifying the final expression in (8) yields the claimed formula, completing the proof. Theorem 34 is the final piece needed to compute NVol(∇ PQ G ) for any outerplane graph whose 2-connected components satisfy the conditions of Theorem 34. Corollary 35. Let G be any outerplane graph on [N] such that each block with at least three vertices is obtained from C N by a sequence of applications of the subdivision recurrence and the triangle recurrence. Label the components of G by G 1 , . . . , G k and let B i,1 , . . . , B i,b i be the blocks of G i . Then NVol(∇ PQ G ) = k i=1 b i j=1 2 1+S(B i,j ) F ∈F(B i,j ) deg B i,j ( * * ) (v F ).(9) The graphs satisfying the conditions needed in Corollary 35 form a proper, but large, class of outerplane graphs. Experimental data suggests that the formula is, in fact, true for all outerplane graphs, but a proof eludes the authors. Conjecture 36. For any outerplane graph G, Equation (9) holds. the electronic journal of combinatorics 27 (2020), #P00 Beyond outerplanarity Outerplanar graphs form a large class of graphs but are far from the class of planar graphs, let alone all graphs. For example, even though there are about 56.7 × 10 9 labeled outerplanar graphs on 10 vertices, these account for only approximately 1.76% of all labeled planar graphs on 10 vertices [10, Sequences A098000, A066537]. Because of the difficulty in computing NVol(∇ PQ G ) for all graphs, a natural next step would be to consider graphs that are not-quite-outerplanar. Toward this end, we use the following alternate characterization of outerplanar graphs. This is a direct analogue of Kuratowski's theorem, allowing one to study graphs G that contain no subdivision of K 5 or K 3,3 but may contain a subdivision of K 4 or K 2,3 . In this case, a formula for \|D(G)\| remains elusive, although we do have the following partial result. We use the notation Proof. If (c 1 , . . . , c N ) ∈ D(K 2,N −2 ), then c 1 + c 2 = k for some 0 k N − 1. All possible choices of c 1 , c 2 are part of a D(K 2,N −2 )-draconian sequence except for (c 1 , c 2 ) ∈ {(N − 1, 0), (0, N − 1)} since these are the only two resulting in sequences not satisfying the corresponding draconian inequalities. However, for the moment, we will include these in our calculations for algebraic ease. In order to satisfy the D(K 2,N −2 )-draconian inequalities we need the subsequence c ′ = (c 3 , . . . , c N ) to be a weak composition of N − 1 − k using 0s, 1s, and 2s such that there is at most one 2. This leads to two cases: if c ′ contains a 2, then there must be N − 3 − k copies of 1 and k copies of 0. A simple counting argument gives (N − 2) N − 3 k such possibilities. On the other hand if c ′ does not contain any 2s, then there must be N − 1 − k copies of 1 and k − 1 copies of 0. There are N −2 k−1 such possibilities. Adding the values from these two cases and summing over all k yields N −1 k=0 (k + 1) (N − 2) N − 3 k + N − 2 k − 1 . The reader may verify that this simplifies to 2 N −4 (N 2 − N + 6). Subtracting the two compositions where (c 1 , c 2 ) ∈ {(N − 1, 0), (0, N − 1)} and applying Theorem 8 gives us our final formula. Question 39. What is NVol(∇ PQ K M,N ) for arbitrary M, N? Notice that the formula in Proposition 38 cannot be written in the form of (9). Thus, a general formula for planar graphs will require refining the techniques of Section 3 or separate tools altogether. A second important class of graphs which are planar but not outerplanar is the class of wheel graphs W N = K 1 ∨ C N . We conjecture the following. Conjecture 40. For all N 3, NVol(∇ PQ W N ) = 3 N − 2 N + 1. This conjecture has been verified for all 3 N 13. Wheels were examined in detail in [7] within a related, but distinct, context from ∇ PQ W N . We hope to uncover similarly rich structure in the present setting. It may be useful to recognize that 3 N − 2 N + 1 = 2S(N + 1, 3) + S(N + 1, 2) + S(N + 1, 1), where S(n, k) denotes the Stirling number of the second kind. Remark 41. In the time since this article was first prepared, Conjecture 40 has been proven by Ohsugi and Tsuchiya [12]. Finally, we give another broad class of graphs which contains all outerplanar graphs but not all planar graphs. Strictly speaking, these graphs will allow for repeated edges, but as repeating an edge in G does not affect ∇ PQ G , we need not worry about that case. Following [8], first consider the directed graphs formed in the following way. Begin with a single edge and designate one vertex the source and another vertex the sink. This is an example of a two-terminal series-parallel graph. All other two-terminal series-parallel graphs are those formed by applying one of the following operations to two existing two-terminal series-parallel graphs G and H with sources g and h and sinks g ′ and h ′ , respectively, 1. parallel composition: produce a new graph P(G, H) by identifying g with h and g ′ with h ′ . The source of P(G, H) is g ∼ h and its sink is g ′ ∼ h ′ . 2. series composition: produce a new graph S(G, H) by identifying g ′ with h. The source of S(G, H) is g and its sink is h ′ . A graph G is a series-parallel graph if there are two vertices x, y such that, when designating x as the source and y as the sink, G can be obtained through a sequence of applications of P(·, ·) and S(·, ·) when starting with a disjoint union of edges. Series-parallel graphs are of interest in computer algorithms, as recognizing them is difficult but not intractable. For our purposes, they are of interest because their recursive structure suggests that they may be good candidates for computing NVol(∇ PQ G ). In fact, we have already seen an example of a series-parallel graph: K 2,N −2 is the parallel composition of N − 2 copies of P 3 , each of which is a series composition of two edges. We ask the following question broadly, and would be interested in seeing answers to even nontrivial subclasses which are not outerplanar. Question 42. What is NVol(∇ PQ G ) for a series-parallel graph G? Subdivision recurrence (see Theorem 20). Let G be a 2-connected graph and let e = uv be an edge. Denote by G : e the graph obtained by subdividing e. If deg G (u) = 2 and the neighbors of u are neighbors of each other, then NVol(∇ PQ G:e ) = 2 NVol(∇ PQ G ) + NVol(∇ PQ G\e ). Triangle recurrence (see Theorem 28). Let G be any connected graph and let e = uv be an edge with deg G (u) = 2. If deg G (v) = 2 or if the neighbors of u are neighbors of each other, then For a positive integers M, N, let K N denote the complete graph on [N] and let K M,N denote the complete bipartite graph with partite sets [M] and [N ] = {1, . . . , N }. Let N G (v) denote the set of vertices of G adjacent to v. Keeping this notation in mind, we may now begin in earnest. Figure 1 : 1A graph G, left, and its corresponding bipartite graph D(G), right. Definition 3 . 3Let G be a simple graph on [N]. Define D(G) to be the subgraph of K N,N with edges {i, i} for each i ∈ [N] and {i, j} and {j, i} for each edge ij in G. Lemma 6 . 6For any simple graph G, G is connected if and only if D(G) is connected. the electronic journal of combinatorics 27 (2020), #P00 Let G be any graph. Then \|P − D(G) ∩ Z N \| = \|D(G)\|. Theorem 8. For any connected graph G on [N], NVol(∇ PQ G ) = \|D(G)\|. 3} and N D(G) (4) = {2, 4}. Theorem 8 tells us that NVol(∇ PQ G ) that V (B) = [M], and that V (B ′ ) = {1, M + 1, . . . , N}. We claim that the map f : D(B) × D(B ′ ) → D(G) which sends (c 1 , c 2 , . . . , c M ), (c ′ 1 , c ′ M +1 , . . . , c ′ N ) to (d 1 , . . . , d N ) = (c 1 + c ′ 1 , c 2 , . . . , c M , c ′ M +1 , . . . , c ′ N ) is a well-defined bijection. For notational convenience set c = (c 1 , c 2 , . . . , c M ) and c ′ = (c ′ 1 , c ′ M +1 , . . . , c ′ N ). Since c ∈ D(B) and c ′ ∈ D(B ′ ), we know c i = M − 1 and which is a contradiction to d being D(G)-draconian. Therefore the D(B)-draconian inequalities for c all hold, and c ∈ D(B). An analogous argument shows c ′ ∈ D(B ′ ), proving f is a bijection. This implies \|D(B)\|\|D(B ′ )\| = \|D(G)\|; applying Theorem 8 completes the proof. Lemma 16 . 16Let G be any connected graph on [N] and let e = uv be an edge. If c ∈ D(G), then α(c) ∈ D(G : e) where α(c) = (c, 1). Moreover, α is an injection. Proof. Let c ∈ D(G). By Remark 9 we may assume that e = {N − 1, N}. Showing that α is an injection is routine, so we focus mainly on showing α(c) ∈ D(G : e). Let c = (c 1 , . . . , c N ). Since c 1 + · · · + c N = N − 1, the sum of entries of α(c) is N. By construction, N D(G) (i) = N D(G:e) (i) for i = 1, . . . , N − 2, N D(G:e) (i j ) contains N + 1 and all elements of T with the potential exception of the vertex in D(G : e) corresponding to the neighbor of N − 1 other than N + 1, which is unique since we have assumed deg G (N − 1) = 2. From this we conclude For the remaining case, suppose γ(c) = γ(d) where γ(c) falls under the conditions of 1(b) and γ(d) falls under the conditions of 2 Fix a particular edge e of a 2-connected graph G for which one of the endpoints has degree 2 in G. Let A G (e), B G (e), and C G (e) be the set of D(G : e)-draconian sequences constructed from α, β, and γ in Lemmas 16, 17, and 18, respectively.Lemma 19. Let G be a 2-connected graph with an edge e = uv such that deg G (u) = 2 and the neighbors of u are neighbors of each other. The sets A G (e), B G (e), and C G (e) are pairwise disjoint.Proof. We continue to use the convention that e = {N − 1, N} and deg G (N − 1) = 2. We will also make the assumption that the other neighbor of N − 1 in G is N − 2.By comparing the values of c N +1 , it is clear that A G (e) ∩ B G (e) = ∅ and A G (e) ∩ C G (e) = ∅. Thus we only need to focus on B G (e) ∩C G (e). In fact, since γ is an injection, we only need to consider elements of C G (e) that fall under the conditions of 1(b) or 2(b) of Lemma 18. Suppose that β(c) = γ(c ′ ) where c = (c 1 , . . . , c N ) and c ′ = (c ′ 1 , . . . , c ′ N )are sequences with c ∈ D(G \ e) and c ′ ∈ D(G). If γ(c ′ ) were to be constructed by the conditions of 1(b) in Lemma 18, it would follow that c = c ′ since, in this case, β j=1 NN j=1D(G:e) (i j ) ∪ N D(G:e) D(G:e) (i j ) , where the inequality follows from Lemma 16 and the last equality comes from recognizing that N D(G:e) (N − 1) ⊆ N D(G:e) (N − 2) ∪ N D(G:e) (N + 1), hence N D(G:e) (N − 1) may be freely dropped from the expression. Finally, if i k−1 < N − 2, then (c) both (d 1 , . . . , d N −2 , d N −1 −1, d N −1, 2) / ∈ D(G:e) and (d 1 , . . . , d N −2 , d N −1 , d N − 1) ∈ D(G \ e). Therefore, if d N −1 = 0, then condition 3(c) holds.Next suppose d N −1 = 2. Condition 3(c) clearly cannot hold since this condition requires d N −1 1, so we must show that either 3(a) or 3(b) holds. Suppose that condition 3(a) does not hold, that is,suppose (d 1 , . . . , d N −2 , 1, d N ) / ∈ D(G \ e).Showing that this sequence is in D(G) can be done directly repeating our by-now-usual strategies, so the sequence is in D(G) \ D(G \ e).Toshow that (d 1 , . . . , d N −2 , 0, d N , 2) / ∈ D(G : e), observe that (d 1 , . . . , d N −2 , 1, d N ) / ∈ D(G \ e)implies there is some inequality of the form then add 2 to the left side of (6) and 1 = {N} to the right side; the conclusion is the same. Thus, if condition 3(a) does not hold, then condition 3(b) does hold.For the case of when d N −1 = 1, the first part of condition 3(b) clearly holds. Verifying that (d 1 , . . . , d N −2 , 0, d N ) ∈ D(G) is now routine, so either condition 3(a) holds or 3(b) holds.We have shown that, regardless of value of d N +1 , one of the three conditions holds, henced ∈ A G (e) ∪ B G (e) ∪ C G (e) and D(G : e) = A G (e) ∪ B G (e) ∪ C G (e). By Lemma 19, this union is disjoint, so \|D(G : e)\| = \|A G (e) ⊎ B G (e) ⊎ C G (e)\| = \|A G (e)\| + \|B G (e)\| + \|C G (e)\| = 2\|D(G)\| + \|D(G \ e)\|. Applying Theorem 8, the result is proven. Example 21. Consider C 3 = ([3], {12, 13, 23}) and let e = 13; there are six D(C 3 replaces the edge 13 with edges 34 and 14 to obtain C 4 . By the subdivision recurrence, D(C 4 ) = D(C 3 : e) = A C 3 (e) ⊎ B C 3 (e) ⊎ C C 3 (e). Following the definitions of α, β, and γ we obtain A C 3 (e) = {(2, \|D(C 3 )\| = 3 · 2 1 and \|D(C 4 )\| = 4 · 2 2 . [N] and edges {1, 2}, {2, 3},. . . ,{N − 1, N},{1, N}, and we will subdivide the edge e = {N − 1, N}. Let c ∈ D(C N ) so that γ ′ (c) / ∈ D(C N : e) and c / ∈ D(C N \ e). Setting c = (c 1 , . . . , c N ), we have γ(c) = (γ 1 , . . . , γ N +1 ) = (c 1 , . . . , c N −2 , c N −1 + N D(C N :e) (i j ) right away. If i k−1 < N − 1, then neither N nor N + 1 is a neighbor of any of i 1 , . . . , i k−1 in either D(C N ) or D(C N : e), so Lemma 24 . 24Let G be any connected graph on [N] and e any edge. If c ∈ D(G), then α △ (c) ∈ D(G△e) where α △ (c) = (c, 1). Moreover, α △ is injective. Lemma 25. Let G be a connected graph on [N] and let e = uv be any edge. If c ∈ D(G), γ △ (c) = (c 1 , . . . , c N −2 , c N −1 , c N + 1, 0), then showing it is D(G△e)-draconian is entirely analogous to the proof of Lemma 17. Otherwise, γ △ (c) = (c 1 , . . . , c N −2 , c N −1 − 1, c N , 2). Being in this case means that (c 1 , . . . , c N −1 , c N + 1, 0) ∈ B △ (e). Hence, we know c N −1 1, so all entries of γ △ are nonnegative, and we also know Lemma 27 . 27Let G be a graph having an edge e = uv with deg G (u) = 2. The sets A △ G (e), B △ G (e), and C △ G (e) are pairwise disjoint. As in Section 3.1, the previous four lemmas imply A △ G (e) ⊎ B △ G (e) ⊎ C △ G (e) ⊆ D(G△e). The reverse inclusion again holds under certain restrictions, establishing what we call the triangle recurrence.. We present the proof below, deferring portions of it to two lemmas afterward. Theorem 28 (Triangle Recurrence). Let G be any connected graph and let e = uv be an edge with deg G (u) = 2. If deg G (v) = 2 or if the neighbors of u are neighbors of each other, then NVol(∇ PQ G△e ) = 3 NVol(∇ PQ G ). Proof. As usual we assume V (G) = [N], e = {N − 1, N}, and deg G (N − 1) = 2. We will further assume that the other neighbor of N − 1 in G is N − 2. Lemmas 24, 25, and 26 show that A △ G (e) ∪ B △ G (e) ∪ C △ G (e) ⊆ D(G△e), so we must show the reverse inclusion holds. Figure 2 : 2Two graphs G 1 (left) and G 2 (right).Thus, \|D(G△e)\| = \|A △ G (e)\| + \|B △ G (e)\| + \|C △ G (e)\| = 3\|D(G)\|. Finally, by Theorem 8, we obtain NVol(∇ PQ G△e ) = 3 NVol(∇ PQ G ).Example 29. Let C 3 be the 3-cycle as in Example 21 and again choose e = 13. The D(G△e)-draconian sequences are formed from the disjoint union of the three setsA △ C 3 (e) = {(2, Lemma 31 . 31Let G be any connected graph on [N] for which e = uv with deg G (u) = 2 and the neighbors of u are neighbors of each other. If (d 1 , . . . , d N , 0) ∈ D(G△e), then (d 1 , . . . , d N ) − e u ∈ D(G) or (d 1 , . . . , d N ) − e v ∈ D(G).Proof. As usual we assume V (G) = [N], e = {N − 1, N}, and deg G (N − 1) = 2. We will further assume that the other neighbor of Lemma 32 . 32Let G be any connected graph on[N] for which e = uv with degG (u) = deg G (v) = 2. If (d 1 , . . . , d N , 0) ∈ D(G△e), then (d 1 , . . . , d N )−e u ∈ D(G) or (d 1 , . . . , d N )− e v ∈ D(G).Proof. As usual we assume V (G) = [N], e = {N − 1, N}, and, this time, deg G (N − 1) = deg G (N) = 2. If the neighbors of N − 1 are neighbors of each other, then we are done by Lemma 31. So, we assume that the neighbors of N − 1 are nonadjacent. We also may assume that the other neighbor of N − 1 in G is N − 2 and the other neighbor of N in G is N − 3. If (d 1 , . . . , d N −2 , d N −1 − 1, d N ) ∈ D(G), then we are done. Otherwise, we will show Figure 3 : 3A graph G (gray) with its weak dual G ( * ) superimposed (left, dashed) and with its extended weak dual G ( * * ) superimposed (right, dotted). where each component is a 2-connected subgraph of G. Notice as well that F(H) ⊆ F(G) with \|F(H)\| = \|F(G)\| − 1 and that Theorem 37 ([ 3 , 373Theorem 10.24]). A graph is outerplanar if and only if contains no subdivision of K 4 or K 2,3 as a subgraph. K 0 M,N to denote the complete bipartite graph with partite sets [0, . . . , M − 1] and [M, M + N − 1]. Proposition 38. For all N 3,NVol(∇ PQ K 2,N−2 ) = 2 N −4 (N 2 − N + 6) − 2. Theorem 5 ([14, Theorem 12.2]). Let G be a graph on [N] for which D(G) is connected and let Notation, background, and translating to draconian sequences Before we prove our results, we will establish assorted notation that will be needed throughout this work. Additional notation will be introduced as needed. First, if e is the electronic journal of combinatorics 27 (2020), #P00 the electronic journal of combinatorics 27 (2020), #P00 The authors would like to thank Florian Kohl, Joakim Jakovleski, Qizhe Pan, and the anonymous referee for their detailed feedback. Their comments greatly improved the quality of this article. Integer-point enumeration in polyhedra. Matthias Beck, Sinai Robins, Undergraduate Texts in Mathematics. David AustinNew YorkSpringerComputing the Continuous DiscretelyMatthias Beck and Sinai Robins. Computing the Continuous Discretely. Undergrad- uate Texts in Mathematics. Springer, New York, second edition, 2015. Integer-point enumeration in polyhedra, With illustrations by David Austin. 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Alex Postnikov, Victor Reiner, Lauren Williams, Doc. Math. 13Alex Postnikov, Victor Reiner, and Lauren Williams. Faces of generalized permuto- hedra. Doc. Math., 13:207-273, 2008. . Alexander Postnikov, Permutohedra, Beyond, Int. Math. Res. Not. IMRN. 6Alexander Postnikov. Permutohedra, associahedra, and beyond. Int. Math. Res. Not. IMRN, (6):1026-1106, 2009.	[]
[ "ON IRREDUCIBLE CHARACTERS OF THE IWAHORI-HECKE ALGEBRA IN TYPE A", "ON IRREDUCIBLE CHARACTERS OF THE IWAHORI-HECKE ALGEBRA IN TYPE A" ]	[ "Naihuan Jing ", "Ning Liu " ]	[]	[]	We use vertex operators to compute irreducible characters of the Iwahori-Hecke algebra of type A. Two general formulas are given for the irreducible characters in terms of those of the symmetric groups or the Iwahori-Hecke algebras in lower degrees. Explicit formulas are derived for the irreducible characters labeled by hooks and two-row partitions. Using duality, we also formulate a determinant type Murnaghan-Nakayama formula and give another proof of Ram's combinatorial Murnaghan-Nakayama formula. As applications, we study super-characters of the Iwahori-Hecke algebra as well as the bitrace of the regular representation and provide a simple proof of the Halverson-Luduc-Ram formula.The algebra H n (q) becomes a Frobenius algebra under the non-degenerate associative bilinear form B(h 1 , h 2 ) = χ(h 1 h 2 ), where χ( h∈Hn(q) a h h) = a 1 . The algebra H n is semisimple[5,24]and isomorphic to C[S n ], so its irreducible representations are also labeled by partitions of n. Let χ λ be the irreducible character of H n (q) associated with partition λ ⊢ n. Ram [18] used the quantum Schur-Weyl duality to prove the Frobenius type character formula for H n (q) in terms of one-row Hall-Littlewood functions and Schur symmetric functions (see also[14,13]). He also gave the q-analogue Murnaghan-Nakayama formula for H n (q) as an iterative combinatorial rule to compute the irreducible characters in terms of those corresponding to smaller partitions. Since Ram's work, there have been various discussions and generalizations on the rule, cf. [23, 8, 17, 2, 19, 21] etc. A nice computer algebra program [3] on Hecke algebras is also available. All these 2010 Mathematics Subject Classification. Primary: 20C08, 17B69; Secondary: 05E10.developments have shown that the q-Murnaghan-Nakayama rule remains the most practical algorithm to compute the irreducible characters of H n (q) (also see Starkey's rule[22,2]). Nevertheless, how to effectively carry out the computation deserves further study; getting new formulas and reformulating the known ones and related structures may also help practical computation and offer new perspective.The goal of this paper is to offer an efficient and practical method to compute irreducible characters of H n (q) using vertex operators. In the vertex operator approach to symmetric functions [9, 10], Hall-Littlewood and Schur symmetric functions are expressed as simple products of vertex operators.Unlike the usual raising operators in symmetric functions which are often not well-defined, the vertex operators obey nice algebraic structures, so the irreducible character values can be studied in terms of the vertex operators and their dual operators. Using this idea, we first derive two general formulas to compute irreducible character values of the Hecke algebra: the first one computes the character in terms of those of the symmetric group of lower degrees, while the second one reduces the computation to lower degree characters of the Hecke algebra. We emphasize that both general formulas are different from the q-Murnaghan-Nakayama formula. Thanks to these new formulas, we are able to give explicit compact formulas of the irreducible H n (q)-characters labeled by hooks and two-row partitions. The formulas are then utilized to give a simple proof of the q-analogue Berele-Regev formula for H n (q), which was first obtained [25] by using the iterative Murnaghan-Nakayama formula.Using the same strategy we formulate a determinant type Murnaghan-Nakayama rule for H n (q) by exploiting the Jacobi-Trudi rule. The determinantal version implies the combinatorial Murnaghan-Nakayama rule easily. Finally we compute the bitrace of the regular representation of the Iwarhoti-Hecke algebra, originally computed by Halverson, Luduc and Ram [6] using Roichman's formula. Our	10.1016/j.jalgebra.2022.01.020	[ "https://arxiv.org/pdf/2104.06067v1.pdf" ]	233,219,472	2104.06067	4cc70effeab6b7815b80930bb037cbfdf22b4ca2	ON IRREDUCIBLE CHARACTERS OF THE IWAHORI-HECKE ALGEBRA IN TYPE A 13 Apr 2021 Naihuan Jing Ning Liu ON IRREDUCIBLE CHARACTERS OF THE IWAHORI-HECKE ALGEBRA IN TYPE A 13 Apr 2021arXiv:2104.06067v1 [math.QA] We use vertex operators to compute irreducible characters of the Iwahori-Hecke algebra of type A. Two general formulas are given for the irreducible characters in terms of those of the symmetric groups or the Iwahori-Hecke algebras in lower degrees. Explicit formulas are derived for the irreducible characters labeled by hooks and two-row partitions. Using duality, we also formulate a determinant type Murnaghan-Nakayama formula and give another proof of Ram's combinatorial Murnaghan-Nakayama formula. As applications, we study super-characters of the Iwahori-Hecke algebra as well as the bitrace of the regular representation and provide a simple proof of the Halverson-Luduc-Ram formula.The algebra H n (q) becomes a Frobenius algebra under the non-degenerate associative bilinear form B(h 1 , h 2 ) = χ(h 1 h 2 ), where χ( h∈Hn(q) a h h) = a 1 . The algebra H n is semisimple[5,24]and isomorphic to C[S n ], so its irreducible representations are also labeled by partitions of n. Let χ λ be the irreducible character of H n (q) associated with partition λ ⊢ n. Ram [18] used the quantum Schur-Weyl duality to prove the Frobenius type character formula for H n (q) in terms of one-row Hall-Littlewood functions and Schur symmetric functions (see also[14,13]). He also gave the q-analogue Murnaghan-Nakayama formula for H n (q) as an iterative combinatorial rule to compute the irreducible characters in terms of those corresponding to smaller partitions. Since Ram's work, there have been various discussions and generalizations on the rule, cf. [23, 8, 17, 2, 19, 21] etc. A nice computer algebra program [3] on Hecke algebras is also available. All these 2010 Mathematics Subject Classification. Primary: 20C08, 17B69; Secondary: 05E10.developments have shown that the q-Murnaghan-Nakayama rule remains the most practical algorithm to compute the irreducible characters of H n (q) (also see Starkey's rule[22,2]). Nevertheless, how to effectively carry out the computation deserves further study; getting new formulas and reformulating the known ones and related structures may also help practical computation and offer new perspective.The goal of this paper is to offer an efficient and practical method to compute irreducible characters of H n (q) using vertex operators. In the vertex operator approach to symmetric functions [9, 10], Hall-Littlewood and Schur symmetric functions are expressed as simple products of vertex operators.Unlike the usual raising operators in symmetric functions which are often not well-defined, the vertex operators obey nice algebraic structures, so the irreducible character values can be studied in terms of the vertex operators and their dual operators. Using this idea, we first derive two general formulas to compute irreducible character values of the Hecke algebra: the first one computes the character in terms of those of the symmetric group of lower degrees, while the second one reduces the computation to lower degree characters of the Hecke algebra. We emphasize that both general formulas are different from the q-Murnaghan-Nakayama formula. Thanks to these new formulas, we are able to give explicit compact formulas of the irreducible H n (q)-characters labeled by hooks and two-row partitions. The formulas are then utilized to give a simple proof of the q-analogue Berele-Regev formula for H n (q), which was first obtained [25] by using the iterative Murnaghan-Nakayama formula.Using the same strategy we formulate a determinant type Murnaghan-Nakayama rule for H n (q) by exploiting the Jacobi-Trudi rule. The determinantal version implies the combinatorial Murnaghan-Nakayama rule easily. Finally we compute the bitrace of the regular representation of the Iwarhoti-Hecke algebra, originally computed by Halverson, Luduc and Ram [6] using Roichman's formula. Our Introduction Let C(q) be the field of rational functions in the variable q. The Iwahori-Hecke algebra H n (q) is the unital associative algebra over C(q) generated by generators T 1 , T 2 , . . . , T n−1 subject to the relations T i T j = T j T i , if \|i − j\| > 1, (1.1) T i T i+1 T i = T i+1 T i T i+1 , (1.2) T 2 i = (q − 1)T i + q. (1.3) method is a straightforward computation using the vertex operator techniques. The structure of the paper is as follows. Section 2 discusses how to treat irreducible characters by the vertex operator realization of Schur and Hall-Littlewood symmetric functions. We express all irreducible characters of the Hecke algebra of type A n−1 as matrix coefficients of vertex operators in [9]. Based on this we derive two general formulas to compute all irreducible characters of the Iwahori-Hecke algebra, including one to express the characters in terms of those of the symmetric group in lower degrees. We derive compact formulas for the irreducible characters corresponding to hooks and two-row partitions (see (2.31)-(2.32)). In Section 3, we first use the general vertex operator formula to formulate a determinant type Murnaghan-Nakayama formula for H n (q), which gives another proof of the combinatorial one. Our current approach to the problem is based upon the idea of dual vertex operators developed in [9] and [11], which was first used in [10] on Schur's Q-functions. Vertex operators and character values χ λ µ (q) Let Λ be the ring of symmetric functions in the x n (n ∈ N) over the integers. In this paper we mostly work with the ring Λ F over the field F = Q(t) and view it as a graded ring under the natural degree. The ring Λ has several linear bases indexed by partitions. A partition λ = (λ 1 , λ 2 , . . .) is a weakly decreasing sequence of non-negative integers. The sum \|λ\| = i λ i is called the weight and the number of nonzero parts is called the length l(λ). A partition λ of weight n is denoted by λ ⊢ n, and the set of partitions is denoted by P. When the parts of λ are arranged in increasing order, so λ = (1 m 1 2 m 2 . . .) where m i = Card{λ j = i \| 1 ≤ j ≤ l(λ)} is the multiplicity of i in λ. λ is called a strict partition if m i ≤ 1. When the finite sequence λ = (λ 1 , λ 2 , . . .) of nonnegative integers is an ordered or not necessarily weakly decreasing one such that i λ i = n, λ is called a composition of n = i λ i , denoted as λ \|= n. The length l(λ) is the number of nonzero parts. A partition λ is visualized by its Young diagram: the set of nodes (or boxes situated at) (i, j) ∈ Z 2 + such that 1 ≤ j ≤ λ i . If λ is a diagram, then an inner corner of λ is a node (i, j) ∈ λ whose removal still leaves the diagram as that of a partition. The conjugate partition λ ′ = (λ ′ 1 , . . . , λ ′ λ 1 ) corresponds to the reflection of the Young diagram along the diagonal. For each r > 1, let p r = x r i be the rth power-sum. Then p λ = p λ 1 p λ 2 · · · p λ l (λ ∈ P) form a Q-basis of Λ Q . Let s λ be the Schur function associated with the partition λ, then s λ 's form an orthonormal basis Λ under the inner product p λ , p µ = δ λµ z λ (2.1) where z λ = i≥1 λ i m i (λ)!. The function p n acts on Λ as a multiplication operator, and its the dual operator is the differential operator p * n = n ∂ ∂pn . Note that * is Q(t)-linear and anti-involutive satisfying (2.2) p n u, v = u, p * n v for u, v ∈ Λ. We now recall the vertex operator realization of the Schur symmetric functions [10]. Let S(z) and the dual vertex operator S * (z) be the linear maps: Λ −→ Λ[[z, z −1 ]] defined by S(z) = exp   n≥1 1 n p n z n   exp   − n≥1 ∂ ∂p n z −n   = n∈Z S n z n , (2.3) S * (z) = exp   − n≥1 1 n p n z n   exp   n≥1 ∂ ∂p n z −n   = n∈Z S * n z −n . (2.4) We use the convention to index the components by their degrees. The operators S n ∈ End(Λ) are the Bernstein vertex operators realizing the Schur functions. The dual operators S * n ∈ End(Λ), introduced in [9], also realize the Schur functions. The following relations will be useful in our discussion. Proposition 2.1. [9] (1) The components of S(z) and S * (z) obey the following commutation relations: S m S n + S n−1 S m+1 = 0, (2.5) S * m S * n + S * n+1 S * m−1 = 0, (2.6) S m S * n + S * n−1 S m−1 = δ m,n . (2.7) (2) For any composition µ = (µ 1 , . . . , µ k ), the product S µ 1 · · · S µ k .1 = s µ is the Schur function labeled by µ. In general, s µ = 0 or ±s λ for a partition λ such that λ ∈ S l (µ + δ) − δ. Here δ = (l − 1, l − 2, . . . , 1, 0), where l = l(µ). Moreover, S −n .1 = δ n,0 , S * n .1 = δ n,0 , (n ≥ 0). Let q n = q n (x; t) be the generalized homogeneous symmetric function defined by 8) and q n = 0 if n < 0. Denote q λ = q λ 1 q λ 2 · · · q λ l for any partition λ, then the set {q λ } forms a basis of Λ Q(t) . Their dual operators with respect to the inner product (2.1) are defined by 9) and q * n = 0 if n < 0. In particular, q * n .1 = δ n,0 for n ≥ 0. For n ≥ 0, by (2.8) and (2.4), we have Q(z) = exp ∞ n=1 1 − t n n p n z n = n≥0 q n z n ,(2.Q * (z) = exp ∞ n=1 (1 − t n ) ∂ ∂p n z −n = n≥0 q * n z −n ,(2.q n = λ⊢n 1 z λ (t) p λ , (2.10) S * −n .1 = λ⊢n (−1) l(λ) z λ p λ , (2.11) where z λ (t) = i≥1 i m i (λ) m i (λ)! 1−t λ i , and z λ = z λ (0). Let H n (q) be the Iwahori-Hecke algebra of type A (see (1.1)-(1.3)). Let w = s i 1 · · · s i k be a reduced expression of w ∈ S n , where s i = (i, i + 1). We define T w = T i 1 · · · T i k , which is well-defined and independent from the choice of reduced expressions. Then H n (q) has a linear basis consisting of T w , w ∈ S n . For σ r = (12 · · · r) ∈ S n in cycle notation, let T σr be the corresponding element in H n (q). For any composition µ = (µ 1 , . . . , µ l ) of n, let σ µ = σ µ 1 × · · · × σ µ l ∈ S µ 1 × · · · × S µ l ֒→ S n , we define T σµ = T σµ 1 · · · T σµ l . Let φ : H n (q) −→ End(V ) be an irreducible representation associated with partition λ. In general the character χ λ (T ) = Tr(φ(T )) is no longer a function of conjugacy classes of S n . However, it is known that all irreducible character values at elements T w are determined by their values at the element T σµ [4]. For this reason, we will denote χ λ (T σµ ) = χ λ µ (q). Denoteq r (t) = t r t−1 q r (t −1 ) and letq µ (t) =q µ 1 (t)q µ 2 (t) · · ·q µ l (t) = t \|µ\| (t−1) l(µ) q µ (t −1 ) . We have the following Frobenius type formula for the characters of H n (q) from [18]. Proposition 2.2. [18] The irreducible character χ λ of H n corresponding to λ is determined bỹ q µ (q) = λ⊢n χ λ µ (q)s λ . (2.12) where µ ⊢ n and s λ is the Schur function associated with partition λ. Therefore, χ λ µ (q) = q µ (q), s λ = q \|µ\| (q − 1) l(µ) q µ (q −1 ), S λ .1 , (2.13) and we are going to compute g λ µ (t) := q µ (t), S λ .1 in the following. Proposition 2.3. For any m ∈ Z + , n ∈ Z S * n q m = q m S * n + (1 − t) m k=1 q m−k S * n−k , (2.14) q * m S n = S n q * m + (1 − t) m k=1 S n−k q * m−k . (2.15) Proof. The usual vertex operator calculus gives that S * (z)Q(w) = Q(w)S * (z) z − tw z − w , (2.16) Q * (w)S(z) = S(z)Q * (w) w − tz w − z , (2.17) where the rational functions are expanded at w = 0 and z = 0 respectively. The relations then follow by comparing coefficients of z −n w m in (2.16) and z n w −m in (2.17) respectively. For two compositions λ, µ, we say λ ⊂ µ if λ i ≤ µ i for all i ≥ 1. In this case, we write λ − µ = (λ 1 − µ 1 , λ 2 − µ 2 , . . .) \|λ\| − \|µ\|. For each partition λ = (λ 1 , . . . , λ l ), we define that λ [i] = (λ i+1 , · · · ,q * k S λ .1 = τ \|=k (1 − t) l(τ ) S λ−τ .1 (2.19) S * k q µ = τ ⊂ µ \|τ \| ≥ k (1 − t) l(τ ) q µ−τ S * k−\|τ \| .1 = n i=k τ ∈C µ i (1 − t) l(τ ) q µ−τ S * k−i .1, (2.20) where C µ k {τ \|= k \| τ ⊂ µ}. Proof. We argue by induction on l(λ) for the first relation. The initial step is clear. Assume that (2.19) holds for any partition with length < l(λ), it follows from Proposition 2.3 that q * k S λ = S λ 1 q * k S λ 2 · · · S λ l .1 + (1 − t) k i=1 S λ 1 −i q * k−i S λ 2 · · · S λ l .1 = τ \|=k (1 − t) l(τ ) S λ 1 S λ [1] −τ .1 + k i=1 τ \|=k−i (1 − t) l(τ )+1 S λ 1 −i S λ [1] −τ .1 = τ \|=k (1 − t) l(τ ) S λ−τ .1. The other relation (2.20) is shown similarly. Example 2.5. Let λ = (321), µ = (2 2 1 2 ) be two partitions. g λ µ (t) = q 2 q 2 q 1 q 1 , S 3 S 2 S 1 .1 = (1 − t) 2 q 2 q 1 q 1 , S 3 S 1 .1 + S 2 S 2 .1 + S 2 S 1 S 1 .1 = 2(1 − t) 4 q 1 q 1 , S 1 S 1 .1 + S 2 .1 = 4(1 − t) 5 q 1 , S 1 .1 = 4(1 − t) 6 . To compute the characters of the Hecke algebra, we collect some simple facts (see [12]). Lemma 2.6. For partitions λ, µ ⊢ n, λ⊢n (−1) l(λ) z λ (t) =      t n − t n−1 if n ≥ 1 1 if n = 0 (2.21) λ⊢n 1 z λ (t) =      1 − t if n ≥ 1 1 if n = 0 (2.22) χ (1 n ) µ (1) = p µ , S (1 n ) .1 = (−1) n−l(µ) (2.23) χ (n) µ (1) = p µ , S n .1 = 1. (2.24) The following formulas are well-known [18]. They are also obtained as special cases in our formulas for the hook and two-row cases. Proposition 2.7. For µ ⊢ n, one has that χ (n) µ (q) = q n−l(µ) , (2.25) χ (1 n ) µ (q) = (−1) n−l(µ) . (2.26) For µ ⊢ n and 1 ≤ i ≤ n, we introduce two sequences of polynomials respectively in t and t −1 : a i (µ; t) = (1 − t) −l(µ) τ ∈C µ i (1 − t) l(µ−τ ) (1 − t −1 ) l(τ ) , (2.27) b i (µ; t) = (1 − t −1 ) −l(µ) τ ∈C µ i (1 − t −1 ) l(µ−τ )+l(τ ) . (2.28) They are fixed by the generating functions shown below. Lemma 2.8. For a partition µ = (µ 1 , . . . , µ r ), we have that (1 − t −1 v) r r i=1 (1 + v + · · · + v µ i −1 ) = \|µ\| i=0 a i (µ, t)v i (2.29) r i=1 (t −1 + v µ i + (1 − t −1 )(1 + v + · · · + v µ i −1 )) = \|µ\| i=0 b i (µ, t)v i (2.30) Proof. Let c i (µ; t ′ ) = (1 − t ′ ) −r τ ∈C µ i (1 − t ′ ) l(µ−τ ) (1 − t −1 ) l(τ ) . Then (1 − t ′ ) r \|µ\| i=0 c i (µ; t ′ )v i = \|µ\| k=0 τ ⊂µ,τ k (1 − t ′ ) l(µ−τ ) (1 − t −1 ) l(τ ) v k = \|µ\| k=0 τ 1 +···+τr=k (1 − t ′ ) r−δτ 1 ,µ 1 −···−δτ r ,µr (1 − t −1 ) r−δ τ 1 ,0 −···−δ τr ,0 v k = (1 − t ′ ) r r i=1 µ i τ i =0 (1 − t ′ ) −δτ i ,µ i (1 − t −1 ) 1−δ τ i ,0 v τ i , which implies the lemma by noting that 1 + n−1 i=1 (1 − t −1 )v i + 1 − t −1 1 − t ′ v n =      (1 − t −1 v)[n] v , t ′ = t t −1 + v k + (1 − t −1 )[n] v , t ′ = t −1 . We remark that a 0 (µ; t) = b 0 (µ; t) = 1 and for any 0 ≤ j ≤ n = \|µ\| a j (µ; t) = (−1) l(µ) a n−j (µ; t −1 ), b j (µ; t) = b n−j (µ; t). Now we give the irreducible character formulas labeled by hook and two-row partitions. Theorem 2.9. For partition µ ⊢ n and k ≥ 0, we have χ (k,1 n−k ) µ (q) = (−1) n−k+l(µ) n i=k a i (µ; q)q i (2.31) Proof. Let l = l(µ). By Theorem 2.4 it follows that g (k,1 n−k ) µ (t) = q µ , S (k,1 n−k ) .1 = S * k q µ , S (1 n−k ) .1 = n i=k \| τ \|= i τ ⊂ µ (1 − t) l(τ ) q µ−τ S * k−i .1, S (1 n−k ) .1 = n i=k \| τ \|= i τ ⊂ µ (1 − t) l(τ ) µ (1) ⊢µ 1 −τ 1 ,··· ,µ (l) ⊢µ l −τ l λ⊢i−k (−1) l(λ) z µ (1) (t) · · · z µ (l) (t)z λ p µ (1) ∪···∪µ (l) ∪λ , S (1 n−k ) .1 = n i=k \| τ \|= i τ ⊂ µ (1 − t) l(τ ) µ (1) ⊢µ 1 −τ 1 · · · µ (l) ⊢µ l −τ l λ⊢i−k (−1) n−k (−1) l(µ (1) ) · · · (−1) l(µ (l) ) z µ (1) (t) · · · z µ (l) (t)z λ =(−1) n−k n i=k τ ∈C µ i t n−i (1 − t −1 ) l(µ−τ ) (1 − t) l(τ ) , where we have used (2.21) and (2.22) in the last equation. The theorem follows by recalling (2.27) and χ λ µ (q) = (−1) l(µ) q n (1 − q) −l(µ) g λ µ (q −1 ). Lemma 2.8 offers practical way to compute (2.31) by expanding the generating function up to certain power of v and then setting v = q. Example 2.10. Let λ = (6, 1 2 ), µ = (2 4 ). Taking terms of v with degree ≥ 6 and then setting v = q, (1 − t −1 v) 4 (1 + v) 4 ≡ (1 − t −1 v) 4 (6v 2 + 4v 3 + v 4 ) ≡ t −4 v 4 · 6v 2 + (t −4 v 4 − 4t −3 v 3 )4v 3 + (t −4 v 4 − 4t −3 v 3 + 6t −2 v 2 )v 4 ≡ 6t 2 − 12t 3 + 3t 4 . Therefore χ (6,1 2 ) (2 4 ) (q) = 6q 2 − 12q 3 + 3q 4 . Theorem 2.11. For partition µ ⊢ n and k ≥ n − k ≥ 0 we have χ (k,n−k) µ (q) = q n−l(µ) (b k (µ; q) − b k+1 (µ; q)), (2.32) [n/2] i=0 χ (n−i,i) µ (q) = q n−l(µ) b [n/2] (µ; q). (2.33) Proof. By the same argument in the proof of Theorem 2.9 we see that g (k,n−k) µ (t) = S * k q µ , S n−k .1 is equal to n i=k τ ∈C µ i (1 − t) l(τ ) µ (1) ⊢µ 1 −τ 1 ,··· ,µ (l) ⊢µ l −τ l λ⊢i−k (−1) l(λ) z µ (1) (t) · · · z µ (l) (t)z λ = τ ∈C µ k (1 − t) l(µ−τ )+l(τ ) − τ ∈C µ k+1 (1 − t) l(µ−τ )+l(τ ) , which implies the first result by (2.28). The second one is clear. Example 2.12. Let λ = (4, 2), µ = (3, 2, 1). By Theorem 2.11 3 i=1 (t −1 + v i + (1 − t −1 )[i] v ) − v −1 3 i=1 (t −1 + v i + (1 − t −1 )[i] v ) = (v − v −1 )(t −1 + v 2 + (1 − t −1 )(1 + v))(t −1 + v 3 + (1 − t −1 )(1 + v + v 2 )). The coefficient of v 4 is (1 − t −1 )(2 − t −1 ). Therefore χ (42) (321) (q) = q 3 (b 4 (q) − b 5 (q)) = q 3 (2 − 3q −1 + q −2 ) = 2q 3 − 3q 2 + q. Using the similar argument, we can compute the general character values. For a partition-valued function µ = (µ (1) , . . . , µ (r) ), we define (2.34) z µ (t) = z µ (1) (t) · · · z µ (r) (t), and call \|µ\| = i \|µ (i) \| the weight of µ and l(µ) = i l(µ (i) ) the length of µ. Given a composition τ = (τ 1 , . . . , τ r ), we denote by µ ⊢ τ the partition-valued function µ = (µ (1) , . . . , µ (r) ) such that µ (i) ⊢ τ i . Clearly \|µ\| = \|τ \|. The following general formula expresses the Hecke algebra character in terms of irreducible characters of the symmetric groups of lower degrees. Theorem 2.13. For λ, µ ⊢ n, the irreducible character χ λ µ (q) is given by χ λ µ (q) = 1 (q − 1) l(µ) n i=λ 1 q i \|τ \| = i τ ⊂ µ (1 − q −1 ) l(τ ) ν⊢µ−τ ρ⊢i−λ 1 (−1) l(ν)+l(ρ) z ν (q)z ρ χ λ [1] ν∪ρ . (2.35) Proof. Let l(µ) = l. As in the proof Theorem 2.9 we have that χ λ µ (q) = q n (q − 1) l S * λ 1 q µ , S λ [1] .1 = q n (q − 1) l n i=λ 1 \| τ \|= i τ ⊂ µ (1 − q −1 ) l(τ ) ν⊢µ−τ ρ⊢i−λ 1 (−1) l(ρ) z ν (q −1 )z ρ χ λ [1] ν∪ρ = 1 (q − 1) l n i=λ 1 \| τ \|= i τ ⊂ µ q i (1 − q −1 ) l(τ ) ν⊢µ−τ ρ⊢i−λ 1 (−1) l(ρ)+l(ν) z ν (q)z ρ χ λ [1] ν∪ρ . We remark that one obtains another general formula for χ λ µ (q) based on the transition matrix from the elementary symmetric function S * −m .1 to the generalized symmetric functions q ρ . Using the definition of Q(z), it is easy to see that (2.36) S * (tz).1 = Q(z)S * (z).1 Then we have the generalized Newton's formula: (2.37) S * −m .1 = 1 t m − 1 (q 1 S * −m+1 .1 + q 2 S * −m+2 .1 + · · · q m−1 S * −1 .1 + q m ), which leads to the decomposition: S * −m .1 = ρ⊢m C m,ρ (t)q ρ . The first few terms are given by S * −1 .1 = q 1 t − 1 , S * −2 .1 = q 2 1 (t 2 − 1)(t − 1) + q 2 t 2 − 1 , S * −3 .1 = q 3 1 (t 3 − 1)(t 2 − 1)(t − 1) + (t + 2)q 2 q 1 (t 3 − 1)(t 2 − 1) + q 3 t 3 − 1 , S * −4 .1 = q 4 1 (t 4 − 1)(t 3 − 1)(t 2 − 1)(t − 1) + (t 2 + t + 2)q 3 q 1 (t 4 − 1)(t 3 − 1) + (t 2 + 2t + 3)q 2 q 2 1 (t 4 − 1)(t 3 − 1)(t 2 − 1) + q 2 2 (t 4 − 1)(t 2 − 1) + q 4 t 4 − 1 . The following result can be shown similarly as Theorem 2.13. Theorem 2.14. For λ, µ ⊢ n. Then the irreducible character χ λ µ (q) is given by χ λ µ (q) = q λ 1 (q − 1) l(µ) n i=λ 1 \|τ \| = i τ ⊂ µ (1 − q −1 ) l(τ ) ρ⊢i−λ 1 (q − 1) l(µ−τ )+l(ρ) C i−λ 1 ,ρ (q −1 )χ λ [1] (µ−τ ) * ∪ρ (q), where C m,ρ (t) is the transition coefficient from S * −m .1 to q ρ and (µ − τ ) * is the rearranged partition obtained from µ − τ . Applications of the hook and two-row formulas Let V = V0 V1 be a Z 2 -graded vector space over C(q) with basis {v 1 , v 2 , · · · , v a } for the even subspace V0 and basis {v a+1 , v a+2 , · · · , v a+b } for the odd subspace V1. Let π be the endomorphism of V ⊗ V by (v k ⊗ v l )π =            (−1) \|v k \|\|v l \| v l ⊗ v k + (q − 1)v k ⊗ v l , k < l (−1) \|v k \| (q+1)+q−1 2 v k ⊗ v l , k = l (−1) \|v k \|\|v l \| tv l ⊗ v k , k > l,(3.1) and let π i be the endomorphism of V ⊗n by letting π acting on the (i, i + 1) factor and identity elsewhere, i = 1, · · · , n − 1. It is known [16] that the map Φ q,n a,b : H n (q) −→ End C(t) (V ⊗n ) defined by T i −→ π i gives rise to a representation of H n (q) called the sign q-permutation representation. Let ϕ n a,b be the character of Φ q,n a,b and let s a,b (λ) be the number of (a, b)-semi standard tableaux of shape λ [20]. Then ϕ n a,b decomposes as follows [25]: ϕ n a,b = λ∈H(a,b;n) s a,b (λ)χ λ where H(a, b; n) = {λ ⊢ n \| λ a+1 ≤ b} and χ λ is the irreducible character of H n (q) labeled by λ ⊢ n. We consider two special cases a = b = 1 and a = 2, b = 0. Then ϕ n 1,1 = n−1 k=0 χ (n−k,1 k ) (3.2) ϕ n 2,0 = [n/2] k=0 (n − 2k + 1)χ (n−k,k) (3.3) Theorem 3.1. For µ ⊢ n with l(µ) = l, we have that ϕ n 1,1 (T γµ ) = n−1 i=0 χ (n−i,1 i ) µ (q) = (−1) n−l 2 l−1 l i=1 [µ i ] −q , (3.4) ϕ n 2,0 (T γµ ) = [n/2] i=0 (n − 2i + 1)χ (n−i,i) µ (q) = q n−2l(µ) l i=1 (1 + q + µ i (q − 1)). (3.5) Proof. It follows from the hook formula (2.31) that n−1 i=0 χ (n−i,1 i ) µ (q) = n−1 i=0 (−1) l+i n j=n−i a j (µ; q)q j =(−1) n+l−1 [n/2] j=1 a 2j−1 (µ; q)q 2j−1 =(−1) n−l n j=0 (−1) j − 1 2 a j (µ; q)q j =(−1) n−l 2 l−1 l i=1 [µ i ] −q , where we have used Lemma 2.8 and the identity n i=0 a i (µ; q)q i = 0. Similarly, using the two-row formula (2.32) and (2.30) we have that [n/2] i=0 (n − 2i + 1)χ n−i,i µ (q) = [n/2] i=0 (n − 2i + 1)q n−l (b n−i (µ; q) − b n−i+1 (µ; q)) =q n−l n i=0 b i (µ; q) =q n−2l(µ) l i=1 (1 + q + µ i (q − 1)). We remake that Zhao [25] obtained a negated version of (3.4) as a q-analog Berele-Regev formula [1] by the Murnaghan-Nakayama rule. Determinant type Murnaghan-Nakayama rule The Murnaghan-Nakayama rule is an iterative formula to compute the Hecke algebra characters. The combinatorial rule was proved by Ram [18] using the Frobenius formula and the Hall-Littlewood symmetric functions. In this section, we formulate a determinant type Murnaghan-Nakayama rule and use it to compute the characters and also give a new proof of the Murnaghan-Nakayama rule. For two partitions µ ⊂ λ, the set-theoretic difference θ = λ − µ is called a skew diagram denoted by λ/µ. A subset ξ of θ is connected if any two squares in θ are connected by a path in ξ. The connected components, themselves skew diagrams, are by definition the maximal connected subsets of θ. A skew diagram λ/µ is a vertical (resp. horizontal) strip if each row (resp. column) contains at most one box. A skew diagram θ is a border strip if it contains no 2 × 2 blocks of squares and is connected (see [15]). If the border strip θ = λ − µ has m connected components (ξ 1 , ξ 2 , . . . , ξ m ), we define the weight of θ by wt(θ; t) = (t − 1) m−1 m i=1 (−1) r(ξ i )−1 t c(ξ i )−1 . where the r(ξ i ) (resp. c(ξ i )) means the number of rows (resp. columns) in the border strip ξ i . Usually we refer θ as a broken border strip to emphasize the non-connectedness of θ. Let λ ⊢ n. A Young tableau of shape λ is an assignment of numbers 1, 2, . . . , n into the dots of the Young diagram. A tableau is standard if its rows and columns are increasing sequences. Let f λ be the number of standard tableau of shape λ and let λ − be a partition obtained by removing an inner corner of λ, then [20] f λ = λ − f λ − . (4.1) Recall from (2.19), for a fixed partition λ with l = l(λ) we have that q * k S λ .1 = τ1, . . . , τ l ≥ 0 \|τ \| = k (1 − t) l− δ τ i ,0 S λ−τ .1. (4.2) For any composition τ k, S λ−τ .1 = sg(σ)S µ .1 if λ − τ + δ is strict and µ = σ(λ − τ + δ) − δ for some permutation σ ∈ S l , otherwise S λ−τ .1 = 0. Note that µ must be a partition ⊂ λ with weight \|λ\| − k if S λ−τ .1 = 0. In fact, note that µ i = λ σ(i) − τ σ(i) + i − σ(i) for some permutation σ that sends λ − τ + δ into partition µ. If j = σ(i) ≥ i, then λ i − µ i = λ i − λ j + τ j − i + j ≥ 0. If j = σ(i) < i, we claim that λ i − µ i ≥ 0 otherwise λ i − µ i = λ i − λ j + τ j − i + j < 0, then (λ j − τ j , λ j+1 − τ j+1 , . . . , λ i − τ i ) + (i − j, . . . , 1, 0) = (λ j − τ j + i − j, . . . , λ i − τ i ). Observe that (λ i − τ i ) − (λ j − τ j + i − j) < 0 so σ cannot send i to j, which is a contradiction! Conversely, for any µ ⊢ \|λ\| − k and any σ ∈ S l , the l-tuple (λ 1 − µ σ(1) − 1 + σ(1), λ 2 − µ σ(2) − 2 + σ(2), · · · , λ l − µ σ(l) − l + σ(l)) is a composition of k provided that λ i − µ σ(i) + i − σ(i) ≥ 0. Then sg(σ)S µ .1 = S λ−τ .1 for τ i = λ i − µ σ(i) − i + σ(i). Therefore (4.2) can be rewritten as: q * k S λ .1 = µ σ ∈ S l λi − i − µ σ(i) + σ(i) ≥ 0 sgn(σ)(1 − t) l− δ λ i −i−µ σ(i) +σ(i),0 S µ .1 = (1 − t) l µ det(δ λ i −i≥µ j −j (1 − t) −δ λ i −i,µ j −j )S µ .1 where the sum is over all partitions µ ⊂ λ such that \| λ/µ \|= k. For partitions µ ⊂ λ and \|λ/µ\| = k, consider the l(λ) × l(λ)-matrix M (λ/µ; t) = (δ λ i −i≥µ j −j (1 − t) −δ λ i −i,µ j −j ) where δ a≥b = 1 if a ≥ b and δ a≥b = 0 otherwise. Then we have proven the following theorem. q * k S λ .1 = µ (1 − t) l det M (λ/µ; t)S µ .1 (4.3) where the sum is over all partitions µ ⊂ λ such that \| λ/µ \|= k. The matrix M (λ/µ; t) has the following properties: (i) if M (λ/µ; t) ij = 1 1−t , then all entries in the southwest region to the (i, j)-entry are zero; (ii) if M (λ/µ; t) ij = 0, then all entries down the jth column are also zero; and (iii) if M (λ/µ; t) ij = 1, then all entries in the northeast region to the (i, j)-entry are also 1. et's divide λ/µ into several cases: Case 1 : If λ/µ is a k-border strip with the initial box at the rth row, then M (λ/µ; t) has the following form:                     1 1−t · · · 1 1 1 · · · 1 1 . . . . . . . . . . . . . . . · · · . . . . . . 0 · · · 1 1−t 1 1 · · · 1 1 0 · · · 0 1 1 · · · 1 1 0 · · · 0 1 1−t 1 · · · 1 1 0 · · · 0 0 1 1−t · · · 1 1 . . . . . . . . . . . . . . . . . . . . . . . . 0 · · · 0 0 0 · · · 1 1−t 1                     l×l =   M 0 * 0 M 1   , det(λ/µ; k) = (1 − t) −r+1 ( −t 1 − t ) l−r where M 0 is an (r − 1) × (r − 1) upper-triangular matrix with 1 1−t on the diagonal and M 1 is a (l − r + 1) × (l − r + 1) quasi upper-triangular matrix with 1 on the diagonal and 1 1−t on the (lower) secondary diagonal. For simplicity we denote M 1 by M (ξ; t), where ξ is the border strip obtained from λ/µ by removing the initial empty boxes. Case 2 : If λ/µ is connected and contains a 2 × 2 block of boxes, then det(λ/µ; t) = 0. This can be seen as follows. Without loss of generality, suppose ht(λ/µ) = l(λ) − 1 and the 2 × 2 block appear at the r-th and (r + 1)-th rows (1 ≤ r ≤ l − 1), then M (λ/µ; t) has the following form: M (λ/µ; t) =                           r − 1 r 1 1 · · · 1 1 · · · 1 1 1 1−t 1 · · · 1 1 · · · 1 1 0 1 1−t · · · 1 1 · · · 1 1 . . . . . . . . . . . . . . . . . . . . . . . . r 0 0 · · · 1 1−t 1 · · · 1 1 r + 1 0 0 · · · 0 1 · ·0 0 0 0 0 · · · 1 1−t 1                           , det(λ/µ; t) = 0. Case 3 : If λ/µ has two connected components ξ 1 , ξ 2 , then M (λ/µ; t) is a 2 × 2 upper-triangular block matrix with the diagonal blocks corresponding to ξ 1 , ξ 2 . If one of the components contains a 2×2-block of boxes, then det(λ/µ; t) = 0. Suppose each ξ i is a border strip and there are r i unoccupied rows immediately above ξ i on the border of λ, then M (λ/µ; t) has the following form: M (λ/µ; t) =   M 1 * 0 M 2   , det M i = (1 − t) −r i +1 ( −t 1 − t ) ht(ξ i )−1 , where each M i is a 2 × 2 block diagonal matrices specified in case one. In general, if λ/µ has m connected components ξ 1 , ξ 2 , · · · , ξ m , then det M (λ/µ; t) = 0 unless each ξ i is a border strip. Then M (λ/µ; t) =         M 1 * · · · * 0 M 2 · · · * 0 . . . · · · . . . 0 0 . . . M s         where the diagonal blocks are either 1 1−t or the quasi upper-triangular matrix M (ξ j ; t), i = 1, 2, . . . , s; j = 1, 2 . . . , m. It is clear that the number of 1 1−t on the diagonal is equal to Card{λ i = µ i \| i = 1, . . . , l} = l − j r(ξ j ) and the order of M (ξ j ; k) is equal to r(ξ j ), j = 1, 2, . . . , m. Therefore, det(λ/µ; t) = ( 1 1 − t ) l− r(ξ i ) m i=1 det(ξ i ; t) = ( 1 1 − t ) l− r(ξ i ) m i=1 ( −t 1 − t ) r(ξ i )−1 = (1 − t) m−l m i=1 (−t) r(ξ i )−1 . Thus we have proved the following result. q * k S λ .1 = µ (1 − t) C(λ/µ) C(λ/µ) i=1 (−t) r(ξ i )−1 S µ .1 (4.4) where the sum runs through all partitions µ ⊂ λ such that λ/µ is a k-broken border strip. Here ξ 1 , . . . , ξ C(λ/µ) are the connected components of λ/µ. g λ (n) (t) =      0, if λ is not a hook (1 − t)(−t) n−m , if λ = (m, 1 n−m ). Similarly, when k = 1, then q * 1 S λ .1 = (1 − t) λ − S λ − .1, where λ − runs through all partitions obtained from λ by removing an inner corner. Therefore, g λ (1 n ) (t) = (1 − t) λ − g λ − (1 n−1 ) (t). By (4.1) it follows that, g λ (1 n ) (t) = (1 − t) n f λ where f λ is the number of standard tableau of shape λ. We now present a new proof of the Murnaghan-Nakayama rule for H n (q) [18] (cf. [7,8]). where the sum is over all partitions λ ⊃ µ such that λ/µ is a k-broken border strip. Proof. By Corollary 4.2 and duality, it is enough to check that (1 − t) m m i=1 (−t) r(ξ i )−1 = t k−1 (1 − t)wt(λ/µ; t) where λ/µ is a k-broken border strip with m connected components ξ 1 , . . . , ξ m . Suppose \| ξ i \|= k i , then i k i = k. Since ξ i is a border strip, r(ξ i ) + c(ξ i ) = k i + 1, i = 1, 2, . . . , m. Therefore, t k−1 (1 − t)wt(λ/µ; t) = (1 − t) m t k−m m i=1 (−1) r(ξ i )−1 t 1−c(ξ i ) = (1 − t) m m i=1 (−1) r(ξ i )−1 t k i −c(ξ i ) = (1 − t) m m i=1 (−t) r(ξ i )−1 . The bitrace of the regular representation The Hecke algebra H n (q) is a (H n (q), H n (q))-bimodule under the left and right regular actions, and the two actions mutually commute with each other. As a result (5.1) H n (q) = λ⊢n V λ ⊗ V λ where V λ (resp. V λ ) is the irreducible left (resp. right) H n (q)-module labeled by λ. Following [6], we introduce the bitrace of the regular representation of H n (q) by defining for any compositions λ, µ \|= n btr(λ, µ) = ρ⊢n χ ρ λ (q)χ ρ µ (q). (5.2) This is a q-deformation of the second orthogonality relation between irreducible characters of the symmetric group. It is known that btr(λ, µ)\| q=1 = δ λ,µ z λ . Halverson, Luduc and Ram [6] proved the following result by using Roichman's formula [19] for the Hecke algebra. We will give an elementary proof using the technique developed in this paper. where M = (m ij ) run through all r×s nonnegative integral matrices such that a m ia = λ i (1 ≤ i ≤ r) and b m bj = µ j (1 ≤ j ≤ s) and wt(M ) = m ij =0 (q − 1) 2 [m ij ] q 2 . By the Frobenius formula of the Hecke algebra H n (q) it is readily seen that btr(λ, µ) = q λ (q),q µ (q) . Let H λ µ (t) = q λ (t), q µ (t) , then brt(λ, µ)(q) = q 2n (q−1) l(λ)+l(µ) H λ µ (q −1 ). Using vertex operators and (2.8)-(2.9), we have the following relation: Q * (z)Q(w) z − t 2 w z − tw = Q(w)Q * (z) z − tw z − w , (5.4) where the rational functions are power series in negative powers of z. Comparing coefficients of z −n w m we immediately obtain the following commutation relation. For simplicity we denote for integer k > 0 (k) t = (t − 1) t 2k − 1 t + 1 = (t − 1) 2 [k] t 2 and make the convention that (k) t = 0 for k < 0 and (0) t = 1. The following result is easily shown by induction, and also can be used as an inductive definition of (k) t . Lemma 5.3. Let k be a positive integer, then we have Proof. Use induction on k + n. The initial step is clear. Assume the identity holds for q * k ′ q µ such that k ′ + \|µ\| < k + n, then q * k q µ 1 q µ 2 · · · q µ l =q µ 1 q * k q µ 2 · · · q µ l + (1 − t) i≥1 q µ 1 −i q * k−i q µ 2 · · · q µ l + (t − 1) 1 − t + (t − 1) k i=1 (k − i) t t i = (k) t .i≥1 q * k−i q µ 1 −i q µ 2 · · · q µ l t i =q µ 1 τ \|=k τa≥1 (τ a ) t q µ [1] −τ + (1 − t) i≥1 τ \|=k−i τa≥1 (τ a ) t q µ 1 −i q µ [1] −τ + (t − 1) i≥1 τ \|=k−i τa≥1 (τ i ) t q µ 1 −i−τ 1 q µ [1] −τ [1] t i . The first summand is τ \|=k τa≥1 (τ a ) t q µ−τ with τ 1 = 0. Replacing i + τ 1 by j in the third summand and pulling (τ 1 ) t = (j − i) t forward, we simplify the second and the third summands as follows: (1 − t) i≥1 τ \|=k−i τa≥1 (τ a ) t q µ 1 −i q µ [1] −τ + (t − 1) i≥1 j≥i (j − i) t t i τ \|=k−j τa≥1 (τ a ) t q µ 1 −j q µ [1] −τ =(1 − t) i≥1 τ \|=k−i τa≥1 (τ a ) t q µ 1 −i q µ [1] −τ + j≥1 (t − 1) j i=1 (j − i) t t i τ \|=k−j τa≥1 (τ a ) t q µ 1 −j q µ [1] −τ = j≥1 (j) t τ \|=k−j τa≥1 (τ a ) t q µ 1 −j q µ [1] −τ (by Lemma 5.3). Combining this with the first summand, we see the total sum is τ \|=k τa≥1 (τ a ) t q µ−τ . The following is an immediate consequence of Theorem 5.4. This provides a simple proof of Theorem 5.1 due the fact that t 2k (k) t −1 = (k) t , and q i = 0 for i < 0. λ l ), i = 0, 1, . . . , l (2.18) So λ [0] = λ and λ [l] = ∅. Next, we give the main results. Theorem 2 . 4 . 24For partitions λ, µ ⊢ n and integer number k, Theorem 4. 1 . 1Let λ = (λ 1 , . . . , λ l ) be a partition and k be a positive integer. Then one has Corollary 4. 2 . 2Let λ be a partition and k a positive integer. Then one has that Remark 4 . 3 . 43If k = \|λ\| = n in Corollary 4.2, then q * k S λ .1 = 0 unless λ is a hook. In this case, q * n S (m,1 n−m ) .1 = (1 − t)(−t) n−m . So for all λ ⊢ n ≥ 1, Theorem 5 . 1 . 51Let λ = (λ 1 , λ 2 , . . . , λ r ) and µ = (µ 1 , µ 2 , . . . , µ s ) be two compositions of n. Proposition 5 . 2 . 52For any m, n ∈ Z + , as operators on Λ C[t,t −1 ] q * n q m = q m q * n + (1 − t) . 4 . 4Let k ∈ Z + and µ n a composition, thenq * k q µ = τ \|=k τa>0(τ a ) t q µ−τ . (5.7) Corollary 5. 5 . 5Let λ, µ \|= n with l(λ) = r and l(µ) = s. sum is over all r × s nonnegative integer matrices M = (τ (i) a ) 1≤i≤r,1≤a≤s such that the row sums are λ 1 , . . . , λ r and the column sums are µ 1 , . . . , µ s AcknowledgmentsThe work is partially supported by Simons Foundation grant No. 523868 and NSFC grant No.11531004. Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras. A Berele, A Regev, Adv. Math. 642A. Berele, A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superal- gebras, Adv. Math. 64(2) (1987), 118-175. The character table of the Iwahori-Hecke algebra of the symmetric group: Starkey's rule. M Geck, C. R. Acad. Sci. Paris Sér. I. Math. 329M. 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Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA Email address: [email protected] School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China Email address: [email protected]	[]
[ "Super-resolution in recovering embedded electromagnetic sources in high contrast media", "Super-resolution in recovering embedded electromagnetic sources in high contrast media" ]	[ "Habib Ammari ", "Bowen Li ", "Jun Zou " ]	[]	[]	The purpose of this work is to provide a rigorous mathematical analysis of the expected super-resolution phenomenon in the time-reversal imaging of electromagnetic (EM) radiating sources embedded in a high contrast medium. It is known that the resolution limit is essentially determined by the sharpness of the imaginary part of the EM Green's tensor for the associated background. We first establish the close connection between the resolution and the material parameters and the resolvent of the electric integral operator, via the Lippmann-Schwinger representation formula. We then present an insightful characterization of the spectral structure of the integral operator for a general bounded domain and derive the pole-pencil decomposition of its resolvent in the high contrast regime. For the special case of a spherical domain, we provide some quantitative asymptotic behavior of the eigenvalues and eigenfunctions. These mathematical findings shall enable us to provide a concise and rigorous illustration of the super-resolution in the EM source reconstruction in high contrast media. Some numerical examples are also presented to verify our main theoretical results.	10.1137/20m1313908	[ "https://arxiv.org/pdf/2001.07116v1.pdf" ]	210,839,439	2001.07116	1b1e197da1876cfec9c109216005af678b84389d	Super-resolution in recovering embedded electromagnetic sources in high contrast media 20 Jan 2020 Habib Ammari Bowen Li Jun Zou Super-resolution in recovering embedded electromagnetic sources in high contrast media 20 Jan 2020 The purpose of this work is to provide a rigorous mathematical analysis of the expected super-resolution phenomenon in the time-reversal imaging of electromagnetic (EM) radiating sources embedded in a high contrast medium. It is known that the resolution limit is essentially determined by the sharpness of the imaginary part of the EM Green's tensor for the associated background. We first establish the close connection between the resolution and the material parameters and the resolvent of the electric integral operator, via the Lippmann-Schwinger representation formula. We then present an insightful characterization of the spectral structure of the integral operator for a general bounded domain and derive the pole-pencil decomposition of its resolvent in the high contrast regime. For the special case of a spherical domain, we provide some quantitative asymptotic behavior of the eigenvalues and eigenfunctions. These mathematical findings shall enable us to provide a concise and rigorous illustration of the super-resolution in the EM source reconstruction in high contrast media. Some numerical examples are also presented to verify our main theoretical results. Introduction In this work, we study the potential super-resolution phenomenon when using the time-reversal imaging method to reconstruct the EM sources embedded in general media with high refractive indices. Among the various imaging algorithms, the time-reversal approach is one of the most direct and simplest ones. Its principle is to exploit the reciprocity of wave propagation. Intuitively, we retrace the path of the wave observed in the far field backwards in chronology to find the location of its generating source [37,36,19,20]. For a far-field imaging system using the timereversal method, we know from the Helmholtz-Kirchhoff integral that its resolution is limited by the imaginary part of the Green's function of the wave equations associated with the background medium [12,13]. It is connected with the so-called Abbe diffraction limit (half of the operating wavelength) via the concept of full width at half maximum (FWHM) [4,8]. In a more precise way, the sharper the imaginary part of the Green's function, the smaller the full width at its half maximum and the smaller scale the imaging system can resolve. Over the past several decades, intensive efforts have been made to explore the potential of breaking the diffraction limit in two-fold: generating a better raw images, and recovering the finer details of raw images by post-imaging processes. In this work, our discussion shall be restricted to the first procedure, that is, how to physically improve the resolution by obtaining the better a priori information. The Abbe diffraction limit actually results from the fact that the information about subwavelength details of the profile is carried out by the evanescent components of the scattered field that is basically unmeasurable in the far field [15,16], see also Proposition 3.18. To break the resolution barrier, we may need to capture the subwavelength information. It has been demonstrated in many different settings that using resonant media is a promising and feasible choice, e.g., the plasmonic nanoparticles [10,11,3], the bubbly media [5,4], the Helmholtz resonators [12], and the high contrast media [7,13,2]. Under specific circumstances, these resonant media can excite the resonances and serve as an amplifier that increases the strength of the subwavelength information of the sources encoded in the measured data. In general, they are mathematically equivalent to eigenvalue problems [13,5,10]. It was demonstrated in [10] that the surface plasmon resonance can be treated as an eigenvalue problem of the Neumann-Poincaré operator, which was further used to analyze the imaginary part of the Green's function and the possibility of achieving the super-resolution by using plasmonic nanoparticles. For the bubbly media, it was shown in [4] that the super-focusing of acoustic waves can be obtained at frequencies near the Minnaert resonance. The inverse source problem was investigated in [13] for the Helmholtz equation and the super-resolution was explained based on the resonance expansion of the Green's function associated with the medium with respect to the generalized eigenfunctions of the Riesz potential K k D (cf. (2.1)). As a complement of the work [13], the imaging of the target of high contrast was studied in [2] for the Helmholtz system and the experimentally observed super-resolution was illustrated via the concept of scattering coefficients. In this work, we consider the three-dimensional EM wave governed by the full Maxwell equations, and, with the help of an electric integral operator T k D , a solid mathematical foundation is provided for the the expected super-resolution phenomenon in the time-reversal reconstruction of EM sources embedded in a high contrast medium. We also develop some analytical tools very different from the acoustic cases to discuss several critical issues that were not covered in [13,2]. The contributions of this work are three-fold. Firstly, we derive the Lippmann-Schwinger equation to reveal the relations between the medium (shape and refractive indices) and its associated EM Green's tensor (cf. (2.10)), of which the explicit formula is not available. It is worth emphasizing that this derivation is not as trivial and standard as one might think, and, in fact, our arguments and analysis are very different from the ones in [13] for the Helmholtz equation and are much more involved. The main difficulty in our case arises from the strong singularity of the EM Green's tensor so the standard approach (see, e.g., [21,13]) that works for the functions with L 2 -regularity is not applicable. To deal with this problem, we deliberately choose a smooth cutoff function to separate the singular part from the Green's tensor G so that the remaining regular part can be represented by the Lippmann-Schwinger equation. Since the singular term is explicitly constructed, our decomposition (see Theorems 2.1 and 2.2) may also have potential applications in the numerical computation of G. Secondly, as we shall demonstrate, the mechanism underlying the super-resolution in resonant media is closely related to the spectral analysis of T k D , which is still far from being complete. For the case of the electric permittivity being smooth enough on the whole space, the integral operator involved in the Lippmann-Schwinger equation is compact and well-studied [21,22]. When the material coefficients have jumps across the medium interfaces, the integral operator is not compact and its spectral study is largely open. In [24], the authors investigated the essential spectrum of the integral operators arising from the EM scattering on the Lipschitz domain in two dimensions and gave a relatively complete characterization in various cases, which extended their earlier results in [22,23] where only the smooth domain was considered. We refer the readers to [35,18] for the numerical study of the spectrum of EM volume integral operators. To explore the spectral properties of the integral operator T k D in three dimensions, we first show that all the eigenvalues of T k D , except −1, of which the corresponding eigenspace consists of the nonradiating sources, lie in the upper-half plane of C; see Theorem 3.2. Then, by using the Helmholtz decomposition of L 2 -vector fields, we obtain a characterization of the essential spectrum of T k D in a more concise and constructive manner than the existing ones [23,24]. Combining the characterization with the analytic Fredholm theory, we further characterize its eigenvalues of finite type, and give the relation among these eigenvalues, the eigenvalues (point spectrum) and the essential spectrum in Theorem 3.7. To the best of our knowledge, it is the first time that the relations between the various types of spectra of T k D are clearly characterized in the literature. These results, along with the fundamental properties of Riesz projections, allow us to write the pole-pencil decomposition of the resolvent of T k D . After that, we present more quantitative results for the case of a spherical domain. We rigorously establish the asymptotic forms of the eigenvalues of the integral operator, and prove that these complex eigenvalues are rapidly tending to the real axis in Theorem 3.17. We also observe that along these eigenvalue sequences, there is a localization phenomenon for the associated eigenfunctions [29,33], with a mathematical illustration provided in Theorem 3. 19. In Appendix B, we provide another possible perspective to investigate the spectral properties of T k D by regarding it as a quasi-Hermitian operator. Our third contribution is that by applying the pole-pencil decomposition to the Lippmann-Schwinger representation of the Green's tensor, we write the resonance expansion (eigenfunction expansion) for the imaginary part of the Green's tensor, and find that both eigenvalues and eigenfunctions are responsible for the super-resolution in the reconstruction of the EM embedded sources in the high contrast setting. Precisely, the localized eigenfunctions are highly oscillating and can encode the subwavelength information of the sources. Such information is further amplified when the high contrast approaches some resonant values, and then is back-propagated to reconstruct the subwavelength details of the sources. The remainder of this work is organized as follows. In Section 2, we first give a brief review of the resolution of the time-reversal method for the inverse source problem and then derive the Lippmann-Schwinger representation of the EM Green's tensor. In Section 3, we investigate the spectral structure of the involved volume integral operator on a general domain (cf. (2.2)) and obtain the pole-pencil decomposition of its resolvent near the small regular value. We then proceed to provide more quantitative analysis of spectral properties for the spherical domain. With these mathematical findings, we provide a full explanation for the super-resolution in high contrast media in Section 4. In addition, we will present the numerical evidences in the case of a spherical region to validate our main theoretical results. Some details and other useful and interesting results are given in Appendices A, B and C. We shall use some standard notations for the Sobolev spaces (see [32]) throughout this work. For a vector x ∈ R 3 , we denote its transport by x t and its polar form by (\|x\|,x) withx := x/\|x\| ∈ S 2 , where S 2 is the two dimensional unit sphere in R 3 . We denote the inner product and outer product for two vector u, v ∈ R 3 by u t · v and u × v respectively. We also need the tensor product operation ⊗ of two vectors, i.e., given two vectors u ∈ R n and v ∈ R m , u ⊗ v is a n × m matrix given by (u ⊗ v) ij = u i v j . And we always let vector operators act on matrices column by column. For a Banach space X and its topological dual X ′ , we introduce the dual pairing l, x X := l(x). We use ⊕ ⊥ to denote the orthogonal sum in a Hilbert space, while the direct sum in a Banach space is denoted by ⊕. Resolution of imaging EM embedded sources In this section, we shall first introduce the time-reversal reconstruction of EM sources embedded in a high contrast medium and then review its resolution analysis. The main purpose of this section is to work out the explicit relation between the resolution limit and the contrast between the refractive indices of the dielectric inclusion and its surrounding medium. Let us start with the introduction of some notation, definitions and conventions in this work. We consider a dielectric inclusion D embedded in the free space R 3 , where D is a bounded connected open set with a smooth boundary ∂D and the exterior unit normal vector ν. We assume the refractive index n(x) ∈ L ∞ (R 3 ) of the form: n(x) = 1 + τ χ D (x) , where τ ≫ 1 is a positive real constant and χ D is the characteristic function of D. Let k and k τ := k √ 1 + τ be the wave numbers in the free space and in the medium D, respectively. Then we introduce the fundamental solution of the differential operator −(∆ + k 2 ) in R 3 : g(x, y, k) := e ik\|x−y\| 4π\|x−y\| , k ≥ 0. We define the Riesz potential K k D : K k D [ϕ] = D g(x, y, k)ϕ(y)dy for ϕ ∈ L 2 (D, R 3 ) , (2.1) which is a bounded linear operator from L 2 (D, R 3 ) to H 2 loc (R 3 , R 3 ) . This further allows us to introduce the electric volume integral operator T k D : T k D [ϕ] = (k 2 + ∇div)K k D [ϕ] ∈ H loc (curl, R 3 ) for ϕ ∈ L 2 (D, R 3 ) , (2.2) which satisfies ∇ × ∇ × T k D [ϕ] − k 2 T k D [ϕ] = k 2 ϕχ D in R 3 , (2.3) in the variational sense, together with the outgoing radiation condition: \|x\| ∇ × T k D [ϕ](x) ×x − ikT k D [ϕ](x) → 0 as \|x\| → ∞ . (2.4) We say that a L 2 -vector field E solving the homogeneous Maxwell equations is radiating if it satisfies the radiation condition (2.4) in the far-field, and of which we define the far-field pattern E ∞ (x) ∈ L 2 T (S 2 ) by the asymptotic form: E ∞ (x) = e ik\|x\| \|x\| E ∞ (x) + O 1 \|x\| 2 as \|x\| → ∞ . (2.5) The following surface integral operators are also needed: S k ∂D [ϕ] = ∂D g(x, y, k)ϕ((y)dσ(y), K k, * ∂D [ϕ] = ∂D ∂ ∂ν x g(x, y, k)ϕ((y)dσ(y) for ϕ ∈ H − 1 2 (∂D) . (2.6) We recall the normal trace formula for the gradient of S k ∂D : γ n ∇S k ∂D [ϕ] = 1 2 + K k, * ∂D [ϕ](x) , x ∈ ∂D ,(2.7) where γ n [·] = ν t · · is the normal trace mapping which is well-defined on the space H(div, D). For the case where the density function ϕ in S k ∂D is the tangent vector fields from H −1/2 T (div, ∂D), we denote the operator by A k ∂D instead in order to avoid any confusion. When k = 0, we omit the superscript k in the above definitions for simplicity, e.g., we write S ∂D for S 0 ∂D . We are now ready to state the inverse source problem of our interest in this work, and analyze the resolution of the time-reversal reconstruction of the EM embedded sources. Consider the following forward source problem associated with the medium D: ∇ × ∇ × E(x) − k 2 n(x)E(x) = f (x) , x ∈ R 3 , E satisfies the outgoing radiation condition (2.4) , (2.8) where f ∈ L 2 (D, R 3 ) is the electric radiating source in the sense that E has a nontrivial far-field pattern [1]. The corresponding inverse source problem is aimed at reconstructing the source f by using the electric field data E meas (x) collected on the far-field measurement surface ∂B(0,R), where the radiusR is large enough and B(0,R) contains D. In the distribution sense, the measured data E meas (x) on ∂B(0,R) can be written as E meas (x) = D G(x, y, k)f (y)dy , x ∈ ∂B(0,R) , (2.9) where G(x, y, k) is the Green's tensor of Maxwell's equations for the inhomogeneous background, defined by ∇ × ∇ × G(x, y, k) − k 2 n(x)G(x, y, k) = δ(x − y)I , x ∈ R 3 , y ∈ R 3 \∂D ,(2.10) such that each column of G satisfies the outgoing radiation condition (2.4). Here, I is the 3 × 3 identity matrix. The existence of G can be rigorously justified by the boundary integral equations (cf. (2.16)-(2.17)). In our following representation, G(x, y, k) will usually occur with a unit polarization vector p ∈ S 2 , i.e., G(x, y, k)p, physically denoting the electric field generated by the point dipole source δ(x − y)p located at y, and we will not give descriptions for the other similar notations if there is no ambiguity. To re-emit the measured field E meas (x) in (2.9) back to the source, we multiply it by G (phase conjugation is the frequency domain counterpart of time reversal), which immediately leads us to the imaging functional: I(z) = ∂B(0,R) G(z, x, k)E meas (x)dσ(x) , (2.11) where z is any sampling point taken from the sampling region Ω which is a bounded domain satisfying D ⊂ Ω ⊂ B(0, R). The resolution of the above imaging functional is a standard consequence of the following corollary of the well-known Helmholtz-Kirchhoff identity [20,30]: for any p, q ∈ S 2 , k ∂B(0,R) (G(ξ, x, k)q) t · G(ξ, z, k)pdσ(ξ) = q t · ImG(x, z, k)p + O 1 R , ∀ x, z ∈ Ω\∂D . (2.12) To see this, we substitute (2.9) into (2.11), and then readily obtain from (2.12) that for an arbitrary probing direction q ∈ S 2 , it holds that q t · I(z) = ∂B(0,R) q t · G(z, x, k)E meas (x)dσ(x) = D ∂B(0,R) q t · G(z, x, k)G(x, y, k)f (y)dσ(x)dy = 1 k D q t · ImG(z, y, k)f (y)dy + O 1 R , where we have used the reciprocity of the Green's tensor: G(x, y, k) t = G(y, x, k). Thus, we have that I(z) can be approximated byÎ (z) = 1 k D ImG(z, y, k)f (y)dy , z ∈ Ω , whenR tends to infinity. To investigate the properties ofÎ, it suffices to consider the imaginary part of the Green's tensor (with a polarization vector p): ImG(z, z 0 , k)p , z 0 ∈ D , p ∈ S 2 , which is proportional to the raw image I(z) of the point dipole source f (y) = δ z0 (y)p asymptotically. It is worth emphasizing that ImG, unlike the acoustic case, is anisotropic in the sense that q t ·ImGp may present different features for different probing directions q ∈ S 2 and polarization directions p ∈ S 2 , and hence yields a direction dependent diffraction barrier. But we can still expect a better resolution in the image of f obtained from the approximate functionalÎ(z), if ImG(z, z 0 , k)p exhibits subwavelength peaks. To figure out how the high contrast τ influences the behavior of the imaginary part of the Green's tensor, the Lippmann-Schwinger formulation may be adopted, as it was suggested in [13] for the acoustic case. However, it is not a trivial task to derive the Lippmann-Schwinger equation here as in [13] due to the strong singularity of the current Green's tensor G(x, y, k) associated with the Maxwell equations for the inhomogeneous background. We observe that ImGp does not satisfy the outgoing radiation condition (2.4) although it obeys ∇ × ∇ × ImG(x, y, k)p − k 2 n(x)ImG(x, y, k)p = 0 , x ∈ R 3 , y ∈ R 3 \∂D . Thus, we need to to deal directly with G(z, z 0 , k)p that solves the equation: ∇ × ∇ × G(z, z 0 , k)p − k 2 n(x)G(z, z 0 , k)p = δ z0 (z)p , z 0 ∈ D, z ∈ R 3 \∂D , (2.13) or equivalently, ∇ × ∇ × [G(z, z 0 , k) − G 0 (z, z 0 , k)] p−k 2 [G(z, z 0 , k) − G 0 (z, z 0 , k)] p = k 2 τ χ D G(z, z 0 , k)p , z 0 ∈ D, z ∈ R 3 \∂D , (2.14) where G 0 (x, y, k) := I + 1 k 2 ∇div g(x, y, k)I (2.15) is the Green's tensor of Maxwell equations for the free space with wave number k. By (2.3) and (2.14), the integral equation for G may be formally formulated as G(z, z 0 , k)p − G 0 (z, z 0 , k)p = τ T k D [G(·, z 0 , k)p] (z), z ∈ D . Nevertheless, there is a strong singularity of G(z, z 0 , k) near z 0 (cf. (2.18)), resulting in the fact that G(z, z 0 , k)p / ∈ L 2 (D, R 3 ) and the evaluation of T k D [G(·, z 0 , k)] (z) makes no sense. To address this issue, we need an a priori information on the singularity of Green's tensor G, which we shall observe from the boundary integral equation for G. With the help of the integral operator A k ∂D introduced earlier in this section, we assume that G(x, y, k)p has the following ansatz: for y ∈ D, G(x, y, k)p = G 0 (x, y, k τ )p + ∇ × A kτ ∂D [φ](x) + ∇ × ∇ × A kτ ∂D [ψ](x) , x ∈ D , ∇ × A k ∂D [φ](x) + ∇ × ∇ × A k ∂D [ψ](x) , x ∈ R 3 \D ,(2.16) and for y ∈ R 3 \D, G(x, y, k)p = ∇ × A kτ ∂D [φ](x) + ∇ × ∇ × A kτ ∂D [ψ](x) , x ∈ D , G 0 (x, y, k)p + ∇ × A k ∂D [φ](x) + ∇ × ∇ × A k ∂D [ψ](x) , x ∈ R 3 \D . (2.17) The densities φ, ψ ∈ H −1/2 T (div, ∂D) in (2.16) and (2.17) can be found by solving a boundary integral equation built via the trace formulas related to A k ∂D [9,6]. By (2.16), we readily see that near z 0 ∈ D, G(z, z 0 , k)p has the same singularity as G 0 (z, z 0 , k τ )p in the sense that G(z, z 0 , k)p − G 0 (z, z 0 , k τ )p ∈ L 2 (D, R 3 ) . (2.18) We are now prepared to derive the Lippmann-Schwinger representation of G in terms of T k D and τ . The key idea here is to split G into a singular term with compact support in D and a regular remainder, and then establish the integral equation for the regular part instead. To do so, we construct a smooth cutoff function χ z0 (z) with a compact support in D satisfying χ z0 (z) ≡ 1 on a small ball B(z 0 , r) ⊂ D , and define g(z, z 0 , k) := χ z0 (z)g(z, z 0 , k) , z ∈ R 3 , (2.19) which helps us to separate the singularity indicated in (2.18) locally. It follows that ∇ z div z ( g(z, z 0 , k)p) is a distribution on R 3 with its support and singular support, respectively, given by the compact set supp( χz 0 ) and the single point {z 0 }. We now write G(z, z 0 , k)p as G(z, z 0 , k)p = G 0 (z, z 0 , k)p − τ k 2 τ ∇ z div z ( g(z, z 0 , k)p) + V (z, z 0 , k)p , z ∈ R 3 ,(2.20) where V (·, z 0 , k)p\| D defined by the above formula is an L 2 -vector field, by (2.18) and (2.19). Substituting (2.20) back into (2.13), we can find, by a direct computation, that V (z, z 0 , k)p satisfies ∇ × ∇ × V (z, z 0 , k)p − k 2 n(z)V (z, z 0 , k)p = τ k 2 χ D (z)(G 0 (z, z 0 , k)p − 1 k 2 ∇ z div z ( g(z, z 0 , k)p)) ,(2.21) where we have used the fact that G 0 is the fundamental solution to the homogeneous Maxwell equations and a simple but important observation that k 2 n(x) τ k 2 τ ∇ z div z ( g(z, z 0 , k)p) = τ ∇ z div z ( g(z, z 0 , k)p) , z ∈ R 3 . The above observation also suggests the reasons why it is necessary to restrict the singularity in the domain D. Note that the source term in the right-hand side of (2.21) is an L 2 -vector field. We define a matrix function G(z, z 0 , k) := G 0 (z, z 0 , k) − 1 k 2 ∇ z div z ( g(z, z 0 , k)I) , z, z 0 ∈ D . (2.22) Then the corresponding Lippmann-Schwinger equation for V p reads as follows: V (z, z 0 , k)p = τ T k D [ G(·, z 0 , k)p + V (·, z 0 , k)p](z) , z ∈ D . If 1 − τ T k D is invertible (as we shall see in Theorem 3.2, this is always the case for a high contrast τ ), we further have V (z, z 0 , k)p = (1 − τ T k D ) −1 (τ T k D − 1 + 1)[ G(·, z 0 , k)p](z) = (1 − τ T k D ) −1 [ G(·, z 0 , k)p](z) − G(z, z 0 , k)p , z ∈ D . (2.23) Then it follows from the decomposition (2.20), the definition of G in (2.22) and the relation k τ = k √ 1 + τ that G(z, z 0 , k)p = G(z, z 0 , k)p + ( 1 k 2 − τ k 2 τ )∇ z div z ( g(z, z 0 , k)p) + V (z, z 0 , k)p = G(z, z 0 , k)p + 1 k 2 τ ∇ z div z ( g(z, z 0 , k)p) + V (z, z 0 , k)p , z, z 0 ∈ D . Combining this decomposition with (2.23), we arrive at the main result of this section. Theorem 2.1. The Green's tensor of the Maxwell equations (2.13) with a polarization vector p ∈ S 2 , has the following representation: G(z, z 0 , k)p = 1 k 2 τ ∇ z div z ( g(z, z 0 , k)p) + (1 − τ T k D ) −1 [ G(z, z 0 , k)p](z) , z, z 0 ∈ D ,(2.24) where g and G are given by (2.19) and (2.22), respectively. In the above construction, the definitions of g and G depend on the position of z 0 and the explicit choice of the cutoff function χ z0 (z). If we re-define g and G in (2.19) and (2.22) as g(z, z ′ , k) = χ z0 (z)g(z, z ′ , k), z ∈ R 3 , z ′ ∈ B(z 0 , r) , (2.25) and G(z, z ′ , k) = G 0 (z, z ′ , k) − 1 k 2 ∇ z div z ( g(z, z ′ , k)I), z ∈ R 3 , z ′ ∈ B(z 0 , r) ,(2.26) respectively, and revisit the proof of Theorem 2.1 carefully, we can find the same representation of G(z, z ′ , k)p as the one in (2.24) for z ∈ D and z ′ ∈ B(z 0 , r) but with g and G replaced by the ones in (2.25) and (2.26). More generally, given an arbitrary compact subset D ′ of D, we may replace the cutoff function χ z0 (z) in (2.25) by another smooth cutoff function χ D ′ such that χ D ′ (z) ≡ 1 on a small neighborhood of D ′ . Then, by a very similar argument as above, we can derive an improved variant of Theorem 2.1. Theorem 2.2. Given a compact subset D ′ of D, let g be given by (2.25) with χ z0 (z) replaced by the smooth cutoff function χ D ′ (z) associated with D ′ , and let G be defined as in (2.26) with the newly defined g. Then the following decomposition of the Green's tensor G(z, z ′ , k) (cf. (2.10)) holds, G(z, z ′ , k) = 1 k 2 τ ∇ z div z ( g(z, z ′ , k)I) + (1 − τ T k D ) −1 [ G(·, z ′ , k)](z) , z ∈ D, z ′ ∈ D ′ . (2.27) We can clearly see from (2.27) (or (2.24)) how the high contrast τ affects the behavior of G. In the high contrast regime, i.e., τ ≫ 1, the first term of (2.27) involves the contrast τ in an explicit way, and we can find that its imaginary part is of order τ −1 and thereby negligible since Im g(z, z ′ , k) is a sufficiently smooth function. At the same time, the second term in (2.27) is strongly influenced by the property of operator (τ −1 − T k D ) −1 . If there are some poles of the resolvent of T k D near τ −1 , we may expect that the term (1 − τ T k D ) −1 [ G(·, z ′ , k)](z) blows up and hence ImG exhibits a sharper peak than the one in the homogeneous space. These observations lead us to the investigations of the spectral structure as well as the resolvent of T k D in the next section, which serves as the mathematical preparations for a complete study of the possibility of achieving the super-resolution in high contrast media in Section 4. Spectral analysis of the volume integral operator For a bounded linear operator A on a complex Banach space, we denote by σ(A) its spectrum, by σ p (A) its eigenvalues (point spectrum), and by (λ − A) −1 the resolvent, which is an analytic operator-valued function defined on the resolvent set ρ(A) := C\σ(A). We refer to the elements in ρ(A) as the regular values of A. We have seen in Section 2 that the resolution limit in the EM inverse source problem is closely related to the behaviour of the resolvent (λ − T k D ) −1 near the small regular value τ −1 ≪ 1. Spectral structure In this subsection, we are going to first consider the distribution of eigenvalues of T k D and then give characterizations of the essential spectrum and eigenvalues of finite type (their definitions will be given after Corollary 3.3). These results are fundamental to the pole-pencil decomposition of the resolvent (λ − T k D ) −1 that shall be derived in Section 3.2. We start with an easily observed but quite important lemma for our later use. Lemma 3.1. For the integral operator T k D defined by (2.2), we have 0 / ∈ σ p (T k D ). Moreover, the eigenvalue equation (λ − T k D )[ϕ] = 0 has nontrivial solutions for some λ ∈ C (i.e., λ ∈ σ p (T k D ) ) if and only if the following transmission problem has a nontrivial radiating solution u ∈ H loc (curl, R 3 ), ∇ × ∇ × u − k 2 u = k 2 λ uχ D in R 3 . (3.1) In this case, the solution u to (3.1), restricted on D, is an eigenfunction of T k D associated with λ. Proof. Suppose (λ, ϕ) is the eigenpair of T k D , i.e., T k D [ϕ] = λϕ, ϕ = 0, which directly yields, by (2.3), (∇ × ∇ × −k 2 )T k D [ϕ] = (∇ × ∇ × −k 2 )λϕ = k 2 ϕχ D in R 3 . (3.2) We readily see that if λ = 0, then ϕ = 0 on D, from which it follows that 0 / ∈ σ p (T k D ) and λ in (3.2) does not vanish. Since ϕ is the eigenfunction of T k D with eigenvalue λ, we can write the right-hand side of (3.2) as k 2 T k D [ϕ/λ]χ D and then conclude that T k D [ϕ] is a nontrivial solution of (3.1). Conversely, if u is a nontrivial solution of (3.1), by the uniqueness of a solution to the Maxwell source problem and (2.3), we have u = T k D [u/λ], which also implies that u\| D is an eigenfunction of T k D associated with λ. We denote the interior wave number k √ 1 + λ −1 in (3.1) by k λ . Here and throughout this work, we consider the principal branch of √ · with the branch cut given by (−∞, 0]. It should be stressed that the equation (3.1) is defined on the whole space R 3 and understood in the variational sense. This fact immediately yields ∇ × u ∈ H loc (curl, R 3 ), and hence ∇ × u ∈ H 1 loc (R 3 , R 3 ) by noting that div(∇ × u) = 0 and making use of the embedding theorem (cf. [14, Theorem 2.5]). These facts can also be verified by the integral representation of u, i.e., u = T k D [u/λ]. We now give the first main result of this subsection, concerning an a priori characterization of the distribution of the eigenvalues and eigenspaces of T k D . The proof follows a similar spirit of the one for proving the uniqueness of a solution to the direct acoustic scattering problem (cf. [21, Theorem 2.14]) but pays a special attention to the ranges of the eigenvalues and the topology of the domain. Theorem 3.2. For a bounded smooth domain D, we have that if λ ∈ σ p (T k D )\{−1}, then Imλ > 0. Suppose that R 3 \D is connected. We have that if λ = −1 is an eigenvalue of T k D , then the associated eigenspace must be contained in ∇H 1 0 (D). Proof. We assume that u ∈ H loc (curl, R 3 ) is a radiating solution to (3.1), or equivalently, the following system:    ∇ × ∇ × u − k 2 λ u = 0 in D , ∇ × ∇ × u − k 2 u = 0 in R 3 \D , [ν × u] = 0, [ν × ∇ × u] = 0 on ∂D ,∇ × ∇ × u ·ū − k 2 u ·ūdx = B(0,R)\D \|∇ × u\| 2 − k 2 \|u\| 2 dx + ∂B(0,R)x × ∇ × u ·ūdσ(x) − ∂D ν × ∇ × u ·ūdσ(x) = B(0,R)\D \|∇ × u\| 2 − k 2 \|u\| 2 dx − ik ∂B(0,R) \|u\| 2 dσ(x) + O 1 R − ∂D ν × ∇ × u ·ūdσ(x) , (3.4) where we have used the radiation condition (2.4) and the fact that ∇ × u ∈ H 1 loc (R 3 , R 3 ). By taking the imaginary parts of both sides of (3.4) and letting R tends to infinity, we have Im ∂D ν × ∇ × u ·ūdσ(x) = −k S 2 \|u ∞ \| 2 dσ(x) ≤ 0 . (3.5) Here, u ∞ is the far-field pattern of u given by (2.5). We now consider the field inside the domain. Similarly, with the help of an integration by parts over D and the first equation in (3.3), we obtain − D \|∇ × u\| 2 − k 2 λ \|u\| 2 dx = ∂D ν × ∇ × u ·ūdσ(x) ,(3.6) and its imaginary part Im ∂D ν × ∇ × u ·ūdσ(x) = Im D k 2 1 λ \|u\| 2 dx . (3.7) Noting that Imλ −1 = −Im(λ/\|λ\| 2 ) ≥ 0, we readily have Im ∂D ν × ∇ × u ·ūdσ(x) = 0 , by (3.5) and (3.7), since the tangential traces of u and ∇ × u are continuous. Then, we see from the above formula and (3.5) that the far-field pattern u ∞ vanishes, and thus u vanishes in the unbounded connected component of R 3 \D by Rellich's lemma (cf. [21, Theorem 6.10]). Therefore, it follows that ν × u = 0 , ν × ∇ × u = 0 on Γ 0 , (3.8) where Γ 0 is the boundary of the unbounded component of R 3 \D. To complete the proof, let us first consider the simple case: λ = −1, where the interior wave number k λ does not vanish. The desired result that u = 0 in D directly follows from (3.8) and the Holmgren's Theorem (cf. [21, Theorem 6.5]). We now consider the other case where λ = −1 under the condition that R 3 \D is connected. In this case, we only have ∇ × u = 0 in D, i.e., u ∈ H 0 (curl0, D), from (3.6) and the observation Γ 0 = ∂D. Recalling (A.2), we have the following characterization of H 0 (curl0, D): H 0 (curl0, D) = ∇H 1 0 (D) , since the R 3 \D is connected and thus the corresponding normal cohomology space K N (D) is trivial. Therefore, we can conclude u = ∇p for some p ∈ H 1 0 (D) if λ = −1 is an eigenvalue, and complete the proof. The above theorem does not tell us whether λ = −1 is an eigenvalue or not. However, if we extend an L 2 -field u from ∇H 1 0 (D), or more generally, H 0 (curl0, D), by zero outside the domain D, i.e., χ D u, we can find that it solves the system (3.3) for λ = −1, which indicates that λ = −1 is indeed an eigenvalue of T k D . Thus, we actually have the following corollary. Corollary 3.3. For a bounded smooth domain D, λ = −1 is always an eigenvalue of T k D with the associated eigenspace containing H 0 (curl0, D). If R 3 \D is connected, then the eiganspace is equal to ∇H 1 0 (D). To proceed, we need the following concepts about the spectrum of a bounded linear operator A. We say that λ ∈ σ(A) is an eigenvalue of finite type if and only if λ is an isolated point in σ(A) and the corresponding Riesz Projection P λ : P λ (A) = 1 2πi Γ (z − A) −1 dz, (3.9) is a finite rank operator, where Γ is a Cauchy contour in C enclosing only the eigenvalue λ among σ(A), and the definition does not depend on the choice of Γ. The other concept is the essential spectrum σ ess (A) defined by σ ess (A) = {λ ∈ C ; λI − A is not Fredholm operator} . Inspired by the work [22] where the strongly singular volume integral equation associated with the EM scattering problem was transformed to a coupled surface-volume system involving only weakly singular kernels by introducing an additional variable on the boundary via an integration by parts, here we exploit the Helmholtz decomposition of L 2 -vector fields to obtain another operator matrix similar to the one in [22] but with fully decoupled unknown variables. This newly derived system enables us to see a clear and insightful spectral structure of T k D . We now recall from Theorem A.3 the Helmholtz decomposition of L 2 -vector fields: L 2 (D, R 3 ) = ∇H 1 0 (D) ⊕ ⊥ H 0 (div0, D) ⊕ ⊥ W , (3.10) where W is the function space consisting of H 1 -harmonic functions and H 0 (div0, D) = curl X 0 N ⊕ ⊥ K T (D). Denote by P 0 , P d and P w the projections from L 2 (D, R 3 ) to ∇H 1 0 (D), H 0 (div0, D) and W , respectively. In Appendix A, we show how these subspaces are connected with the divergence, curl and normal trace of a vector field. In particular, we have P 0 u = −∇Sdivu and P w u = γ −1 n γ n (u + ∇Sdivu); see Appendix A for the definitions of operators S and γ −1 n . For our subsequent analysis, we introduce a product space: X := ∇H 1 0 (D) × H 0 (div0, D) × H − 1 2 0 (∂D) , equipped with the norm F X := f 1 L 2 (D) + f 2 L 2 (D) + f 3 H −1/2 0 (∂D) for F = (f 1 , f 2 , f 3 ) ∈ X, which is isomorphic to L 2 (D, R 3 ) via the isomorphism Ξ : f → Ξ[f ] = (P 0 f, P d f, γ n P w f ). By using the isomorphism Ξ, we define an operator T k D on X by T k D := ΞT k D Ξ −1 , (3.11) which is similar to T k D and hence has the same spectral properties as T k D . We remark that the inverse of Ξ is given by Ξ −1 (f 1 , f 2 , f 3 ) = f 1 + f 2 + γ −1 n f 3 . We proceed to consider the spectral analysis of T k D . We first observe that ∇H 1 0 (D) and divergence-free vector fields H(div0, D) are T k D -invariant spaces. In fact, for φ ∈ H 1 0 (D), we have T k D [∇φ] = k 2 ∇K k D [φ] + ∇∆K k D [φ] = −∇φ ,(3.12) which can be verified by using integration by parts with the fact that φ has zero trace on ∂D. On the other hand, by a density argument and the fact that div : L 2 (D, R 3 ) → H −1 (D), we have divT k D [φ] = −divφ for φ ∈ L 2 (D, R 3 ) . By these observations and the definition of T k D (cf. (3.11)), we can write the operator matrix T k D as follows: T k D =   −1 0 0 0 P d T k D P d T k D γ −1 n 0 γ n T k D γ n T k D γ −1 n   . (3.13) To further analyze the properties of T k D , we need to work out explicit formulas for the operators involved in (3.13), which are only defined in an abstract way. To do so, a direct calculation gives us that T k D [ϕ] = k 2 K k D [ϕ] − ∇S k ∂D [ϕ · ν] = k 2 K k D [P d ϕ + P w ϕ] − ∇S k ∂D [γ n P w ϕ] (3.14) holds for ϕ ∈ H(div0, D). Then, we take the normal trace on both sides of (3.14) and find γ n T k D [ϕ] = k 2 γ n K k D [P d ϕ + P w ϕ] − ( 1 2 + K k, * ∂D )[γ n P w ϕ] for ϕ ∈ H(div0, D) ,(3.15) where we have used the normal trace formula (2.7) for ∇S k ∂D . By (3.14) and (3.15), we readily have    P d T k D [·] = k 2 P d K k D [·] , γ n T k D [·] = k 2 γ n K k D [·] on H 0 (div0, D) , P d T k D γ −1 n [·] = k 2 P d K k D γ −1 n [·] − P d ∇S k ∂D [·] , γ n T k D γ −1 n [·] = k 2 γ n K k D γ −1 n [·] − ( 1 2 + K k, * ∂D )[·] on H −1/2 0 (∂D) . (3.16) We are now in a position to prove the following lemma. Lemma 3.4. R k D := T k D − diag(−1, 0, − 1 2 ) is a compact operator on X. Proof. To prove the compactness of R k D on the product space X, it suffices to show that each block in R k D is compact. By the mapping property of K k D and Rellich's lemma for Sobolev spaces, we can obtain that P d K k D and γ n T k D are compact operators from H 0 (div0, D) to H 0 (div0, D) and H −1/2 0 (∂D), namely, the operators (R k D ) 2,2 and (R k D ) 3,2 are compact (cf. (3.16)). Meanwhile, a further fact that K k, * ∂D is compact gives us the compactness of (R k D ) 3,3 = γ n T k D γ −1 n + 1/2 on H −1/2 0 (∂D), by (3.16). To show that (R k D ) 2,3 = P d T k D γ −1 n is compact from H −1/2 0 (∂D) to H 0 (div0, D) , we write it, by using (3.16), as P d T k D γ −1 n [·] = (k 2 P d K k D γ −1 n − P d ∇(S k ∂D − S ∂D ))[·] − P d ∇S ∂D [·] , where the first term is obviously compact, and the second term actually vanishes due to the fact that ∇S ∂D [·] ∈ W . The proof is complete. By Lemma (3.4) and the fact that the essential spectrum is stable under a compact perturbation [28], we directly have the characterization of the essential spectrum [23]: σ ess (T k D ) = σ ess (T k D ) = σ ess (diag(−1, 0, − 1 2 )) = {−1, 0, − 1 2 } , and λ − T k D is an analytic Fredholm operator function with index zero on C\σ ess as a consequence of the definition of essential spectrum and the fact that the Fredholm index ind(λ − T k D ) is a constant on a connected open set. Then, by using the analytic Fredholm theory [28] and Theorem 3.2, we can conclude that (λ−T k D ) −1 is extended to a meromorphic function on C\σ ess (T k D ) with its poles being a discrete and countable bounded set given by σ p (T k D )\σ ess (T k D ), and for some λ 0 ∈ σ p (T k D )\σ ess (T k D ) and λ in a sufficiently small neighborhood of λ 0 , (λ − T k D ) −1 has the following Laurent expansion: (λ − T k D ) −1 = ∞ n=−q(λ0) (λ − λ 0 ) n T n ,(3.17) where T 0 is Fredholm operator with index zero, and T i , −q(λ 0 ) ≤ i ≤ −1, are finite rank operators with q(λ 0 ) being a positive integer. From now on we shall denote the set of all the eigenvalues of finite type of T k D by σ f (T k D ). To better understand this set, we recall the following fundamental property concerning the Riesz projection (cf. [28, Thm 2.2]). Lemma 3.5. For a bounded linear operator A on a Banach space X, let σ be an isolated part of σ(A) and P σ (A) is the associated Riesz projection. Then both imP σ (A) and kerP σ (A) are the invariant subspaces of A with σ(A\| imPσ ) = σ and σ(A\| kerPσ (A) ) = σ(A)\σ. Moreover, X has the direct sum decomposition: X = imP σ (A) ⊕ kerP σ (A). From Lemma 3.5 it immediately follows that σ f (T k D ) is a subset of σ p (T k D ). Conversely, note from (3.17) that for λ 0 ∈ σ p (T k D )\σ ess (T k D ), P λ0 (T k D ) = 1 2πi Γ (λ − λ 0 ) −1 T −1 dλ = T −1 is a finite rank operator. By this fact, together with the definition of eigenvalues of finite type and σ f (T k D ) ⊂ σ p (T k D ), we readily have σ p (T k D )\σ ess (T k D ) = σ f (T k D )\σ ess (T k D ) . (3.18) In fact, we can obtain a sharper version of (3.18) by some further observations. We first note from Lemma 3.1 and Theorem 3.2 that {0, − 1 2 } ⊂ σ p (T k D ) and further that σ p (T k D )\σ ess (T k D ) = σ p (T k D )\{−1} ⊂ {λ ∈ C ; Imλ > 0} . (3.19) To consider the relation between σ f (T k D ) and σ ess (T k D ), we need a general result from [25,Lemma 4.3.17]. Lemma 3.6. Let A be a bounded linear operator, and let λ 0 be an isolated point in σ(A). Then we have λ 0 ∈ σ ess (A) if and only if the Riesz projection P λ0 (A) has an infinite-dimensional range. In particular, we have σ ess (A) σ f (A) = ∅ . This lemma, along with (3.18) and (3.19), allows us to conclude that σ p (T k D )\{−1} = σ f (T k D ) . With all the above arguments, we actually have proved our second main result of this subsection. Theorem 3.7. The spectrum σ(T k D ) is a disjoint union of essential spectrum and eigenvalues of finite type, i.e., σ(T k D ) = σ ess (T k D ) σ f (T k D ) , where σ ess (T k D ) and σ f (T k D ) are given by σ ess (T k D ) = {−1, 0, − 1 2 } , σ f (T k D ) = σ p (T k D )\{−1} ⊂ {λ ∈ C ; Imλ > 0} , and σ ess (T k D ) gives all the possible accumulation points of σ f (T k D ). Furthermore, (λ−T k D ) −1 is a meromorphic function on C\σ ess (T k D ) with a discrete set of poles given by σ f (T k D ). Remark 3.8. This remark is to emphasize the special roles of eigenvalue −1 and its eigenspace, and to connect it with the nonradiating sources. We have observed in Corollary 3. 3 that H 0 (curl0, D) is a T k D -invariant subspace with σ(T k D \| H0(curl0,D) ) = {−1} , which can also be obtained by a direct calculation as in (3.12). In fact, we have T k D [ϕ] = curlK k D [curlϕ] − curlA k ∂D [ν × ϕ] − ϕχ D for ϕ ∈ H(curl, D) . Hence, the space H 0 (curl0, D) also corresponds to the nonradiating sources in the sense that T k D [ϕ] for ϕ ∈ H 0 (curl0, D) vanishes in the far field since T k D [ϕ] = −ϕχ D . A more general version of this fact has actually been included in the proof of Theorem 3.2 implicitly. We have proved therein that if u is the eigenfunction of T k D with eigenvalue −1, then T k D [u] has a vanishing far-field pattern. We refer the readers to [17] for the detailed characterization of nonradiating sources for Maxwell's equations in the homogeneous space. Pole-pencil decomposition To fully understand the structure of (λ − T k D ) −1 , we may need to perform the full expansion of a vector field with respect to eigenfunctions and generalized eigenfunctions of T k D as the one given in [13] for the Helmholtz equation. Nevertheless, such a full expansion does not work here since we do not know whether the set of eigenfunctions and generalized eigenfunctions is complete in the space L 2 (D, R 3 ). To circumvent this technical barrier, we develop a new pole-pencil decomposition (local expansion) in this subsection for the resolvent (λ − T k D ) −1 near the reciprocal of the contrast τ instead, which relies on the concept of eigenvalues of finite type and Theorem 3.7. For our purpose, we define an ε-neighborhood of τ −1 in σ(T k D ): σ := B(τ −1 , ε) ∩ σ(T k D ) ,(3.20) where ε is a given small enough constant. By the fact from Theorem 3.7 that σ f (T k D ) is discrete, we readily see that σ must be a finite set of eigenvalues of finite type of T k D , i.e., σ = ∪ i∈I {λ i } = {λ i ; λ i ∈ B(τ −1 , ε) ∩ σ f (T k D )} , where I ⊂ N is a finite index set. Without loss of generality, we assume that σ is a nonempty set. In view of the facts that ∇H 1 0 (D) is an invariant space of T k D and σ(T k D \| ∇H 1 0 (D) ) = {−1} is disjoint from σ, it suffices to consider the resolvent of the restriction of T k D on H(div0, D) to derive the pole-pencil decomposition of (λ − T k D ) −1 . In the remainder of this subsection, we simply denote T k D \| H(div0,D) by T k D . To proceed, we first note from (3.13) and Lemma 3.4 that Theorem 3.7 still holds with T k D replaced by T k D except σ ess ( T k D ) = {0, −1/2} and σ f ( T k D ) = σ p ( T k D ) . It follows that both σ and its complement ζ := σ( T k D )\σ are closed subsets of σ( T k D ), which allows us to choose a Cauchy contour Γ in ρ( T k D ) around σ separating σ from ζ, and define the Riesz projection corresponding to σ: P σ := 1 2πi Γ (λ − T k D ) −1 dλ = i∈I P λi . (3.21) The Riesz projection corresponding to ζ can be introduced similarly. By Lemma 3.5, H(div0, D) can be decomposed into two invariant subspaces of T k D (and also T k D ): H(div0, D) = imP σ ⊕ kerP σ ,(3.22) with kerP σ = imP ζ , and it holds that σ(T k D \| imPσ ) = σ = ∪ i∈I {λ i }, σ(T k D \| kerPσ ) = σ( T k D )\ ∪ i∈I {λ i } . This decomposition (3.22), along with the Helmholtz decomposition (3.10), gives us the following T k D -invariant subspace decomposition of L 2 -vector fields: L 2 (D, R 3 ) = ∇H 1 0 (D, R 3 ) ⊕ ⊥ (imP σ ⊕ imP ζ ) . On the associated product space: ∇H 1 0 (D, R 3 ) × imP σ × imP ζ , the operator λ − T k D with λ ∈ C has a diagonal representation: diag(λ + 1, λ − T k σ , λ − T k ζ ), where T k σ and T k ζ are shorthand notations of T k D \| imPσ and T k D \| imP ζ respectively. With the help of these notations, we arrive at the following representation of the solution to (λ−T k D )[ϕ] = f for f ∈ L 2 (D, R 3 ) and λ ∈ B(τ −1 , ε)\σ: ϕ = 1 λ + 1 P 0 f + (λ − T k σ ) −1 P σ f + (λ − T k ζ ) −1 P ζ f . (3.23) To further understand the behavior of (λ − T k D ) −1 locally, we recall from the definitions of σ and P σ that imP σ is of finite-dimensional and T k D \| imPσ is an operator acting on a finite-dimensional vector space with eigenvalues {λ i } i∈I . By the Jordan theory to the finite-dimensional linear operator, there exists a basis such that the matrix representation of T k D \| imPσ has a Jordan canonical form, that is, the representation matrix is a block diagonal one consisting of elementary Jordan blocks: J =       λ 1 λ . . . . . . 1 λ       . More precisely, suppose that λ i has geometric multiplicity N i , and then the associated Jordan matrix J λi will have the form: J λi = diag(J 1 λi , · · · , J Ni λi ), where J j λ i , 1 ≤ j ≤ N i are the elementary Jordan blocks. Suppose also that for each Jordan block J j λ i , there is a Jordan chain ϕ j λi := (ϕ j,0 λi , ϕ j,1 λi , · · · , ϕ j,nij −1 λi ), N ∋ n ij ≥ 1, an ordered collection of linearly independent generalized eigenfunctions, such that J j λ i is the representation matrix of T k D restricted on E j λi : T k D \| E j λ i ϕ j λi = ϕ j λi J j λi , where E j λi is the invariant subspace of T k D spanned by the Jordan chain ϕ j λi . Without loss of generality, we assume ϕ j,s λ i L 2 (D) = 1, for i ∈ I, 1 ≤ j ≤ N i , 0 ≤ s ≤ n ij − 1 in the rest of the exposition. With E j λi , we can write the following invariant subspace decomposition of imP σ : imP σ = ⊕ i∈I ⊕ Ni j=1 E j λi . In our notation, the eigenspace corresponding to λ i is spanned by {ϕ j,0 λi } Ni j=1 with dimension N i while the generalized eigenspace is given by ⊕ Ni j=1 E j λi with dimension Ni j=1 n ij (the algebraic multiplicity of λ i ). For vector ϕ ∈ E j λi , denote by (ϕ) ϕ j λ i = ((ϕ) ϕ j λ i (0), (ϕ) ϕ j λ i (1), · · · (ϕ) ϕ j λ i (n ij − 1)) ∈ Rϕ = ϕ j λi · (ϕ) ϕ j λ i := nij −1 k=0 (ϕ) ϕ j λ i (k)ϕ j,k λi . (3.24) With the help of these notions and (3.23), we arrive at the pole pencil decomposition of (λ − T k D ) −1 . Theorem 3.9. The resolvent (λ − T k D ) −1 on B(τ −1 , ε)\σ has the following pole pencil decomposition: (λ − T k D ) −1 [·] = 1 λ + 1 P 0 [·] + i∈I Ni j=1 ϕ j λi · (λ − J j λi ) −1 (P j λi [·]) ϕ j λ i + (λ − T k ζ ) −1 P ζ [·]. (3.25) Here, P j λi := P j i P λi is the composition of projections P j i and P λi , where P j i (i ∈ I, 1 ≤ j ≤ N i ) are finite-dimensional projections from imP λi to E λ j i . By the above theorem, we clearly see that the behavior of (λ − T k D ) −1 is essentially determined by its principal part: i∈I Ni j=1 ϕ j λi · (λ − J j λi ) −1 (P j λi [·]) ϕ j λ i in the sense that it contains all the singularity of (λ − T k D ) −1 on B(τ −1 , ε) while the remainder term (λ + 1) −1 P 0 + (λ − T k ζ ) −1 P ζ is an analytic operator function on B(τ −1 , ε). In fact, if σ has only one element λ i , the principal part here exactly matches the one in the Laurent series of (λ − T k D ) −1 (3.17) near the pole λ i : Ni j=1 ϕ j λi · (λ − J j λi ) −1 (P j λi [·]) ϕ j λ i = −1 n=−q(λi) (λ − λ i ) n T n . (3.26) We also note that (λ − J j λi ) −1 has the following explicit form: (λ − J j λi ) −1 =       (λ − λ i ) −1 (λ − λ i ) −2 · · · (λ − λ i ) −nij (λ − λ i ) −1 . . . . . . . . . (λ − λ i ) −2 (λ − λ i ) −1       , which readily gives us that the order q(λ i ) of the pole λ i is determined by q(λ i ) = max 1≤j≤Ni n ij . (3.27) Hence, we may expect that there is a blow-up of (λ − T k D ) −1 near the pole λ i with order of 1/\|λ − λ i \| q(λi) . In fact, we have the following local resolvent estimate (see Theorem 3.10) directly from (3.17) and the estimate for (λ − J j λi ) −1 : (λ − J j λi ) −1 ≤ C 1 \|λ − λ i \| nij ,(3.28) where λ is in a small neighborhood of λ i and C is a generic constant depending on n ij and the aforementioned neighborhood of λ i . Note that we do not indicate the matrix norm that is used due to the norm equivalence property on a finite-dimensional space. Theorem 3.10. Suppose that B(τ −1 , ε) and σ are given as in (3.20). There exists a constant depending on ε and the pole set σ such that the following estimate holds for f ∈ L 2 (D, R 3 ) and λ ∈ B(τ −1 , ε)\σ, (λ − T k D ) −1 f L 2 (D) ≤ C i∈I 1 \|λ − λ i \| q(λi ) f L 2 (D) , where q(λ i ) is given by (3.27). This subsection ends with two remarks for a further discussion of the resolvent estimate of T k D . Remark 3.11. In [31], the author gives the following bound for the smallest singular value of a n × n Jordan block J with λ being its diagonal elements: ( n + 1 n ) n \|λ\| n n + 1 ≤ min 1≤j≤n s j (J) < \|λ\| n for 0 < \|λ\| < n n + 1 , where s j (A) n j=1 denote the singular values for a general n × n matrix A. The above estimate further gives us a sharper estimate for the induced 2-norm of the resolvent of J j λi than (3.28): (λ − J j λi ) −1 2 = max 1≤j≤nij s j ((λ − J j λi ) −1 ) = 1 min 1≤j≤nij s j ((λ − J j λi )) ≤ ( n ij n ij + 1 ) nij n ij + 1 \|λ − λ i \| nij , when 0 < \|λ − λ j \| ≤ n ij /(n ij + 1). It allows us to derive new local resolvent estimate for T k D : (λ − T k D ) −1 f L 2 (D) ≤ C i∈I Ni j=1 √ n ij ( n ij n ij + 1 ) nij n ij + 1 \|λ − λ i \| nij f L 2 (D) , for a generic constant C and λ ∈ B(τ −1 , ε), which seems to be a little bit shaper than the one in Theorem 3.10 but actually does not provide us new information on the singularity of (λ − T k D ) −1 and its blow-up rate near the regular value τ −1 . Remark 3.12. In general, it is very difficult to obtain a sharp global estimate for the resolvent (λ − T k D ) −1 of the non-selfadjoint and non-compact operator T k D . Nevertheless, by noting that T k D is a quasi-Hermitian operator, we can apply a general result to T k D to obtain its resolvent estimate. We put the detailed analysis and some relevant definitions in Appendix B. We have observed from Theorems 3.2 and 3.7 that τ −1 − T k D is invertible, and then Theorems 3.9 and 3.10 permit us to write (3.29) and to see that the behavior of (τ −1 − T k D ) −1 is indeed significantly influenced by the poles of resolvent of T k D near τ −1 and their associated eigenstructures, as it is suggested at the end of Section 2. (τ −1 − T k D ) −1 ∼ i∈I Ni j=1 ϕ j λi · (τ −1 − J j λi ) −1 (P j λi [·]) ϕ j λ i , Spherical region In view of the formula (3.29), both eigenvalues and eigenfunctions can play a crucial role in the local behavior of (λ − T k D ) −1 near the very small regular value τ −1 , which motivates us to quantitatively investigate the asymptotic behaviors of eigenvalues and eigenfunctions of the operator T k D as λ → 0 to further explore the mechanism lying behind the super-resolution. In this subsection, we consider the spectral properties of T k D on the unit ball D = B(0, 1) in R 3 , where the Mie scattering theory is applicable. We have seen in Lemma 3.1 that solving the eigenvalue equation (λ − T k D )[ϕ] = 0 is equivalent to finding λ and the associated nontrivial radiating solution to the transmission problem: ∇ × ∇ × E − k 2 E = k 2 λ Eχ D . (3.30) In this subsection, we assume λ = −1 so that the wave number k λ = k √ 1 + λ −1 inside the domain will never vanish, see Remark 3.14, also Remark 3.8 for a discussion of the case of λ = −1. By the Mie theory, any solution E of the time-harmonic Maxwell equations ∇ × ∇ × E − k 2 E = 0 in the far field can be represented in the following series form: To establish the equations for eigenvalues λ, we match the Cauchy data (x × E,x × ∇ × E) of (3.31) and (3.32) on the boundary ∂B(0, 1). By the trace formulas of multipole fields (C.5) and (C.6), and recalling that {U m n } and {V m n } is an orthonormal basis of L 2 T (S 2 ), matching Cauchy data reduces the original eigenvalue problem to solving infinite linear systems: E(x) = ∞ n=1 n m=−n γ n,m E T E n,m (k, x) + η n,m E T M n,m (k, x) ,(3.[x × E(x)] = 0 ⇐⇒ γ n,m h(1) n (k) = α n,m j n (k λ ) η n,m H n (k) = β n,m k k λ J n (k λ ) n = 1, 2, · · · , m = −n, · · · , n , and [x × ∇ × E(x)] = 0 ⇐⇒ γ n,m H n (k) = α n,m J n (k λ ) η n,m kh(1) n (k) = β n,m k λ j n (k λ ) n = 1, 2, · · · , m = −n, · · · , n , which can be reformulated into the following independent equations with the undetermined coefficients as unknowns: j n (k λ ) −h (1) n (k) J n (k λ ) −H n (k) α n,m γ n,m = 0 , n = 1, 2, · · · , m = −n, · · · , n , (3.33) and k k λ J n (k λ ) −H n (k) k λ j n (k λ ) −kh (1) n (k) β n,m η n,m = 0 , n = 1, 2, · · · , m = −n, · · · , n . We readily observe that the coefficient matrices in above linear systems do not depend on the index m, and the equation (3.30) has nontrivial solutions for λ ∈ σ p (T k D )\{0} if and only if (3.33) or (3.34) has nonzero solutions for some index n ∈ N + , or equivalently, the determinants of the associated coefficient matrices are zero: h (1) n (k)J n (k λ ) − j n (k λ )H n (k) = 0 , (3.35) or k 2 k 2 λ h (1) n (k)J n (k λ ) − j n (k λ )H n (k) = 0 . (3.36) To proceed, let us focus on the first case, i.e., equations (3.33) and (3.35). We note from the fact that all the zeros of j n (z)(n ∈ N + ), expect the possible point z = 0, are simple [38] that j n (k λ ) and J n (k λ ) cannot vanish simultaneously. Neither can h (1) n (k) and H n (k) by a similar observation. Then all the nontrivial solutions of (3.33) have the form: (α n,m , γ n,m ) = c n,m (α n , γ n ) with α n , γ n = 0 and c n,m ∈ C\{0}. Therefore, for λ such that (3.35) holds for some index n, there is an associated subspace spanned by the eigenfunctions { E T E n,m } m=n m=−n . If the same λ happens to satisfy (3.33) for index n ′ = n or (3.34) for index n ′′ , we can find another (sub)eigenspace spanned by { E T E n ′ ,m } m=n ′ m=−n ′ or { E T M n ′′ ,m } m=n ′′ m=−n ′′ , which is orthogonal to the aforementioned one. Moreover, the geometric multiplicity of λ is the sum of the dimensions of these subspaces, which must be finite, since all the eigenvalues of T k D except −1 are eigenvalues of finite type, see Theorem 3.7. The same arguments can be applied to the system (3.34), as well as the equation (3.36). We summarize the above facts in the following theorem. σ f (T k D ) = σ p (T k D )\{−1} = ∪ ∞ n=1 (σ 1 n ∪ σ 2 n ) . And for each λ ∈ σ f (T k D ), the finite-dimensional eigenspace is spanned by ∪ 2 i=1 ∪ n∈Λi ∪ n m=−n E i n,m (k λ , x), where Λ i , i = 1, 2, is a finite subset of N + such that λ ∈ σ i n for n ∈ Λ i . Here, E i n,m (k λ , x) , i = 1, 2, denote the eigenfunctions E T E n,m (k λ , x) and E T M n,m (k λ , x), respectively. Remark 3.14. As we have seen in Corollary 3.3 and Remark 3.8, the eigenspace of eigenvalue λ = 1 is given by ∇H 1 0 (D), which are the nonradiating sources. For the case of the domain B(0, 1), it is spanned by the gradient of eigenfunctions u n of the Dirichlet Laplacian, that is, ∆u n = −k 2 n u n in B(0, 1) , u n = 0 on ∂B(0, 1) . The explicit formulas of the Dirichlet eigenvalues k n and eigenfunctions u n are available in [26]. It is also worth mentioning that in the above argument, we have actually proved that all of these eigenfunctions: E T E n,m and E T M n,m are the radiating sources, since both solution spaces of (3.33) and (3.34) are one-dimensional and spanned by some vector p ∈ C 2 with non-vanishing components p 1 , p 2 , i.e., p 1 , p 2 = 0. Asymptotic behavior of eigenvalues This subsection is devoted to the understanding of the distribution of eigenvalues in σ i n for i = 1, 2, namely, the eigenvalues of T k D . For this purpose, it suffices to investigate the zeros of f i n (z) for i = 1, 2 on C\{0}, where f i n (z) are introduced by the right-hand side of (3.35) and (3.36) by setting z = k λ , i.e., f 1 n (z) = h (1) n (k)J n (z) − j n (z)H n (k), (3.37) f 2 n (z) = k 2 z 2 h (1) n (k)J n (z) − j n (z)H n (k). (3.38) We readily see from the analyticity of z −n j n (z) on C that f i n for i = 1, 2 and n ∈ N + are analytic on the whole complex plane C, except that f 2 1 (z) is a meromorphic function on C with 0 being its only simple pole. By the symmetry property of j n (z): j n (−z) = (−1) n j n (z) [34], we have J n (−z) = j n (−z) + (−z)j ′ n (−z) = (−1) n j n (z) + (−1) n zj ′ n (z) = (−1) n J n (z) , which directly gives us the following lemma. As a consequence of Lemma 3.15, the zeros of f i n are symmetric with respect to the origin. To obtain an intuition about the behavior of those zeros, we numerically compute the zeros of f i n for i = 1, 2 and different values of n in the right half plane {z ∈ C ; − π 2 < arg(z) ≤ π 2 }, by Muller's method [6]. As we can observe in Figure 1, the zeros of f i n (z) are complex and lie in the lower half-plane. This fact has been theoretically justified by Theorem 3.2. Also the overall magnitudes of their imaginary parts rapidly decrease as the value of n increases. Moreover, it is remarkable to note that for fixed i and n, there is a sequence of zeros of f i n tending exponentially fast to the real axis. It motivates us to investigate the asymptotic behavior of zeros of f i n (z) as \|z\| → ∞. For this, we first consider f 1 n (z) and see the following asymptotics from (C.8) and (C.10) that for \| arg(z)\| < π, f 1 n (z) = h (1) n (k) cos z − nπ 2 − 1 z H n (k) cos z − nπ 2 − π 2 + e \|Imz\| O 1 \|z\| = h (1) n (k) cos z − nπ 2 + e \|Imz\| O 1 \|z\| as \|z\| → ∞ , (3.39) where we have also utilized the fact that both h (1) n (z) and H n (z) do not have real zeros. In view of (C.11), we can find generic positive constants C 1 , C 2 and C 3 depending on n such that \|f 1 n (z)\| ≥ \|h 1 n (k)\| e \|Imz\| − 1 2 − e \|Imz\| C 1 1 \|Imz\| ≥ C 2 , when \|Imz\| ≥ C 3 . Combining the above estimate with the symmetry of the zeros, it readily follows that the zeros of f 1 n (z) must lie in the strip: {z ∈ C ; \|Imz\| ≤ C 3 } . In this region, the remainder term e \|Imz\| O(\|z\| −1 ) in (3.39) converges to zero as \|z\| → ∞. Since all the zeros of the entire function cos(z − nπ 2 ) are real and simple, given by z n,l = (1 + 2l + n)π 2 , l ∈ N , (3.40) we foresee that there are zeros of f 1 n (z) lying near z n,l when \|z\| is large enough, which is indeed the case, by a direct application of Rouché's theorem and inverse function theorem. To see this, we define the entire function g n (z) = h (1) n (k) cos(z − nπ 2 ) on the complex plane C, which has the minimal period 2π in the sense that if α ∈ C satisfies g n (z +α) = g n (z) for all z, then α = 2πm for some integer m. Noting that g ′ n ( z n,l ) = 0 for l ∈ N, by the inverse function theorem, we can find an open neighborhood V n of z n,0 and an open neighborhood W n of the origin such that g(V n + lπ) = (−1) l W n and g is an analytic isomorphism from the neighborhood V n + lπ of z n,l to the neighborhood (−1) l W n of 0 for each l ∈ Z, where we also use the periodicity and symmetry of g n (z) : g n (z + lπ) = (−1) l g n (z), l ∈ Z. We denote by r n the radius of the largest ball contained in V n with center at the z n,0 , and define M n := inf z∈∂B( z n,l ,rn) \|g n (z)\| , which is independent of the value of l. When l is large enough, we can guarantee sup z∈∂B( z n,l ,rn) \|f 1 n (z) − g n (z)\| < M n , by using the asymptotic expansion (3.39). Then, the Rouché's theorem helps us to conclude that in the region B( z n,l , r n ) ⊂ V n + lπ, f 1 n (z) has a simple zero denoted by z n,l . It then directly follows that g n (z n,l ) ∈ (−1) l W n and 0 = g n ( z n,l ) = f 1 n (z n,l ) = g n (z n,l ) + O 1 \|z n,l \| . (3.41) Hence we have, by using (3.41) and the local invertibility of g ′ n , \|z n,l − z n,l \| \|g n (z n,l ) − g n ( z n,l )\| = \|z n,l − z n,l \| \|g n (z n,l ))\| ≤ sup ξ∈(−1) l Wn \|(g −1 n ) ′ (ξ)\| = sup z∈Vn \|g ′ n (z)\| −1 < +∞ , which immediately implies \|z n,l − z n,l \| ≤ C n 1 \|z n,l \| ≤ C n \|l\| −1 , n,l } l∈N denote the zeros with − π 2 < arg(z) ≤ π 2 . Then {z 1 n,l } has the following estimate: \|z 1 n,l − z 1 n,l \| ≤ C n l −1 for all l ∈ N + , (3.43) where { z 1 n,l } is given by (3.40). Recall that what we are truly interested in is λ 1 n,l := k 2 /((z 1 n,l ) 2 − k 2 ) ∈ σ 1 n . We translate the above lemma with respect to z 1 n,l to λ 1 n,l and obtain λ 1 n,l − 4k 2 (1 + 2l + n) 2 π 2 − 4k 2 ≤ C n \|l\| −4 , by applying the mean-value theorem to the one-dimensional function h(t) = k 2 /(( z 1 n,l + t(z 1 n,l − z 1 n,l )) 2 − k 2 ) on [0, 1]. This estimate can be further simplified as follows: λ 1 n,l − 4k 2 (1 + 2l + n) 2 π 2 ≤ C n \|l\| −4 as l → +∞ . For our second case, by a very similar argument applied to zf 2 n (z), which has the same zeros away from the origin as f 2 n (z) and satisfies the following asymptotic form: zf 2 n (z) = k 2 z h (1) n (k) cos z − nπ 2 − H n (k) cos z − nπ 2 − π 2 + e \|Imz\| O 1 \|z\| = −H n (k) cos z − nπ 2 − π 2 + e \|Imz\| O 1 \|z\| as \|z\| → ∞ , (3.44) we can obtain that the zeros {z 2 n,l } of f 2 n (z) in the right half plane satisfy the estimate: z 1 n,2 − z 2 n,l ≤ C n l −1 with z 2 n,l := (2l + n)π 2 for all l ∈ N + , (3.45) and the associated {λ 2 n,l } ⊂ σ 2 n have the asymptotics: λ n,l − 4k 2 (2l + n) 2 π 2 ≤ C n \|l\| −4 as l → +∞ . We now give the main result of this subsection. Theorem 3.17. Let {λ i n,l } l∈N be the eigenvalues in σ i n for i = 1, 2 and n ∈ N + . Then, when l → +∞, the following asymptotic estimates hold, λ 1 n,l − 4k 2 (1 + 2l + n) 2 π 2 = O(l −4 ) , λ 2 n,l − 4k 2 (2l + n) 2 π 2 = O(l −4 ) . (3.46) We refer the readers to Theorem B.2 for an interesting related result. Asymptotic behavior and localization of eigenfunctions Theorem 3.17 has clearly described asymptotic behaviors of the eigenvalues in σ i n , i = 1, 2. We see from (3.46) and (3.29) that when l is large enough, λ 1 n,l and λ 2 n,l will most likely be contained in the ε-neighborhood of τ −1 so that the high-frequency resonant modes E T E n,m (k λ , x), E T M n,m (k λ , x) for the same value of n and m will be excited simultaneously. Via the integral operator T k D , these resonant modes carrying the subwavelength information of the embedded sources can propagate into the far field. In this subsection, instead of considering the vector fields ·)](x), we consider their tangential component measurements for ease of exposition, which can be explicitly represented byx T k D [ E T E n,m (k λ , ·)](x) and T k D [ E T M n,m (k λ ,× T k D [ E T E n,m (k λ , ·)](x) = ik 3 n(n + 1)h (1) n (k\|x\|)U m n (x) 1 0 j n (kr)j n (k λ r)r 2 dr , andx × T k D [ E T M n,m (k λ , ·)](x) = − k n(n + 1) k λ \|x\| H n (k\|x\|)V m n (x) 1 0 J n (kr)J (k λ r) + n(n + 1)j n (kr)j n (k λ r)dr , for \|x\| > 1, by (C.12) and (C.13) in Appendix C.3. These formulas motivate us to define the following two propagating functions, respectively, responsible for the propagation of vector spherical harmonics U m n and V m n : ϕ λ,1 n (kt) := n(n + 1)λj n (k λ t) 0 < t ≤ 1 ik 3 n(n + 1)h (1) n (kt) 1 0 j n (kr)j n (k λ r)r 2 dr t > 1 ,(3.47) and ϕ λ,2 n (kt) :=    iλ √ n(n+1) k λ t J n (k λ t) 0 < t ≤ 1 − k √ n(n+1) k λ t H n (kt) 1 0 J n (kr)J (k λ r) + n(n + 1)j n (kr)j n (k λ r)dr t > 1 . (3.48) Here, to define ϕ λ,i n inside the domain for i = 1, 2, we have used the fact that E T E n,m and E T M n,m are eigenfunctions of T k D with eigenvalue λ. From the definitions (3.47) and (3.48), we readily see that when t > 1, ϕ λ,1 n (resp., ϕ λ,2 n ) is proportional to h (1) n (kt) (resp., H n (kt)), and thereby has the same asymptotic behavior as h (1) n (kt) (resp., H n (kt)) as t → +∞. To understand the roles played by ϕ λ,i n for different orders n in the far-field measurement, we give the result about their asymptotics for large order n. The detailed calculations and estimates are included in Appendix C.3. Proposition 3.18. The following asymptotic estimates uniformly hold for t in a compact subset of (1, +∞), ϕ λ,1 n (t) = O e 2t n+1 k n−1 λ (n + 1) n , ϕ λ,2 n (t) = O ek 2t n−1 k n−2 λ (n − 1) n−3 as n → ∞ ,(3. 49) where we remind that the big-O terms are bounded by constants independent of n but depending on other parameters: the wave number k, the eigenvalue λ, and the compact set for variable t. In view of the exponential decay of propagating functions ϕ λ,i n in (3.49) when n tends to infinity, we have theoretically justified the previously mentioned fact in the introduction that the evanescent part of the radiating EM wave with the fine-detail information of the objects, i.e., the remainder term of the infinite sum in (3.31) from large enough n, is almost negligible in the measured far-field data. It is the low-frequency component: E low (x) = N n=1 n m=−n γ n,m E T E n,m (k, x) + η n,m E T M n,m (k, x) , γ n,m , η n,m ∈ C , \|x\| ≫ 1 that dominates the far-field behavior of the radiating wave E, where N is a given small positive integer . We plot both real and imaginary parts of ϕ λ,i n in Figures 2 and 3 for different values of n and k = 1, from which we can clearly observe that the higher the resonant mode oscillates, the smaller the amplitude is. (a) ϕ λ,1 9 (t). Left: t ∈ (0, 1); right: t∈ (1, 6). (b) ϕ λ,2 9 (t). Left: t ∈ (0, 1); right: t∈ (1, 6). Figure 3: Propagating function ϕ λ,i 9 for the first four λ from σ i 9 , i = 1, 2. First row: real part of ϕ λ,i 9 ; second row: imaginary part of ϕ λ,i 9 , for i = 1, 2. We also note from Figures 2(a) and 3(a) that the imaginary parts of ϕ λ,1 n for different n have very small amplitudes inside and outside the domain, while for the case ϕ λ,2 n , it is the real part. However, it is not a surprising fact, if we recall from Theorem 3.17 that the eigenvalues λ of T k D is near the real axis and there is an additional factor i in the definition of ϕ λ,2 n , compared to ϕ λ,1 n (cf. (C.3) and (C.4)). We next consider the behaviors of the propagating functions ϕ λ i n,l ,i n (kt) for i = 1, 2 inside the domain. To simplify the notations, we re-denote them as follows: ϕ 1 n,l (kt) := ϕ λ 1 n,l ,1 n (kt) = n(n + 1)λ 1 n,l j n (z 1 n,l t) for t ∈ [0, 1] , (3.50) and ϕ 2 n,l (kt) := ϕ λ 2 n,l ,2 n (kt) = iλ 2 n,l n(n + 1) z 2 n,l t J n (z 2 n,l t) for t ∈ [0, 1] . (3.51) By estimates (3.43) and (3.45), the zeros z i n,l , i = 1, 2 have very small imaginary parts when l is large enough (for the case n = 5, Imz i n,l ∼ 10 −8 by numerical simulation, see Figure 1). This indicates that ϕ 1 n,l is almost a real function while ϕ 2 5,l is almost purely imaginary (for the case n = 5, Imϕ 1 n,l ∼ 10 −10 and Reϕ 2 n,l ∼ 10 −11 by numerical simulation, see Figure 2). We plot in Figure 4 the normalized real parts of propagating function ϕ 1 n,l (k\|x\|): Reϕ 1 n,l (k\|x\|) = Reϕ 1 n,l (k\|x\|) max 0≤\|x\|≤1 Reϕ 1 n,l (k\|x\|) , and the normalized imaginary parts of propagating function ϕ 2 n,l (k\|x\|): Imϕ 2 n,l (k\|x\|) = Imϕ 2 n,l (k\|x\|) max 0≤\|x\|≤1 Reϕ 2 n,l (k\|x\|) on a two-dimensional cross-sectional plane of the ball B(0, 1) passing through the origin, for k = 1, n = 5, and different values of l. And we readily see from Figure 4 that for a fixed n, when l tends to infinity, both Reϕ 1 5,l (\|x\|) and Imϕ 2 5,l (\|x\|) present a remarkable localization pattern in the sense that they are highly oscillating, essentially distributed in a small neighborhood of the origin and rapidly attenuated towards the boundary. We now give a qualitative mathematical result to illustrate this localization phenomenon. max t∈[a,1] \|ϕ 1 n,l (kt)\| max t∈[0,1] \|ϕ 1 n,l (kt)\| = O(l −1 ) , max t∈[a,1] \|ϕ 2 n,l (kt)\| max t∈[0,1] \|ϕ 2 n,l (kt)\| = O(l −1 ) as l → ∞ , (3.52) where a is a positive real number from (0, 1). Proof. The proof is direct and simple based on two lemmas in Appendix C. We only give the argument for the first estimate in (3.52). The analysis for the second one can be conducted by the same idea. In fact, by Lemma C.2 and the asymptotic expansion (C.8), we have max t∈[a,1] \|ϕ 1 n,l (kt)\| max t∈[0,1] \|ϕ 1 n,l (kt)\| = max t∈[a,1] \|j n (z 1 n,l t)\| max t∈[0,1] \|j n (z 1 n,l t)\| ≤ C 1 \|z 1 n,l \| −1 max t∈[0,1] \|j n ( z 1 n,l t)\| − C 2 \|l\| −1 , where C 1 and C 2 are some generic constants depending on n. Note that letting l tends to infinity, both { z 1 n,l } and {z 1 n,l } vanish with the rate l −1 . Then the result directly follows from Lemma C.1. Remark 3.20. It is also possible to obtain more subtle estimates for the localization speed under various L p -norm (p ≥ 1) in a similar manner as in [33], where the authors considered the high-frequency localization of Laplacian eigenfunctions under various boundary conditions and norms. However, the detailed discussions are beyond the scope of this work. We intend to investigate this interesting topic in our future work. Applications to super-resolutions in high contrast media We have established the main mathematical results in this work concerning the spectral properties of T k D and the behavior of the resolvent (λ − T k D ) −1 in the high contrast regime, as well as the asymptotic estimates for the eigenvalues and eigenfunctions for a spherical domain. In this section, we shall derive the resonance expansions for the Green's tensor G and its imaginary part ImG, by Theorems 2.2 and 3.9, and use it to explain the expected superresolution phenomenon when imaging the source f embedded in the high contrast medium. We shall also provide the numerical experiments for the case of a spherical region to show the existence of the possible subwavelength peaks of the imaginary part of the Green's tensor. Resonance expansion of Green's tensor To write the resonance expansion for the Green's tensor G, we directly substitute the pole-pencil decomposition in (3.25) into the representation of G in (2.27) with a polarization p ∈ S 2 and then obtain G(z, z ′ , k)p = 1 k 2 τ ∇ z div z ( g(z, z ′ , k)p) + 1 τ + 1 P 0 G(z, z ′ , k)p + 1 τ i∈I Ni j=1 ϕ j λi (z) · (τ −1 − J j λi ) −1 (P j λi G(·, z ′ , k)p) ϕ j λ i + (1 − τ T k ζ ) −1 [P ζ G(·, z ′ , k)p](z) ,(4.1) for z ∈ D and z ′ ∈ D ′ ; see Theorem 2.2 for the definitions of g and G here. To derive the resonance expansion of ImG, we first recall the explicit form of P 0 : −∇Sdiv and formula (2.22), and then have ImP 0 G(z, z ′ , k)p = P 0 Im G(z, z ′ , k)p = −∇ z Sdiv z ImG 0 (z, z 0 , k)p + ∇ z Sdiv z 1 k 2 ∇ z div z Im g(z, z 0 , k)p = − 1 k 2 ∇ z div z (Im g(z, z 0 , k)p) ,(4.2) by noting that div z ImG 0 (z, z 0 , k)p = 0 and S is the inverse of −∆ in the variational sense (cf. (A.1)). In view of (4.2), taking imaginary part of both sides of (4.1) gives us the following resonance expansion of ImG: ImG(z, z ′ , k)p =Im 1 τ i∈I Ni j=1 ϕ j λi (z) · (τ −1 − J j λi ) −1 (P j λi G(·, z ′ , k)p) ϕ j λ i + Im(1 − τ T k ζ ) −1 [P ζ G(·, z ′ , k)p](z) , z ∈ D, z ′ ∈ D ′ ,(4.3) which has a more concise expression than (4.1). Note that the counterpart of the expansion (4.3) for the imaginary part of the free space Green's tensor ImG 0 can be derived from (2.22) and (4.2): ImG 0 (z, z ′ , k)p = 1 k 2 ∇div(Im g(z, z ′ , k)p) + Im(P 0 + P σ + P ξ ) G(z, z ′ , k)p = Im i∈I Ni j=1 ϕ j λi (z) · (P j λi G(·, z ′ , k)p) ϕ j λ i + ImP ξ G(z, z ′ , k)p , z ∈ D, z ′ ∈ D ′ ,(4.4) where we have used the fact that P 0 + P σ + P ξ is the identity operator, and the definition of P σ in (3.21) and the expression (3.24). The first term in the above expansion may be viewed as the high-frequency part of ImG 0 that can encode the subwavelength information of the sources due to the super-oscillatory nature of the generalized eigenfunctions in the Jordan chains ϕ j λi , see Figures 3 and 4. Comparing it with (4.3), we can find that this highfrequency part is amplified by the resolvents of Jordan matrices: (1 − τ J j λi ) −1 when τ −1 is approaching the eigenvalues λ i , i ∈ I. Therefore, the imaginary part of G may display a sharper peak than the one of G 0 for some specified high contrast parameters, and thus help us more accurately resolve subwavelength details. Numerical illustrations In this subsection, we numerically study the imaginary part of the Green's tensor G(x, y, k) corresponding to the spherical medium B(0, 1) with the high contrast τ , as a complement of the analysis and the illustration for the super-resolution provided in the previous subsection. For the sake of simplicity, we let y = 0 and write G(x, k) (resp., G 0 (x, k)) for G(x, 0, k) (resp., G 0 (x, 0, k)). By the addition formula in (C.7) for G 0 and noting that E T E n,m (k, 0) = 0 for n ≥ 1 and E T M n,m (k, 0) = 0 for n ≥ 2, we have G 0 (x, k) = ik 2 1 m=−1 E m (k, x) ⊗ E m (k, 0) , x ∈ R 3 \{0} . Here and throughout this subsection, we simply denote E T M 1,m (resp., E T M n,m ) by E m (k, x) (resp., E m ), for m = −1, 0, 1. As in Section 3.3, via the vector wave functions, we assume that the Green's tensor G with a real polarization vector p ∈ R 3 has the following ansatz: G(x, k)p = G 0 (x, k τ )p + 1 m=−1 a m E m (k τ , x) \|x\| ≤ 1 , 1 m=−1 b m E m (k, x) \|x\| ≥ 1 , (4.5) where a m and b m for m = −1, 0, 1 are complex constants to be determined and linearly depending on p. To proceed, we note that, from (C.5) and (C.6), it follows that To avoid calculating the three coefficients a m (m = −1, 0, 1), we choose a special real polarization vector p:   x × G 0 (x, k τ )p = − 1 √ 2\|x\| H 1 (k τ \|x\|) 1 m=−1 V m 1 (x) E m (k τ , 0) t · p , x ∈ R 3 \{0} , x × ∇ × G 0 (x, k τ )p = − k 2 τ √ 2 h (1) 1 (k τ \|x\|) 1 m=−1 U m 1 (x) E m (k τ , 0) t · p , x ∈ R 3 \{0} .p = p E0(kτ ,0) 2 ∈ R 3 , p = E 0 (k τ , 0)/i ,(4.7) according to two easily verified observations that E m (k, 0), m = −1, 0, 1 are orthogonal vectors with the same l 2 -norms (cf. (C.13)), and E 0 (k τ , 0) has purely imaginary components since Y 0 1 (x) is a real vector function on S 2 . With this specially chosen p, we can simplify (4.6) as follows: x × G 0 (x, k τ )p = i √ 2\|x\| H 1 (k τ \|x\|)V 0 1 (x) , x ∈ R 3 \{0} , x × ∇ × G 0 (x, k τ )p = ik 2 τ √ 2 h (1) 1 (k τ \|x\|)U 0 1 (x) , x ∈ R 3 \{0} . (4.8) Matching the Cauchy data of the field in (4.5) inside and outside the domain on the boundary ∂B(0, 1), we obtain, by using (C.5) and (C.6), that a −1 = a 1 = 0 and b −1 = b 1 = 0, and the following equation for (a 0 , b 0 ): 1 ikτ J 1 (k τ ) − 1 ik H 1 (k) −ik τ j 1 (k τ ) ikh (1) 1 (k) a 0 b 0 = i 2 H 1 (k τ ) ik 2 τ 2 h (1) 1 (k τ ) . Then the solution a 0 to the above equation readily follows (we only need a 0 to investigate the behavior of G inside the domain): a 0 = − k 2 2kτ H 1 (k τ )h(1)1 (k) + kτ 2 H 1 (k)h(1)1 (k τ ) k 2 k 2 τ J 1 (k τ )h(1)1 (k) − j 1 (k τ )H 1 (k) . We regard a 0 as a function of the real variable k τ and plot its absolute value in Figure 5 for k = 1, from which we clearly see that it blows up when k τ hits the real parts of the discrete zeros z 2 1,l of f 2 n (z). Since the spherical harmonics has nothing to do with the contrast τ , in the following, we shall pay attention to the imaginary part of the radial part: φ(k τ , t) = i √ 2t H 1 (k τ t) − a 0 √ 2 ik τ t J n (k τ t), t ∈ [−1, 1] , of the tangential componentx × G(x, k)p of G(x, k)p: x × G(x, k)p = i √ 2\|x\| H 1 (k τ \|x\|)V 0 1 (x) − a 0 √ 2 ik τ \|x\| J n (k τ \|x\|)V 0 1 (x) . We remark that φ(k τ , t) is a one-dimensional function but keeping all the main features of ImG(x, k)p we are interested in; and the radial part of the normal componentx · G(x, k)p has a very similar behavior as φ(k τ , t). From Figure 6(a), where we present Imφ(k τ , t) for different values of k τ , we see that when k τ increases and hits the real parts of z 2 1,l , the imaginary part of Green's tensor become highly oscillating and exhibit a subwavelength peak, and hence the super-resolution can be achieved with the increasing likelihood. When τ tends to infinity, we can even expect the infinite resolvability of the imaging system, by Theorems 3.17 and 3.19. However, we would like to stress that the super-resolution phenomenon can only be expected for discrete values of τ . For those τ taking high values but not near the resonant values, the magnitude of ImG(x, k)p will not be significantly enhanced and have almost the same order of ImG 0 (x, k)p, although it is more oscillatory than the one in the homogeneous space; see Figure 6(b). Concluding remarks In this work, we have considered the time-reversal reconstruction of EM sources embedded in an inhomogeneous background, and tied its anisotropic resolution to the resolvent of a certain type of integral operators T k D via a newly derived Lippmann-Schwinger representation that reveals the close relation between the medium (shape and refractive indices) and its associated EM Green's tensor. We have then investigated the spectral structure of T k D for a bounded smooth domain with a very general geometry and found that all the poles of its resolvent in C\σ ess (T k D ) are eigenvalues of finite type and lie in the upper-half plane with σ ess (T k D ) being all its possible accumulation points. With these new findings, we have derived the pole-decomposition for the resolvent of T k D and obtained the local resonance expansion for the Green's tensor associated with the high contrast medium. More quantitative results about the asymptotic behaviors of eigenvalues and eigenfunctions have been also provided for the case of a spherical domain. As a byproduct of our spectral analysis, we have given a characterization and discussion about the EM nonradiating sources, see Remarks 3.8 and 3.14. Some further interesting spectral results about the operator T k D based on the fact that T k D is a quasi-Hermitian operator have been included in Appendix B. In Section 4, we have applied our new theoretical results to explain the expected super-resolution in the inverse electromagnetic source problem at some discrete characteristic values. It turns out that both eigenvalues and eigenfunctions are responsible for the super-resolution phenomenon in the sense that the eigenfunctions are super-oscillatory and can encode the subwavelength information of the sources; while the eigenvalues serve as an amplifier when they nearly hit the reciprocal of the contrast so that these subwavelength information can be measurable in the far field. We finally remark that our analysis and results can be naturally extended to the Lipschitz domain by noting the facts that the Helmholtz decomposition in Appendix A still holds [14] and that for a selfadjoint operator on a Hilbert space, the essential spectrum is a compact subset of the real line [28]. Appendix A. Helmholtz decomposition of L 2 -vector fields In this section we give a complete review of the Helmholtz decomposition of L 2 -vector fields in a unified manner due to its great significance to our main analysis in the work. For a vector field u, the Helmholtz decomposition provides us a procedure to separate its divergence, curl, and the normal trace information. In the following, we show how to extract these information from a field u by solving some sub-variational problems. Let us first give a more precise description about the geometry of the domain D. We denote by Γ j , 0 ≤ j ≤ J, the connected component of ∂D, in which Γ 0 is the boundary of the unbounded connected component of R 3 \D. And the genus L of ∂D may be nontrivial, i.e., L ≥ 0 (for L ≥ 1, we can construct interior cuts: Then there exists a unique solution ψ 1 := −Sdivu ∈ H 1 0 (D) satisfying (A.1), from which it follows that u − ∇ψ 1 is divergence-free in the distribution sense, and the normal trace γ n is well-defined. Σ i , 1 ≤ i ≤ L contained in D such that D\ ∪ L i=1 Σ i is To obtain the curl part of u, we need to solve a magnetostatics problem. To do so, we introduce the Hilbert space X N := H 0 (curl, D) H(div, D) with the graph norm · XN := · L 2 (D) + div· L 2 (D) + curl· L 2 (D) , and its subspace X 0 N := H 0 (curl, D) H(div0, D). By the well-known de Rham diagram (cf. [32, Section 3.7]), we see that the kernel space of the curl operator in H 0 (curl, D), i.e., H 0 (curl0, D), has the following orthogonal decomposition: H 0 (curl0, D) = ∇H 1 0 (D) ⊕ ⊥ K N (D), (A.2) where K N (D) is the normal cohomology space with the dimension J, given by By Friedrich's inequality (cf. [14,Corollary.3.19]), on the space X N , the seminorm \| · \| XN := curl· L 2 (D) + div· L 2 (D) + J j=1 γ n ·, 1 H 1/2 (Γj ) is equivalent to the graph norm · XN . We now define the following quotient space: X N := X N /K N (D) with the standard quotient norm [u] XN := inf v∈KN (D) \|u + v\| XN , where [u] ∈ X N denotes the equivalent class of u. It is easy to see that the quotient norm has an explicit form: [u] XN = curl[u] L 2 (D) + div[u] L 2 (D) ,(A.v = − J j=1 γ n u, 1 H 1/2 (Γj ) ∇p j ∈ K N (D) such that for the representation element u + v of [u], the term J j=1 \| γ n ·, 1 H 1/2 (Γj ) \| vanishes, which directly leads us to (A.3). Moreover, on the subspace X 0 N := X 0 N /K N (D) the quotient norm reduces to curl· L 2 (D) . We are now ready to consider the following magnetostatic field problem: for f ∈ L 2 (D, R 3 ), find ψ ∈ X N such that    curlcurlψ = curlf in D , divψ = 0 in D , ν × ψ = 0 on ∂D , (A.4) which shall be seen to have a unique solution. Its variational formulation is given by the next lemma. Lemma A.2. The system (A.4) is equivalent to the following variational problem: find ψ ∈ X N such that it holds, for all φ ∈ X N , that Denoting the space defined in (A.6) by X, we then obtain L 2 (D) = div(X/K N (D)) since for all v ∈ L 2 (D), we can find ϕ ∈ H 1 0 (D) such that ∆ϕ = v in the variational sense. By choosing φ ∈ X/K N (D) in (A.5), we readily see divψ = 0, and hence the proof is complete. To show the existence and uniqueness of a solution, we introduce the isomorphism T : is an irrational and divergence-free vector field, i.e., divv = curlv = 0. X ′ N → X N such that for l ∈ X ′ N , T l satisfies l, φ XN = (curlTl, curlφ) L 2 (D) + (divTl, divφ) L 2 (D) , ∀ ϕ ∈ X N , The last step regarding the normal trace is relatively simple by noting the fact that the restriction of normal trace mapping γ n := γ n \| W on W is an isomorphism from W to H −1/2 0 (∂D). To be precise, for φ ∈ H −1/2 0 (∂D), γ −1 n φ is the gradient, which is unique, of a solution to the following Neumann problem: ∆p = 0 in D , ∂p ∂ν = φ on ∂D . By setting φ = γ n v, where v is introduced in (A.8), we can find an element γ −1 n γ n v from W to characterize the normal trace information of v (and also u). However, after we remove the divergence, curl and normal trace component of u, the remaining part: u − ∇ψ 1 − curlψ 2 − γ −1 n γ n v is still nontrivial if the genus L ≥ 1, and it is located in the so-called tangential cohomology space K T (D), defined by K T (D) = {u ∈ H 0 (div, D) ; ∇ × u = 0, divu = 0 in D} , which has dimension L. We remark that there exists a similar characterization as in Lemma A.1 for K T (D). We now summarize the above constructions in the following result, where the L 2 -orthogonal relation can be verified directly. Theorem A.3. L 2 (D, R 3 ) has the following L 2 -orthogonal decomposition: L 2 (D, R 3 ) = ∇H 1 0 (D) ⊕ ⊥ curl X 0 N ⊕ ⊥ W ⊕ ⊥ K T (D) , where ∇H 1 0 (D), curl X 0 N , and W are uniquely determined by divu, curlu, and γ n (u + ∇Sdivu), respectively. Here, the operator S is given by (A.1). Appendix B. T k D as a quasi-Hermitian operator B.1. A global resolvent estimate In this subsection, we provide a resolvent estimate for (λ − T k D ) −1 on ρ(T k D ) by applying a general spectral result from [27]. To do this, We first introduce some notions. We consider the bounded linear operator A acting on a separable Hilbert space H. The imaginary Hermitian component A I and the real Hermitian componet A R are defined as follows: A I = A − A * 2i , A R = A + A * 2 , where A * is the adjoint operator of A in the Hilbert sense. Moreover, we say that an operator A is quasi-Hermitian operator if it is a sum of a selfadjoint operator and a compact one. For such kind of operators, we have a general resolvent bound under the condition (cf. where λ k (A) are the eigenvalues of A counting multiplicity and · HS denotes the Hilbert-Schmidt norm. For our purpose, we write T k D as the sum of T D and N k D := T k D − T D , where T D is known to be a selfadjoint operator. We consider the kernel K N of the integral operator N k D : K N (x, y) := (k 2 + ∇ x div x )(g(x, y, k) − g(x, y, 0)) . It is easy to see that when x approaches y, the kernel has following singularity: K N (x, y) = O 1 \|x − y\| . It directly follows that N k D and its imaginary Hermitian component N k D,I are Hilbert-Schmidt operators. We further note the relation: which helps us to conclude that T k D is a quasi-Hermitian operator satisfying condition (B.1), and thus Theorem (B.1) can be applied. B.2. Decay property and bound of the imaginary parts of eigenvalues Formula (B.3) has suggested us that {Imλ k (A)} is a bounded sequence and tends to zero when k → ∞. Its detailed proof can be found in [27, pp.106-107]. Here we provide a sketch of the main argument for the sake of completeness. For a quasi-Hermitian operator A satisfying condition (B.1), we have the following triangular representation: (Imλ k (A)) 2 < +∞ . A = D + V , We end this appendix with the corresponding result for T k D . Theorem B.2. For the integral operator T k D defined in (2.2), its spectrum σ(T k D ) is contained in a strip in the complex plane: σ(T k D ) ⊂ {z ∈ C ; \|Imz\| ≤ C} for some C , and the imaginary parts of the eigenvalues in the spectrum consists of a 2-power summable sequence, i.e., ∞ i=0 Imλ i (T k D ) 2 < +∞ , λ i ∈ σ f (T k D ) . where j n (t) is the spherical Bessel function of the first kind and order n and J n is given by J n (t) := j n (t) + tj ′ n (t). Then, a direct calculation gives us the tangential traces of the multipole fields:   x × E T E n,m (k, x) = n(n + 1)h J n (kr)J n (k λ r) + n(n + 1)j n (kr)j n (k λ r)dr = − k n(n + 1) k λ \|x\| H n (k\|x\|)V m n (x) 1 0 J n (kr)J (k λ r) + n(n + 1)j n (kr)j n (k λ r)dr . (C. 13) The integrals involved in (C.12) and (C.13) can be explicitly calculated by the Lommel's integrals [38] for n ≥ 1: 1 0 j n (kr)j n (k λ r)r 2 dr = 1 k 2 − k 2 λ [k λ j n (k)j n−1 (k λ ) − kj n−1 (k)j n (k λ )] , (C.14) and 1 0 n(n + 1)j n (kr)j n (k λ r) + J n (kr)J n (k λ r)dr = kk λ 2n + 1 (n + 1) 1 0 j n−1 (kr)j n−1 (k λ r)r 2 dr + n 1 0 j n+1 (kr)j n+1 (k λ r)r 2 dr . (C. 15) We next provide the calculations and estimates for Proposition 3.18. We recall the following asymptotic forms of j n and h (1) n for large n that uniformly hold for z in a compact subset of C away from the origin: n and Stirling's formula (cf. [21, p.30]). For the propagating function ϕ λ,1 n (kt), by (C.12) and (C.14), a direct application of (C.16) gives us, for t from a compact subset of (1, +∞), = O e 2t n+1 \|k λ \| n−1 (n + 1) n . A very similar but more complicated calculation yields the second estimate in (3.49). We omit the details here. The following two lemmas were used for Theorem 3.19. \|f (x)\| for any a ∈ R larger than some fixed a 0 > 0. Moreover, let {a n } be a sequence such that a n → +∞ when n → +∞, then {f (a n x)} are localized near the origin in the sense that lim n→+∞ max x∈[a,1] \|f (a n x)\| max x∈[0,1] \|f (a n x)\| = 0 . Lemma C.2. For j n (z) and J n (z)/z, the following estimates uniformly hold for t ∈ [0, 1], j n (z 1 n,l t) − j n ( z 1 n,l t) = O(l −1 ) , J n (z 2 n,l t) z 2 n,l t − J n ( z 2 n,l t) = 0 is a complex number with Imλ ≤ 0. We shall prove that if λ = −1 (equivalently, k λ = 0), u must be zero everywhere; if λ = −1, then u ∈ ∇H 1 0 (D), provided that the open set R 3 \D is connected. For this purpose, choose an open ball B(0, R) centered at the origin with large enough radius R such thatD ⊂ B(0, R), and multiply both sides of the second equation in the system (3.3) by the test functionū. Then a direct integration by parts on B(0, R nij the coefficients in the expansion of ϕ with respect to the basis {ϕ j,s λi } nij −1 s=0 , i.e., 31) where the complex coefficients γ n,m and η n,m are to be determined and ET E n,m and E T M n,m are vector wave functions defined in the Appendix C.1. Similarly, any solution E to the Maxwell equations ∇ × ∇ × E − k 2 λ E = 0 near 0 has the following representation: m (k λ , x) + β n,m E T M n,m (k λ , x) , (3.32) with undetermined coefficients α n,m , β n,m ∈ C, see (C.3) and (C.4) for the definitions of E T E n,m and E T M n,m . Theorem 3 . 13 . 313Denote by σ 1 n and σ 2 n the sets of λ such that (3.35) and (3.36) holds respectively, then we have the set of eigenvalues of finite type of T k D for a spherical region B(0, 1) is given by Lemma 3 . 15 . 315For f i n (z), n ∈ N + , i = 1, 2 defined by (3.37) and (3.38), the following symmetry properties hold, f 1 n (−z) = (−1) n f 1 n (z) , f 2 n (−z) = (−1) n f 2 n (z) . Figure 1 : 170 zeros of f i n (z) for i = 1 (the first row), i = 2 (the second row), and n = 1, 5, 9 (from left to right) in the right half plane: {z ∈ C ; − π 2 < arg(z) ≤ π 2 }. enough l, where C n denotes a generic constant depending on n and may have different values in the following. Further, considering the fact that a non-constant analytic function on the closure of a bounded domain can only have finite zeros, we arrive at the following result. Lemma 3 . 16 . 316The zeros of f 1 n (z) are symmetric with respect to the origin and contained in the strip: {z ∈ C ; \|Imz\| ≤ C} for some constant C. Let {z 1 Figure 2 : 2Propagating function ϕ λ,i 5 for the first four λ from σ i 5 , i = 1, 2. First row: real part of ϕ λ,i 5 ; second row: imaginary part of ϕ λ,i 5 , for i = 1, 2. 22.8956, 70.4700, 164.8413) from left to right. Figure 4 : 4(a) Normalized real part of ϕ 1 5,l (\|x\|); (b) normalized imaginary part of ϕ 2 5,l (\|x\|) for different values of l on the cross-sectional plane: \|x\| ≤ 1 with x 3 = 0. Theorem 3 . 19 . 319Let {ϕ i n,l }, i = 1, 2 be the sequences of propagating functions defined by (3.50) and (3.51). Then the following asymptotics hold, Figure 5 : 5\|a 0 (k τ )\| as a function of k τ , k τ ∈[1, 50]. = 1 , 1Re(z 2 1,l ) for l = 2, 3, 4, 5, i.e., kτ =1, 7.5944, 10.8119, 13.9949, 17.1626. Figure 6 : 6Imaginary part of φ(k τ , t) for various k τ . simple connected; see [32, Section 3.7]). A typical example of D with L = 1 and J = 1 is a torus with a ball hole. Denote by S : H −1 (D) → H 1 0 (D) the solution operator of the Dirichlet source problem, namely, for l ∈ H −1 (D), Sl ∈ H 1 0 (D) solves the variational problem: Find ψ ∈ H 1 0 (D) such that l, ϕ H 1 0 (D) = (∇ψ, ∇ϕ) L 2 (D) , ∀ ϕ ∈ H 1 0 (D) . (A.1) We remark that S is an isomorphism between H −1 (D) and H 1 0 (D). Note that div : L 2 (D, R 3 ) → H −1 (D) is the adjoint operator of −∇ : H 1 0 (D) → L 2 (D, R 3 ). For u ∈ L 2 (D, R 3 ), we consider (A.1) with l, ϕ H 1 0 (D) := (u, ∇ϕ) L 2 (D) , ∀ ϕ ∈ H 1 0 (D) . K N (D) = {u ∈ H 0 (curl, D) ; ∇ × u = 0, divu = 0 in D}. Moreover, we have the following characterization of K N (D) from [32, Theorem 3.42]. Lemma A.1. K N (D) is spanned by ∇p j , 1 ≤ j ≤ J, where p j ∈ H 1 (D) satisfies ∆p j = 0 in D, and p j = δ j,s on Γ s , 0 ≤ s ≤ J.In addition ∂pj ∂ν , 1 H 1/2 (Γs) = δ j,s , 1 ≤ j ≤ J, and ∂pj ∂ν , 1 H 1/2 (Γ0) = −1. (f, curlφ) L 2 (D) = (curlψ, curlφ) L 2 (D) + (divψ, divφ) L 2 (D) . (A.5) Proof. If ψ is a solution of (A.4), by the first equation in (A.4), then it holds for all φ ∈ H 0 (curl, D) that (f, curlφ) L 2 (D) = (curlψ, curlφ) L 2 (D) . Therefore, by combining it with the fact that divψ = 0, we can directly see that (A.5) holds. Conversely, if (A.5) holds, it suffices to prove that divψ = 0 to conclude the lemma. Recalling (A.2), we have H 0 (curl0, D) H(div, D) = {∇ϕ ; ϕ ∈ H 1 0 (D) with ∆ϕ ∈ L 2 (D)} ⊕ ⊥ K N (D). (A.6) by (A.3) and Riesz representation theorem. We note that curl can be regarded as a continuous mapping from L 2 (D, R 3 ) toX ′ N , by setting curlu, φ XN := (u, curlφ) L 2 (D) , (A.7) which is well-defined since curlφ is independent of the choice of the representative element of [φ]. Then for u ∈ L 2 (D, R 3 ), there is a unique ψ 2 := Tcurlu ∈ X 0 N solving (A.5) or (A.4) with f = u. By the above constructions, we can see that the remaining v of u ∈ L 2 (D, R 3 ): v := u − ∇ψ 1 − curlψ 2 = u + ∇Sdivu − curlTcurlu ∈ L 2 (D, R 3 ) , (A.8) [ 27 , 27Thm.7.7.1]): A I is a Hilbert-Schmidt operator. (B.1) Theorem B.1. Under condition (B.1), the following bound for the norm of (λ − A) −1 holds, (λ − A) such that σ(D) = σ(A), where D is a normal operator and V is a compact operator with σ(V ) = {0} and < +∞ . Then, by using σ(A) = σ(D) and the fact that D is a normal operator, result of series expansions of j n and h Lemma C. 1 . 1Suppose that f (x) is a continuous function on [0, +∞) with f (x) → 0 as x → +∞. z 2 n,l t = O(l −1 ) , (C.17) when l tends to infinity. Here, {z i n,l } and { z i n,l }, i = 1, 2, are the same as the ones in (3.43) and (3.45). 3 ) 3where curl[u] and div[u] are well-defined. Indeed, we can choose Appendix C. Some definitions, calculations and auxiliary results for Section 3.3C.1. Vector wave functionsLet Y m n (x), n = 0, 1, 2, · · · , m = −n, · · · , n, be the spherical harmonics on S 2 . The vector spherical harmonics, which form a complete orthonormal system of L 2 T (S 2 ) [21,Theorem 6.25], are introduced as follows:U m n = 1 n(n + 1) ∇ S Y m n , V m n =x × U m n , n = 1, 2, · · · , m = −n, · · · , n .Define the radiating electric multipole fields in R 3 \{0} for n = 1, 2, · · · and m = −m, · · · , n[32]:where hThe entire electric multipole fields E T E n,m (k, x) and E T M n,m (k, x) can be similarly introduced[32]:We end this section with the addition formula of the Green's tensor G 0 (x, y, k) [21, Theorem 6.29]:C.2. Asymptotic expansions for spherical Bessel functionsWe collect some standard results about asymptotic expansions for j n (z), n ≥ 0. For the complex variable z with \| arg(z)\| < π, the following asymptotics holds[38, p.199],Combining (C.8) with the following recurrence relations of Bessel functions[34,38]:we see the asymptotic form of j ′ n (z):By definition of J n (z), (C.8) and (C.9), it holds thatwhere we have also used the observation:(C.11)C.3. Auxiliary results for propagating functionsIn this section, we first calculate the tangential traces ofx·)](x) on the sphere ∂B(0, \|x\|) with radius \|x\| > 1, where D = B(0, 1). By the addition formula for the Green's tensor (C.7) and the definition of T k D , we have, by using the orthogonality of {U m n } and {V m n },Proof. For the first estimate, we first observe from (C.9) and (C.11) that \|j ′ n (z)\| is bounded by a constant M on the strip:where the constant C ∈ R is chosen such that {z 1 n,l } l∈N lie in (C.18). Then we have, by using the analyticity of j n (z) and the contour integral, \|j n (z n,l t) − j n ( z n,l t)\| = γ j ′ n (ξ)dξ ≤ M \|z n,l − z n,l \|\|t\| , t ∈ [0, 1] , where γ is the segment connecting z 1 n,l t with z 1 n,l t. Combining the above estimate with (3.43), we can conclude that the first estimate in (C.17) holds. For the second estimate, it suffices to note that the derivative of J n (z)/z is entire and satisfies the following asymptotic form:as \|z\| → ∞ , and can also bounded on a strip of the form (C.18) with a different constant C such that it contains the zeros {z 2 n,l } of f 2 n (z). Then, in view of (3.45), the same argument as the previous one allows us to complete the proof. The inverse source problem for Maxwell's equations. Inverse problems. R Albanese, P B Monk, 221023R. Albanese and P. B. Monk. The inverse source problem for Maxwell's equations. Inverse problems, 22:1023, 2006. 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[ "JONES REPRESENTATIONS OF THOMPSON'S GROUP F ARISING FROM TEMPERLEY-LIEB-JONES ALGEBRAS", "JONES REPRESENTATIONS OF THOMPSON'S GROUP F ARISING FROM TEMPERLEY-LIEB-JONES ALGEBRAS" ]	[ "Valeriano Aiello ", "Arnaud Brothier ", "Roberto Conti " ]	[]	[]	Following a procedure due to V. Jones, using suitably normalized elements in a Temperley-Lieb-Jones (planar) algebra we introduce a 3-parametric family of unitary representations of the Thompson's group F equipped with canonical (vacuum) vectors and study some of their properties. In particular, we discuss the behaviour at infinity of their matrix coefficients, thus showing that these representations do not contain any finite type component. We then focus on a particular representation known to be quasi-regular and irreducible and show that it is inequivalent to itself once composed with a classical automorphism of F. This allows us to distinguish three equivalence classes in our family. Finally, we investigate a family of stabilizer subgroups of F indexed by subfactor Jones indices that are described in terms of the chromatic polynomial. In contrast to the first non-trivial index value for which the corresponding subgroup is isomorphic to the Brown-Thompson's group F 3 , we show that when the index is large enough this subgroup is always trivial.MSC 2010: 22D10, 46L37, 20F65 (Primary), 43A35, 05C31, 57R56, 57M25 (Secondary).	10.1093/imrn/rnz240	[ "https://arxiv.org/pdf/1901.10597v1.pdf" ]	119,732,894	1901.10597	dd631c33e4dd7963c524c276ebddf6e9d6bb5d37	JONES REPRESENTATIONS OF THOMPSON'S GROUP F ARISING FROM TEMPERLEY-LIEB-JONES ALGEBRAS Valeriano Aiello Arnaud Brothier Roberto Conti JONES REPRESENTATIONS OF THOMPSON'S GROUP F ARISING FROM TEMPERLEY-LIEB-JONES ALGEBRAS Thompson's groupbinary treecategory of forestsgroup of fractionsunitary representationmatrix coefficientfunction of positive typestabilizercom- mensuratorautomorphismTemperley-Lieb relationsplanar algebrasubfactorJones indexTutte polynomialchromatic polynomialKauffman bracketTQFTCFT Following a procedure due to V. Jones, using suitably normalized elements in a Temperley-Lieb-Jones (planar) algebra we introduce a 3-parametric family of unitary representations of the Thompson's group F equipped with canonical (vacuum) vectors and study some of their properties. In particular, we discuss the behaviour at infinity of their matrix coefficients, thus showing that these representations do not contain any finite type component. We then focus on a particular representation known to be quasi-regular and irreducible and show that it is inequivalent to itself once composed with a classical automorphism of F. This allows us to distinguish three equivalence classes in our family. Finally, we investigate a family of stabilizer subgroups of F indexed by subfactor Jones indices that are described in terms of the chromatic polynomial. In contrast to the first non-trivial index value for which the corresponding subgroup is isomorphic to the Brown-Thompson's group F 3 , we show that when the index is large enough this subgroup is always trivial.MSC 2010: 22D10, 46L37, 20F65 (Primary), 43A35, 05C31, 57R56, 57M25 (Secondary). Introduction and main results R. Thompson's group F is one of the most fascinating countable discrete groups, yet very mysterious for the study of its analytical properties has been challenging experts for decades. We refer to [CFP96] for a nice introduction to the basic facts about F and its relatives T and V . It is known that F is inner amenable [Jo97] and it has the Haagerup approximation property [Fa03], both of which express some weakened form of amenability. However, somewhat surprisingly, despite the many attempts the question about the amenability of F still remains unanswered, along with exactness (and thus weak amenability) and soficity. Several approximation properties for groups are based on suitable asymptotic behaviour of matrix coefficients of unitary representations. For this reason, it is of some Valeriano Aiello acknowledges support of the Swiss National Science Foundation. Roberto Conti acknowledges partial support by Sapienza Università di Roma. Arnaud Brothier was supported by European Research Council Advanced Grant 669240 QUEST and is now supported by a University of New South Wales Sydney starting grant. interest to determine as much as possible about the representation theory of F . Earlier studies of unitary representations of F appear in [Gar12], [DuMe14] and [Ole16]. In this work, we follow the general procedure introduced by V.F.R. Jones in [Jo14,Jo16] where a large family of unitary representations of F (and T ) are built using Jones' planar algebras or, more generally, a category/functor method. Some of these representations admit easy algorithms for computing matrix coefficients leading to new proofs regarding analytical properties of the Thompson's groups [BJ18] but also computation of limits of rotations [BJ18b]. They also provide an explicit connection with the Cuntz algebra in the spirit of the work of Nekrashevych [Ne04]. In this work we consider only F and the very first construction of Jones involving a planar algebra. We take some step towards an understanding of the main features of a class of such representations depending on three different parameters, one selecting a Temperley-Lieb-Jones (planar) algebra and two more determining a normalized element in that algebra expressed as a linear combination of the identity and a nontrivial TLJ-generator. More precisely, given a loop parameter δ ∈ {2 cos(π/n), n = 4, 5, 6, . . .} ∪ [2, ∞) and a normalized 2-box R characterised by two complex parameters a, b satisfying a suitable equation, we obtain a unitary representation π = π δ a,b of F . This representation comes along with a certain vacuum vector Ω and thus a vacuum state. We prove that for any choice of δ and non-zero real parameters a, b as above the corresponding representations do not admit any finite-dimensional subrepresentation and moreover their matrix coefficients do not vanish at infinity. This result, combined with a theorem of Dudko-Medynets [DuMe14], implies that these representations do not contain any finite type components (of type I n and II 1 ). For any loop parameter δ, the Jones-Wenzl idempotent (properly rescaled) gives us a normalized element R of TLJ. Jones called it the chromatic choice because the vacuum state applied to g ∈ F can be expressed in terms of the chromatic polynomial of a certain graph Γ g associated to g evaluated at δ 2 . Jones showed that at δ = √ 2 the stabilizer subgroup of the vacuum vector is nothing but the set of those elements g ∈ F for which the graph Γ g is bipartite. It is a remarkable subgroup F < F called the Jones subgroup, that turns out to be isomorphic to F 3 [GS17, Lemma 4.7.]. Moreover, the representation π for this choice is unitarily equivalent to the quasi-regular representation λ F/ F associated to F < F. In passing, we mention that many other graph and link invariants may be interpreted as vector states associated with unitary representations of both the Thompson's group and the Jones oriented subgroup. We refer, for instance, to [ACJ], where this is shown for F and the HOMFLYPT polynomial, and to [AiCo1,AiCo2] where this is done with different and more elementary (but less powerful) methods. By flipping the two parameters a, b, we obtain a new representation that is unitarily equivalent to λ F/σ F ( F ) , where σ F is the automorphism of F associated to the homeomorphism σ(t) = 1 − t of the unit interval [0, 1]. We prove that F and σ F ( F ) are not quasi-conjugate. By Golan-Sapir [GS17], F is equal to its own commensurator, implying that the quasi-regular representation λ F/ F is irreducible. This, together with a classical argument going back to Mackey, shows that the quasi-regular representations associated to F < F and σ F ( F ) < F are not unitary equivalent, thus providing two unitarily inequivalent irreducible representations π and π • σ F in our family. Choosing another representation π with parameters a and b equal to each other, we get that π and π • σ F are unitary equivalent, implying that π is unitary equivalent neither to π nor to π • σ F , thus ensuring that our family of representations contains at least three distinct classes. Then we give a closer look at the stabilizer subgroups F δ < F of the vacuum vector for the chromatic choice but for any value of δ. We prove that for δ large enough (δ > 2.16) or δ ∈ { √ 3, 2}, the subgroup F δ is actually trivial. Last but not the least, for any choice of π δ a,b with real and non-zero parameters a, b the vacuum state can be expressed in terms of the Tutte polynomial suggesting that similar arguments could prove that the stabilizer of the vacuum is generically trivial. We have not investigated this direction yet. There are some important questions that remain open. (1) Do we have that in most of the cases considered in Section 5 the stabilizer subgroup of the vacuum vector is trivial? So far the only case for which the stabilizer is known to be non-trivial appears when δ = √ 2. (2) Is π always irreducible? This is only known to be true again when δ = √ 2 and we are in the chromatic choice or when we compose this representation with σ F . (3) Does the considered family of unitary representations contains infinitely (perhaps uncountably) many distinct classes? So far, we could only distinguish three mutually inequivalent non-trivial representations. The content of the paper is as follows. In Section 1 we collect some preliminaries to be used throughout the whole text. In Section 2 we discuss some useful facts about unitary equivalence of the Jones representations. In Section 3 we introduce the main class of representations studied in this paper and prove a number of results about their matrix coefficients. In Section 4 we exhibit a pair of quasi-regular Jones representations that are inequivalent. In Section 5 we focus on a natural class of representations related to the chromatic polynomial, and show that for these representations the stabilizer subgroup of the vacuum vector is trivial in many cases. = N ⊂ M 0 = M ⊂ M 1 ⊂ · · · is the Jones tower obtained by iterating the Jones basic construction. It has been axiomatized by Popa as a λ-lattice and later on by Jones as a subfactor planar algebra [P95,Jo99]. Note that Ocneanu gave another axiomatization for finite depth subfactors, i.e. when the N -bimodule tensor category generated by L 2 (M ) has finitely many equivalence classes of irreducible objects [O89]. We are interested in subfactor planar algebras that we briefly define. We refer the reader to [Jo99] for details. A shaded C-planar algebra P = (P ± n ) n≥0 is a collection of finite dimensional C-algebras P ± n on which the operad of shaded planar tangles acts. We assume that P + 0 and P − 0 are one-dimensional. We think of an element of P ± n as a box with 2n boundary points, n on the top and n on the bottom. The distinguished interval (the dollar sign) being at the top left corner of the box, that is in a region shaded by ±. The multiplication is then given by vertical concatenation and thus the unit of P ± n is a diagram with n vertical straight lines. We have a unital inclusion of P ± n into P ± n+1 by adding one vertical straight line to the right of a box. The planar algebra admits two loop parameters δ + , δ − that are the values of a closed circle in P + 0 and P − 0 respectively, where P ± 0 is identified with C. Each P ± n admits two tracial states τ l , τ r that are the left and right traces. The value τ l (x) (resp. τ r (x)) is obtained by connecting each string of the bottom to a string of the top on the left (resp. on the right) of the box and dividing by δ n ± if x ∈ P ± n . If each of them is faithful we say that P is non-degenerate and we say that P is spherical if τ l = τ r . We then write τ := τ l and call it the trace of P. In that case δ + = δ − =: δ, that we call the loop parameter. A subfactor planar algebra is a non-degenerate spherical C-planar algebra. Note that the loop parameter δ of a subfactor planar algebra is the square root of a non-trivial finite Jones index and thus belongs to the set {2 cos(π/n) : n ≥ 4} ∪ [2, ∞), where we have excluded 1 and ∞. A subfactor planar algebra is irreducible if P + 1 and P − 1 are both one-dimensional. Recall that for any δ in the above set there is a unique minimal subfactor planar algebra with loop parameter δ that is called the Temperley-Lieb-Jones planar algebra and that we denote by TLJ(δ) or simply TLJ if the context is clear [TL71,Jo83]. A spanning set of TLJ ± n is given by all planar diagrams of non-crossing curves in a box with n boundary points on the bottom and n on the top. The antilinear involution applied to such diagram is then the vertical symmetry. Note that TLJ is an irreducible subfactor planar algebra. We briefly define the rectangular category of a subfactor planar algebra. We refer the reader to [Jo14] for details and precise definitions. Let P be a subfactor planar algebra and consider its rectangular category Rec(P) whose collection of objects is N = {1, 2, · · · }. The space of morphism from n to m is empty if n + m is odd and otherwise is equal to a copy of P + (n+m)/2 . We think of an element of Rec(P)(n, m) as an element of P + (n+m)/2 that we represent as a box with n points on the bottom and m on the top and where the distinguished interval is placed on the top left corner. The composition of morphism is then obtained by concatenating vertically such diagrams. Note that Rec(P)(n, n) can be canonically identified with P + n as an algebra. We equip the category Rec(P) with a contravariant endofunctor † : Rec(P) → Rec(P) such that †(n) = n and x † , with x ∈ Rec(P)(n, m) identified with an element of P + (n+m)/2 , is obtained by considering the element x * ∈ P + (n+m)/2 and identifying it with a box-diagram with m boundary points on the bottom and n on the top and where the distinguished interval is still on the top left corner. Therefore, x † ∈ Rec(P)(m, n) and we have that (x † ) † = x. This provides a sesquilinear form on Rec(P)(n, m) given by x, y := τ (y † • x) where τ is the trace on P + n . Since P is non-degenerate we have that this form is an inner product and then it gives a structure of Hilbert space for Rec(P)(n, m) since it is complete by finite dimensionality. 1.2. Jones representations of Thompson's group F . 1.2.1. Thompson's group F . We recall some basic facts about Thompson's group F . We refer the reader to [CFP96] for details. The group F is defined as the set of orientation preserving homeomorphisms of the closed unit interval [0, 1] that are piecewise linear with finitely many breakpoints at dyadic rationals and slopes in 2 Z . In this paper, F is endowed with the discrete topology. It is known that F is an ICC (infinite conjugacy classes) countable group. The quotient of F by its commutator subgroup [F, F ] is isomorphic to Z 2 . Moreover, any proper quotient of F is abelian. It also known that an irreducible finite dimensional representation of F must be necessarily of dimension one [DuMe14]. F is finitely presented, as it can be described by A, B \| [AB −1 , A −1 BA], [AB −1 , A −2 BA 2 ] . However, we often consider the equally well-known infinite presentation given by F = x 0 , x 1 , . . . \| x n x k = x k x n+1 for k < n The (left) shift φ : F → F is the homomorphism of F defined by φ(x i ) = x i+1 , i ∈ N 0 = {0, 1, 2, 3, . . .} . 1.2.2. Thompson's group F as a group of fractions. The elements of F admit a nice graphical description. Indeed, any element of F can be described as an equivalence class of pairs of rooted planar binary trees with the same number of leaves. . For instance, the standard generator x 0 is represented by a pair of trees with three leaves. We introduce this diagrammatic approach from a categorical point of view, as described in [Jo16]. See also [Be04, Section 7.2]. A binary forest with n roots and m leaves is an isotopy class of planar diagrams in the strip R × [0, 1], the roots and the leaves being respectively n distinct points in R × {0} and m distinct points in R × {1} joined by straight lines possibly bifurcating from the bottom to the top. We number the roots and the leaves from left to right. A tree is a binary forest with only one root. The composition f • g (or simply f g) of two binary forests f and g is defined when the number of leaves of g equals the number of roots of f . This is then the binary forest obtained by stacking vertically f on the top of g lining up the leaves of g with the roots of f , followed by (vertical) rescaling. In this way, we obtain a category F ≡ F 2 with objects the natural numbers and morphisms between n and m the set F(n, m) of binary forests with n roots and m leaves. Similarly, by using k-ary forests, one gets the category F k . It is useful to introduce a special notation for some selected trees: \| is the tree with only one leaf, while Y denotes the tree with two leaves. We also denote by • the operation of horizontal concatenation of forests, so that if f ∈ F(n, m) and g ∈ F(n , m ) then f • g ∈ F(n + n , m + m ) (f on the left of g), see the example below. Note that the set of trees, denoted by T, is a directed set where s ≤ t if and only if there exists a forest f such that f • s = t. Consider the set of pairs of trees (t, s), where t and s have the same number of leaves, that we mod out by the relation generated by (t, s) ∼ (f • t, f • s) for any composable forest f . We denote by t s or (t, s) the class of the pair (t, s). We define a multiplication on this quotient set by the formula t s · t s = pt qs where ps = qt . It gives a group structure such that the inverse of t s is s t and the neutral element is t t for any tree t. In turn, the group thus obtained is called the group of fractions of the category F and it is isomorphic to F . Graphically we represent an element t s by first drawing the tree t upside down and then s matching up the leaves of the two trees. For instance the generators x 0 and x 1 are depicted below x 0 = , x 1 = . In this diagrammatic approach, the shift φ is described by φ t s = (\| • t) • Y (\| • s) • Y . With this notation, for instance, x 0 = (Y •\|)•Y (\|•Y )•Y . 1.2.3. Jones representations of Thomspon's group F . In [Jo14], Jones constructed a large class of unitary representations for F and also for Thompson's group T . Given a triple (P, M, R) where P is a subfactor planar algebra (or even any non-degenerate C-planar algebra), M a module over the rectangular category of P, and R a certain normalized element of P, we can construct a representation of F . It M is a module over the affine category of P, then we obtain a representation of the larger group T . Jones generalized this construction in a very beautiful way in [Jo16]. Given a well behaved category C he constructed a group of fraction G C . Then any functor Φ : C → Hilb with target the category of Hilbert spaces with isometries for morphisms provides a unitary representation of the group of fractions G C associated to C [Jo16]. We are interested in representations of F given by a triple (P, M, R) where P is a subfactor planar algebra (most of time the TLJ-planar algebra) and M is the regular module over Rec(P). We recall the construction of [Jo14] in this particular case. Fix a subfactor planar algebra P and R ∈ P + 2 . We say that R is normalized if we have the following identity Consider the morphism that we simply denote by R if the context is clear. Consider a binary tree t. Lett be the unique ternary tree which is obtained from t by replacing any binary branching by a ternary branching where the additional branch go straight to the top in the middle. We then replace any branching oft by an instance of R 3 1 which gives us an element Φ(t) ∈ Rec(P)(1,m) wherem := 2m − 1. For example, If f is a binary forest with trees t 1 , · · · , t n , then we construct the ternary forestf =t 1 •\|• t 2 •\|•· · · \|•t n where a trivial tree is added betweent k andt k+1 for any 1 ≤ k ≤ n−1. Then replace any branching off by an instance of R 3 1 . If f ∈ F(n, m), we obtain a morphism Φ(f ) ∈ Rec(P)(ñ,m) whereñ := 2n − 1. For any n ≥ 1 let Φ(n) ⊂ Rec(P)(1, n) be the span of Φ(t) where t runs over all tree with n leaves. Given a tree t with n leaves we put H t := {(t, ξ) : ξ ∈ Φ(n)} that is a copy of Φ(n) indexed by t that we equip with the restriction of the inner product of Rec(P)(1, n). Note that when P is irreducible we have that the inner product of Rec(P)(1, n) is given by x, y := y † • x where Rec(P)(1, 1) is identified with C. Define the quotient space X := {(t, ξ) : t a tree , ξ ∈ Φ(target(t))}/ ∼ where ∼ is the equivalence relation generated by (t, ξ) ∼ (f t, Φ(f ) • ξ) . This quotient space is nothing but the inductive limit of the directed system (H t ) t∈T with inclusion maps ι f t t : H t → H f t , (t, ξ) → (f t, Φ(f ) • ξ). Note that the map ι f t t is an isometry and thus we obtain an inner product on X that makes it a pre-Hilbert space. Let H be its completion and identify H t with a subspace of H for any tree t. We denote by t ξ or even (t, ξ) the equivalence class of (t, ξ) ∈ H t inside H . We can now introduce the Jones representation π : F → U(H ) as the unitary representation densely defined by the formula π t s v ξ := pt Φ(q)ξ where p • s = q • v. In particular, π t s s ξ = t ξ . We write Ω for the unit vector (\|, 1) belonging to H \| ⊂ H . By construction, Ω is a cyclic vector inside H . To emphasize the role of P and R we will sometimes add the subscript P, R or simply R to Φ, π, H , and Ω. Note that we slightly modified the definition of the representation of π with respect to [Jo14]. Indeed, therein the space H is the completion of {(t, ξ) : t ∈ T, ξ ∈ Rec(P)(1, target(t))}/ ∼ and thus Ω is not necessarily cyclic. The reason is that we are mostly interested in the cyclic representation generated by Ω. We denote by ϕ(·) = π(·)Ω, Ω the vector state given by Ω and notice the following equality: (1) ϕ t s = Φ(s), Φ(t) for any element t s ∈ F. We can interpret the representation π in the setting of [Jo16] where Φ defined a functor from the category of forests F to the category of Hilbert spaces with isometries as morphisms. This functor provides a unitary representation (π Φ , H Φ ) which is unitarily equivalent to the representation (π, H ) described above. As an illustration we provide an explicit computation of some values of ϕ for a specific choice of (P, R). Those values will be used in Section 3. Proposition 1.1. Consider a subfactor planar algebra P with loop parameter δ ∈ {2 cos(π/n) : n ≥ 4} ∪ [2, ∞) and two complex numbers a, b satisfying the equation 1 = δ(\|a\| 2 + \|b\| 2 ) +āb + ab. Put R 1 = \| \|, R 2 = ∪ ∩ , and R := aR 1 + bR 2 . Then, R is a normalized element of P that defines a functor Φ and a unitary representation (π, H ) with vacuum vector Ω and associated vector state ϕ. Furthermore, evaluating ϕ on x 0 , x 0 x 1 , x 1 x 0 ∈ F we find the following equalities: ϕ(x 0 ) = 1 + \|ab\| 2 (1 − δ 2 ) ϕ(x 1 x 0 ) = ϕ(x 0 ) 2 ϕ(x 0 x 1 ) = ϕ(x 1 x 0 ) + \|a 2 b\| 2 (δ 2 − 1) 2 δ δ 2 − 1 − \|b\| 2 . Proof. According to our setting, we will repeatedly use the following rules: (2) (to ease the graphical description of the elements of the planar algebra we replace the occurrences of R by a small black disk). It is then easy to check that the normalization condition reads as 1 = δ(\|a\| 2 + \|b\| 2 ) +āb + ab, which is our assumption. Since x 0 = , we compute ϕ(x 0 ) \| = = a + b ā +b(a + bδ) = \|a\| 2 δ + ab + bā + \|b\| 2 (a + bδ)(ā +bδ) \| = 1 + \|ab\| 2 (1 − δ 2 ) \| where we used the normalization condition. Now we consider x 0 x 1 = , x 1 x 0 = . On the one hand, we have ϕ(x 1 x 0 ) \| = = ϕ(x 0 ) = (ϕ(x 0 )) 2 \| , on the other hand, ϕ(x 0 x 1 ) \| = =ā +b =ā        a + b        +b(a + bδ) = \|a\| 2 (aδ + b)(āδ +b) +āb +b(a + bδ) = \|a\| 2 (aδ + b)(āδ +b) + (1 − \|a\| 2 δ) = \|a\| 2 (aδ + b)(āδ +b) + (1 − \|a\| 2 δ)ϕ(x 0 ) \| = \|a\| 2 \|aδ + b\| 2 + (1 − \|a\| 2 δ)(1 + \|ab\| 2 (1 − δ 2 )) \| The conclusion now follows by lengthy but straightforward computations, using again the normalization condition. First results and observations on the Jones representations 2.1. Equivalent representations. In this section P is a subfactor planar algebra and R is a normalized element of P. We prove that the representation we get does not change up to unitary equivalence if we multiply R by a phase. Proposition 2.1. Consider the Jones representation (π R , H R ) associated to a couple (P, R). If z is a complex number of modulus one, then zR is still normalized in P providing a representation π zR that is unitary equivalent to π R . Proof. For any tree t we write H R,t and H zR,t the t-component subspace of H R and H zR respectively. Observe that Φ R (n) = Φ zR (n) for any n ≥ 1 and thus we can identify H R,t and H zR,t for any tree t. Given a tree t with n leaves we put U t : H R,t → H zR,t that is the multiplication by z n−1 . Observe that Φ zR (f ) • U t = U f t • Φ R (f ) for any forest f implying that the family of maps (U t ) t∈T defines a map U : H R → H zR . It is easy to see that U is a unitary transformation which intertwines π R and π zR . The next proposition points out that we can apply a planar algebra automorphism to R without changing the unitary class of the representation. Proposition 2.2. Consider a couple (P, R) and α ∈ Aut(P) an automorphism of the planar algebra P. Then the representations π R and π α(R) are unitarily equivalent. Proof. Note that S := α(R) is normalized since α commutes with the action of the tangles and thus defines a representation (π S , H S ). Denote by H R,t and H S,t the tcomponent subspaces of H R and H S respectively for each tree t. Consider the map U t : H R,t → H S,t , (t, ξ) → (t, α(ξ)), where we identify H R,t and H S,t with subspaces of the planar algebra P. The system of maps (U t ) t∈T agrees with the inductive structures of (H R,t ) t∈T and (H S,t ) t∈T and thus defines a map U : H R → H S . It turns out that U is a unitary transformation satisfying Ad(U ) • π R = π S . Note that the TLJ-planar algebra does not have any non-trivial automorphisms. However, it has some symmetries that behave almost as automorphisms. Fix a loop parameter δ and consider the TLJ-planar algebra TLJ := TLJ(δ). Let D(m, n) be the set of rectangular TLJ-diagrams with m boundary points on the bottom and n on top. The map that consists of a symmetry with respect to a vertical line on D(m, n) (that we extend linearly to Rec(TLJ)(m, n)) induces an involution σ TLJ : Rec(TLJ) → Rec(TLJ) defined on the rectangular category associated to the planar algebra TLJ . We also need the homeomorphism x → 1−x of the unit interval and the induced order two automorphism σ F of the Thompson's group F given by the formula σ F (g)(x) = 1 − g(1−x) for x ∈ [0, 1] and g ∈ F (when F is viewed as a subgroup of the homeomorphism group of [0, 1]). If σ is the involution on the set of forests that consists of a symmetry with respect to a vertical line, then σ F s t = σ(s) σ(t) . For instance, it is not difficult to see that σ F (x 0 ) = x −1 0 . The automorphism σ F is not inner, see for instance [Br96, Theorem 1] for a thorough analysis of Aut(F ) and also [Heo,Section 4], where it is shown that σ F is not inner in the group von Neumann algebra L(F ). Consider a normalized element R ∈ TLJ. Proposition 2.3. The map σ TLJ induces a unitary transformation u σ : H R → H σ TLJ (R) given by (t, ξ) → (σ(t), σ TLJ (ξ)), mapping the vacuum vector Ω R into the vacuum vector Ω σ TLJ (R) and such that Ad(u σ ) • π R = π σ TLJ (R) • σ F . Proof. One can check that Φ σ TLJ (R) (σ(f )) = σ TLJ (Φ R (f )) for any forest f . Moreover, the maps σ and σ TLJ are multiplicative with respect to vertical concatenation. This implies that u σ is well defined. It is easy to see that u σ is a unitary transformation whose inverse maps (s, η) ∈ H σ TLJ (R) to (σ(s), σ TLJ (η)) ∈ H R . Consider a vector (t, ξ) ∈ H σ TLJ (R),t and an element g = s r ∈ F . We have that Ad(u σ ) • π R (g) t ξ = u σ π R (g) σ(t) σ TLJ (ξ) = u σ π R ps pr qσ(t) Φ R (q)σ TLJ (ξ) = u σ ps Φ R (q)σ TLJ (ξ) = σ(ps) σ TLJ • Φ R (q)ξ = σ(ps) Φ σ TLJ (R) (σ(q))ξ where pr = qσ(t). On the other hand, we have that π σ TLJ (R) (σ F (g)) t ξ = π σ TLJ (R) σ(s) σ(r) t ξ = π σ TLJ (R) σ(p)σ(s) σ(p)σ(r) σ(q)t Φ σ TLJ (R) (σ(q))ξ = σ(ps) Φ σ TLJ (R) (σ(q))ξ . This concludes the proof by a density argument. We can provide a similar statement for any subfactor planar algebra, but with an anti-unitary. The proof is similar to the one given above and we leave it to the reader. Proposition 2.4. Let P be a subfactor planar algebra, R a normalized element, with its associated functor Φ R and unitary representation (π R , H R ). Define the following antilinear involution of the rectangular category: J : x $ n m → x $ n m and put S := J(R). Then the map J is multiplicative, J(Φ R (f )) = Φ J(R) (σ(f )) for any forest f , and the element S is normalized. Moreover, J induces an anti-unitary transformation U J : H R → H S densely defined as U J ((t, ξ)) := (σ(t), J(ξ)) for any (t, ξ) ∈ H R,t . Finally, we obtain the following equality Ad(U J ) • π R = π S • σ F . In general, it would also make sense to restrict a representation π R of F as above to some notable subgroups of F . Remark 2.5. The subgroup F σ := {g ∈ F : σ F (g) = g} is isomorphic to F itself since it corresponds to the maps g : [0, 1] → [0, 1] of F whose graph is symmetric with respect to the point (1/2, 1/2). We have the following isomorphism: α : F → F σ , s t → (s • σ(s)) • Y (t • σ(t)) • Y . Consider the unitary representation (π, H ) associated to a normalized element R of an irreducible subfactor planar algebra P. We define a new unitary representation of F as follows: π σ := π • α. Consider the vacuum vector Ω ∈ H and observe that π σ (g)Ω, Ω = π(g)Ω, Ω π(σ F (g))Ω, Ω , g ∈ F. Note that we use the fact that the 1-box space of P is one-dimensional. It follows from [Dix, Proposition 2.4.1., p. 37] that π σ π ⊗ (π • σ F ). 2.2. Some properties of the Jones representations. The shift endomorphism of the Thompson's group F is the map φ F defined as φ F (x i ) = x i+1 for any i ≥ 0 for the usual presentation of F , see Section 1.2. It can be diagrammatically defined as follows: if t is a tree with n leaves we consider φ(t) ∈ F(1, n + 1) that is a new tree equal to the composition of the unique tree with two leaves Y with the forest with the trivial tree on the left and the tree t on the right, i.e. φ(t) = (\| • t) • Y . If g ∈ F is described by the pair of trees (t, s), then we observe that φ F (g) is described by the pair of shifted trees (φ(t), φ(s)). We drop the subscript F for φ when it is clear from the context. Proposition 2.6. If g ∈ F is non-trivial, then the sequence g 0 = g, g n+1 := φ(g n ), n ≥ 0 tends to infinity in F (for the Fréchet filter). Consider a Jones representation π constructed with an irreducible P subfactor planar algebra ( in particular, the one box space P + 1 is one dimensional), then the vacuum vector state ϕ is invariant under the shift of F . In particular, π is not contained in any direct sum of copies of the left regular representation as long as there exists some non-trivial h ∈ F with ϕ(h) = 0. Proof. If g is a non-trivial element, then it is described by a reduced pair of different trees. It is then easy to see that φ(g) is still described by a reduced pair of trees but with one more leaf. This implies that the number of leaves of the reduced pair associated with g n := φ n (g) tends to infinity and thus g n tends to infinity. The invariance of the vacuum vector state readily follows from the normalization property of R and the irreducibility condition on P. Indeed, identify the one box space P + 1 with C via the trace and consider an element g = t s ∈ F. We have that Φ(t) * • Φ(s) = = ϕ(g) \| Then ϕ(φ(g)) \| = = ϕ(g) = ϕ(g) \| Suppose that there exists a non-trivial h ∈ F such that ϕ(h) = 0. Then, the vacuum vector state does not tend to zero at infinity since ϕ(φ n (h)) = ϕ(h) = 0. Recall that any vector in the carrier Hilbert space of λ ⊕∞ , the infinite direct sum of copies of the left regular representation of F , tends to zero at infinity. Therefore, π is not contained in λ ⊕∞ . In particular, it follows that a Jones representation with a non-trivial stabilizer of the vacuum vector is never contained in the regular representation of F . Note that such a π can still be unitarily equivalent to a quasi-regular representation, as it was shown in [Jo14], cf. Section 5. In Section 3.3 we will show that for any unitary representation constructed with TLJ and two real parameters a, b we have ϕ(x 0 ) = 0 for x 0 ∈ F in all the cases but one. 3. Family of representations constructed with the TLJ planar algebra 3.1. First observations. Fix a loop parameter δ ∈ {2 cos(π/k) \|k = 4, 5, 6, . . .} ∪ [2, +∞) and consider the Temperley-Lieb-Jones planar algebra TLJ := TLJ(δ) with this parameter. A normalized element R of TLJ is a linear combination of the identity tangle R 1 = \| \| and a multiple of the Jones projection R 2 = ∪ ∩ . Observe that if R = aR 1 + bR 2 ∈ TLJ, where a, b ∈ C, then R is normalized if and only if (3) (a, b) ∈ C δ := {(x, y) ∈ C × C \| δ(\|x\| 2 + \|y\| 2 ) + xȳ +xy = 1} . Note that the curve C δ is invariant under various operations such as the flip of coordinates, the simultaneous complex conjugation of both the coordinates and the simultaneous multiplication of both the coordinates by a phase factor. Given (a, b) ∈ C δ , we write π δ a,b and ϕ δ a,b for the associated Jones representation and vacuum vector state. We first observe that if a or b is equal to zero, then we obtain the trivial representation. The proof is rather obvious but we include it in order to illustrate the formalism. Proposition 3.1. Consider (a, b) ∈ C δ such that a or b is equal to zero and let (π, H ) be the associated representation. Then H is one dimensional and π is the trivial representation. Proof. Assume that b = 0. Up to a multiplication by a phase we can assume that a = δ −1/2 . Write Φ R the functor associated with R. We claim that Φ R (t) = δ −(n−1)/2 \| • ∪ •n−1 for any tree t with n leaves. The proof can be done by induction. The cases n = 1 and 2 are trivial. Suppose that the claim is true for n − 1 and let t be a tree with with n leaves. Then t may be seen as a composition of a tree t ∈ F 2 (1, n − 1) and a forest f ∈ F 2 (n − 1, n), with Φ R (f ) we obtain that Φ R (t) = Φ R ( t ) • Φ R (f ) = δ −(n−1)/2 \| • ∪ •n−1 , which proves the claim. It follows that each space H t is one dimensional, as well as the inductive limit H . Now consider a group element g = (t, s) ∈ F described by a pair of trees with n leaves and observe that ϕ(g) = Φ(s)Ω, Φ(t)Ω = δ −(n−1)/2 \| • ∪ •n−1 , δ −(n−1)/2 \| • ∪ •n−1 = 1. Therefore, π is the trivial representation. The case a = 0 can be handled in a similar way but it also follows at once from Proposition 3.2. The following result is an immediate consequence of Proposition 2.3. Proposition 3.2. For each δ, we have π δ a,b • σ F ∼ = π δ b,a and ϕ δ a,b • σ F = ϕ δ b,a for all (a, b) ∈ C δ . In particular, for any δ as above, there exists only one class π δ + := π δ a,a of unitary representations of F with a = b, that can be obtained by taking a = 1/ 2(δ + 1) 1/2 . By symmetry, the class of π δ + is not changed if we compose it with σ F . Likewise, there exists only one class π δ − with a = −b, that can be obtained by taking a = 1/ 2(δ −1) 1/2 , again still invariant under σ F up to unitary equivalence. For the time being, it is unclear whether π δ + and π δ − are unitarily equivalent. As one can expect, apart from the trivial representation, all the other representations have trivial kernel. Proposition 3.3. The representation π δ a,b is faithful for any (a, b) ∈ C δ with a = 0 = b. Proof. If π δ a,b had a non-trivial kernel then, by [CFP96,Theorem 4.3], the quotient group by this kernel would necessarily be abelian and thus ϕ(xy) = ϕ(yx) for all x, y ∈ F. However, thanks to Proposition 1.1, the equality ϕ(x 0 x 1 ) = ϕ(x 1 x 0 ) leads to \|b\| 2 = δ δ 2 −1 and, similarly, ϕ σ F (x 0 x 1 ) = ϕ σ F (x 1 x 0 ) leads to \|a\| 2 = δ δ 2 −1 , which are easily seen to be incompatible with the normalization condition and the fact that δ > 1. 3.2. Matrix coefficients which tend to zero. Throughout the rest of this section we consider δ ∈ {2 cos(π/k) \| k = 4, 5, 6, . . .} ∪ [2, +∞) as above and a real couple (a, b) ∈ C δ such that a = 0 = b. Note that C δ R := C δ ∩ R 2 is an ellipse whose axes are rotated by π/4 in the plane, with semi-major and semi-minor axes given by 1/ √ δ − 1 and 1/ √ δ + 1, respectively. Set P := TLJ(δ) the TLJ-planar algebra with loop parameter δ and consider R := R a,b that is a normalized element of P 1 . We will show that there exists a sequence γ n ∈ F such that lim n ϕ(γ n ) = 0 when a, b are real. In fact, the sequence (γ n ) n does not depend on the choice of the parameters (δ, a, b). Consider the pair of trees s n , t n ∈ F(1, n) for n ≥ 2 defined as follows. Let s 2 be the unique tree with one root and two leaves and s n+1 = f n • s n where f n is the unique forest with n roots and n + 1 leaves such that its n-th root is connected to its n-th and n + 1-th leaves. We define t n as the reflection of s n with respect to a vertical axis. For example, s 4 = , t 4 = . Consider the Thompson's group element γ n := sn tn . Note that γ n = x n−1 0 for the usual presentation of Thompson's group F , see Section 1.2. Proposition 3.4. We have that lim n ϕ(γ n ) = 0. Proof. Since the TLJ-planar algebra is irreducible, we can identify the algebra of (1,1)rectangular labelled tangles Rec(P)(1, 1) with the algebra of complex numbers. In particular, we obtain that ϕ(γ n ) = Φ(s n ) * • Φ(t n ), that we denote by d n . We introduce a sequence of tangles e n ∈ Rec(TLJ(δ))(1, 1) C that is derived from the sequence (d n ) n such that e 2 = R R R * , e 3 = R R R R * R * , etc. By expanding the very top R * -box in the diagram of d n and using (2) we obtain d n+1 =b(bδ + a)d n +āe n . Similarly, by expanding the top left R-box in the diagram of e n we obtain that e n+1 = a(āδ +b)e n + bd n+1 = (\|a\| 2 δ + ab + ab)e n + \|b\| 2 (bδ + a)d n . Therefore, it holds Av n = v n+1 where A := b(bδ + a) a \|b\| 2 (bδ + a) \|a\| 2 δ + ab + ab and v n = d n e n . We obtain that ϕ(γ n+2 ) is the first component of the vector A n v 2 . Denote by r ± = tr(A) ± √ ∆ 2 the roots of the characteristic polynomial χ A (t) = t 2 − tr(A)t + det(A), and put ∆ = (tr(A)) 2 − 4 det(A) for its discriminant. Recall that we are considering only the real case and thus the curve C δ R = {(x, y) ∈ R × R \| δ(x 2 + y 2 ) + 2xy = 1}. There exists an invertible matrix S such that A = S r + 0 0 r − S −1 or A = S r + 1 0 r + S −1 and accordingly A n = S r n + 0 0 r n − S −1 or A n = S r n + nr n−1 + 0 r n + S −1 for n ≥ 1. This means that if \|r ± \| < 1, then lim n A n v 2 = 0 and thus lim n ϕ(γ n ) = 0. Claim 1: If ∆ < 0, then \|r ± \| < 1. First of all we observe that r + = r − , thus det(A) = r + r − = \|r + \| 2 = \|r − \| 2 ≥ 0. Without loss of generality we may suppose that det(A) = 0 and that a ∈ R + , a ≥ \|b\|. We see that det(A) = ab(bδ + a)(aδ + b) = (b 2 δ + ab)(a 2 δ + ab) = x(1 − x) where x := a(aδ + b) > 0. Since det(A) > 0, we get that 0 < 1 − x, which in turn implies that 0 < det(A) < 1. Therefore, \|r ± \| < 1. Claim 2: If ∆ ≥ 0 and tr(A) ≥ 0, then \|r ± \| < 1. We claim that χ A (1) > 0. Recall tr(A) = 1 + ab and that det(A) = ab(aδ + b)(bδ + a) = ab(abδ 2 + 1 − ab) = ab(1 + ab(δ 2 − 1)) . Then, χ A (1) = 1 − (1 + ab) + det(A) = ab(1 + ab(δ 2 − 1) − 1) = (ab) 2 (δ 2 − 1) > 0. Note that ab ≤ 1/2 since 1 = δ(a 2 + b 2 ) + 2ab and thus tr(A) = 1 + ab ≤ 3/2 < 2 implying that r − = tr(A)− √ ∆ 2 ≤ tr(A) 2 < 1. Since tr(A) and ∆ are non-negative we have that r − , r + are real numbers such that r + = \|r + \| \|r − \|. Moreover, χ A (t) is negative if and only if r − ≤ t ≤ r + . Since r − < 1 and χ A (1) > 0 we necessarily have that r + < 1 and thus \|r ± \| < 1. Claim 3: If tr(A) < 0, then ∆ < 0 implying that \|r ± \| < 1 by Claim 1. Assume that tr(A) < 0. Since tr(A) = 1 + ab we have that ab < −1 and thus (ab) 2 > \|ab\| > 1. Using the formula det(A) = ab(1 + ab(δ 2 − 1)) we obtain: ∆ = tr(A) 2 − 4 det(A) = 1 + (ab) 2 + 2ab − 4ab(1 + ab(δ 2 − 1)) ≤ 1 − 2ab − 3(ab) 2 since δ 2 ≥ 2 < 0 since (ab) 2 > \|ab\| > 1. This proves Claim 3. Altogether, we obtain that \|r ± \| < 1 for every cases implying that lim n ϕ(γ n ) = 0. We obtain the main result of this section. Theorem 3.5. Let (π, H ) be the unitary representation of F constructed from the planar algebra TLJ(δ) and a real couple (a, b) ∈ C δ R with a = 0 = b. Then, for any ε > 0 and any finite dimensional subspace K ⊂ H there exists γ ∈ F such that \| π γ η, ζ \| < ε η ζ for any η, ζ ∈ K . In particular, π does not contain any finite dimensional subrepresentation. Proof. We claim that for any ε > 0 and any n ≥ 1 there exists m ≥ n and forests p, q ∈ F(n, m) such that Φ(p) * • Φ(q) = c id P 2n−1 with \|c\| < ε. Consider the sequence of elements γ k ≡ x k 0 = s k t k with trees s k , t k as in Proposition 3.4. Set p k = s k • \| •n−1 and q k = t k • \| •n−1 . We have that Φ(p k ), Φ(q k ) ∈ F(ñ, 2ñ − 1), whereñ := 2n − 1. Then Φ(q k ) * • Φ(p k ) = Φ(g k ) * • Φ(f k ) • \| •ñ−1 = π γ k Ω, Ω \| •ñ = ϕ(γ k ) \| •ñ , where we used the fact that dim P 1 = 1 and hence, for all x ∈ P 1 , x = τ l (x) τ l being the normalized left trace of the planar algebra P. This proves the claim since the sequence (ϕ(γ k )) k tends to zero by Proposition 3.4. We first prove the statement when K is equal to the t-subspace H t with t a given tree. Set γ := pt qt ∈ F such that Φ(p) * •Φ(p) = c id with \|c\| < ε and consider η = t v , ζ = t w ∈ H t . We obtain \| π γ η, η \| = \| Φ(q)v, Φ(p)w \| = \|c\|\| η, ζ \| < ε η ζ . Now, fix 0 < ε < 1 and a finite dimensional subspace K ⊂ H . Let η 1 , · · · , η k be an orthonormal basis of K . By density there exists a tree t and a collection of unit vectors η 1 , · · · , η k in H t such that η i − η i < ε/k for any 1 ≤ i ≤ k. By our previous argument, there exists γ ∈ F such that \| π γ η, ζ \| < ε η ζ for any η, ζ ∈ H t . Consider some unit vectors η, ζ ∈ K and expand them in the given orthonormal basis, namely η = k i=1 a i η i and ζ = k j=1 b j η j . Put η = k i=1 a i η i and ζ = k j=1 b j η j in H t . We observe that η − η , ζ − ζ < ε and thus η , ζ < 1 + ε. It follows that \| π γ η, ζ \| ≤ \| π γ (η − η ), ζ \| + \| π γ η , (ζ − ζ ) \| + \| π γ η , ζ \| ≤ η − η + η ζ − ζ + ε η ζ ≤ ε + ε(1 + ε) + ε(1 + ε) 2 < 7ε. This concludes the proof of the first statement of the theorem. Assume that K is the carrier space of a finite dimensional subrepresentation of π. By the first assertion there exists γ ∈ F such that \| π γ η, ζ \| < 1/2 η ζ for any η, ζ ∈ K . If η is a unit vector of K , we have that sup{\| π γ η, ζ \| : ζ ∈ K unit vector } = 1 which contradicts the previous inequality. In particular, we see that π does not contain any one-dimensional representation and thus it does not contain the trivial representation π δ 1/ √ δ,0 , see Proposition 3.1. Remark 3.6. Let δ be as above. If (a t , b t ) is a path in C δ R parametrized 0 ≤ t ≤ 1 and approaching (1/ √ δ, 0) when t tends to zero, then the net π t := π δ at,bt tends to the trivial representation in the Fell topology. Therefore, 1 F is weakly contained in the direct sum ⊕ t j π t j for any countable subsequence (t j ) j tending to zero. If the space of equivalence classes of π t is finite, then we can find a sequence t j tending to zero such that each π t j is unitary equivalent to a single representation π = π δ a,b with a = 0 = b. Then 1 F is weakly contained in the infinite direct sum of π and thus 1 F is weakly contained in π. Therefore, if 1 F is not weakly contained in π δ a,b , then there are infinitely many pairwise non-unitarily equivalent representations in the class of Jones representations {π δ a,b : (a, b) ∈ C δ R }. 3.3. Matrix coefficients which do not tend to zero. We show that for any vector state and choice of real parameters a, b there exists a sequence of group elements tending to infinity such that the corresponding coefficients do not tend to zero. We start by showing that the vacuum vector state does not vanish everywhere outside the identity when a, b are reals, which entails that none of the associated representations is contained in a multiple of the left regular representation of F . Proposition 3.7. Consider a real pair (a, b) ∈ C δ R and its associated representation π := π δ a,b and vector state ϕ := ϕ δ a,b . Then, we have that π ⊂ λ ⊕∞ . Proof. By Proposition 2.6 it is enough to exhibit an element g ∈ F such that ϕ(g) = 0. In all but one case the element is the generator x 0 . Indeed, assume that ϕ(x 0 ) = 0. Proposition 1.1 tells us that ϕ(x 0 ) = 1 + \|ab\| 2 (1 − δ 2 ) implying that a = 0 = b. We obtain that b = ±1 a √ δ 2 −1 . If b = 1 a √ δ 2 −1 , then the normalization condition gives us a 2 + 1 a 2 (δ 2 − 1) δ + 2 √ δ 2 − 1 = 1. If X := a 2 , then we obtain the equation: P (X) := δX 2 + 2 − √ δ 2 − 1 √ δ 2 − 1 X + δ δ 2 − 1 = 0. But the discriminant of P is equal to − 3(δ 2 −1)+4 √ δ 2 −1 δ 2 −1 < 0. Hence, P does not have any real root, a contradiction. On the other hand, if b = −1 a √ δ 2 −1 , by means of similar computations and also using the constraints on the values of δ, one can show that ϕ(x 0 ) = 0 precisely when δ = √ 2 and a = 2 1/4 , b = −2 −1/4 or a = 2 −1/4 , b = −2 1/4 . In the former case, we have that one half of ϕ(g) coincides with the chromatic polynomial of Γ(g) evaluated at 2 for all g ∈ F (cf. Prop 5.2) and thus ϕ(x 0 x 1 ) = 1. In the latter case we get ϕ(σ F (x 0 x 1 )) = 1, see Propositions 3.2 and 2.1. Consider the constant C := sup κ∈F \e \|ϕ(κ)\| and recall that it is strictly positive by Proposition 3.7. Proposition 3.8. For any vectors η, ζ ∈ H we have the inequality: lim sup γ∈F \| π γ η, ζ \| ≥ C sup θ∈F \| π θ η, ζ \|. Proof. Consider unit vectors η, ζ ∈ H and ε > 0. We fix β ∈ F such that \| π β η, ζ \| ≥ sup θ∈F \| π θ η, ζ \| − ε. By density we can assume that there exists a tree t with n leaves such that π β η and ζ belong to H t . Fix α ∈ F \ e such that \|ϕ(α)\| ≥ C − ε and consider some trees s, r such that α = r s . Denote by α i the i-th shift of α and consider some trees s i , r i satisfying α i = r i s i . Put r i,n := r i • \| •n−1 the forest with r i and n − 1 trivial trees on its right and γ i = r i,n •t s i,n •t ∈ F where s i,n is defined similarly to r i,n . Observe that \| π γ i β η, ζ \| = \| Φ(s i,n )π β η, Φ(r i,n )ζ \| = \| ([Φ(r i ) * • Φ(s i )] • \| •n−1 )(π β η), ζ \| = \|ϕ(α i )\|\| π β η, ζ \| = \|ϕ(α)\|\| π β η, ζ \| ≥ (C − ε)(sup θ∈F \| π θ η, ζ \| − ε) ≥ C sup θ∈F \| π θ η, ζ \| − 2ε. Since α = e we have that α i tends to infinity which implies that γ i β tends to infinity. We obtain that lim sup γ∈F \| π γ η, ζ \| ≥ C sup θ∈F \| π θ η, ζ \| − 2ε for any ε > 0 which finishes the proof. The left regular representation of an infinite group satisfies that lim n ψ(g n ) = 0 for any vector state ψ and any sequence (g n ) n which goes to infinity. This fact together with Proposition 3.8 imply the following. Theorem 3.9. For any of a, b ∈ C δ R the left regular representation of F is not contained inside π δ a,b . Moreover, π δ a,b does not admit any coefficient vanishing at infinity. Note that the proof of Proposition 3.8 still works for any representation π arising from an irreducible subfactor planar algebra. Furthermore, if this representation satisfies sup κ∈F \e \|ϕ(κ)\| > 0 then we also have the conclusion of Theorem 3.9. Remark 3.10. In [DuMe14] it is shown that the only finite factor representations of F are (the multiples of ) the regular representation and one dimensional representations given by characters on the quotient by the commutator subgroup of F . Therefore, Theorems 3.5 and 3.9 imply that the representations π δ a,b , with a, b non-zero real numbers and any δ, do not contain any finite factor subrepresentations. In the next section we will consider one case where the representation π δ a,b is known to be irreducible and thus of type I ∞ . Two representations that are not unitarily equivalent Consider the loop parameter δ = √ 2 and the unitary representation (π, H ) associated to the normalized Jones-Wenzl idempotent R = 2 1/4 R 1 − 2 −1/4 R 2 . The subgroup of F that fixes the vacuum vector Ω in this representation is the Jones subgroup F introduced in [Jo14]. Consider the involution σ : t → 1 − t of [0, 1] and its associated group isomorphism σ F : F → F, g → σgσ. We writeπ := π • σ F and note thatπ is unitary equivalent to πR withR := −2 −1/4 R 1 + 2 1/4 R 2 by Remark 3.2. The aim of this section is to show that π andπ are not unitary equivalent. We expect that the family of unitary representations {π δ a,b , δ, a, b} contains infinitely many classes of unitary representations but so far π andπ are the first nontrivial representations that we are able to distinguish directly up to unitary equivalence. This also implies that π andπ cannot be unitary equivalent to any representation π δ a,a or π δ a,−a i.e. when a = b or a = −b Jones showed that the representation π is unitary equivalent to the quasi-regular representation λ F/ F : F 2 (F/ F ) implying thatπ and λ F/σ F ( F ) are unitary equivalent. Consider two subgroups A, B < G. They are called commensurable if A ∩ B has finite index both in A and in B. Moreover, they are called quasi-conjugate if there exists g ∈ G such that A and gBg −1 are commensurable. Finally, recall that the commensurator Comm G (A) is a subgroup of G equal to the set of g ∈ G such that A ∩ gAg −1 has finite index both in A and in gAg −1 . The following theorem is due to Mackey, see [Ma76] or [BdH97]. Golan and Sapir proved that the commensurator Comm F ( F ) is equal to F [GS17, Corollary 3] (and thus F is not almost normal in F ). This implies that Comm F (σ F ( F )) = σ F ( F ) as well. Therefore, the representations π andπ are unitary equivalent if and only if F and σ F ( F ) are quasi-conjugate in F . We recall a characterization of F due to Golan and Sapir [GS17]. For any dyadic rational t ∈ (0, 1) there exists a unique n ≥ 0 and a 1 , · · · , a n ∈ {0, 1} such that t = n i=1 a i 2 i + 1 2 n+1 . We write t = .a 1 · · · a n 1. Define S as the set of dyadic rationals t = .a 1 · · · a n 1 such that the set of 1 ≤ i ≤ n satisfying a i = 1 is even. Let us characterize in a similar way our subgroup σ F ( F ) < F . Lemma 4.3. The subgroup σ F ( F ) < F is the stabilizer of σ(S). Moreover, σ(S) is the set of t = .a 1 · · · a n 1 with n ≥ 0, a 1 , · · · , a n ∈ {0, 1} such that the set of 1 ≤ i ≤ n satisfying a i = 0 is even. Proof. Consider a dyadic rational t ∈ (0, 1). It can be written as t = n i=1 a i 2 i + 1 2 n+1 with a 1 , · · · , a n ∈ {0, 1} and n ≥ 0. Observe that σ(t) = 1 − t = ∞ j=1 1 2 j − n i=1 a i 2 i + 1 2 n+1 = n i=1 1 − a i 2 i + 0 2 n+1 + ∞ j=n+2 1 2 j = n i=1 1 − a i 2 i + 1 2 n+1 . Therefore, σ(.a 1 · · · a n 1) = . (1 − a 1 ) · · · (1 − a n )1 which implies the desired characterization of σ F (S). We need a technical lemma to show that F and σ F ( F ) are not quasi-conjugate. Lemma 4.4. There exists d ∈ F ∩σ F ( F ) satisfying lim n→∞ d n (t) = 0 for every t ∈ (0, 1). Proof. Consider the pair of trees s := (\| • Y • \| • \|) • (Y • \| • \|) • (Y • \|) • Y and t := (\| • \| • Y • \|) • (\| • \| • Y ) • (\| • Y ) • Yd(t) =              .000a if t = .0a .0010a if t = .10a .0011a if t = .1100a .01a if t = .1101a .1a if t = .111a , where a is any sequence of 0 and 1 with finitely many 1. From this description it is easy to check that d stabilizes both the sets S and σ(S). Moreover, we have that lim n→∞ d n (t) = 0 for any t ∈ (0, 1). We are now able to prove the main result of this section. The proof follows a similar strategy developed by Golan and Sapir in [GS17]. Theorem 4.5. The subgroups F and σ F ( F ) of F are not quasi-conjugate. In particular, the representations π andπ are not unitary equivalent. Proof. We start by proving that the index I := [ F : F ∩ σ F ( F )] is infinite. Consider the element c := (x 0 x 1 ) −1 that is in F where x n , n ≥ 0 is the classical set of generators of F . If I is finite, then there exists r ≥ 1 such that c r ∈ σ F ( F ). Recall from [GS17, Remark 4.1.3] that c(t) = .00a if t = .0a .011a if t = . 110a . Set N (t) := \|{i : a i = 0}\| if t = .a 1 · · · a n 1 and note that N (c r (t)) = N (t) + r if t = .0a N (t) + r − 1 if t = . 110a . This implies that c r does not stabilizes σ(S) for any r = 0 and thus c r / ∈ σ F ( F ) for any r = 0. Therefore, I = ∞. Fix g ∈ F and consider the index I g := [σ F ( F ) : σ F ( F ) ∩ g F g −1 ] . If g ∈ F , then I g = I = ∞ by the previous argument. Assume that g / ∈ F . Consider d ∈ F ∩ σ F ( F ) as in Lemma 4.4. Assume that I g is finite. Then there exists r ≥ 1 such that d r ∈ g F g −1 . Since g is not in F there exists t ∈ (0, 1) that is not in S such that g(t) is in S. Since d is in F we have that d m (g(t)) ∈ S for any m ∈ Z. By [GS17, Lemma 4.14] we have that there exists ε > 0 such that g −1 (S ∩ (0, ε)) is a subset of S. Since d satisfies the hypothesis of Lemma 4.4, there exists n ≥ 1 such that d nr (g(t)) ∈ S ∩ (0, ε) and thus g −1 (d nr (g(t))) ∈ S. Since t / ∈ S we obtain that g −1 d nr g / ∈ F implying that g −1 d r g / ∈ F , a contradiction. Therefore, I g is infinite for any g ∈ F implying that F and σ F ( F ) are not quasi-conjugate. Our discussion at the beginning of this section implies that the representations π and π are not unitary equivalent. We have proved that π and π • σ F are not unitary equivalent. The remark done after Proposition 3.2 tells us that when a = b then the Jones representation π δ a,a is unitary equivalent to π δ a,a • σ F . Since π does not share this property we obtain the following corollary. Corollary 4.6. Consider any δ equal to the square root of a non-trivial Jones index and the representation π δ a,a where a = (2 + 2δ) −1/2 . Then π δ a,a , π and π • σ F are mutually unitarily inequivalent. Remark 4.7. We have seen in Remark 3.6 that if each non-trivial representation π δ a,b does not weakly contain the trivial representation, then there are necessarily infinitely many classes of unitary representations in this family. If a representation weakly contains the trivial one then it is amenable (which would always be the case if the group F were amenable). Looking at a quasi-regular representations such as λ F/ F , then it weakly contains the trivial representation if and only if the homogenous space F/ F is amenable which, in this case, is equivalent to saying that the representation λ F/ F is amenable. Even in this specific situation we do not know if 1 F is weakly contained in λ F/ F , which is an interesting open problem. The stabilizer subgroup of the vacuum vector In this section we restrict our attention to a subfamily of Jones representations constructed with the planar algebra TLJ. Fix a loop parameter δ and put R = δ δ 2 − 1 1/2 R 1 − 1 δ 3 − δ 1/2 R 2 that is a scalar multiple of the Jones-Wenzl idempotent. Denote by (π, H , Ω) the associated representation with its vacuum vector and put t := δ 2 . We write F Ω = F Ω (t) the subgroup of elements of F that stabilize the vacuum vector. We recall a formula that describes the vacuum vector state in term of the chromatic polynomial. It makes use of a well known argument that appears in [Ka88, Section 2] in a somewhat different setting (see also [FK09]). The corresponding statement for loopsided planar algebras can be found in [Jo14, Section 5.2] (cf. [Jo99,p. 28]). For the convenience of the reader we include a short proof that fits well with our formalism. Recall that the chromatic polynomial of a graph evaluated at a natural number n is equal to the number of proper n-colorations of the graph and is the only one satisfying this condition. Notation 5.1. If G is a graph and t is a complex number, then we denote by G(t) its chromatic polynomial evaluated in t. Proposition 5.2. Consider a group element g of the Thompson's group F described by the pair of trees T ± having n leaves. Then π g Ω, Ω = Γ(t) t(t − 1) n−1 , where Γ = Γ(T + , T − ) is the graph associated to T ± described in [Jo14] and t = δ 2 . Proof. Fix g, n, T ± , and Γ := Γ(T + , T − ) as in the statement. Denote by A the graph obtained as usual by concatenating vertically T − on the bottom and T + on top. Recall that Γ has n vertices and 2(n − 1) edges. The construction of Γ gives us a bijection between the edges of Γ and the vertices of A. We associate to A the tangle T A having 2(n − 1) interior discs with 4 boundary points corresponding to each vertices and having one string on top and on the bottom. For any subset S ⊆ E(Γ) consider the element B S ∈ TLJ 1 where we plug in − 1 √ t ∪ ∩ in the interior of each disc of T A corresponding to an edge of S and \| \| for the others and do not remove any closed curve. We write Z(B S ) for the scalar value of B S which consists of t loop(S)/2 , where loop(S) is the number of closed curves of B S . Let Γ S be the subgraph of Γ with the same vertex set as Γ and edge set equal to S ⊂ E(Γ). Write k(S) for the number of connected components of Γ S and F (S) for the number of regions of Γ S . Note that by Euler Formula we have that (4) n − \|S\| + F (S) = 1 + k(S). The element B S may be shaded (by convention the left part of B S is not coloured and thus the right side is coloured). Observe that the number of uncoloured (resp. coloured) regions of B S is equal to k(S) (resp. F (S)). Then it is not difficult to see that (5) k(S) + F (S) = loop(S) + 2 since B S is a diagram with one vertical string and some closed curves. By definition, we have that on F (here, T Γ(T + ,T − ) (x, y) denotes the Tutte polynomial) as the vacuum coefficient function of π = π δ a,b , namely π g Ω, Ω = T g (x, y) , g ∈ F , when x < 1, y < 1 (so that x + y < 0) for a = δ (x+y)(y−1) , b = − (y−1) δ(x+y) , and when x > 1, y > 1 for a = δ (x+y)(y−1) , b = (y−1) δ(x+y) . With x, y as above, if in addition x = y, the associated representation of F is invariant under composition with σ F (up to unitary equivalence). Indeed, in this case one has a = −b or a = b, respectively, and from Proposition 2.1. The previous formulae can be rewritten directly in terms of a and b. For any loop parameter δ and (a, b) ∈ C δ R satisfying ab = 0 we can express the vacuum state in terms of the Tutte polynomial as follows: π δ a,b (g)Ω, Ω = T g x, y , g ∈ F, where x = (ab) −1 − δb/a − 1 and y = δb/a + 1. This provides a friendly combinatorial description of the vacuum state. It also suggests that the techniques developed in this section could be adapted to any non-degenerate choice of the real parameters a, b. π g Ω, Ω = S⊆E(Γ) √ t t − 1 n−1 Z(B S ) = S⊆E(Γ) √ t t − 1 n−1 √ t loop(S) −1 √ t \|S\| = S⊆E(Γ) (−1) \|S\| t (loop(S)+n−1−\|S\|)/2 (t − 1) n−1 = S⊆E(Γ) (−1) \|S\| t ( Remark 5.3. We point out another particular choice of the parameters which yields a vacuum vector state with a geometrical interpretation [Jo14]. For n = 3, 4, 5, . . . set A := e iπ(1±1/n)/2 and δ := −A 2 − A −2 = 2 cos(π/n). In this setting, if one choses x = A/(−A 2 − A −2 ) 1/2 and y =Ā/(−A 2 − A −2 ) 1/2 , that is R = A/(−A 2 − A −2 ) 1/2 \| \| + A/(−A 2 − A −2 ) 1/2 ∪ ∩ , then it can be easily verified that R is normalized and that the vacuum vector state is equal (up to normalization) to the Kauffman bracket of a certain link. To be more precise, we have ϕ δ x,y (g) = L(T + , T − ) (−A 2 − A −2 ) n−1 where g = g(T + , T − ) ∈ F for some trees T + , T − with n leaves and L(T + , T − ) is the knot/link associated to g produced with Jones' construction relevant for A being a root of unity [Jo14, Section 5.3]. We recover the description of the stabilizer subgroup F Ω (t) < F of the vacuum vector Ω that is the g ∈ F such that Γ(t) = t(t − 1) n−1 where g = (T + , T − ), Γ = Γ(T + , T − ), and T ± has n leaves. Note that if t = 2, the subgroup F Ω is the Jones subgroup F studied in Section 4 and is equal to the set of g = (T + , T − ) having that Γ = Γ(T + , T − ) is bipartite as it was proven by Jones. We are interested in studying this subgroup F Ω (t) for the other values of t. We will prove that in many cases this subgroup is trivial. Remark 5.4. Note that since π is a unitary representation and Ω is of norm one we have that (6) \|Γ(t)\| t(t − 1) n−1 for any Thompson graph Γ (the graph associated to any group element g = (T + , T − ) ∈ F ) with n vertices and for any index {4 cos(π/k) 2 : k 4} ∪ [4, ∞). As far as we know it is an open question whether this inequality remains true for t ≥ 2 and for any connected planar graph Γ with n vertices. In particular, the subgroup F Ω (t) is trivial if for any Thompson graph Γ with n vertices that is not a tree we have that \|Γ(t)\| < t(t − 1) n−1 . Here are some well known and easy observations. Proposition 5.5. (1) If equation (6) is true for any connected planar graphs for a fixed real t > 2, then the subgroup F Ω (t) is trivial. (2) If \|Γ(t)\| > 0 for any connected planar graph for a fixed real t, then the subgroup F Ω (t) is trivial. (3) Equation (6) is true for any natural numbers k = t ≥ 2 and any connected planar graphs. (4) There are at most countably many values of t such that F Ω (t) is non-trivial. Proof. Proof of (1). Consider a Thompson graph Γ with vertices 1, · · · , n. Note that Γ = Λ + e + f + n, where Λ is obtained from Γ by removing the last vertex n that has degree one or two and we wrote e, f the edges. Note that Γ = (Γ − e) − Γ/e = (t − 1)Λ − Λ + Λ/(ij) = (t − 2)Λ(t) + Λ/(ij) where i = target(e), j = target(f ). The inequality implies that \|Γ(t)\| (t − 2)t(t − 1) n−2 + t(t − 1) n−3 < t(t − 1) n−1 since t > 2. In particular, π(g)Ω, Ω = 1 when the Thompson graph Γ associated to g is not a tree (up to parallel edges). But this means that g = 1 and thus the subgroup is trivial. Proof of (2). Consider t such that it is known that Γ(t) > 0 for any planar graph. Fix a connected planar graph Γ that is not a tree. Consider a connected planar graph Γ. It contains a spanning tree T and thus Γ = T + (e 1 , · · · , e l ) where e i are some edges of Γ. We prove the result by induction on l. Inizialization, l = 1: we have Γ = T +e = T −T /e. Therefore, Γ(t) < T (t). Suppose the result true for l, assume Γ = T + (e 1 , · · · , e l , e l+1 ), and put Λ = T + (e 1 , · · · , e l ). Then Γ = Λ + e l+1 = Λ − Λ/e l+1 . The same argument gives that Γ(t) < Λ(t) < T (t). Proof of (3). Consider a connected planar graph Γ, a natural number k 2, and assume that Γ is not a tree. Consider a spanning subtree T ⊂ Γ with the set vertex set. Observe that the set of k-coloring of Γ is a subset of the k-coloring of T which implies the inequality. Proof of (4). Consider a non-trivial group element g ∈ F and its associated graph Γ. If g can be expressed by a pair of trees with n leaves, then π t g Ω, Ω = Γ(t) t(t−1) n−1 , where π t is the representation associated with t = δ 2 that we considered all along this section. Therefore, g ∈ F Ω (t) if and only if Γ(t) = t(t − 1) n−1 . It is well known that only a tree with n leaves has its chromatic polynomial equal to the polynomial x(x − 1) n−1 . Moreover, if g is non-trivial, then its associated graph Γ is not a tree. Therefore, the set X g := {t > 2 : g ∈ F Ω (t)} is necessarily finite since it is equal to some roots of a non-zero polynomial. We obtain that X = g∈F,g =1 X g is a countable set that is equal to the set of t such that F Ω (t) is non-trivial. Birkhoff and Lewis proved that any planar graph has no real roots in [5, ∞) and thus is strictly positive on this half-line implying that F Ω (t) is trivial for t > 5 by Proposition 5.5. We present an elementary proof giving a slightly smaller lower bound. Theorem 5.6. Equation (6) is true for any connected planar graph and any real number t > 4.63. Moreover, the inequality is strict if Γ is not a tree implying that F Ω (t) is trivial for any t > 4.63. We start with a useful lemma. Lemma 5.7. Consider a graph Λ and k ≥ 1 vertices of Λ written {1, · · · , k}. Let 0 be a new vertex and let {(01), · · · , (0k)} be some edges from 0 to j, 1 ≤ j ≤ k. Denote by Λ k the graph obtained by adding 0 and those edges to Λ. We identify the graph Λ with its chromatic polynomial evaluated in t. By removing-contracting edges we obtain the following formula for any k: (7) Λ k = (t − k)Λ + i<j Λ/(ij) − i<j<k Λ/(ijk) + i<j<k<l Λ/(ijkl) − · · · , where Λ/(ij) is the graph Λ quotiented by (ij) that is an edge between i and j. We write Λ/(ijk) the quotient by (ij) and (jk) and so on. In particular, Proof. This can be easily proved by induction on k and by using the graph L k = Λ + ((1k), (2k), · · · , (k + 1, k)). Proof of the Theorem. Fix t > 4.63. We prove the theorem by induction on the number of vertices n. It is easy to prove it for n = 1, 2, 3, 4. Assume it is true for any k n and consider a connected planar graph Γ with n vertices. Since Γ is planar it admits a vertex of degree smaller or equal to 5 by Euler formula. Therefore, there exists a planar graph Λ with n − 1 vertices such that Γ = Λ k with 1 k 5. First assume that Λ is connected. Then any quotient of Λ is also connected and thus satisfied the inequality of the theorem. We have that Λ k t − k t(t − 1) n−2 + kt(t − 1) n−2 + k 2 t(t − 1) n−3 + · · · . We can show that for our t > 4.63, each of those polynomials evaluated in t is strictly smaller than t(t − 1) n−1 when k = 2, 3, 4, 5. If k = 1 then Γ = (t − 1) Λ (t − 1)t(t − 1) n−2 . Moreover, the inequality is strict if Λ is not a tree which is equivalent to have that Γ is not a tree. Suppose now that Λ is not connected. Since Γ is connected we have that k 2 and Γ is a union of connected graphs Γ 1 , · · · , Γ j , j k with one common vertex. We can prove by induction on j that Γ = ( t−1 t ) j−1 j i=1 Γ i . Moreover, j i=1 n i = n + j − 1 where n i is the number of vertices of Γ i . Applying the inequality to each Γ i we obtain that \|Γ\| t(t − 1) n−1 with equality if each of the Γ i is a tree. Each Γ i is a tree implies that Γ is a tree. This finishes the proof of the theorem. Remark 5.8. We know that the Jones subgroup F = F Ω (2) is non-trivial and that F Ω (t) is trivial for t = 3, 4 and t > 4.63. Thomassen and Perrett have recently proved that Γ(τ + 2) > 0 for all planar graphs Γ (see e.g. [P16, Theorem 6.16, p. 100]) which implies that F Ω (τ + 2) is trivial where τ = 1+ √ 5 2 = 2 cos(π/5) is the golden ratio. It seems very likely that all subgroups F Ω (t) are trivial except the first one with t = 2. However, except for t = τ + 2 and t = 3 we have been unable to rule out the discrete series of t = 4 cos(π/k) 2 , k ≥ 5. The next Jones index after 2 is τ 2 = τ + 1 and then 3. Unfortunately, we cannot apply the argument for τ + 2 to the value τ + 1 as there are elements g ∈ F with an edge e of Γ(g) such that Γ(g)/e is the complete 4-graph K 4 , and hence (Γ(g)/e)(τ 2 ) = τ 2 (τ 2 − 1)(τ 2 − 2)(τ 2 − 3) < 0. An example of this sort is given by e Γ(g) = are k vertical edges on the left and n − k − 2 vertical edges on the right. Consider the transformation ∼: F 2 → F 3 defined by replacing any binary branching with a ternary branching and adding a vertical straight line between two consecutive roots. Thus we see that t ∈ F 3 (1, 2n − 3) andf ∈ F 3 (2n − 3, 2n − 1), wheref has the formf= . . . . . . with 2k straight lines on the left and 2n − 2k − 4 on the right. Now, Φ R (f ) = . . . . . . is a morphism with 2k + 1 straight lines on the left and 2n − 2k − 4 on the right. By the induction hypothesis Φ R (t ) is a straight vertical line followed by n − 2 cups. Since there is an odd number of straight lines to the left of the cup of Theorem 4. 1 . 1If Comm G (A) = A and Comm G (B) = B, then the quasi-regular representations λ G/A and λ G/B are unitary equivalent if and only if A and B are quasiconjugate. The subgroup F is the stabilizer of S for the usual action F [0, 1], i.e. F = {g ∈ F : gS = S}. This defines an element of the Thompson's group g = t s ∈ F . It sends the standard dyadic partition {[0, }. It can be defined by its action on dyadic rationals t ∈ (0, 1) as follows: (− 1 ) 1\|S\| t (2k(S)−2)/2 (t − 1) n−1 by Euler Formula (4). This implies the result via the Birkhoff and Whitney formula (see e.g. [Bo98, Chapter X]) which claims that the chromatic polynomial of Γ is equal to S⊂E(G) (−1) \|S\| X k(S) . The next result should be compared to [AiCo1, Theorem 4.2]. With δ, T ± , n as above, if x, y lie on the hyperbola (x − 1)(y − 1) = δ 2 , by a similar argument one can indeed get the Tutte function T g (x, y) := T Γ(T + ,T − ) (x,y) (x+y) n−1 1 . 1Preliminaries 1.1. Subfactors and planar algebras. A subfactor N ⊂ M is a unital inclusion of type II 1 factors. Its Jones index [M : N ] is the Murray-von Neumann dimension of L 2 (M ) as a left N -module where L 2 (M ) is the Gelfand-Naimark-Segal Hilbert space associated to the unique faithful normal tracial state of M [Jo83]. The celebrated index rigidity theorem of Jones claims that the range of Jones indices is exactly equal to the following set {4 cos(π/n) 2 : n ≥ 3} ∪ [4, ∞]. standard invariant of N ⊂ M where [M : N ] is finite is the lattice of relative commutants M i ∩ M j for i ≤ j and where M −1The Graph polynomials and link invariants as positive type functions on Thompson's group F , accepted for publication in J. Knot Theory Ramifications. V Aiello, R Conti, 10.1142/S0218216519500068arXiv:1510.04428V. Aiello, R. Conti, Graph polynomials and link invariants as positive type functions on Thompson's group F , accepted for publication in J. Knot Theory Ramifications. doi: 10.1142/S0218216519500068 , arXiv:1510.04428 V Aiello, R Conti, 10.1007/s11785-018-0866-6arXiv:1603.03946The Jones polynomial and functions of positive type on the oriented Jones-Thompson's groups F and T , accepted for publication in Complex Analysis and Operator Theory. V. Aiello, R. Conti, The Jones polynomial and functions of positive type on the oriented Jones- Thompson's groups F and T , accepted for publication in Complex Analysis and Operator Theory. doi: 10.1007/s11785-018-0866-6 , arXiv:1603.03946 The Homflypt polynomial and the oriented Thompson's group. Quantum Topol. V Aiello, R Conti, V F R Jones, 9V. Aiello, R. Conti, V. F. R. Jones The Homflypt polynomial and the oriented Thompson's group. Quantum Topol. 9 (2018), 461-472. Thompson's group F. J Belk, arXiv:0708.3609Cornell UniversityPhD ThesisJ. Belk. Thompson's group F. PhD Thesis (Cornell University), arXiv:0708.3609. On groups of PL-homeomorphisms of the real line. R Bieri, R Strebel, American Mathematical SocietyR. Bieri, R. Strebel. On groups of PL-homeomorphisms of the real line. American Mathemat- ical Society, 2016. Graduate Texts in Mathematics. B Bollobás, Springer-Verlag184New YorkModern graph theoryB. Bollobás, Modern graph theory. Graduate Texts in Mathematics, 184. Springer-Verlag, New York, 1998. The chameleon groups of Richard J. Thompson: automorphisms and dynamics. M Brin, Inst. Hautes Etudes Sci. Publ. Math. 84M. Brin, The chameleon groups of Richard J. Thompson: automorphisms and dynamics. Inst. Hautes Etudes Sci. Publ. Math. 84 (1996), 5-33. A Brothier, V F R Jones, arXiv:1805.02177On the Haagerup and Kazhdan properties of R. Thompson's groups. A. Brothier, V. F. R. Jones. On the Haagerup and Kazhdan properties of R. Thompson's groups, arXiv:1805.02177 (2018). A Brothier, V F R Jones, arXiv:1807.06215Pythagorean representations of Thompson's groups. A. Brothier, V. F. R. Jones. Pythagorean representations of Thompson's groups, arXiv:1807.06215 (2018). Constructing irreducible representations of discrete groups. M Burger, P De La Harpe, Proc. Indian Acad. Sci. (Math. Sci.). 107M. Burger, P. de la Harpe, Constructing irreducible representations of discrete groups, Proc. Indian Acad. Sci. (Math. Sci.) 107, 3 (1997), 223-235. Introductory notes on Richard Thompson's groups. J W Cannon, W J Floyd, W R Parry, Enseign. Math. 42J. W. Cannon, W. J. Floyd, W. R. Parry, Introductory notes on Richard Thompson's groups. Enseign. Math. 42 (1996), 215-256. . J Dixmier, C , North-Holland Publishing Co15Amsterdam-New York-OxfordNorth-Holland Mathematical LibraryJ. Dixmier, C * -algebras. North-Holland Mathematical Library, Vol. 15. North-Holland Pub- lishing Co., Amsterdam-New York-Oxford, 1977. Finite factor representations of Higman-Thompson's groups. A Dudko, K Medynets, Groups Geom. Dyn. 8A. Dudko, K. Medynets, Finite factor representations of Higman-Thompson's groups. Groups Geom. Dyn. 8 (2014), 375-389. Proper isometric actions of Thompson's groups on Hilbert space. D S Farley, Int. Math. Res. Not. 45D. S. Farley, Proper isometric actions of Thompson's groups on Hilbert space. Int. Math. Res. Not. 45 (2003), 2409-2414. Tutte chromatic identities from the Temperley-Lieb algebra. P Fendley, V , Geom. Topol. 13P. Fendley, V. Krushkal, Tutte chromatic identities from the Temperley-Lieb algebra, Geom. Topol. 13 (2009), 709-741. Analogs of principal series representations for Thompson's groups F and T . Indiana Univ. L Garncarek, Math. J. 61L. Garncarek, Analogs of principal series representations for Thompson's groups F and T . Indiana Univ. Math. J. 61 (2012), 619-626. subgroup of R. Thompson's group F. G Golan, M Sapir, On Jones, J. Algebra. 470G. Golan, M. Sapir, On Jones' subgroup of R. Thompson's group F, J. Algebra 470, (2017), 122-159. Non-inner amenability of the Thompson's groups T and V. U Haagerup, K K Olesen, J. Funct. Anal. 27211U. Haagerup, K. K. Olesen, Non-inner amenability of the Thompson's groups T and V , J. Funct. Anal. 272 (2017), no. 11, 4838-4852. On the Thompson's group factor. J Heo, J. Operator Th. 53J. Heo, On the Thompson's group factor, J. Operator Th. 53, 1 (2005), 185-195. . P Jolissaint, Moyennabilité Intérieure Du Groupe F De Thompson, C. R. Acad. Sci. Paris Sér. I Math. 325P. Jolissaint, Moyennabilité intérieure du groupe F de Thompson, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), 61-64. Index for subfactors. V F R Jones, Invent. Math. 72V. F. R. Jones, Index for subfactors. Invent. Math. 72 (1983), 1-25. V F R Jones, arXiv:math/9909027Planar algebras, I. V. F. R. Jones, Planar algebras, I. arXiv:math/9909027. Some unitary representations of Thompson's groups F and T. V F R Jones, J. Comb. Algebra. 11V. F. R. Jones, Some unitary representations of Thompson's groups F and T. J. Comb. Algebra 1 (2017), no. 1, 1-44. A no-go theorem for the continuum limit of a quantum spin chain. V F R Jones, Comm. Math. Phys. 357V. F. R. Jones, A no-go theorem for the continuum limit of a quantum spin chain. Comm. Math. Phys. 357 (2018), 295-317. Statistical mechanics and the Jones polynomial. L Kauffman, Contemp. Math. 78L. Kauffman, Statistical mechanics and the Jones polynomial, Contemp. Math. 78 (1988), 263-297. The theory of unitary group representations. G Mackey, The University of Chicago PressG. Mackey, The theory of unitary group representations, The University of Chicago Press (1976). Cuntz-Pimsner algebras of group actions. V Nekrashevych, J. Operator Theory. 52V. Nekrashevych, Cuntz-Pimsner algebras of group actions, J. Operator Theory 52 (2004), 223-249. Graphs determined by polynomial invariants. M Noy, Theoret. Comput. Sci. 307M. Noy, Graphs determined by polynomial invariants. Theoret. Comput. Sci. 307 (2003), 365-384. A Ocneanu, Quantized groups, string algebras, and Galois theory for algebras. D. Evans & M. TakesakiCambridgeCambridge University PressOperator Algebras and ApplicationsA. Ocneanu, Quantized groups, string algebras, and Galois theory for algebras. In D. Evans & M. Takesaki (Eds.), Operator Algebras and Applications (London Mathematical Society Lecture Note Series, pp. 119-172). Cambridge: Cambridge University Press. (1989) Analytic aspects of the Thompson's groups. K K Olesen, Department of Mathematical Sciences, Copenhagen UniversityPhD ThesisK. K. Olesen, Analytic aspects of the Thompson's groups. PhD Thesis, Department of Math- ematical Sciences, Copenhagen University (2016). Roots of the chromatic polynomial. T J Perrett, Technical University of Denmark (DTUPhD ThesisT. J. Perrett, Roots of the chromatic polynomial, PhD Thesis, Technical University of Denmark (DTU), 2016. An axiomatization of the lattice of higher relative commutants of a subfactor. S Popa, Invent. Math. 120S. Popa, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math. 120 (1995), 427-445. From skein theory to presentations for Thompson's group. Y Ren, arXiv:1609.04077Journal of Algebra. 498Y. Ren, From skein theory to presentations for Thompson's group, Journal of Algebra 498 (2018): 178-196. arXiv:1609.04077. Relations between the 'percolation' and 'colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the 'percolation' problem. N Temperley, E Lieb, Proc. R. Soc. 332N. Temperley, E. Lieb, Relations between the 'percolation' and 'colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the 'percolation' problem, Proc. R. Soc. 332 (1971), 251-280. Case Postale 64, 1211 Genève 4, Suisse E-mail address: valerianoaiello@gmail. School of Mathematics and Statistics. Section de Mathématiques Université de Genève 2-4 rue du Lièvre ; University of New South WalesRoom 6107 E-mail address: arnaud.brothier@gmailSection de Mathématiques Université de Genève 2-4 rue du Lièvre, Case Postale 64, 1211 Genève 4, Suisse E-mail address: [email protected] School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia, The Red Centre, East Wing, Room 6107 E-mail address: [email protected] https://sites.google.com/site/arnaudbrothier/ Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma. Italy E-mail address: [email protected] di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy E-mail address: [email protected]	[]
[ "Prepared for submission to JHEP Probing Black Hole Microstate Evolution with Networks and Random Walks", "Prepared for submission to JHEP Probing Black Hole Microstate Evolution with Networks and Random Walks" ]	[ "Anthony M Charles [email protected] \nDepartment of Physics\nLeinweber Center for Theoretical Physics\nUniversity of Michigan\n450 Church Street48109-1020Ann ArborMIUSA\n\nInstitute for Theoretical Physics\nKU Leuven\nCelestijnenlaan 200DB-3001LeuvenBelgium\n", "Daniel R Mayerson [email protected] \nDepartment of Physics\nLeinweber Center for Theoretical Physics\nUniversity of Michigan\n450 Church Street48109-1020Ann ArborMIUSA\n\nInstitut de Physics Théorique\nUniversité Paris Saclay CEA\nCNRS\nF-91191Gif-sur-YvetteFrance\n" ]	[ "Department of Physics\nLeinweber Center for Theoretical Physics\nUniversity of Michigan\n450 Church Street48109-1020Ann ArborMIUSA", "Institute for Theoretical Physics\nKU Leuven\nCelestijnenlaan 200DB-3001LeuvenBelgium", "Department of Physics\nLeinweber Center for Theoretical Physics\nUniversity of Michigan\n450 Church Street48109-1020Ann ArborMIUSA", "Institut de Physics Théorique\nUniversité Paris Saclay CEA\nCNRS\nF-91191Gif-sur-YvetteFrance" ]	[]	We model black hole microstates and quantum tunneling transitions between them with networks and simulate their time evolution using well-established tools in network theory. In particular, we consider two models based on Bena-Warner three-charge multi-centered microstates and one model based on the D1-D5 system; we use network theory methods to determine how many centers (or D1-D5 string strands) we expect to see in a typical late-time state. We find three distinct possible phases in parameter space for the late-time behaviour of these networks, which we call ergodic, trapped, and amplified, depending on the relative importance and connectedness of microstates. We analyze in detail how these different phases of late-time behavior are related to the underlying physics of the black hole microstates. Our results indicate that the expected properties of microstates at late times cannot always be determined simply by entropic arguments; typicality is instead a highly non-trivial, emergent property of the full Hilbert space of microstates.	10.21468/scipostphys.8.5.077	[ "https://arxiv.org/pdf/1812.09328v2.pdf" ]	119,201,351	1812.09328	b49b240066a01c22565af63b2a0da9e93c6b5e59	Prepared for submission to JHEP Probing Black Hole Microstate Evolution with Networks and Random Walks 16 May 2019 Anthony M Charles [email protected] Department of Physics Leinweber Center for Theoretical Physics University of Michigan 450 Church Street48109-1020Ann ArborMIUSA Institute for Theoretical Physics KU Leuven Celestijnenlaan 200DB-3001LeuvenBelgium Daniel R Mayerson [email protected] Department of Physics Leinweber Center for Theoretical Physics University of Michigan 450 Church Street48109-1020Ann ArborMIUSA Institut de Physics Théorique Université Paris Saclay CEA CNRS F-91191Gif-sur-YvetteFrance Prepared for submission to JHEP Probing Black Hole Microstate Evolution with Networks and Random Walks 16 May 2019Contents 1 Introduction and Summary 2 1.1 From Microstates to Network Models 5 1.2 Summary of Results 7 We model black hole microstates and quantum tunneling transitions between them with networks and simulate their time evolution using well-established tools in network theory. In particular, we consider two models based on Bena-Warner three-charge multi-centered microstates and one model based on the D1-D5 system; we use network theory methods to determine how many centers (or D1-D5 string strands) we expect to see in a typical late-time state. We find three distinct possible phases in parameter space for the late-time behaviour of these networks, which we call ergodic, trapped, and amplified, depending on the relative importance and connectedness of microstates. We analyze in detail how these different phases of late-time behavior are related to the underlying physics of the black hole microstates. Our results indicate that the expected properties of microstates at late times cannot always be determined simply by entropic arguments; typicality is instead a highly non-trivial, emergent property of the full Hilbert space of microstates. Introduction and Summary String theory is expected to somehow resolve puzzles arising in general relativity involving black holes, such as the origin of their entropy and the information paradox. The fuzzball programme argues that the resolution of such puzzles is that extended stringy objects alter the horizon structure of black holes drastically from its classical expectation. In this picture, the black hole horizon should be seen as an effective geometry, averaged over the individual "fuzzballs" that actually make up the states of the black hole [1][2][3][4][5][6][7]. Microstate geometries are fuzzballs that can be constructed and studied as classical solutions in the supergravity limit of string theory; these are smooth, horizonless solutions with the same asymptotic charges as the black holes. These microstate geometries intrinsically live in dimensions larger than four and have non-trivial topological cycles that are supported by fluxes. This non-trivial topological structure allows for smooth supersymmetric soliton solutions that support charges and mass [8,9]. For the three-charge supersymmetric black hole, many microstate geometries have already been constructed in the literature. These include the multi-centered Bena-Warner solutions [2] and the more recent superstrata [10][11][12], which are themselves a generalization of the two-charge D1-D5 Lunin-Mathur supertubes [1,13]. Assuming supersymmetry simplifies the search for microstate geometries, as the relevant BPS equations are often more tractable than the equations of motion themselves. 1 Importantly, these supersymmetric states are void of actual dynamics and so they necessarily avoid the question: how can smooth black hole microstates form in a dynamical process? A gravitational collapse of a shell of matter will form a horizon well before any curvatures or quantum effects are expected to be large, which seems to be in contradiction to the fuzzball proposal where no horizon should form. A resolution to this puzzle, proposed in [24][25][26], is that the large phase space at horizon scales is the crucial ingredient that invalidates the usual classical intuition and renders quantum effects large at the horizon scale. The logic is that a collapsing shell of matter can quantum tunnel into a fuzzball microstate before it forms a horizon. The tunneling amplitude to form any one particular fuzzball is O(e −S ), exponentially suppressed by the would-be black hole entropy. However, when the collapsing shell reaches its putative horizon scale, the available number of microstates is incredibly large, namely e S . The two exponentials can cancel, and the tunneling probability to go to any microstate ends up being O (1). The result is that the collapsing shell of matter will necessarily quantum tunnel into a fuzzball instead of forming a horizon. These arguments are very general, but they are not based on any explicit calculations involving actual fuzzballs or microstate geometries. The first (and so far, only) concrete calculation of a formation rate of black hole microstates by quantum tunneling was performed in [27]. There, the authors considered forming smooth multi-center microstate geometries by starting with a collapsing shell of branes and repeatedly tunneling off a small amount 1 See [14][15][16][17][18][19][20][21] for explicit constructions of non-supersymmetric microstate geometries. It is also possible to construct non-supersymmetric microstates by adding a non-supersymmetric probe onto a supersymmetric background [22,23]. of charge from the shell onto a new center. This iterative tunneling procedure is depicted schematically in figure 1. By treating the tunneling branes in this picture as probe branes, the tunneling amplitudes can be computed explicitly from the probe brane action. The final result is that the tunneling amplitude to end up in a final state with N centers goes like Γ ∼ exp −N −β ,(1.1) where β is some positive exponent of order one. 2 The conclusion is that the tunneling rate is enhanced for larger values of N , and so it is easier to form a microstate geometry with a large number of centers. Of course, this only tells us what to expect when the collapsing shell of matter first tunnels into a microstate. After this initial tunneling, it is still possible for tunneling between various microstates to occur as time goes on. A broader question we can ask is the following: what do we expect to see after the collapsing shell of matter "settles down"? That is, if we wait for a long enough time for the system to end up in some kind of equilibrium configuration, what microstate should we expect the system to be in? The calculations of [27] are not enough to answer this question, because they only consider particular formation paths. It may be relatively easy to form a microstate with a large number of centers along these paths, but if there are many more few-centered microstates available (with formation paths leading to them) in the phase space, then the complete picture may result in being more likely to end up with a small number of centers after all. To have such a complete picture of the late-time dynamics of the microstates, we should take into account all possible microstates (including possible fuzzballs that do not have a classical microstate geometry description, or that do not have one in the duality frame the evolution started out in [28,29]), their degeneracies, and their interactions (in the form of quantum tunneling between each other). This very naturally leads us to consider a network, as depicted in figure 2, where every node in the network corresponds to a particular microstate, every edge corresponds to an allowed tunneling path, and these edges are weighted by the corresponding tunneling rate between the nodes. We can then use the full array of well-developed tools in network theory [30] to understand the late-time behavior of such a network. For example, the latetime relative importance of nodes within a network is directly related to the eigenvector centrality of the network, as we will discuss in section 2. It is important to note that we cannot feasibly construct all possible microstate geometries (let alone non-classical fuzzballs) explicitly, let alone compute the transition rates between each pair. Therefore, the networks we construct in this paper will necessarily be models, in the sense that we will try to capture important general features of the microstates, while leaving out some of the more intricate details of the actual microstate geometries. Our models will have a number of a priori unknown parameters that will parametrize how the degeneracies of the microstates and the tunneling rates between them depend on certain properties of the microstates. By exploring the phase space of these parameters, we will be able to make general statements about which microstate properties are important in determining the late-time behaviour of the black hole microstate evolution. The rest of the paper is organized as follows. In the next subsection 1.1, we introduce the three network models we will consider and discuss how they capture features of certain classes of black hole microstates in string theory. In subsection 1.2, we briefly summarize our main results. In section 2, we provide an overview of how network theory methods and random walks can be used to understand the dynamics of quantum systems. In sections 3, 4, and 5, we present a detailed description of each of our three network models (respectively), use network theory methods to understand their time evolution, and then give a detailed discussion of the results. Finally, we conclude in section 6 with some important discussions regarding all three models. We give more technical details on the network and random walk methods used throughout this paper in appendix A, and in appendix B we present a number of explicit calculations involving black hole microstates which were used to inform various features of our network models. From Microstates to Network Models Here, we introduce the three models of microstate networks we will study in this paper, and we discuss how they are inspired by existing classes of black hole microstate solutions in string theory. Models 1 and 2: Multi-centered black hole microstates Our goal for the first two models is to model the dynamics of five-dimensional three-charge, smooth, multi-centered microstate geometries [2], as reviewed in appendix B. These can be viewed as microstates of the five-dimensional three-charge supersymmetric BMPV black hole. Since we want non-trivial dynamics, we want to consider adding a small amount of non-extremality to these microstates in order to excite them away from BPS limit. The slightly non-extremal microstates obtained in this way should then be interpreted as microstates of a near-extremal BMPV black hole. Note that the amount of non-extremality we add to the microstate is not a tuneable parameter; we consider it to be infinitesimal to avoid large backreaction on the BPS microstates. It is not known how many ways there are to add a small amount of non-extremality to a generic multi-centered black hole microstate geometry. 3 One possibility we can imagine is to give one or more of the centers small velocities relative to the others. Another possibility is to "wiggle" the bubbles themselves and excite oscillation modes of the topological bubbles between the centers. From these considerations, it seems very likely that different supersymmetric microstates have a different number of ways to excite them above extremality. We will allow for this possibility by taking into account a degeneracy factor for each microstate that depends on its properties. The only dynamics that we allow in our models are the quantum tunneling transitions between different microstate geometries. Heuristically, the transitions we want to allow should be thought of as taking some or all charge from a center and tunneling it either onto a new center or an existing center, similar in spirit to the explicit tunneling calculations performed in [27]. This means any single transition will either leave the number of centers unaltered or change the number by one. The Bena-Warner multi-center microstate geometries are complicated solutions in fivedimensional supergravity. The charges of the centers and their positions must satisfy the non-linear bubble equations, which become increasingly complicated to solve when the number of centers increases. Restrictions are often placed on the centers to facilitate solving the bubble equations, such as putting them all on a single line. For our networks, we will use simple models that capture some of the important qualitative physics of these microstate geometries without having to actually construct all relevant supergravity solutions. 4 3 One can add probes to supersymmetric backgrounds that will break supersymmetry [22,23,31,32], but even a counting of all such possible non-extremal probes has not been done. 4 Given how complicated the Bena-Warner multi-centered geometries are, one might wonder if it is actually possible to find solutions that are related by the "splitting" transitions discussed above. We give a proof of concept of this in appendix B.2 by explicitly constructing two solutions to the bubble equations that have the same asymptotic charges and only differ by the centers that have undergone this splitting. Our first model is a very minimal model of black hole microstates where we only consider the number of centers N that each microstate has. We will not assume any particular configurations of these centers, such as restricting them to be on a line. The degeneracy of each microstate (i.e. the number of ways to lift the microstates off extremality, as discussed above) can then only depend on N , while the tunneling rate between states can depend on the number of centers of the initial and final states. Our second model has more features to allow for richer physics. To construct a microstate, we fix a total charge Q of the black hole and divide this over a variable number N of centers. For simplicity, we will only consider one type of charge in the system. We will further assume that the centers are all on a line. A microstate in model 2 is thus determined by giving an ordered list {Q 1 , Q 2 , . . . Q N } of charges of each of the centers such that N i=1 Q i = Q; this is called a composition of the integer Q. The degeneracy and transition rates are more intricate in model 2, as they can depend on the details of the charge distribution within the microstates. Note that the microstates in this model are in one-to-one correspondence with compositions of Q, and there are 2 Q−1 such compositions. Model 3: the D1-D5 system In model 3, we want to understand dynamics of microstates in the D1-D5 system. Specifically, we will consider type IIB string theory on R 4,1 ×S 1 ×T 4 with N 1 D1-branes wrapping the S 1 and N 5 D5-branes wrapping S 1 × T 4 . This system is well-understood in many contexts. 5 In particular, the low-energy world-volume dynamics of this system are described by a sigma model CFT whose target space is a deformation of the symmetric product (T 4 ) N /S N , where N = N 1 N 5 . The Ramond ground states of this CFT are holographically dual to the smooth Lunin-Mathur supertubes [1,13]. We will find the "string gas" picture of the D1-D5 ground states the most useful, where these ground states can be seen as a gas of strings winding the S 1 with total winding number N . If there are N w strings with winding number w, then we must have ∞ w=1 wN w = N . (1.2) Furthermore, there are 8 bosonic and 8 fermionic modes that each string can be in: N w = µ N w,µ + µ N w,µ ,(1.3) where the sum over µ is over the 8 bosonic modes (so N w,µ = 0, 1, 2, . . .) and the sum over µ is over the 8 fermionic modes (so N w,µ = 0 or 1). For a given N w , there are Ω(N w ) distinct ways of dividing the strings into the 8 bosonic and 8 fermionic modes, where: 6 Ω(k) = 8 l=0 8 l k − l + 7 7 . (1.4) In model 3, we will consider ground states in the D1-D5 system with a slight amount of non-extremality added. Using the string gas picture described above, we will model these states by unordered collections {w 1 , w 2 , . . .} of winding numbers, with i w i = N . We will associate the correct winding mode degeneracies (1.4) to each microstate, but unlike in models 1 and 2 we will not associate an additional degeneracy related to the number of ways to add the slight non-extremality to the microstate. Note that an unordered collection {w 1 , w 2 , . . .} corresponds to a partition of the integer N ; asymptotically, the number of partitions p(N ) scales as p(N ) ∼ exp π 2N/3 . Of course, the total number states we are considering in model 3, including the degeneracies (1.4), is precisely (by construction) the number of D1/D5 ground states, which scales as ∼ exp 2π √ 2N . We will allow transitions where a single string of winding w can split into two smaller strings with windings w a , w b (with w a + w b = w), or the reverse where two strings with windings w a , w b combine into a larger string of winding w = w a + w b . The transition rate for either of these processes will then depend non-trivially on the initial and final windings w a , w b and w = w a + w b . Summary of Results Late-Time Behavior Phase Ergodic Trapped Amplified 6 To understand this formula, note that l denotes the number of fermionic excitations turned on (which is limited to 8). The first factor is the combinatorial factor associated with distributing the l fermionic excitations into 8 possible bins. The second factor is the combinatorial factor for dividing the k − l bosonic excitations into 8 possible bins, with the possibility of putting multiple excitations into the same bin; i.e. the number of weak compositions of k − l into 8 parts. In section 1.1, we have introduced three different network models of black hole microstates. (They will be further specified in sections 3.1, 4.1, and 5.1, respectively.) Our main goal in this work is to use network theory tools to study the time-evolution of these models. Crucially, the methods we will use do not assume the validity of equilibrium statistical mechanics, which allows for non-trivial and interesting late-time behavior. Our main results are shown in table 1. We find that there are, broadly speaking, three different types of late-time behavior that our microstate networks exhibit: • Ergodic behavior. All microstates are approximately equally likely at late times, and so the probability distribution of microstates is determined entirely by the microstate degeneracies and not the details of tunneling rates between states. Random walks on the networks in this regime are able to move around the entire network freely. • Trapped behavior. At late times, the probability distribution is restricted to only a small subnetwork of the full microstate network. The transition rates between microstates determine precisely what subnetwork is relevant. Random walks are effectively restricted to move only on this subnetwork. • Amplified behavior. The most degenerate microstates comprise the most highlyconnected nodes on the network. At late times, the system is much more likely to be on these highly-connected nodes than any others. Random walks are likely to stay on this highly-connected subnetwork, but excursions to other states are allowed. Model 1 shows only ergodic behavior, while model 3 shows only amplified behavior. Model 2 can demonstrate either ergodic or trapped behavior, depending on where in parameter space we are. Interestingly, the cross-over between these two types of behaviors is very sharp and sudden, indicating a phase transition in parameter space of the network's latetime behavior. As emphasized earlier, the main question we want to answer is what do we expect to see at late times as these black hole microstate systems evolve? If the system exhibits ergodic late-time behavior, this means that all microstates are (approximately) equally likely and so we can determine properties of a typical state simply by counting the number of states with that property. For example, in model 1 we can look at the number of microstates with a given number of centers N , and whichever value of N maximizes this degeneracy will be the number of centers we expect a typical late-time state to have. If the system exhibits amplified late-time behavior, then these most degenerate (with respect to the number of centers or number of winding strings) microstates are also the most connected in terms of tunneling paths, and so this the system will favor these highly-degenerate states even more strongly. In the trapped phase, though, we cannot simply tabulate all microstates to understand typicality; the system at late times is forced to be in one of only a small subset of black hole microstates. This behavior is surprising because it indicates that entropic arguments are not sufficient to explain the dynamics of the system, despite its large ground-state degeneracy of microstates. We will elaborate on the impliciations of this trapped late-time behavior for Bena-Warner microstates in section 6. The main takeaway from all of this is that the late-time dynamics of black hole microstates have a rich and interesting structure to them. We find that there is no simple way to express what "typicality" in black hole microstates means; it depends intricately on the full details of the Hilbert space of microstates. Moreover, the network theory methods presented in this work give an effective way to probe microstate dynamics, and we find a number of intriguing results that are worthy of further exploration. Network Theory and Quantum Tunneling Consider a quantum mechanical system with a discrete number of accessible states. Quantum fluctuations will generically give rise to non-zero tunneling probabilities between different states. The dynamics of the system are described by its Hamiltonian, which can be used to compute how a particular initial state (or probabilistic superposition of states) tunnels into other states over time. However, these kinds of computations are difficult to perform in many cases, in particular for systems with a very large number of states. In addition, these methods fail if we only know estimates of the tunneling rates between states and not the full Hamiltonian. A historically successful approach that evades some of these difficulties is to treat the time evolution of the system as a stochastic process, where time is discretized, and at each discrete time slice the system evolves randomly to another state according to the tunneling amplitudes that are available to the current state [35][36][37]. The dynamics of this stochastic system are then understood very naturally through the lens of network theory. Specifically, we can view these systems as directed networks where the nodes are the available states in the system and the edges between nodes are weighted by the corresponding tunneling amplitudes. Many properties of the underlying quantum system can then be understood quantitatively by analyzing properties of this corresponding network. In particular, this network-theoretic approach has led to many significant developments in physics-related fields, including percolation [38], protein interactions [39], quantum cosmology [40,41], tunneling in the string landscape [42], and brain function [43], to name a few. In this paper, we will be interested in asking the following question: what state do we generically expect the system to be in? That is, as the quantum system tunnels and evolves over time to some kind of equilibrium configuration, how can we compute the probability that the system is in any particular one of its many accessible states? These are hard problems to tackle for generic quantum systems, due to both the difficulty of numerically evolving the Schrödinger equation as well as the sensitivity of this evolution to initial conditions [44,45]. However, network theory gives us a whole slew of powerful tools designed to answer exactly these sorts of questions. In particular, we will investigate eigenvector centrality and random walks of networks as tools to probe the late-time dynamics of quantum systems. Eigenvector Centrality Concretely, let's consider a system with N distinct accessible states labelled by i = 1, . . . , N , each with an associated degeneracy ω(i); for example, we could consider a system with N distinct energy levels and ω(i) possible ways for the system to be in each energy level i. We denote the tunneling rate from state i to state j as Γ(i → j). Note that we will allow for self-transitions Γ(i → i) as well. This system can be represented by a network, as shown in figure 3, where the nodes of the network are the states and the edges are the tunneling amplitudes. The adjacency matrix A of this network is defined as the N ×N matrix whose elements A ij are the edge weights of the graph (i.e. the transition rates Γ(i → j)), weighted by the degeneracy of the starting and ending nodes (i.e. the degeneracy of the initial and final states). That is, Γ ( 2 → 1 ) Γ ( 2 → 3 ) Γ ( 3 → 2 ) Γ( 4 → 2) Γ(3 → 4) Γ(2 → 2) Γ(1 → 1)A ij = ω(i)Γ(i → j)ω(j) . (2.1) The degree d i of a node i is the sum of all outgoing adjacency matrix elements from that node: d i = j A ij = j ω(i)Γ(i → j)ω(j) . (2.2) We can also define the transfer matrix T, whose elements T ij are the probability for the system to tunnel from state i to state j. This probability is the edge weight A ij , multiplied by an overall constant of proportionality chosen such that the total probability of transitioning from any given node is one. We therefore set T ij = A ij d i = ω(i)Γ(i → j)ω(j) k ω(i)Γ(i → k)ω(k) . (2.3) Let p(t) be a vector whose components p i (t) are the probability to find the system in state i at a discrete time t. For stochastic systems, this probability evolves according to the transfer matrix: p(t + 1) = p(t)T . (2.4) As t → ∞, the system will approach a steady state configuration p ∞ that is a fixed point of the time evolution such that p ∞ = p ∞ T . (2.5) That is, p ∞ is the left eigenvector of T with an eigenvalue of one. Importantly, T is a column-stochastic matrix (i.e. each of its columns sums to one), and so all of its eigenvalues are guaranteed to have magnitude \|λ\| ≤ 1. The eigenvector centrality of a matrix is defined to be the left eigenvector with the largest eigenvalue, which means that the eigenvector centrality of the transfer matrix is a left eigenvector with an eigenvalue of one. This is precisely the criterion for p ∞ to be a fixed point of time evolution in (2.5), and so p ∞ is the eigenvector centrality of T. Therefore, by computing the eigenvector centrality of the network, we immediately know what the steady-state configuration of the system is at late times. An analytic expression for p ∞ can easily be obtained when the tunneling amplitudes Γ(i → j) are symmetric under exchange of i and j, which is the case in our models 1 & 3. Physically, this can be interpreted as considering ensembles of states that are at approximately the same energy and so the tunneling amplitude between any two states is the same in both directions, i.e. no irreversible relaxation processes occur in addition to or in tandem with tunneling processes. The eigenvector centrality of such a network is then given exactly by [46,47] p ∞,i = d i k d k = j ω(i)Γ(i → j)ω(j) k,l ω(k)Γ(k → l)ω(l) . (2.6) Notice that this expression is independent of the initial conditions in the network. No matter which state (or what probabilistic superposition of states) the system begins in, it always evolves to a late-time steady state given by (2.6), which only uses information about the degeneracies of each state and the transition amplitudes between states. Random Walks on Networks One problem with using the analytic result (2.6) for the network centrality is that it requires computing the degree of every node on the network. For very large networks this kind of computation can be computationally expensive and unfeasible to do. Another issue is that the centrality only tells you the behavior of the system in the t → ∞ limit; it doesn't capture any of the finite-time behavior of the network. In these cases, we can instead understand the evolution of the system by performing a random walk on the network. 7 In a random walk, if at a discrete time t the system is at node i, then at the next time step t + 1 the system moves randomly to a neighboring node according to the probabilities in the transfer matrix T. Once we have done a sufficiently large number of such time iterations, we can tally up what fraction f i of steps were spent in node i. If the random walk were to run for an infinite amount of time, the fraction f i would converge precisely to the steady-state probability p ∞,i . For finite-time random walks, the fraction f i will serve as a good estimate of p ∞,i , as long as the random walk has run for longer than the characteristic relaxation time of the network [46,48]. For a more detailed discussion of random walk convergence, see appendix A. Random walks are often more efficient to compute than the actual centrality because they rely on only local neighborhoods of nodes and not the full network. For example, consider the network depicted in figure 4, where the highly-connected nodes only comprise a small subnetwork of the full network. Performing a random walk on such a network will typically only require computing the transfer matrix elements on the highly-connected subnetwork, whereas the centrality (2.6) requires computing all entries of the transfer matrix. Other Network Properties We have so far only discussed methods for determining the late-time behavior of a quantum mechanical system. However, this only scratches the surface of the wide array of networktheoretic tools that can be used to gain insight into non-trivial properties of quantum systems. For example, community detection algorithms can be used to look for the presence of highly-connected subnetworks, which can be thought of as subspaces of the full Hilbert space whose dynamics are approximated by truncating the full Hilbert space onto the subspace [49][50][51][52]. Additionally, more refined versions of the eigenvector centrality can be constructed by modifying the transfer matrix in particular ways; these generalized eigenvector centralities can give insight into late-time behavior when features like damping, sources and sinks, and random noise are present [53,54]. All in all, we believe that this network-based approach to quantum mechanics is a fruitful topic to explore for a wide range of physical systems. Model 1: N Centers Setup In our first model, as discussed in section 1.1, we will model multi-centered black hole microstates very minimally. Every microstate in our model will be imbued with only one property: the number N of centers that it has. We will set a cut-off on how many centers a microstate can have by demanding that 1 ≤ N ≤ N max for some maximum number of centers N max . We want to associate a degeneracy to each microstate with N centers, related to the number of ways to add a slight non-extremality excitation to the given microstate (as also discussed in section 1.1). Our model for the degeneracy of black hole microstates is 8 ω(N ) = N β , (3.1) where β is a tuneable numerical parameter. If the most important contribution to the degeneracy function comes from the number of ways to "wiggle" bubbles in a multicentered solution, larger bubbles should give a larger degeneracy; since larger bubbles typically arise when there are fewer centers, we would then expect β ≤ 0. On the other hand, if the most important contribution to the degeneracy comes from the configurational entropy of the N centers (i.e. rearranging them in space) or from adding small velocities to the centers, then a larger number of centers would give a larger degeneracy and we would expect β ≥ 0. We will consider both situations for the parameter β. We model the transition rate between two microstate solutions by Γ(N → N ) = exp −γ min(N, N ) δ , for \|N − N \| ≤ 1 , (3.2) where γ and δ are some numerical parameters. Quantum tunneling rates are exponentially suppressed, so it is natural to demand γ ≥ 0. We also expect that it should be easier for tunneling to occur when there are more centers present, since then the bubbles are smaller and thus give rise to lower potential barriers. (This is also congruent with the results of [27], which found a higher tunneling rate for a larger number of centers.) We therefore will choose δ ≤ 0 in order for the tunneling rate to be suppressed for small N . Note also that we take the minimum of N and N in order to guarantee that the tunneling rate is symmetric when tunneling between N and N . Importantly, the only transitions allowed are N → N = N, N ± 1. Since the only property of the microstates we are looking at is the number of centers, any two microstates with the same number of centers will appear identical. So, the probability of going from an N -center microstate to any microstate with N centers must be weighted by the degeneracies ω(N ), ω(N ) of the initial and final microstates. The probability P (N → N ) to tunnel from a microstate with N centers to one with N centers is therefore given by P (N → N ) = ω(N )Γ(N → N )ω(N ) n ω(N )Γ(N → n)ω(n) , (3.3) where the normalization is chosen such that all probabilities sum to one. Note also that any constant prefactors in the degeneracies and transition rates will cancel in this expression; it is only the relative differences in degeneracies and transition rates that affect the late-time behavior. The dynamics of this model can be captured by the simple network shown in figure 5. There are N max nodes in the network, each labeled by the number of centers of the microstates they describe. The edges are directed and represent allowed transitions in the model. The weight of each edge is the corresponding probability for that transition to occur. The adjacency matrix A of this network has elements A ij = ω(i)Γ(i → j)ω(j), 1 2 Γ(1 → 2) Γ(2 → 1) Γ(1 → 1) Γ(2 → 2) 2 3 Γ(2 → 3) Γ(3 → 2) Γ(2 → 2) Γ(3 → 3) N max Γ(N max → N max ) · · · Figure 5: A network representation for model 1. Each node corresponds to a different value of the number of centers N and each (directed) edge is weighted by the probability to tunnel from one value of N to another. while the transfer matrix T has elements T ij = P (i → j) (using (3.3)). Note that i and j range from 1 to N max , so A and T are N max × N max square matrices. Moreover, their definitions here are consistent with the general formulas presented in section 2. The eigenvector centrality p ∞ (i.e. the left eigenvector of T with an eigenvalue of one) determines the late-time behavior of the system. In particular, the late-time probability of being in a node with N centers is simply the N th component of p ∞ . We can also compute the expected value N of the number of centers at late times via N = n n p ∞,n . (3.4) Results Now that we have established the setup of our model, we now want to explore how the numerical parameters of the model affect the late-time behavior of the black hole microstates. Specifically, we will look at how the parameters affect the eigenvector centrality of the our network model and make conclusions about what kind of microstate we expect to be in at late times. We will consider the effects of the degeneracy parameter β introduced in (3.1) and the transition rate parameters γ, δ introduced in (3.2). Note that, for this simple network, we can simply calculate the analytic and exact late-time probability vector as given in (2.6), and thus do not need to actually perform explicit random walks on this network. Degeneracy Dependence We first want to analyze how the degeneracy ω(N ) of microstates affects the eigenvector centrality. A plot of the degeneracy versus the number of centers for various values of β is given in figure 6. When β = 0, the degeneracy is uniform and there are an equal number of microstates for any value of N . As β is tuned below zero, though, the degeneracy is slanted towards microstates with a small number N of centers. We would therefore expect that setting β close to zero makes the eigenvector centrality uniform, since there are an equal number of states for all values of N , while tuning β to be more negative corresponds to shifting the centrality towards smaller values of N . Making β positive would also simply push the centrality towards larger values for N . Our intuition is confirmed by the eigenvector centrality of our system; a plot of the eigenvector centrality for multiple values of β and N max = 20 are shown in figure 7, for now setting δ = 0 (and γ = 1). We see that we can smoothly tune the eigenvector centrality to be pushed entirely to small values of N by making β more negative. This trend persists when δ = 0 as well (still with γ = 1). Heat plots of N for a range of β and δ values are given in figure 8. 9 No matter what value of δ, γ, and N max we look at, very negative values of β push N close to one while values of β close to zero push N close to N max /2. Moreover, the crossover between these two regions is smooth and continuous, with no sharp phase transitions appearing. Transition Rate Dependence One immediately striking fact about the centrality results given in figure 8 is that the transition rate is much less impactful than the degeneracy in determining the centrality of the network. Nonetheless, the transition rate still affects the centrality in a non-trivial way. The functional form is the same for all three transitions, so for the purpose of understanding the transition rate we will first just look at the N → N + 1 transition. Plots of the transition rate Γ(N → N + 1) versus N for various values of δ are shown in figure 9. When δ = 0, the transition rate becomes independent of N and N and thus uniform. The centrality is therefore determined entirely by β when δ = 0. As δ is first tuned below zero, the transition rate becomes non-uniform; instead, it increases with N . This means that transitions are more likely between microstates with higher numbers of centers, and so we would expect the centrality to be shifted towards higher values of N . As we continue to tune δ below zero, though, this effect becomes less pronounced; the transition rate is mostly uniform in N for very negative values of δ. For very negative values of δ, then, the centrality is once again determined entirely by β. We can thus conclude that δ should push the system to larger values of N when δ is negative, though this effect should diminish as δ becomes very negative. Again, our intuitive picture is confirmed by plotting the centrality for β = 0 and varying δ in figure 10. We can also look at how N varies with δ for different values of β. Some plots of this are shown in figure 11. These plots again demonstrate precisely the behavior we expected from our analysis of the transition rates plotted in figure 9. δ = -2.0 δ = -1.6 δ = -1.2 δ = -0.8 δ = -0.4 δ = 0.0 The behavior we have observed above is generic, in the sense that it holds for any value of β or γ. The effect γ has is to make the peak in the N versus δ plots more pronunced. 2 1 0 δ 4 6 8 10 N β = 0.0 β = -0.2 β = -0.4 β = -0.6 β = -0.8 β = -1.0 (a) γ = 1. 2 1 0 δ 4 6 8 10 12 N β = 0.0 β = -0.2 β = -0.4 β = -0.6 β = -0.8 β = -1.0 (b) γ = 2. As we can see from figure 11, when γ is tuned below zero the peaks become sharper and more defined. This can also be seen from the heat map plots in figure 8, where the contours are clearly sharper for the γ = −2 heat maps than the γ = −1 heat maps. Once again, though, this effect is small compared to the effect β has on the centrality. Conclusions The main conclusion we can derive from our analysis of this network is that the latetime behavior of the microstates is dominated by their degeneracy ω(N ) (as opposed to their transition rates Γ(N → N )). The eigenvector centrality of the network is primarily determined by the parameter β in the degeneracy, with values of β close to zero making the centrality uniform in N and more negative (resp. positive) values of β pushing the centrality to be larger for microstates with smaller (resp. larger) N . The values of γ and δ in the transition rates give rise to small modulation of on top of these effects; in particular, δ negative (but not too negative) pushes the centrality to be slightly larger for larger values of N , while negative values of γ makes this δ-effect more relevant. Another noticeable feature we found is that the physics of the network is "smooth", in the sense that the parameters β, γ, and δ can be tuned continuously with no sudden spikes or jumps in the centrality. In particular, the parameters can be tuned as needed to make the expected value N at late times whatever value we want. This will not be true in model 2 below. One could wonder if the degeneracy function ω(N ) could have a different functional dependence on N than we have considered in (3.1). We have also considered an exponential degeneracy function of the form: ω(N ) = exp γ N β , (3.5) where we considered γ = ±1 and β ∈ (−1, 1). We found that the physics of such a degeneracy function is qualitatively the same as the polynomial degeneracy (3.1) and thus follows the same qualitative behaviour as already discussed in this section. In order to interpret the degeneracy dominance of the network, we can recall the interpretation of our network and its parameters. The degeneracy function ω(N ) represents the number of slightly non-extremal N -center microstates, while the transition rate Γ(N → N ) represents the quantum tunneling probability between such microstates (such as calculated in [27]). With this picture in mind, the degeneracy dominance of our results indicates that the counting of non-extremal microstates is much more important than the details of the tunneling interactions between the microstates. As we have mentioned, the counting of such non-extremal microstates depends on the counting of the initial (BPS) microstates as well as the number of ways one can add a slight non-extremality to a given microstate. We interpret the dynamics of the microstates in this model as being near-BPS states whose collective dynamics are ergodic, in the sense that, given enough time, any microstate will eventually tunnel into any other given one. At any given time, the system is (approximately) equally likely to be in any microstate. Model 2: N Centers with Charge Setup In our second model, we want to add more features to our previous model of multi-centered black hole microstates. As discussed in section 1.1, we will consider microstates where all centers lie on a line, as depicted in fig. 12. We will model this microstate as having a total charge Q that is distributed among each of the centers. 10 Thus, we can view the corresponding microstates as ordered partitions (i.e. compositions) of the total charge Q; each microstate has a number of centers N , as well as a set of charges {Q i } concentrated at each of the centers that sum up to the total charge Q. We will also require that each center contains at least one unit of charge and that all charges are positive; this implies the maximum possible number of centers is N max = Q. Figure 12: A sample charged black hole microstate. This one has three centers, each with an associated charge, as well as topological bubbles stretched between each adjacent center. Q 1 Q 2 Q 3 Like we did with model 1 (and as discussed in section 1.1), we model the possible degeneracy of adding non-extremality to a microstate with the degeneracy function: ω(N, {Q i }) = N i=1 αQ β i , (4.1) where α and β are some numerical parameters. If this degeneracy is dominated by the number of ways to excite or "wiggle" the topological bubbles between centers, then the degeneracy should be greater for microstates with larger bubbles (and thus larger charges), and so we would expect β ≥ 0. On the other hand, if the degeneracy is dominated by the number of ways to add small velocities to the centers, then the microstates with a large number of centers (and thus smaller charges) will be more degenerate, which requires β ≤ 0. To cover both cases, we will consider both cases for β. The transitions between microstates in this model can be pictured as breaking off an amount of charge from a certain center and tunneling it onto an adjacent center or tunneling it to create a brand new (adjacent) center. (We will not allow charge to tunnel elsewhere, i.e. we will not allow charge to "hop" over existing centers.) Our model for the tunneling rate is: Γ N, {Q i } → N , {Q i } = exp −γ Q δ T Q λ L Q λ R , (4.2) for some tunable parameters γ, δ, and λ. Q T is the amount of charge that has broken off the original center and tunnels away. Q L,R are measures of how much charge sits to the left and right, respectively, of the center that the charge tunnels from. If the charge tunnels away from the i th center, these are given by Q L = max(Q L , 1),Q L = Q i−1 + ωQ i−2 + ω 2 Q i−3 + . . . = i−1 j=1 ω j−1 Q i−j , Q R = max(Q R , 1),Q R = Q i+1 + ωQ i+2 + ω 2 Q i+3 + . . . = N −i j=1 ω j−1 Q i+j , (4.3) where ω is a parameter that encodes how far-ranging the electromagnetic force is. We require that 0 ≤ ω ≤ 1, which ensures that centers closer by will have a larger effect on the tunneling rate. Since Q L , Q R only depend on the initial (and not the final) microstate in the tunneling process, the tunneling rate (4.2) is not symmetric under interchange of initial and final states. Note also that if there are multiple transitions possible from one microstate to another, then we take all of them into account simultaneously -the total tunneling rate is then simply the sum over the tunneling rates of all these possible transitions. It is interesting to make contact between our transition rate (4.2) and the explicit tunneling rates calculated in specific multicentered microstates in [27]. Of course, our simplistic model does not capture the intricacies of the actual multi-center microstate solutions. Nevertheless, from [27], it is clear that the physical choice for the parameter δ is δ = 1. In appendix B.3, we make an attempt to also extract very rough estimates for the values of λ and ω from explicit microstate tunneling calculations; the result is λ ≈ −0.18 and ω ≈ 0.37. Although these values for the parameters could be argued to be the most physically relevant, we will still consider varying these parameters in order to explore the full parameter phase space of our model. Finally, we note that [27] assumes (but does not explicitly construct or verify) that it is possible to construct many microstates with different numbers of centers N that have the same asymptotic charges at infinity 11 . In appendix B.2, we give a proof of concept that it is possible to "split" a center into multiple centers while keeping all asymptotic charges fixed by constructing an explicit example of such a splitting. The dynamics of model 2 can be encoded into a network where each node is represented by an ordered partition (i.e. composition) of the total charge, while the edges represent the allowed transitions; an example is shown in figure 13. Importantly, the number of compositions of Q is 2 Q−1 and thus scales exponentially with Q; generating the entire explicit network for Q 20 becomes computationally unfeasible. Instead, we will perform dynamic random walks that only generate nodes as needed on the network as the random walk progresses. We perform the random walk until its behavior has converged to a steadystate behavior; see appendix A for details. Then, as discussed in section 2.2, we can use the fraction of steps spent in each node to numerically estimate the late-time behavior of our model. Once we fix the model parameters and perform a random walk, there are two main pieces of information we can extract from the random walk after it has converged: how many centers the node had at each time step, and how charge was distributed among these centers. For example, consider the case where α = γ = δ = 1, β = λ = 0, and Q = 20. One random walk performed with these parameters is given in figure 14. On the left in this figure is the probability for the random walk to be in a state with a particular number of centers, plotted in blue. This probability is calculated simply by tabulating the random walk results and counting the fraction of steps spent at each value of N . The superimposed red line in this plot is the number of microstates that have a particular number of centers N , normalized by the total number of states. The probability distribution will match this exactly when all microstates were equally likely; we will refer to this as ergodic behavior (precisely as in model 1), since all nodes in the network will eventually be visited by a random walker with (approximately) equal probability. {1, 1, 1, 1} {1, 2, 1} {2, 1, 1} {1, 1, 2} {2, 2} {3, 1} {1, 3} On the right of this figure is a plot of the three largest charges present at each time step in the random walk, plotted in red, orange, and yellow (in descending order). These can be useful to determine whether or not a random walk is trapped at a particular node, because we can end up in situations where the number of total centers is unchanging but charges are nonetheless tunneling between the centers. In the case shown in figure 14, the random walk probability is very close to the number of states as a function of N , with an expected value of N ≈ 11. In the plot above we have set α = 1 and β = 0, so this red curve is only representing how many compositions of the charge Q have one center, how many have two centers, etc. We can modify this number through the degeneracy function ω (which depends on α and β), which models how many additional ways there are to lift each of these states away from extremality. Additionally, we note from the right-hand side of figure 14 that the three largest charges tend to stay below Q i = 8, with relatively few excursions to very large charges., This matches the suppression of large-charge states in the degeneracy. We can therefore conclude for this set of parameters that the system is demonstrating ergodic behavior; there are no significant departures of the late-time probability distribution from the degeneracy. Of course, this is just a single example, meant to demonstrate how our random walk results are tabulated. In the next section, we will use many different random walk results to form broad conclusions about our model. Results Now that our methodology is clear, we will look at random walk results throughout our parameter space. We will consider the effects of varying the degeneracy parameters α, β introduced in (4.1), and the transition parameters γ, δ, λ, ω introduced in (4.2) and (4.3). We first want to investigate how the late-time behavior depends on the transition rate (4.2). We will therefore first fix α = 1 and β = 0 in order to set the degeneracy to be unity for all nodes. The new features of this model are the parameters λ and ω, which encode how much resistance the tunneled charge experiences from nearby centers. We will first focus on understanding these parameters by fixing γ = δ = 1, with Q = 20. A number of random walks for different values of λ and ω are shown in 15. For low λ, the random walks demonstrate mostly ergodic behavior, with the random walk probability matching the degeneracy very well. A caveat to this is that increasing ω seems to push the true peak slightly to the right of the degeneracy peak. Nonetheless, the behavior of the system is still mostly ergodic. As λ increases, though, the system begins to depart drastically from this ergodic behavior. For λ = 0.8, we can see that the random walk starts to favor microstates with small numbers of centers and larger charges. The plot of the three largest charges is still fluctuating, though, indicating that transitions between these few-center states still occur. At λ = 1, though, these transitions stop occurring. The random walk very quickly becomes locked in or trapped in a state with around N = 4 centers, and very few transitions from that state occur. This indicates that at λ = 1, the system enters a trapped phase with much more rigid and constant behavior than the ergodic phase. Crucially, this seems to be true for all three values of ω in figure 15. To investigate this trapped behavior further, we can determine N as a function of λ for a number of different random walks, as depicted in figure 16. In these, we can see that N is around half of the total charge, as expected for an ergodic phase, for λ < 1. There is a small increase in N as λ increases in this range, but only a small one (on the order of one center). As λ gets very close to 1, though, the system enters the trapped phase and N ≈ 4. This critical behavior is robust, in the sense that it is relatively unaffected by changing the values of ω and δ. Additionally, the plots shown have Q = 20, but the same features appear for larger values of Q as well. We can also consider varying the parameter δ (which was kept constant at δ = 1 above). It is most convenient to illustrate this with the heat plots shown in figure 17, which give N for a range of λ, δ, and ω values. From these plots, we can immediately see that the phase transition at λ = 1 is present for any values of δ and ω. δ has a very small effect on the results; larger values of δ push N to be slightly larger or smaller, depending on if λ > 0 or λ < 0, respectively. This is consistent with our results from section 3 where the similar parameter δ also only provided a small modulation to the late-time behavior. We have so far only investigated how the late-time behavior of model 2 depends on the transition rate (4.2). We now need to account for how this depends on the degeneracy (4.1). Since ω ∝ α N , increasing α will push the random walks to peak at higher values of N , while decreasing it pushes the random walk to peak at lower values of N . The degeneracy dependence on β is a little more complicted; to understand what it does, consider figure 18, where we plot the degeneracy of all microstates with total charge Q = 20 and with a particular number of centers N as a function of N . The degeneracy has an extremum at intermediate values of N ; when β < 0, this peak gets pushed lower and so the intermediate states become relatively disfavored, while for β > 0 the peak becomes greater and they become relatively more favored. β will therefore control the width of the peak in our random walk results, with narrow and wide peaks corresponding to β > 0 and β < 0, respectively. We considered these effects and studied random walk behavior for different values of α and β, but their only effect was to alter the random walk probability in smooth, continuous behavior similar to what we saw from degeneracy effects in section 3. That is, the parameters α and β can be smoothly tuned to shift the location and width of the degeneracy peak as desired. No matter what we set α and β to, though, it is the transition rate parameter λ that affects how closely the actual centrality matches the ergodic prediction. We will discuss the physics of the λ-dependence in more detail in the next two subsections. Phase Transition The most distinct feature in our results is an apparent phase transition, where the system goes from an ergodic phase with N ∼ Q/2 to a trapped phase with N ≈ 4 as soon as λ 1. Intuitively, it is straightforward to see that there should be these two types of phases. For \|λ\| 1, the transition rate are largely independent of Q L and Q R , so the transition rates between states are approximately uniform. This leads us to a distribution where N is related primarily to the degeneracy of the system. On the other hand, when λ 1, it becomes much harder for charge to tunnel off of centers if they are close to an area of large charge concentration, as depicted in figure 19. At large λ, these highly-charged areas act as charge sinks, as charge can easily tunnel onto the sink but is very unlikely to tunnel off of it. The long-time behavior of the system is to end up in microstates with very few total centers. Note that the larger λ is, the larger the asymmetry is of the transition rates between the initial and final states; this asymmetry could be interpreted physically as an additional irreversible relaxation process that happens immediately after the (reversible) tunneling process, leading to a total transition rate that is asymmetric between the initial and final states. What is surprising, though, is how sudden the transition from ergodic to trapped behavior is in parameter space. Instead of having a smooth, gentle transition from one phase to another, the transition is sharp and sudden. Moreover, the random walk numerics are stable in the cross-over regime, indicating that this is not simply a numerical artifact arising in our methodology. It would be interesting to investigate this feature further, although (as we have stressed before) the large size of our networks make analytic analyses difficult. One interesting thing to note about the trapped phase of our system is that the network is not locked into one particular microstate. Explicit random walk results (see e.g. figure 15) show that the number of centers and amount of charge on each center fluctuate, but much less than the random walk fluctuations in the ergodic phase. Statistical fluctuations are effectively restricted such that tunneling only occurs among the centers with N ≈ 3 − 4 rather than the whole network; we can effectively truncate our full network to just this subnetwork when analyzing dynamics in the trapped phase. The phase transition behavior we observe is present across all of parameter space in our model as long as ω is in the range 0.001 ≤ ω ≤ 0.6. For ω > 0.6, the system becomes stuck; the tunneling rate suppression due to adjacent charges becomes so large that the random walk state is trapped at whatever its initial microstate is. This should not be interpreted as a physical effect. Instead, it indicates that the spectrum of the transfer matrix is highly degenerate such that the network relaxation time is very large; as such, finite-time random walks cease to be a good approximation of the late-time behavior of our network. It seems likely that the eigenvector centrality will still give trapped behavior in this regime, although verifying this is computationally difficult. Smaller λ Behavior Another interesting feature of our results is that, as long as λ is below the critical value of the phase transition, the expected number of centers increases monotonically with λ. This effect is mild (see e.g. figure 16 where N increases by one or two on the range 0 ≤ λ ≤ λ c ), but it is nonetheless persistent when varying the other transition rate parameters. This indicates that there is some variance possible in N in the ergodic phase of the system; it can be tuned slightly away from Q/2 by small changes in the transition rate. It is important to note that the intuitive picture we used in section 4.2.1 to explain the phase transition predicts that N should decrease monotonically with λ; from that perspective, this observed (opposite) behavior at small λ is unexpected. In addition, since this feature is present only in the ergodic phase, it seems likely that one cannot come up with a physical justification for this effect on a truncated subspace of our network. This feature therefore serves as good example of how our network-theoretic approach can lead to emergent phenomena that only become apparent when we consider the dynamics of all states in the theory at once. Conclusions Our main result in model 2 is that there is an apparent phase transition in parameter space of the late-time behavior of the microstates. This phase transition is intimately related to the tunneling rates between the microstates, and occurs when the tunneling rate parameter λ hits the critical value λ c ≈ 1. For λ < λ c , the system is in an ergodic phase (similar to that of model 1): the degeneracy of the microstates is far more important than the interactions between them; the system is equally likely to be in any microstate at any given time; and the model parameters can be tuned to change late-time behavior in a smooth, continuous way. For λ > λ c , the system is in a trapped phase where the late-time behavior is completely dominated by microstates with very few centers and no excursions to other microstates are allowed. This trapped phase demonstrates a regime in parameter space where the details of the transition rates (and not the degeneracy) determine the late-time behavior of the network. We found no such phase in model 1; it is only by adding in more intricate details of charge interactions between centers that this phase appears. The physical interpretation of these results is as follows. In model 2, we have considered charged microstates whose centers are distributed along a line. When the electromagnetic interactions between the centers are weak, it is easy for charges to move between the centers, and so it is very easy for microstates to tunnel into one another. The system's dynamics are thus ergodic, and all microstates are equally likely to occur. However, if the electromagnetic interactions are sufficiently strong, the microstates with very few centers suddenly become bound states that are very unlikely to tunnel off charge. These microstates therefore become long-lived and metastable, breaking the ergodicity of the system and dominating the time evolution of the black hole microstate system. See also sec. 6 for a discussion relating this behavior to meta-stable black hole glassy physics. Model 3: The D1-DSystem Setup In model 3, we will model the dynamics of the D1-D5 system with N 1 D1-branes and N 5 D5-branes, as discussed in section 1.1. We fix N , where N = N 1 N 5 is the product of the number of D1-and D5-branes. A microstate in this model will be completely given by an unordered collection of integers {w 1 , w 2 , . . .} with i w i = N , i.e. each microstate corresponds to a partition of the integer N . Each integer w i in the partition can be thought of a string with winding number w i in the string gas picture, where we have in mind the picture where a ground state in the D1-D5 system can be seen as a gas of strings with total winding number N . (For more details, see section 1.1.) If there are N w strings with winding number w present in a given microstate, then the degeneracy Ω associated to each winding number is the number of ways to divide the strings into bosonic and fermionic modes (1.4), which we repeat here for clarity: As opposed to models 1 and 2, we will not imbue an extra degeneracy factor to microstates to account for the possible degeneracy associated to adding a small amount of non-extremality to the microstate; such a generalization could certainly be done in a future iteration of the model. Transitions are allowed between microstates where a long string with winding w splits into two smaller strings with windings w a , w b (with w = w a + w b ), as well as the reverse transition where two smaller strings with windings w a , w b combine into one larger string with winding w = w a + w b . We model the transition rate between these states as: Ω(N w ) = 8 l=0 8 l N w − l + 7 7 = 16 315 (N w ) 7 + 64 45 (N w ) 5 + 352 45 (N w ) 3 + 704 105 (N w ) .Γ({w 1 , w 2 , . . .} → {w 1 , w 2 , . . .}) = exp −γ w δ [min(w a , w b )] λ 1 [max(w a , w b )] λ 2 , (5.3) with tuneable parameters γ, δ, λ 1 , λ 2 . Note that this transition rate includes both the splitting transition (where w ∈ {w 1 , w 2 , . . .} and w a , w b ∈ {w 1 , w 2 , . . .}) and the combining transition (where w ∈ {w 1 , w 2 , . . .} and w a , w b ∈ {w 1 , w 2 , . . .}), and is thus manifestly symmetric between these two processes. We will encode the details of this model into a network where each node is an unordered set of winding numbers that total to N , and the edges represent the allowed splitting and combining transitions. One such example with N = 5 is shown in figure 20. The number of nodes in the network is simply the number of partitions of the integer N , which is known to asymptote to p(N ) ∼ 1 4N √ 3 exp π 2N/3 , for N 1 . (5.4) This exponential growth of nodes means that generating the whole network explicitly is computationally unfeasible for N 10. So, just as we did with model 2, we will perform dynamic random walks that generate nodes as needed until the random walk has converged to a steady state. The fraction of steps the random walk spends at each node will then serve as a numerical estimate for the late-time behavior of the system, as discussed in appendix A. Results In models 1 and 2, we concerned ourselves mainly with discussing the (late-time) expected value of the number of centers. An analogous quantity that we can consider in model 3 is the total number of distinct strings N s , given by a sum over winding numbers N s = w N w , (5.5) where we remember that w wN w = N is kept fixed. Additionally, in model 2 we investigated the evolution of the centers with the largest charges; here, we will analgously consider the evolution of the strings with the largest winding numbers. We can now perform random walks and make plots of individual random walk results for different values of the parameters δ, λ 1 , λ 2 in the transition rate (5.3). A sample of these random walk results are shown in figure 21. These random walks seem to suggest that the actual values of the parameters δ, λ 1 , λ 2 do not actually influence the resulting graph for N s very much -at most, the peak of N s can be shifted a slight amount, on the order of ∆N s ≈ 1 − 2. The (three) largest string sizes also do not seem to depend on the parameter values; they each stay bounded above by N/2. We have confirmed this independence of the random walk results on the parameters with a more thorough investigation, exploring the entire space of varying the parameters δ, λ 1 , λ 2 between −2 and 2. 12 We have also tried fixing different values of N and γ, but the random walk behavior is qualitatively the same as for the N = 20, γ = 1 case shown above. In other words, the functional form of the transition rate (5.3) does not seem to matter much for the dynamics of our model. In principle, we could have even chosen an entirely different functional form for our transition rate, and we would still see the same random walk peak centered right at the degeneracy peak. The speed of the convergence of the random walk does depend somewhat on the parameter values. For example, if we increase the parameter δ to be very large (e.g. δ 2), then it takes a long time for large strings to split into smaller strings. However, once they have split into smaller strings, they will not recombine again as it is much easier for the smaller strings to split into even smaller strings, and so on. The result is that, for such large values of δ, the random walks converge much slower, but they still will converge in the end to approximately the same graphs as those depicted in figure 21. A second phenomenon that we notice in the graphs of figure 21 is that the actual graph of the number of strings N s follows the degeneracy function (the red line) but does not exactly match it; rather, the graph's peak is taller and narrower than that of the degeneracy function. This is a non-trivial feature which is indicative that the structure of the network in this model is such that the states around the degeneracy peak are highly connected. In fact, these states are so highly connected that, although a random walker can go to any node in the network, they are much more likely to be at these states than at states with much lower or higher values of N s . This results in the network centrality looking like an amplified version of the degeneracy peak. We will call this behaviour the amplified phase, to distinguish it from what we called the ergodic phase in models 1 and 2, where the centrality graph exactly followed the degeneracy graph (see e.g. figure 14) and all states were approximately equally likely. In the amplified phase we are seeing here in model 3, individual states that sit at the degeneracy peak are actually more likely than other individual states (as opposed to the ergodic phase). Conclusion Our main result from this section is that model 3 exhibits late-time behavior that is functionally very different than that of model 1 or model 2. Not all microstates are equally probable, but nor are there any microstates in the full network that are completely irrelevant. Instead, our model gives rise to an "alignment" of sorts, where the most degenerate microstates are also the ones that have the most connected structure in terms of available transitions. This gives rise to the amplified behavior we saw in figure 21. We cannot simply restrict the full Hilbert space of states to these highly degenerate states, but nonetheless they are dominant in determining the dynamics of the system. These highly degenerate states are the ones in which the total number of strings is slightly above N/2, where N = N 1 N 5 is the product of the number of D1-and D5-branes. The "typical" state in this picture of the D1-D5 system is therefore a somewhat diffuse state with many strings that each have a small winding number. The bulk dual of the D1-D5 states are the Lunin-Mathur supertubes [13], which (from a six-dimensional perspective) have long but finite AdS 3 throats; the length of the throat is determined by the winding numbers of the component strings in the D1-D5 system. In particular, for states with many component strings that each have a small winding number, the length of the throat becomes very large. These microstate geometries are also the ones that contribute most of the entropy to the black hole. Our network results therefore tell us that the typical black hole microstates are in fact these entropically-favored geometries that look like black holes up until very close to the horizon scale. Another important feature of our results is that the structure of the network was important for understanding late-time behavior, but the values of the network edge weights turned out to be irrelevant. Intuitively, this tells us that the details of which D1-D5 system states can interact with one another are important, but the actual details of the relative strengths of different interactions are largely irrelevant. This behavior can be understood in the following way. The full D1-D5 gauge theory has a critical point along its RG flow, at which point it is accurately described by an SCFT [1,33,34,55]. Field theories with such conformal critical points are known to exhibit universality near these critical points, in the sense that their behavior is determined entirely by the details of the breaking of the conformal symmetry [56,57]. The string gas states we consider are BPS states at the conformal fixed point, but with some small amount of non-extremality added in order to add dynamics. Intuitively, then, we are probing the theory slightly away from the critical fixed point with relevant deformations. The universal behavior we see is consistent with this intuitive picture, and gives us an indication that our model is a good description of the D1-D5 system in this nearly-conformal regime. Discussion In this section, we discuss certain aspects of our models and their results. We discuss the possible interpretation of the evolution of microstates in our network as "shedding" angular momentum, and we note the link between the trapped phase of model 2 and "glassy" black hole physics. We also touch on various aspects and/or caveats of our analysis with respect to distinguishability of microstates. We end with some comments on future applications of our network theory techniques. Angular momentum. Physically, the evolution of the network should be thought of as successive tunneling steps in the evolution of the microstate. These tunneling steps are each associated with a change in the properties of the state, such as its energy, its angular momentum, etc. In particular, (as in model 2) if all of our microstates are microstates where all N centers are on a line, then typically microstates with larger N will have larger angular momentum than those with smaller N . So, for the trapped phase of model 2, it seems plausible that the microstates are driven to shed their angular momentum as they evolve over time until they reach a stable point at low angular momentum. This shedding of angular momentum is an irreversible relaxation process that is represented by the asymmetry between initial and final states of the transition rates in model 2. It would be interesting to account explicitly for angular momentum in the details of our models in order to investigate this more thoroughly. The interpretation of the evolution of model 2 as shedding angular momentum is very reminiscent of the discussion in [29]. 13 There, a classical instability of microstate geometries found in [58] was interpreted as an entropic transition driving (atypical) microstates with large angular momentum to (typical) microstates with smaller angular momentum (for which supergravity ceases to be a good approximation). It is interesting that the instability and evolution there is classical whereas the tunneling transitions we model here are intrinsically quantum. However, the flexible nature of the network model may imply that a very similar model to model 2 with similar evolutions and phases could be used to approximately describe the classical evolution of microstate geometries under this classical instability. It would be very interesting to pursue this further to understand the relation between the dynamics of model 2 and this classical instability evolution. Glassy black holes. In the trapped phase of model 2, we found that it was possible for random walks on our microstate network to get stuck in long-lived microstates with very few centers. This is very reminiscent of the viewpoint in previous work on glassy black hole physics [32,[59][60][61] (see also the "supergoop" or "string glasses" of [60]), where one can view a non-extremal multi-centered black hole microstate as a long-lived metastable state, much like glass is. Moreover, one can even find explicit examples of possible ergodicity breaking [60], which is very remniscent of the phase transition between ergodic and trapped behavior we found. The evolution of our network into these trapped states can be interpreted as the evolution of a microstate into local minima of the Hamiltonian that correspond to long-lived (but not absolutely stable) states. It would be interesting to apply the network techniques we have used here to study the evolution of these multi-particle glassy black hole systems in more detail. It would also be informative to understand the generality of the glassy trapped phase, as that is not immediately clear from our analysis. Model 2 contains glassy phases for particular regions of its parameter space; model 1 is not complex enough to allow for such a glassy phase; and model 3 is "too connected" to allow for such a phase. Glassy phases in network models can likely be related to the existence of sparsely connected communities (as discussed in section 4.2.1) and could possibly be studied using community detection algorithms on networks (see also section 2.3); it would be elucidating to analyze the precise conditions that microstate network models need to satisfy to admit such glassy phases. Caveat: N is not a semi-classical observable. In models 1 and 2, we largely focused on the quantity N , the number of centers in a given microstate. An important caveat to mention is that this is not a particularly good semi-classical observable, since the number of centers in a given microstate geometry is certainly not a locally measurable quantity. Strictly speaking, it is a global feature of the spacetime in the same sense that the presence of a horizon is a global feature that no local (or even finite) measurement can determine precisely. As such, our results for the expected value of N in the evolution of black hole microstates do not necessarily translate directly into statements about possible observations of such microstates. Similarly, the number of strings N s that we studied in model 3 is also not a local observable. Nevertheless, N and N s are interesting quantities in microstates; the network approach we have used is ideal to study them as we only need to input global features into the network models. Distinguishibility of microstates. Recently, Raju and Shrivastava have argued [62] that the distinguishibility of individual black hole microstates from the thermal average geometry (i.e. the black hole) is exponentially suppressed and hard to measure. (An important earlier work in a similar vein is [63]. See also [64][65][66][67] for other work on distinguishing microstates.) We do not address these arguments here as we do not directly discuss distinguishibility between microstates or their thermal average. However, we would like to emphasize that [62] assumes a statistical ensemble and thus a thermodynamic equilibrium (i.e. one without time evolution). As we have mentioned above, our results should be thought of more as understanding the time evolution of black hole microstates, in particular leaving open the possibility of glassy, non-equilibrium evolution. And, as we mentioned above, N is not a semi-classical observable, so the arguments of [62] do not obviously directly apply to N either. Other microstate models. In this paper, we have created models based on 5D multicentered Bena-Warner microstates and D1-D5 Lunin-Mathur supertube states. An obvious interesting expansion of the scope of these models would be to create and consider models based on the D1-D5-P "superstrata" microstate geometries of [10][11][12]. Other interesting, related systems include the LLM bubbling geometries [68] that are thought to be microstates of incipient black holes [69][70][71], and fuzzy D-brane geometries in the BFSS matrix model [72] that are thought to be microstates of Schwarzschild black holes in various dimensions [73][74][75][76][77]. A first step towards understanding their dynamics would be computing tunneling rates between microstates. Tunneling and meta-stable states in LLM geometries have been discussed in prior works [27,78], but for the other examples mentioned the tunneling rates between any geometries have not yet been studied. It would nonetheless be interesting to investigate these other models further. Another future direction of research would be to extend our network analysis to account for more general classes of Bena-Warner microstates. In model 2, we constructed a network of multi-centered microstate geometries where all charges are placed on a line; the states in this network model should be thought of as compositions of the total charge Q of the black hole, of which there are 2 Q−1 such compositions. For large Q, this implies an entropy of S model 2 ∼ Q, which is less than the scaling of the three-charge black hole entropy S BH ∼ Q 3/2 . This scaling is also less than the scaling of the total number of Bena-Warner multi-centered microstate geometries, which has been argued to be S Bena-Warner ∼ Q 5/4 [79]. These differences in entropy scaling are unsurprising, since our model only describes solutions where all centers are on a line instead of distributed over R 3 . It would be interesting to see if the late-time behavior we observed in model 2, especially the phase transition, persists when we look at tunneling transitions between these more general Bena-Warner geometries. Larger applicability of our methodology. Finally, we wish to reiterate that the network theory methods we used in this paper are very general and can be applied to a wide range of quantum mechanical systems. The networks we considered in this paper were constructed to model tunneling between smooth, multi-centered black hole microstates or between different states in the D1-D5 system. However, as shown in section 2.1, there is a very precise way in which networks can be used to model the evolution of generic quantum systems with connected states. Moreover, the main observable we looked at in our models was the late-time probability distribution, but we could easily broaden our scope and study other observables. As discussed in section 2.3, there is a rich literature of network theory tools that can be used to probe other dynamical features of these systems. We are optimistic that the methods presented in this work can be used to better understand tunneling dynamics in other areas of theoretical high energy physics and string theory. We are so far only aware of one such application in string theory, where networks are used to study dynamics in the string landscape [42], but we look forward to seeing more novel applications in the future. A Random Walks and Convergence In this section, we will explain the dynamical random walk method used in sections 4 and 5 to determine when the random walk has converged to a suitably steady-state that approximates the analytic eigenvector centrality well. A.1 Theoretical Bounds on Random Walk Convergence Suppose that our system is initially in the state p(0), the probability vector whose components p i (0) are the probability to be at node i at the start of the random walk. The transfer matrix T of the network generates stochastic time evolution of this state such that p(t) = p(0)T t . (A.1) That is, the components p i (t) of this probability vector are the probability that a random walker is at node i after t discrete time steps. We are interested in the late-time behavior of our network, though, and so we want the eigenvector centrality p ∞ that is a fixed point of the time evolution such that p ∞ = p ∞ T. This is often difficult to compute analytically, though, as it may require explictly inverting the transfer matrix, which can be unfeasible for networks with a large number of nodes. Instead, we want to approximate the analytic result by performing finite-time random walks. This requires a precise understanding of the relaxation time of the network, or the time it takes for p(t) to closely approximate p ∞ . Let {λ i } be the eigenvalues of the transfer matrix, ordered from largest to smallest magnitude such that \|λ 1 \| ≥ \|λ 2 \| ≥ . . . ≥ \|λ N \|. The corresponding orthonormal left eigenvectors {v i } of the transfer matrix form a complete basis, so we can decompose the initial state of the random walk as p(0) = N i=1 c i v i , (A.2) for some constants c i . The probability vector after t discrete time steps is then given by p(t) = N i=1 c i v i T t = N i=1 c i λ t i v i . (A.3) The transfer matrix is a stochastic matrix, and so we are guaranteed that all of its eigenvalues have magnitude \|λ i \| ≤ 1 and that the largest eigenvalue is λ 1 = 1. This means that v 1 is the eigenvector centrality, and so p(t) = c 1 p ∞ + N i=2 c i λ t i v i . (A.4) As long as \|λ 2 \| < 1, the contribution to this from the eigenvectors v i for i > 1 will become suppressed by successive powers of λ i . In the limit where t → ∞, we can ignore these other contributions entirely, and we are left with lim t→∞ p(t) = p ∞ . (A.5) The rate of convergence is determined by the magnitude of the eigenvalues of the transfer matrix. If they have small magnitude, then their contribution to the random walk state becomes suppressed very quickly. If they have magnitude close to one, though, then this convergence happens very slowly. It is therefore the parameter \|λ 2 \| that controls the convergence rate of this random walk. Convergence, roughly speaking, is achieved after a number of steps t such that \|λ 2 \| t 1 . (A.6) The closer \|λ 2 \| is to one, the longer it takes for the random walk to approximate the analytic eigenvector centrality result. The difference 1 − \|λ 2 \| is referred to as the spectral gap of the network; a large spectral gap ensures a fast relaxation time for the network. The spectral gap is intimately related to the community substructure of the network. If the network contains multiple highly-connected subnetworks that are minimally connected to one another, the transfer matrix would become approximately block diagonal, with each block corresponding to transition rates between nodes of one of the subnetworks. The transfer matrix would then have multiple eigenvalues with \|λ\| ≈ 1, each corresponding to a steady-state configuration that lives on only one of the subnetworks. Since the subnetworks are minimally connected to one another, it is very unlikely at each time step for a random walker to hop between the subnetworks, and thus the random walk ceases to be a good simulation of the global structure of the network. Intuitively, then, we expect the spectral gap to be smaller for networks with more of these disjoint communities. Unfortunately, actually computing eigenvalues for most networks generically requires inverting the transfer matrix and is therefore just as difficult as analytically computing the eigenvector centrality. In some cases, there are power methods that can be used to approximate these eigenvalues that are more efficient than inverting the entire matrix, but even these computations are unfeasible for the networks in model 2 and model 3, since the number of nodes of these networks is exponentially large. Instead, we can set bounds on the magnitude of λ 2 using results from spectral theory. One such tool for bounding this magnitude is the Cheeger constant h of the network [80]. For a balanced, directed network, this is given by h = min S h(S) , h(S) =       i∈S, j∈S A ij min i∈S d i , i∈S d i       , (A.7) where S is a subnetwork,S is its complement, and the Cheeger constant is minimized over all choices for S. Intuitively, the subnetwork S that minimizes h(S) will be one such that S andS are as sparsely connected as possible (minimizing the numerator of (A.7)) while being roughly even in size (maximizing the denominator). The Cheeger constant is related to the spectral gap of the theory by the Cheeger inequality [81]: h 2 2 < 1 − \|λ 2 \| ≤ 2h . (A.8) For networks that allow for very sparsely-connected bipartitions, h will be small, so \|λ 2 \| will be close to one, and the random walk will take a long time to converge. The Cheeger constant is much simpler to compute than λ 2 , since it doesn't require inverting large matrices, and since we can obtain an estimate for h just by using particular nice choices of S. To see this in action, let's first consider model 1. For the range of parameters where β ≤ 0, δ ≤ 0, and γ ≥ 0, the Cheeger constant corresponds to h(S) for the subnetwork S = {1, . . . , N max /2}. Importantly, on the relevant subset of parameter space γ = 1, −1 ≤ β ≤ 0, and −2 ≤ δ ≤ 0, we find the following numerical bounds on h: N max = 10 : h ≥ 0.071 , N max = 20 : h ≥ 0.034 , N max = 50 : h ≥ 0.013 . (A.9) That is, h is always positive on our parameter space, so the Cheeger inequality tells us that 1 − \|λ 2 \| > 0, and thus the random walk on this network is guaranteed to converge eventually. Additionally, the Cheeger inequality can be used to estimate how quickly the random walk converges, because it gives us the relation (1 − 2h) t ≤ \|λ 2 \| t < 1 − h 2 2 t . (A.10) This expression gives a lower-bound and an upper-bound on the number of steps required to ensure that \|λ 2 \| t 1. For example, in the case of N max = 10, if we want \|λ 2 \| t ≈ 0.01 as our convergence condition, we can plug (A.9) into (A.10) and compute that we would need somewhere between t ≈ 30 and t ≈ 1840 steps in order for the random walk to converge. We can do a similar analysis for model 2. For simplicity, we will look at the simple case where the degeneracies ω and the transition rates Γ are all set to one. For networks with Q ≤ 20, we find that the Cheeger constant corresponds to the subnetwork S that contains all nodes with N ≤ Q/2 − 1. This matches our intuitive notion that h(S) is minimized for a bipartition that splits the network into roughly two equal-size parts. The corresponding Cheeger constants for some values of Q are given below: These are all not too small, indicating that convergence is relatively easy. As expected, convergence becomes harder for larger Q simply because of the exponential growth of microstates with Q. Even for the Q = 20 network, though, we can use (A.10) to estimate that after t = 10, 000 steps, the finite-time probability approximates the late-time probability with an error of \|λ 2 \| t ≤ 7 × 10 −50 . Of course, this analysis was done in the special case where ω = Γ = 1 for simplicity. However, the results are not changed drastically by introducing parameter dependence. We can therefore trust that the random walks that we do in section 4 (see e.g. figure 15) are good approximations of the late-time behavior of model 2, since they are typically run for more than t = 10, 000 steps. We can also compute the Cheeger constant for model 3, although due to its similarities to model 2, the results are very similar. The upshot is that all of our models have good theoretical convergence rates, and and thus random walks (as long as they run for a sufficiently long time) will eventually converge to the analytic centrality. A.2 Dynamic Random Walk Method Now that we have established that the random walk probability p(t) on the networks considered in this paper will converge to the analytic late-time result p ∞ after a sufficiently large (but finite) number of steps, we need to determine how many steps to actually make the random walk go through. Instead of setting a hard cut-off on the number of steps, we instead use a dynamic random walk method that tests for convergence as the random walk progresses and stops when it has converged within some small error threshold. The details of this dynamical method are as follows. Let f i (t) be the fraction of steps spent in node i for a random walk that has run for t discrete time steps. If f i (t + 1) is significantly different from f i (t), we cannot expect the random walk results to be stable, so we cannot expect f i (t) to be close to the true probability p i (t). If the random walk has run for a large number of steps and this fraction is stable, then it will serve as a good approximation. Moreover, since p i (t) approaches p ∞,i at late times, we can conclude that f i (t) ≈ p ∞,i if enough time steps have been run for enough steps such that \|λ 2 \| t 1 and the random walk is stable. In models 2 and 3, we are primarily interested in the probability that the system is in a group of microstates with a particular value of N , where N is the number of centers in model 2 or the number of strings in model 3. So, we do not actually have to have strict convergence for each individual microstate fraction f i (t). Instead, we will define f (N, t) = i\|n(i)=N f i (t) , p ∞ (N ) = i\|n(i)=N p ∞,i , (A.12) as the random walk fraction and analytic probabilities, respectively, for the system to be in any state i with n(i) = N centers or strings, respectively. We will therefore only have to test for a more limited version of convergence, where f (N, t) is not significantly different from f (N, t + 1), in which case we conclude that the random walk serves as a good approximation for the analytic result, i.e. f (N, t) ≈ p ∞ (N ). Our results will look similar whether we test for convergence in N or in each individual node, but convergence in N is much faster to achieve than convergence for each individual node. To concretely test for this convergence, we will set a number of bins n b to put the data into such that each bin contains ∆t = t n b data points. The coarse-grained average of the random walk fraction f (N, t) within each bin is then given by f (N, t) bin n = 1 ∆t n∆t t=(n−1)∆t f (N, t) . (A.13) This coarse-grained fraction is a better way to study convergence, because all random walks will inherently have some small fluctuations over time as they rarely stay at the same node at any consecutive time steps. A random walk will have converged when all of these coarsegrained fractions are comparable. Of course, we should not necessarily include the first few bins, since those are the ones that are sensitive to the initial conditions of the random walk. Instead, we will mark bin n c as the one we start comparing from, and then we look at the last n b − n c + 1 bins. We will define the absolute error E abs (N, t) and the relative error vector E rel (N, t) of these bins to be as follows: E abs (N, t) = max n=nc,...,n b \| f (N, t) bin nc − f (N, t) bin n \| , E rel (N, t) = max n=nc,...,n b f (N, t) bin nc − f (N, t) bin n f (N, t) bin nc . (A.14) That is, the absolute error computes the maximum absolute change in f (N, t) among these bins, while the relative error computes the maximum fractional difference of f (N, t) among these bins 14 . These quantities serve as an estimate for how much the random walk results could be affected by running it for another ∆t steps. The random walk is deemed to have converged to a particular threshold when we find that for all allowed values of N . That is, the coarse-grained absolute and relative errors associated with f (N, t) have to be below a specified threshold for all values of N simultaneously before the random walk is ended. In sections 4 and 5, we tested our random walks for convergence by coarse-graining the random walk fractions into n b = 20 bins. We also set n c = 11 in order to compare the coarse-grained averages in the last ten bins. Additionally, the error thresholds we used were Σ abs = Σ err = 10 −4 . This means that if our random walk converged after t = 20, 000 steps, we would observe a change of at most one part in 10 −4 difference in our results if we ran the random walk for another ∆t = 1000 steps. Tightening the error threshold beyond these values did not end up changing our results significantly in any way. Moreover, increasing the number of coarse-grained bins made the random walk take longer to converge, but with the same results. The error parameters we used led to efficient, accurate results that converged well after the relaxation time of the network, which leads us to conclude that these choices are reasonable to use. B Explicit Black Hole Microstate Calculations In this appendix, we present some explicit calculations pertaining to black hole microstate geometries that are relevant for the setup of the network models (as presented in section 1.1). We first review the construction of the Bena-Warner five-dimensional microstate geometries in supergravity, including the bubble equations, and then we go on to show an explicit construction of microstate splitting. We end with a numerical estimation of tunneling rate parameters in model 2 using properties of a number of explicit microstate geometries. B.1 Supergravity Setup and Multi-center Solutions The solutions we will consider are supersymmetric solutions in 5D minimal supergravity coupled to two vector multiplets. It contains vectors A I (with corresponding field strengths F I ) and scalars y I (using the conventions of [2]): S 5 = 1 16πG 5 5 R 5 − Q IJ 5 dy I ∧ dy J − Q IJ 5 F I ∧ F J − 1 6 C IJK F I ∧ F J ∧ A K , (B.1) where I = 1, 2, 3. The vector multiplet kinetic matrix is Q IJ = 1 2 (y I ) −2 δ IJ (B.2) The scalars obey the restriction 1 6 C IJK y I y J y K = 1 , (B.3) with C IJK = \| IJK \| . (B.4) The metric, scalars and gauge fields of supersymmetric solutions with a timelike Killing vector have the form [82,83]: ds 2 5 = −(Z 1 Z 2 Z 3 ) −2/3 (dt + k) 2 + (Z 1 Z 2 Z 3 ) 1/3 ds 2 4 , (B.5) y I = (Z 1 Z 2 Z 3 ) 1/3 Z I , (B.6) A I = −Z −1 I (dt + k) + B I . (B.7) The forms k and B (I) and the warp factors Z I are supported on and only depend on the 4D Gibbons-Hawking (GH) base space, which has metric: ds 2 4 = V −1 (dψ + A) 2 + V ds 2 3 (R 3 ) , 3 dA = −dV, (B.8) with V a harmonic function on R 3 . 15 The solution is completely determined by 8 harmonic functions (V, K I , L I , M ) on R 3 [84,85] which enter the fields as: B I = V −1 K I (dψ + A) + ξ I , dξ I = − 3 dK I Z I = L I + 1 2 D IJK V −1 K J K K k = µ(dψ + A) + ω , µ = 1 6 V −2 C IJK K I K J K K + 1 2 V −1 K I L I + M , 3 dω = V dM − M dV + 1 2 (K I dL I − L I dK I ) . (B.9) If the harmonic functions have sources at N centers at coordinates r i in R 3 , then we have: V = N i=1 v i \| r − r i \| , M = m 0 + N i=1 m 0,i \| r − r i \| , (B.10) K I = N i=1 k I i \| r − r i \| , L I = 1 + N i=1 I,i \| r − r i \| . (B.11) The only free parameters are the KK monopole charges v i and dipole charges k I i ; smoothness of the solutions at the different centers r i fixes the sources of L I and M : I,i = − 1 2 C IJK k J i k K i v i , m i = 1 2 k 1 i k 2 i k 3 i q 2 i ∀i (no sum) . (B.12) Five-dimensional Minkowski asymptotics requires N i=1 v i = 1 and fixes the constants of the harmonic functions by V \| ∞ = K I \| ∞ = 0, L I \| ∞ = 1 and M \| ∞ = m 0 with m 0 = − 1 2 3 I=1 N i=1 k I i , . (B.13) The charges v i , k I i and the positions of the centers r i cannot be chosen arbitrarily: to ensure that the solution does not have closed timelike curves (CTCs), the so-called bubble equations must be satisfied for each center i [2]: j =i k 1 j v j − k 1 i v i k 2 j v j − k 2 i v i k 3 j v j − k 3 i v i v i v j r ij = −2 m 0 v i + 1 2 3 I=1 k I i , (B.14) where r ij ≡ \| r i − r j \| is the distance between centers i and j. The bubble equations give N − 1 independent constraints on the variables v i , k I i , r i ; the sum of the bubble equations vanishes when (B.13) holds. The physical, asymptotic charges are normalized as Q I ≡ 1 4π 2 Q IJ 5 F J = −2C IJK jk J jk K j v j ,k I j ≡ k I j − v j k k I k , (B.15) 15 5D solutions with a GH base have a natural interpretation upon KK reduction along the GH fibre ψ as 4D multi-center solutions. which gives asymptotically Z I = Q I /ρ 2 for the radius ρ in standard polar coordinates on a constant time slice at infinity. In 5D, there are also two angular momenta. A convenient parametrization for these is given by J R , J L with: J R = 4 3 C IJK ik I ik J ik K i v 2 i . (B.16) The expression for J L is a bit more involved. In the special case where all centers are on a line and ordered from 1 to N , the magnitude of J L is given by: J L = 4 3 C IJK 1≤i≤j≤N v i v j k I j v j − k I i v i k J j v j − k J i v i k K j v j − k K i v i . (B.17) B.2 Explicit Splitting of Multi-centered Microstates In this section, we want to construct an explicit example of a N -center solution where one (or a few) centers "split" into multiple centers, creating a N > N center solution, and where all asymptotic charges are kept fixed. This will serve as proof of principle that such a splitting of centers in a multicentered solution is at the very least a physical possibility and does not necessarily require changing the asymptotic charges. Note that [27] considers the formation of an N -centered solution by considering intermediate N -center solutions and the transitions between them. However, the transitions considered there do not necessarily keep the asymptotic charges fixed; in fact, the angular momentum for the two species of solutions considered in [27] differs for varying N . Along the way, we will also find expressions which quantify how much it is possible to separate the contributions of different centers (or small collections of centers) to the total asymptotic charges of the solution. Let us first develop a general framework that is useful to consider when "splitting" microstates of different numbers of centers. We will generically consider a multi-centered solution that consists of N + 2m centers. The N centers with i = 1, . . . , N are considered to be a "black hole blob" [86] with: N i=1 v i = +1, (B.18) and we define:Q I = −2C IJK N i=1k J ik K i v i ,k I i = k I i − v ik I 0 , k I 0 = N i=1 k I i ,(B.19) wherek I i and thusQ I is gauge invariant (due to (B.18)).Q I can be interpreted as the charge that the N center "blob" contributes to the total charge Q I in (B.15) of the complete N + 2m center solution. The angular momenta of the blob are: J R = 4 3 C IJK N i=1k I ik J ik K i v 2 i , (B.20) supertubes is not enough; there is also a cross-term C IJK d J 1 d K 2 present in the expression for the charge, which comes from the charge generated by the dipole-dipole interaction between the two supertubes. We also give the expressions for Q I , J R for m = 4 where the 3rd, resp. 4th supertube has identical charges to the 1st, resp. 2nd supertube: Q I =Q I + 2C IJK (d J 1 f K 1 + d J 2 f K 2 ) + C IJK (d J 1 d K 1 + d J 2 d K 2 ) + 4C IJK d J 1 d K 2 , (B.31) J R =Ĵ R + 2(d I 1 + d I 2 )Q I + 2(j R,1 + j R,2 ) + C IJK d I 1 d J 1 (d K 1 + 2f K 1 ) + C IJK d I 2 d J 2 (d K 2 + 2f K 2 ) + C IJK d I 1 d J 2 (4f K 1 + 4f K 2 + 6d K 1 + 6d K 2 ). (B.32) Now there are cross-terms between different kinds of supertubes as well as between the pairs of identical supertubes; care must be taken to identify the correct combinatorial factor that these cross-terms appear with. The m = 2 and m = 4 expressions given above in ( have not yet appeared elsewhere in the literature to the best of our knowledge. Now, to give a proof of principle that it is possible to "split" a multi-centered solution from N centers to a solution with N > N centers with exactly the same asymptotic charges, we will consider an explicit case of a solution with 7 centers (a blob of 3 centers and 2 identical supertubes) to 11 centers (a blob of 3 centers and 2 pairs of identical supertubes). Both the 7-and 11-center configurations have all centers on the z-axis and are Z 2 symmetric around z = 0, which implies J L = 0. We have checked that both solutions satisfy the bubble equations 17 and are manifestly free of CTCs everywhere. The pairs of centers (2,1) and (6,7) are the identical supertubes which we take to be the initial m = 2 supertubes; each of these will split into two new supertubes. The "black hole blob" is given by the middle three centers (3,4,5). The relevant parameters of the initial supertubes and the black hole blob are: This solution has exactly the same Q I , J R (andQ,Ĵ R , k 0 associated to the middle "blob" of centers (5,6,7)) as the 7-center one above, per construction. Each of the two (identical) supertubes from the 7-center solution have split into two identical pairs of two supertubes { (8,9), (10, 11)} and {(4, 3), (2, 1)}. The two supertubes (8,9) and (4, 3) have parameters d 1 , f 1 and the two supertubes (10, 11) and (2, 1) have parameters d 2 , f 2 , given by: B.3 Estimating parameters in Model 2 In this section, we will get a rough estimate of the tunneling rate parameters λ and ω in (4.2) of model 2 by fitting those parameters to the tunneling rate in a particular class of multi-centered solutions. The solutions we will consider have N centers on a line where each GH center has alternating charge ±1: The asymptotic charges in this background are all equal and are given by: Q I = −4 j v j (−v j Nk +k) 2 = 4k 2 (N 2 − 1), (B.40) sok is a parameter that is determined if we are given a fixed Q I , N . A graphical representation of this solution is given in fig. 22. We will only consider N = 4k + 3 for some integer k, so that we can put the middle center at the origin and retrieve a Z 2 -symmetric solution. The positions of all other centers are then completely determined by the bubble equations. The tunneling probability to tunnel a supertube carrying charge Q off of a center i to the adjacent center j was calculated in [27] to be: Γ ∼ exp (−B) , B ∼ Q r ij , (B.41) + − + + − + − + − + − + − + − middle blob dipoles dipoles r i+1,i+2 r i−1,i Figure 22: The black hole microstate of N centers on a line with vi = (−1) i−1 and k I i = −viNk +k; positive and negative values of v are drawn in blue and red, respectively. The centers organize themselves into dipoles, except the "middle blob" of three centers. The distance between each pair of centers that makes up a dipole has a tendency to be much smaller than the distance between the dipoles. The distance r d,avg (i) as defined in (B.42) is the average of the distances ri−1,i and ri+1,i+2 where r ij is the (coordinate) distance (in R 3 ) between the two centers i and j. 18 This immediately implies δ = 1 (since r ij does not depend on the supertube's charge Q). The distance r ij is determined by the interaction of the fluxes on the centers through the bubble equations and should therefore be related to the parameters λ and ω in the tunneling rate (4.2) of model 2. Let us try to find a rough estimate for these parameters by investigating how r ij changes according to how much charge is to the left and right of it. We will consider solutions with N = 15, 19,23,27,31,35,39,43,47,51,55,59,63,67 centers. Within these solutions we will compute r d,avg (i), the average distance between the dipole made out of centers i and i + 1 and its neighboring dipoles, as depicted in figure 22. That is, r d,avg (i) ≡ 1 2 (r i−1,i + r i+1,i+2 ) . (B.42) Since the i-th and (i + 1)-th centers form a dipole together, we are essentially considering the average of the distances between that dipole and the two neighboring dipoles. To make these computations tractable, we will just consider i = 3, 5, 7. In terms of the network parameters, this distance should be given by so that Q L is the sum over all dipole charges Q d to the left of center i; each successive dipole has an extra attenuating factor ω. Q R similarly involves a sum over contributions from all dipoles to the right of the dipole (i, i + 1), but it also includes the contribution from the middle blob of three centers with charge Q mid , giving: Q R (i) = mid(i)−1 k=0 Q d ω k + ω mid(i)   Q mid + N −3 4 k=1 Q d ω k   , (B.45) mid(i) = N − 3 4 − i − 1 2 + 1 (B.46) For the solutions we are considering, we have: Q d = −4Nk 2 , Q mid = (N 2 − 6N + 1)k 2 . (B.47) We can now equate r (est) d,avg (i) of (B.43) to r d,avg (i) of (B.42) by fitting the parameters a, λ, ω. In principle, we have 42 = 14 × 3 datapoints (14 different values for N and three for i). However, for a given ω, we delete any datapoints where Q L Q R < 0 as such points would not make sense. The best-fit (using a log − log fit) gives as parameter values (usinĝ k = 10 andã ≡ log(ak 4λ )): One should tread extremely careful when trying to attach actual physical meaning to these results. The model that we have used to determine λ and ω is very rough, and it is wholly too simple to capture the intricacies of the actual interplay due to the bubble equations between intercenter distances and the charges (as indeed the relatively high value for 1 − R 2 of the above fit indicates). Moreover, we have only considered a limited, very special class of multi-centered microstate solutions in determining the parameters. Nevertheless, our results seem to indicate that λ ≈ −0.18 is an approximation for the physical interactions of our multi-centered microstates; in particular, λ is negative. We also have indications that the most physically relevant value of ω is ω ≈ 0.37, which gives an indication of how much the effects of charges on the intercenter distance attenuate as the charge gets further from the centers considered. Figure 1 : 1Two sample formation paths to tunnel into a multi-centered microstate. Figure 2 : 2A schematic network of microstates, labelled by their numbers of centers, with many different tunneling transitions between states possible. Figure 3 : 3A sample of how a stochastic system can be represented with a network. Nodes correspond to states and directed edges correspond to allowed transitions; the edges are correspondingly weighted by the transition rate. Figure 4 : 4A network with a highly-connected subnetwork, indicated in red. Random walks on such networks typically do not require traversing the entire network. Figure 6 : 6The degeneracy ω(N ) versus N for a range of values of β, with Nmax = 20. Figure 7 : 7The eigenvector centrality for different values of β when δ = 0 and Nmax = 20. ) γ = 2, N max = 50 Figure 8 : 8Heat plots showing N for a range of β and δ values. No matter what values of γ and Nmax are chosen, the δ and β dependence remains roughly the same. Figure 9 : 9The transition rate Γ(N → N + 1) versus N for a range of values of δ, with γ = 1. Figure 10 : 10The eigenvector centrality for different values of δ when β = 0, γ = 1, and Nmax = 20. Figure 11 : 11Plots of N versus δ for different values of β, with Nmax = 20. {4} Figure 13 : {4}13The network in model 2 when the total charge is Q = 4. (Unlike depicted in this simplified network figure, note that the transitions between states are not always symmetric, see (4.2).) Figure 14 : 14A sample random walk done in model 2. On the left of each plot is the random walk probability to have a particular number of centers, with the fractional number of microstates with a particular number of centers plotted in red. On the right is a plot of the three largest charges present at each step in the random walk, shown in red, orange, and yellow. Figure 15 : 15Random walk results for different values of ω and λ; other parameters are fixed such that Q = 20, α = 1, β = 0, γ = 1, and δ = 1. Figure 16 : 16Plots of N versus λ for different values of ω, with Q = 20. Figure 17 : 17Heat maps of N as a function of λ and δ, for different values of ω, with α = 1, β = 0, γ = 1, and Q = 30. Figure 18 : 18The degeneracy ω(N, {Qi}) versus N for a range of values of β, with α = 1 and Q = 20. Figure 19 : 19A cartoon of how λ 1 affects tunneling such that typical microstates have only a few centers. degeneracy associated to the microstate is then the product of all such winding number degeneracies: ω({w 1 , w 2 , . . .}) Figure 20 : 20The network in model 3 when the total winding number is N = 5. Figure 21 : 21Random walk results for different values of δ, λ1, λ2. On the left of each plot are the probabilities: the random walk probability to be in a microstate with Ns strings is plotted in blue, while the fractional degeneracy of microstates is in red. On the right of each plot we display the three largest winding numbers of the strings in red, orange, and yellow in descending order, respectively, at each time step in the random walk. In all graphs, N = 20 and γ = 1. Q = 20 : h = 0.15 . (A.11) E abs (N, t) ≤ Σ abs and E rel (N, t) ≤ Σ rel , (A.15) B.26)-(B.28) and (B.29)-(B.30) are generalizations of the m = 1 expressions of [86] and associated to this solution are: Q I = 19200, J R = 5.376 × 10 6 . (B.33) −40,Q I = 3200,Ĵ R = 3.84 × 10 5 . (B.34) choose N odd so that i v i = +1. The three k I i charges for each center are all equal and given by:k I i = −v i Nk +k, (B.38)so that i k I i = 0 and the physical flux between two centers isΠ I ij = (v j − v i )k. (B.39) avg (i) = a [Q L (i)Q R (i)] λ , between the fitted network theory values and the explicit microstate values is depicted in figure 23. For this log − log fit, we have 1 − R 2 ≈ 0.24. The fitted network theory results (red) compared to the actual data points (blue). avg (i) − r d,avg (i)\|/r d,avg (i) between the fitted points and data points. Figure 23 : 23Comparison of the data points and the fit. Table 1 : 1Summary of our main results for the late-time behavior of each microstate model. This work was supported by the U.S. Department of Energy under grant DE-SC0007859. DRM is supported by the ERC Starting Grant 679278 Emergent-BH. AMC is supported in part by the KU Leuven C1 grant ZKD1118 C16/16/005, the National Science Foundation of Belgium (FWO) grant G.001.12 Odysseus, and by the European Research Council grant no. ERC-2013-CoG 616732 HoloQosmos. Specifically, β = 3/2 and β = 0.93 for the two types of microstates considered in[27]. For relevant overviews and discussions of the D1-D5 CFT, see e.g.[1,33,34]. For a good review of random walks on networks, see e.g.[30,46]. We have also considered an exponential degeneracy function of the form ω(N ) = exp γ N β for γ = ±1 and β ∈ (−1, 1). As we discuss in section 3.3, this functional form of the degeneracy function or (3.1) gives the same qualitative results (when β, β have the same sign). We only plot negative values of β in the heat plots; for negative values of β, the (blue) trend of the heat plots offig. 8simply continues on to the left. The fact that each center contributes a well-defined, separatable amount to the total asymptotic charge is necessarily an oversimplifying assumption of our model that is not quite correct in an actual Bena-Warner multi-centered solution. For example, in appendix B.2, we derive the formulae (B.26)-(B.28) and (B.29)-(B.30); these are explicit examples showing that the contribution to the total asymptotic charges of e.g. individual supertubes cannot be separated entirely -there are always "cross-terms" between different supertubes in the expressions for the asymptotic charges. Note that also the assumption that all centers have positive charge is an oversimplification; centers are allowed to have negative charge in actual multicentered solutions. In fact, especially in the non-scaling solution family of[27], the asymptotic angular momentum is not constant for microstates of different N . Beyond this range, the random walk runs into numerical convergence problems. Nonetheless, the results that we have looked at beyond this range demonstrate the exact same parameter-independence. We thank A. Puhm for bringing this to our attention. Note that we want to understand both types of errors, because the absolute and relative errors become more important for larger and smaller values of f (N, t), respectively. It is also straightforward to obtain expressions for JL if the centers are all on a line (also in the m = 4 case), but we will not need those expressions here. In the 7-and 11-center solutions below, all quantities are given only to within a given precision; the bubble equations were solved to a much greater precision than given. In fact, in[27], it was derived that B ∼ \|d\| rij where d is the dipole of the supertube. This is related to the charge carried by the supertube as d ∼ Qk −1 . We are keepingk fixed which implies (B.41). AcknowledgmentsWe thank Iosif Bena, Zachary Charles, Emil Martinec, Ruben Monten, Andrea Puhm, and Bert Vercnocke for useful discussions; we are particularly grateful to Andrea Puhm and Bert Vercnocke for detailed comments on an early draft version of this paper. We also especially wish to thank John K. Golden for collaboration in the early stages of this project. We thank Xiao-yue Sun and his collaborators for remarks on v1 of this paper.and if the centers are all on a line:The 2m other centers are divided into m "supertube pairs". The k-th supertube consists of the centers numbered c (k)It will be useful to define the (gauge-invariant) quantities d I k , f I k for each supertube:We also define:If there is only one supertube, m = 1, the charges Q I , J R , J L (where the latter is only valid if the centers are on a line, with the supertube to the right of the blob) can be written as:These expressions were first given in[86]. 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[ "The effect of oscillator and dipole-dipole interaction on multiple optomechanically induced transparency in cavity optomechanical system OPEN", "The effect of oscillator and dipole-dipole interaction on multiple optomechanically induced transparency in cavity optomechanical system OPEN" ]	[ "Jin-Lou Ma \nInstitute of Theoretical Physics\nLanzhou University\n730000LanzhouChina\n", "Lei Tan [email protected] \nInstitute of Theoretical Physics\nLanzhou University\n730000LanzhouChina\n", "Qing Li \nInstitute of Theoretical Physics\nLanzhou University\n730000LanzhouChina\n", "Huai-Qiang Gu \nSchool of Nuclear Science and Technology\nLanzhou University\n730000LanzhouChina\n", "Wu-Ming Liu \nBeijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n" ]	[ "Institute of Theoretical Physics\nLanzhou University\n730000LanzhouChina", "Institute of Theoretical Physics\nLanzhou University\n730000LanzhouChina", "Institute of Theoretical Physics\nLanzhou University\n730000LanzhouChina", "School of Nuclear Science and Technology\nLanzhou University\n730000LanzhouChina", "Beijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina" ]	[ "ScieNtific RepoRTs \|" ]	We theoretically investigate the optomechanically induced transparency (OMIT) phenomenon in a Ncavity optomechanical system doped with a pair of Rydberg atoms with the presence of a strong control field and a weak probe field applied to the Nth cavity. It is found that 2N − 1 (N < 10) numbers of OMIT windows can be observed in the output field when N cavities couple with N mechanical oscillators and the mechanical oscillators coupled with different even-or odd-labelled cavities can lead to diverse effects on OMIT. Furthermore, the ATS effect appears with the increase of the effective optomechanical coupling rate. On the other hand, two additional transparent windows (extra resonances) occur, when two Rydberg atoms are coupled with the cavity field. With DDI strength increasing, the extra resonances move to the far off-resonant regime but the left one moves slowly than the right one due to the positive detuning effect of DDI. During this process, Fano resonance also emerges in the absorption profile of output field.In atomic systems, electromagnetically induced transparency (EIT) 1-3 is induced by quantum interference effects or Fano-interactions 4 due to the coherently driving atomic wavepacket with an external control laser field. The OMIT, a phenomenon analogous to the EIT, was predicted theoretically firstly 5,6 and then verified experimentally 7,8 in a cavity optomechanical system which is caused by the destructive quantum interference between different pathways of the internal fields. More recently, the study of OMIT has attracted much attention. For instance, the single-photon routers 9 , the ultraslow light propagation 10 , the quantum ground state cooling 11 , the precision measurement 12 , the Brillouin scattering induced transparency and non-reciprocal light storage 13,14 , the optomechanically induced amplifcation 15 , the effective mass sensing 16 , control of photon propagation in lossless media 17 , optomechanically induced stochastic resonance 18 and chaos transfer and the parity-time-symmetric microresonators 19 . In addition, tunable EIT and absorption 20 , polariton states 21 and transition from blockade to transparency 22 in a circuit-QED system have also been studied. On the other hand, the studies on the OMIT have been extended to double-and multi-optomechanically induced transparency 23 by integrating more optical or mechanical modes. It has been reported that multiple OMIT windows may occur in the atomic-media assisted optomechanical system 24-26 , multi-resonators optomechanical system 27 , optomechanical system with N membranes 28 , two coupled optomechanical systems 29 and the multi-cavity optomechanical system 30 . In particular, achieving multi-OMIT phenomenon shows many practical applications for the multi-channel optical communication and quantum information processing, which motivate the further investigation on such OMIT.Currently, a hybrid cavity optomechanical system containing atoms has attracted much attention. The additional control of atomic freedom can lead to rich physics resulted from the enhanced nonlinearities and the strengthened coupling strength, which can also provide an coherent optical controlled method to change the width of the transparency window 25,26,31 , multistability of OMIT 32 and switch from single to double and multiple	10.1038/s41598-018-32506-y	null	52,824,244	1803.02701	96dddec6f20eba7e02b947979dbc46203233c3a4	The effect of oscillator and dipole-dipole interaction on multiple optomechanically induced transparency in cavity optomechanical system OPEN 2018 Jin-Lou Ma Institute of Theoretical Physics Lanzhou University 730000LanzhouChina Lei Tan [email protected] Institute of Theoretical Physics Lanzhou University 730000LanzhouChina Qing Li Institute of Theoretical Physics Lanzhou University 730000LanzhouChina Huai-Qiang Gu School of Nuclear Science and Technology Lanzhou University 730000LanzhouChina Wu-Ming Liu Beijing National Laboratory for Condensed Matter Physics Institute of Physics Chinese Academy of Sciences 100190BeijingChina The effect of oscillator and dipole-dipole interaction on multiple optomechanically induced transparency in cavity optomechanical system OPEN ScieNtific RepoRTs \| 814367201810.1038/s41598-018-32506-yReceived: 3 May 2018 Accepted: 5 September 20181 Correspondence and requests for materials should be addressed to L.T. ( We theoretically investigate the optomechanically induced transparency (OMIT) phenomenon in a Ncavity optomechanical system doped with a pair of Rydberg atoms with the presence of a strong control field and a weak probe field applied to the Nth cavity. It is found that 2N − 1 (N < 10) numbers of OMIT windows can be observed in the output field when N cavities couple with N mechanical oscillators and the mechanical oscillators coupled with different even-or odd-labelled cavities can lead to diverse effects on OMIT. Furthermore, the ATS effect appears with the increase of the effective optomechanical coupling rate. On the other hand, two additional transparent windows (extra resonances) occur, when two Rydberg atoms are coupled with the cavity field. With DDI strength increasing, the extra resonances move to the far off-resonant regime but the left one moves slowly than the right one due to the positive detuning effect of DDI. During this process, Fano resonance also emerges in the absorption profile of output field.In atomic systems, electromagnetically induced transparency (EIT) 1-3 is induced by quantum interference effects or Fano-interactions 4 due to the coherently driving atomic wavepacket with an external control laser field. The OMIT, a phenomenon analogous to the EIT, was predicted theoretically firstly 5,6 and then verified experimentally 7,8 in a cavity optomechanical system which is caused by the destructive quantum interference between different pathways of the internal fields. More recently, the study of OMIT has attracted much attention. For instance, the single-photon routers 9 , the ultraslow light propagation 10 , the quantum ground state cooling 11 , the precision measurement 12 , the Brillouin scattering induced transparency and non-reciprocal light storage 13,14 , the optomechanically induced amplifcation 15 , the effective mass sensing 16 , control of photon propagation in lossless media 17 , optomechanically induced stochastic resonance 18 and chaos transfer and the parity-time-symmetric microresonators 19 . In addition, tunable EIT and absorption 20 , polariton states 21 and transition from blockade to transparency 22 in a circuit-QED system have also been studied. On the other hand, the studies on the OMIT have been extended to double-and multi-optomechanically induced transparency 23 by integrating more optical or mechanical modes. It has been reported that multiple OMIT windows may occur in the atomic-media assisted optomechanical system 24-26 , multi-resonators optomechanical system 27 , optomechanical system with N membranes 28 , two coupled optomechanical systems 29 and the multi-cavity optomechanical system 30 . In particular, achieving multi-OMIT phenomenon shows many practical applications for the multi-channel optical communication and quantum information processing, which motivate the further investigation on such OMIT.Currently, a hybrid cavity optomechanical system containing atoms has attracted much attention. The additional control of atomic freedom can lead to rich physics resulted from the enhanced nonlinearities and the strengthened coupling strength, which can also provide an coherent optical controlled method to change the width of the transparency window 25,26,31 , multistability of OMIT 32 and switch from single to double and multiple Results Theoretical model and Hamiltonian. The 1D MCOS under consideration is shown in Fig. 1. The Nth cavity of the cavity optomechanical arrays is coherently driven by a strong control field of frequency ω c and a weak probe laser field of frequency ω p . N optomechanical cavities are labelled as 1, 2, …, N. The frequencies of jth cavity and jth mechanical oscillator are denoted by ω j and ω mj , respectively. The coupling strength between jth cavity and jth mechanical oscillator is g mj , and g n is the hopping rate between nth and (n + 1)th cavities ≠ n N ( ) . In addition, a pair of DDI ladder-type three level Rydberg atoms are assisted in the ith cavity. The Rydberg atoms of our system may chose Cesium (Cs) atoms, the fine-structure states \|6S 1/2 , F = 4〉 and \|6P 3/2 , F′ = 5〉 can be regarded as the ground state \|g〉 and the intermediate state \|e〉, respectively, while the correspond Rydberg state \|r〉 is assumed as 70 S 1/2 41 . As for the first Rydberg atom, the frequency of control field ω c is coupled to the \| 〉 ↔ \| 〉 e r transition with a Rabi frequency Ω and a frequency detuning Δ r . The ith cavity field drives the \| 〉 ↔ \| 〉 g e transition with strength g and the frequency detuning Δ e . In brief, the second Rydberg atom is assumed to be excited in the Rydberg state The pair of ladder-type three-level Rydberg atoms interact with each other and one of Rydberg atoms is excited in the Rydberg state during the process of interaction. g and Δ e are the coupling strength and the frequency detuning of the transition \| 〉 ↔ \| 〉 g e, respectively. Ω and Δ r are the Rabi frequency and the frequency detuning of the transition \| 〉 ↔ \| 〉 e r, respectively. In addition, V(R) is the DDI strength between two Rydberg atoms. ScieNtific RepoRTs \| (2018) 8:14367 \| DOI: 10.1038/s41598-018-32506-y and coupled with the first Rydberg atom by DDI in the ith cavity (1 ≤ i ≤ N) due to the long lifetime (τ ≥ 100 μs) of the Rydberg state. As explained in refs [42][43][44] , this configuration has an experimental feasibility when the radius of the blockade is smaller than the interatomic distance of a pair of Rydberg atoms, then they can be excited to the Rydberg state simultaneously and their interactions are utilized via van-der-Waals type of DDI. The total Hamiltonian H of the hybrid cavity optomechanical system in the rotating-wave frame can be written as = + + + + H H H H H H ,(1)c m a i n i nt where the first four terms describe the Hamitonians of the optical cavity, the mechanical oscillator, the two Rydberg atoms and the input fields, with the expressions as following ∑ ∑ ω σ σ ω σ ε ε = Δ = = Δ + Δ + Δ + = − + − . = = − Δ Δ † † † † H cc H bb H H i c c i c e c e ,, ( ) ,( ) ( ) (2) c j N j j j m j N mj j j a eee e r rr rg rr in c N N p N i t N i t 1 1 (1) ( 1)(2) The optical modes are described as an annihilation (creation) operator c j ( † c j ) of the jth cavity field, and † b j (b j ) is the creation (annihilation) operator of the jth mechanical resonator. Δ j = ω j − ω c is the detuning of the jth cavity field from the control field, and Δ = ω p − ω c represents the detuning between the probe field and the control field. Δ e = ω eg − ω j , Δ r = ω re − ω p , and ω μν represents the frequency of the atomic transition between the level \|μ〉 and level \|ν〉 (μ, ν = g, e, r). σ µ ν ≡ \| 〉 〈 \| µν k kk ( ) is the projection (μ = ν) or transition (μ ≠ ν) operator of the kth (k = 1, 2) Rydberg atom. Moreover, the Hamiltonian of the input fields includes the Hamiltonian of the control field and probe field. ε c is the control field amplitude and ε p is the probe field amplitude. The last term of Eq. (1) describes the system's interaction Hamiltonian, ∑ ∑ σ σ σ σ = + − + + Ω + + . + . = − + + = † † † † H g c c c c g c c b b gc H c V R ( ) ( ) ( ) ( ) ( )(3) int n N n n n n n j N mj j j j j er i eg r r rr 1 1 1 1 1 (1) ( 1) (1) (2) In Eq. (3), the first term corresponds to the hopping between the two adjacent cavities and g n is the intercavity tunneling strength. The second term describes the interaction between the jth cavity and the mechanical oscillator via the radiation pressure and g mj is the coupling strength. One of the Rydberg atoms interacted with the control field and ith cavity field is listed in the third term, respectively. V(R) is the DDI strength between two Rydberg atoms which is described as the last term, and R is the distance between two Rydberg atoms which can be controlled at different ranges by the separate optical traps 43 . The dynamical equation. The Heisenberg-Langevin equatons for the operators can be obtained based on the Hamiltonian (1). Using the the factorization assumption (mean field approximation), viz, 〈QC〉 = 〈Q〉 〈C〉 5,45 , the equations of the mean value of the operators can be given by For two Rydberg atoms trapped in the ith cavity case If one puts the Rydberg atoms into the Nth cavity, Eq. (5) should be replaced by κ ε ε κ κ γ ω σ γ σ σ σ σ σ γ σ σ σ σ γ σ σ σ σ 〈 〉 = − + Δ 〈 〉 − 〈 〉 + + + 〈 〉 〈 〉 + 〈 〉 〈 〉 = − + Δ 〈 〉 − 〈 〉 + 〈 〉 + 〈 〉 〈 〉 + 〈 〉 ≠ 〈 〉 = − + Δ 〈 〉 − 〈 〉 + 〈 〉 〈 〉 + 〈 〉 〈 〉 = − + 〈 〉 + \|〈 〉\| 〈 〉 = − + Δ 〈 〉 + 〈 〉 − 〈 〉 〈 〉 − Ω〈 〉 〈 〉 = − + + Δ 〈 〉 + 〈 〉〈 〉 − Ω〈 〉 〈 〉 = − + Δ + − Δ 〈 〉 + 〈 〉〈 〉 + Ω 〈 〉 − 〈 〉 ∼ ∼ ∼ − − − Δ − − + i i i S i ig c i () ( ) , ( ) ( ) ( ) , 1, , , ( ) ( ) , ( ) , ( ) ( ) , ( ) , ( ) ( ) , (4)κ σ 〈 〉 = − + Δ 〈 〉 − 〈 〉 + 〈 〉 − 〈 〉 + 〈 〉 〈 〉 + 〈 〉 ≠ . ∼ − − +  † c i c i g c g c ig ig c b b i N ( ) ( ) ( ) , 1,(5)κ σ 〈 〉 = − + Δ 〈 〉 − 〈 〉 + 〈 〉 〈 〉 + 〈 〉 − 〈 〉. ∼  † c i c ig c ig c b b ig ( ) ( )(6)κ ε ε σ 〈 〉 = − + Δ 〈 〉 − 〈 〉 + + − 〈 〉 + 〈 〉 〈 〉 + 〈 〉 ∼ − − − Δ  † c i c ig c e ig ig c b b ( ) ( ) ,(7)N N N N N N c p i t ge mN N N N 1 1 where κ j and γ mj are introduced phenomenologically to denote the dissipation of the jth cavity, and the decay rate of the jth mechanical oscillator, respectively. S = V(R) with σ = 1 rr (2) due to reason that the second Rydberg atom are assumed to be excited to the Rydberg state during the interaction process with the first Rydberg atom. γ μν (μ, ν = g, e, r) is the decay rate of transition between the level \|μ〉 and the level \|ν〉. In addition, λ ∆ = ∆ − ∼ g j j mj j . The general form of λ j will be given in the following. In order to obtain the steady-state solutions, which are exacted for the control field in the parameter ε c and corrected to the first order in the parameter ε p of the probe field. As the probe field is much weaker than the control field, then the average value of the operator O can be approximately written by using the ansatz 46 δ 〈 〉 = + = + + . − − Δ + Δ O O O t O O e Oe ( ) (8) i t i t where O describes the steady-state value of the operator O governed by the control field, but δO(t) is proportional to the weak probing field, which gives rise to the Stokes scattering and the anti-Stokes scattering of light from the strong control field. Subsequently, substituting Eq. (8) into Eqs. (4-7), one can obtain the steady-state solutions of the Heisenberg-Langevin equations. Because O is independent of time, and δO(t) of the same order as ε p depends on the time but remains much smaller than O, one can separate the equations into two parts. One part is irrelevant of time and the other one is related to the time. Assuming that the cavity optomechanical system 47-51 evolves in the resolved sideband regime, e.g., κ ω  j m j , then the Stokes part, the low sidebands and off-resonant one, can be ignored i.e., O + ≈ 0 in Eq. (8), only the anti-Stokes scattering survives in the hybrid system. Thus, all elements of O − can be obtained as follows using the above ansatz, κ ε κ κ σ γ σ σ σ σ γ σ σ σ σ γ σ σ σ σ σ κ γ = − − − + + = − − − + + ≠ = − − − + + − ≠ = − − + − − Ω = − − + + − Ω = − − + + + Ω − = − − − + = − − + − − − − − Δ − − − − − + − − − − − − + − − − − − − − − − − − − − − − − − − ⁎ ix c i g c e i G b ix c ig c g c iG b n i N ix c ig c g c iG b ig i N ix ig c i ix ig c c i ix ig c c i ix c ig c iG b ix b iG c 0 ( ) , 0 ( ) ( ) , 1, , , 0 ( ) ( ) , 1, , 0 ( ) ( ) , 0 ( ) ( ) , 0 ( ) ( ) ( ), 0 ( ) , 0 ( ) ,(9)γ ω ≡ + = \| \| + . ⁎ b b g c 2 (10) j j j mj mj j mj mj 2 2 2 The output field. The response of the system can be detected by the output field at the probe frequency, which can be expressed as follows via the standard input-output theory of the cavity 52 , ε ε ε κ + + = 〈 〉. − Δ − Δ e e c 2 (11) out p i t p i t c N N , Therefore, one can express the total output field as ε ε ε κ ε χ χ = + = = + . −  c i 1 2 (12) T out p p N N p p p , , Here, χ p = Re(ε T ) and χ ε =  Im( ) p T denote the in-phase and out-of-phase quadratures of the output field associated with the absorption and dispersion, respectively. The OMIT is the phenomenon of the simultaneously vanishing absorption and dispersion. These two quadratures of the output field can be measured via the homodyne technique 52 . Using Eq. (9), − c N , can be easily obtained, then the expression of the output field ε T is given in a constructive form, ScieNtific RepoRTs \| (2018) 8:14367 \| DOI:10.1038/s41598-018-32506-y ε κ κ = = + − + − − − + + − +   c B 2 2 ,(13)T NN N N g B , N N g N B i A g i B g B 1 2 1 2 2 1 2 2 1 2 1 where κ = − + = … γ \| \| − B ix j N ( 1, , ) j j j G ix mj mj j 2 . In the above equation, the first line of the denominator describes two cavities with decay rates κ N and κ N−1 are coupled through the coupling strength g N−1 . Second line of the denominator describes the interaction of two cavities with decay rates κ N−1 and κ N−2 and the coupling strength is g N−2 and so on. It is obvious that each line of the denominator contains an interaction term denoted by an effective coupling G mj between the mechanical oscillator and the cavity. Analytically, we note that when G mj = 0, the mechanical oscillator is not coupled with jth cavity. Moreover, the extra term A in the B i line represents the interaction of the cavity field with the pair of Rydberg atoms including DDI, and its general form is shown in Eq. (14) in the following with G gc e i is the effective coupling strength between the Rydberg atom and the cavity field. Certainly, when one traps the atoms in the first cavity, this term will appear in the last line. If Rydberg atoms are localized in the Nth cavity, it will emerge in the first line of the denominator. Q = (γ gr + iΔ r + iS) (γ ge + iΔ e ) + Ω 2 , γ = Δ + − Δ + + γ γ γ + Δ + Δ + +Δ +Ω P i S ( ) r e er Ge i i i S i ( ) ( ) ( ) = . γ σ σ σ σ γ γ          − +      +     + − − + − −      − +      + Ω γ γ − Ω Ω − A(14) g i x ix ix From Eq. (14), it can be found that the output field depends on c j of the jth cavity and the population σ gg σ ( ) rr of the ground (Rydberg) state, which can be determined by solving Eq. (9) for all O. Note that there are four kinds of direct interactions in the system: the coupling between the adjacent cavities, the interaction between the Without Rydberg atoms. In this section, we first focus on the multiple OMIT phenomenon emerged due to the interaction between the cavity field and the mechanical oscillators without the Rydberg atoms. The parameters are ω mN /g mj = 20, γ mN /g mj = 0.001, κ N−1 /g mj = 0.002, κ N /g mj = 2, and we assume = = G g c / 1 mN mj j . The optomechanical coupling parameter g mj = 1 kHz is based on the realistic cavity optomechanical system 7 . For simplicity, the following absorption analysis of the output field are restricted to a hybrid system with four cavities. The generalization to a large number of cavities case can be made according to the same method mentioned based on Eqs. (9)(10)(11)(12)(13)(14). ( 2 1) (2 1) Firstly, Fig. 2 illustrates the absorption Re(ε T ) of the output field as a function of x/κ N for four cavities. In detail, Fig. 2(a) describes only one mechanical oscillator coupled to the first cavity. The mechanical oscillators are coupled to the first and second cavities are shown in Fig. 2(b). Figure 2(c) corresponds to three mechanical oscillators coupled to cavity 1, 2 and 3, respectively. Figure 2(d) depicts four mechanical oscillators coupled to four cavities. The dips of the absorption line correspond to the transparency windows of the output field. From Fig. 2, it can be found that the number of transparency windows adds one with the increase of the mechanical oscillator in turn, which is determined by the infinity denominator of Eq. (13) corresponding to the appearance of the coupling parameters g N−1 and G mN in the denominators. When the hybrid system has N cavities coupled with N mechanical oscillators one by one without considering the effects of the outside environment, the sum of transparency windows adds to 2N − 1. Thus, MCOS becomes transparent to the probing field at 2N − 1 different frequencies, which are the destructive interferences between the input probing field and the anti-Stokes fields generated by the interactions of the coupling cavity field within the multiple cavities and the interactions between the coupling cavity field and the mechanical oscillators. However, when N becomes large and each cavity couples with its bath, numerical results show that the multiple transparency windows of this system become more and more opaque. Therefore, what we are concerned only the small (N < 10) system in the realistic experiment. The origin of the multiple OMIT windows can be explained by the quantum interference effects between different energy level pathways, and the energy level configurations of the hybrid system consisted of N cavities coupled with N mechanical oscillators are presented in Fig. 3. The excited pathway of the probe field is quantum interfering with different coupling pathways G mj (j = 1, …, N) of the control field and the tunneling pathways g i (i = 1, …, N). Therefore, the sum of the quantum interference pathways is 2N − 1 for N cavities and N mechanical oscillators. In addition, those pathways of the destructive quantum interference are formed via the optomechanical interaction and the tunneling, which lead to 2N − 1 transparency frequencies of the output field under the condition of ε T ≈ 0 at extremum points. To further explore the characteristics of the OMIT arising from the interaction of the mechanical oscillators, we plot the absorption Re(ε T ) of the output field as a function of x/κ N for one and two coupled oscillators cases. The case without the mechanical oscillator coupling is also shown for comparison in Fig. 4. Due to the destructive interference between the pathways of the mechanical oscillator and the cavity field, the system will add a new transparency window if the first cavity is coupled with a mechanical oscillator, which is shown in Figs 2(a) and 4(a). However, comparing Fig. 4(b) with 4(a), it can be found that the third labelled mechanical oscillator just broaden the central absorptive peak. On the other hand, Fig. 4(c) describes the coupling between the mechanical oscillator and 2nd cavity. Figure 4(d) describes that the mechanical oscillators interact with 2nd and 4th cavity, Fig. 4(a), it can be found that the even-labelled mechanical oscillators does not change the number of the transparency window for both case, only contributes to broaden the central absorptive dip compared to the case of without mechanical oscillator coupling. Note that, although all the mechanical oscillators are identical, they can still lead to different quantum interference pathways. The numerical calculation shows that, if one enlarges the numbers of the cavities and the odd-(even)-labelled mechanical oscillators, the results are similar with the ones mentioned above. In detail, for the odd-labelled case, the number of the transparency windows only adds one compared with the case of without mechanical oscillator coupling no matter how many mechanical oscillators are coupled with the cavities. And the increased odd-labelled mechanical oscillators only change slightly width of the central absorptive peak. While for the even-labelled ones, the increased oscillator only alter the width of the central absorptive peak or dip. These behaviors can be analyzed from Eq. (13). The equation of ε T ≈ 0 has N − 1 different roots without the coupled mechanical oscillator at the extremum points. For odd-(even)-labelled oscillator coupled with its cavity, ε T ≈ 0 has at most N(N − 1) different roots. Furthermore, when only odd-or even-labelled oscillators are coupled with the cavities, we also find that increasing the effective optomechanical rate G mN , the central absorptive peak or dip will be remarkable broadened. As for the broadened central absorptive dip, the phenomenon of the destructive interference is weakened with the increase of the central absorptive dip of the output field, and the consequent EIT-Autler Townes splitting (ATS) crossover or ATS 53 can occur. Due to the splitting of energy levels resulting from the strong field-driven interactions, identifying OMIT or EIT with ATS has been detailedly investigated in toroidal microcavity system 54 and the circuit circuit quantum electrodynamics system 55,56 . With Rydberg atoms. In the proceeding section, we have considered the variation of the multi-OMIT without the Rydberg aotms. Now, we shall investigate the multi-OMIT in the present system in which two Rydberg atoms are trapped in ith(i = 1, …, N) cavity and interact with the cavity field, and explore the effects of DDI on the OMIT. The parameters γ rr /g mj = γ gr /g mj = γ ee /g mj = γ er /g mj = 0.001, Ω/g mj = g/g mj = 1. The other parameters are same as the ones in the previous section. In order to simplify the model and highlight the effect of the Rydberg atoms in the ith cavity, we just only consider one mechanical oscillator which interacts with the ith cavity as others do not affect the behavior of Rydberg atom directly in principle. In general, the maximal DDI strength is of the order of gigahertz 57 . Figure 5(a-d) describe one mechanical oscillator interacts with the 1st cavity and the Rydberg atoms are also trapped in the same cavity with different DDI strength for four cavities. In Fig. 5(a), when DDI strength is zero, one can find that two extra symmetric transparency windows (extra resonances) appear on both sides of the central absorptive peak compared to the (4(c)), we can find that the DDI only impacts on the width of central absorptive dip or peak when the DDI strength is large. Therefore, the large DDI strength of Rydberg atoms has slight influence on the output field. On the other hand, from Eq. (13), one can find that the DDI strength can adjust the effective detunings x gr and x er , which makes the OMIT be sensitive to the DDI strength. As we all know, with the change of effective detuning, the extra OMIT windows can move and become a Fano line shape 58 . Then the extra narrow OMIT window, a analogue to EIT, evolves into a Fano resonance in the output field of the hybrid optomechanical system with the increase of DDI strength between two Rydberg atoms. In Figs 5 and 6, we discuss the influences of DDI strength and the mechanical oscillator coupling strength in the absorption of the output field. But we only consider the factor of DDI strength in Figs 7 and 8. Compared Figs 5, 6 with 7, 8, it can be found that the same behavior of the output filed appears except the slight differences in the position and width of the transparency windows compared with the cases of wihtout mechanical oscillator coupling. In detail, there are two additional transparency windows for weak DDI strength. When V(R) becomes more and more greater, two extra windows move and become Fano resonance till the right extra resonance of the absorption profile disappears gradually and the left extra resonance approaches the central absorptive peak. Note that, the system reduces to a coupled cavity system assisted a two-level atom in the large range DDI strength 30 . Because the influence of the coupled Rydberg atoms resembles a mechanical oscillator as mentioned above. If the positions of the atoms is different, the different numbers of the transparency window appear as shown in Figs 7(d) and 8(d). This result may be explained as follows. When DDI strength between Rydberg atoms is relatively weak, it is obvious that the second excited Rydberg atom does not shift the level of the first one. The system is regarded as a coupled cavity interacted with both a mechanical resonator and ladder-type Rydberg atoms. Due to the transitions \| 〉 ↔ \| 〉 g e and \| 〉 ↔ \| 〉 e r of the Rydberg atom in the hybrid system, additional interference pathways appear. Therefore, two additional OMIT windows in the absorption profile are observed. With the increase of DDI strength, the Rydberg blockade suppresses the excitation of the first atom and makes the OMIT condition be no longer fulfilled for the first atom. Then the first atom acts as a two-level atom which couples resonantly to the probe field. Conclusion and Discussion In summary, we have studied the OMIT of the MCOS. For the case without Rydberg atoms trapped in the cavity, the MCOS system has been demonstrated the generation of 2N − 1 (N < 10) OMIT windows for the output field, when N cavities interact with N mechanical oscillators, respectively. But the odd-and even-labelled oscillators will lead to different effects, if the odd-labelled oscillators are presented, only one extra OMIT emerges in the absorption profile by the quantum interference. In contrast, the increased even-labelled mechanical oscillators just broaden the central absorptive dip or peak. Under these circumstances, the corresponding transparency window can change from OMIT to ATS by increasing the effective optomechanical rate. On the other hand, when two Rydberg atoms are trapped in the ith cavity with weak DDI and the cavity is coupled with a mechanical oscillator, two extra OMIT windows can be observed. In addition, two extra OMIT windows would gradually move to the far off-resonance regime with the DDI strength increasing. The right extra resonance moves faster with the increase of the DDI strength. But the right one vanishes with great DDI strength. Furthermore, Fano resonances also appear with the changes of DDI strength. In experiment, one possible scheme is the toroidal microcavity-tapered optical fiber system coupled with Rydberg atoms. Firstly, the effect of OMIT in a single optical nanofiber-based photonic crystal optomechanical cavity has been engineered in the experiments 54,59 . Further, a two-color optical dipole trap has also come true by using the red-and blue-detuned evanescent light fields near the optical nanofiber. This method can allow the Rydberg atoms to be prepared at a few hundred nanometers from the nanofiber surface and coupled with the ith photonic crystal cavity 41,60 . And a series of nanofibers acted as a 1D coupled cavity array has been realized experimentally 61 , which is extended to lattices of coupled resonators with Rydberg atoms 62 . Therefore, combined with the above experiments, the multi-cavity optomechanical system with two Rydberg atoms trapped in one cavity may be realizable with the present-day or near-term technology. Figure 1 . 1Schematic diagram of the multi-cavity optomechanical system. (a) N cavities connect through hopping rates g n . A pair of Rydberg atoms are put into the ith cavity. (b) provide the equations in the resolved sideband regime, the detuning parameters are set as Δ = Δ with x er = Δ − Δ r − S and x gr = Δ − Δ r − Δ e − S. x j = Δ − ω mj is the detuning from the center line of the sideband. describe the effective optomechanical coupling rate of the jth cavity and they are equal. By solving the equations for O of the mechanical oscillators, one can obtain λ ω Figure 2 . 2The absorption Re(ε T ) as a function of x/κ 4 for four cavities. The subplot (a) corresponds to one mechanical oscillator coupled to cavity 1, the subplot (b) describes two mechanical oscillators coupled to cavity 1 and 2, respectively. The subplot (c) shows three mechanical oscillators coupled to cavity 1, 2 and 3. The subplot (d) illustrates four mechanical oscillators coupled to cavity 1, 2, 3 and 4.ScieNtific RepoRTs \| (2018) 8:14367 \| DOI:10.1038/s41598-018-32506-y cavities and the oscillators, the interactions of the cavities with the Rydberg atoms and the DDI between the Rydberg atoms, which make the expressions of c j , σ gg and σ rr become very complicated, then it is too difficult to give concrete forms. Fortunately, the values of c j only affect the width of the OMIT windows30 . When one focuses on the numbers of the OMIT window by numerical computation, G mj and G e can be valued by any reasonable and convenient value. Same argument, we also assume that the average σ = 1 gg and σ = 0 rr . Besides, to benefit more OMIT windows as many as possible, the system works in the weak dissipative regime, i.e, κ κ γ ≥  g , j N j mj gr er ge / / / . Without loss of generality, it is assumed that the parameters of the system are chosen as follows. For the mechanical oscillator, , the detunings from the center line of the sidebands are the same = Figure 3 . 3Energy level structure of the multi-cavity optomechanical system coupled with multi-oscillator. The number state of photons and phonons are denoted by n j and m j . The tunneling parameter between \|n 1 , …, n N ; m 1 , m 2 , … m N 〉 and \|n 1 , n 2 + 1, … n N ; m 1 , m 2 , … m N 〉 is g i , the coupling strength between \|n 1 , …, n i + 1, … n N ; m 1 , …, m i , … m N 〉 and \|n 1 , …, n i , … n N ; m 1 , …, m i + 1, … m N 〉 is G mj . Figure 4 . 4The absorption Re(ε T ) as a function of x/κ 4 for four cavities. The subplot (a) corresponds to no mechanical oscillators coupled to cavities, the subplot (b) describes two mechanical oscillators coupled to cavity 1 and 3, respectively. The subplot (c) shows one mechanical oscillator coupled to cavity 2, and the subplot (d) illustrates two mechanical oscillators coupled to the 2nd and 4th cavity, respectively.ScieNtific RepoRTs \| (2018) 8:14367 \| DOI:10.1038/s41598-018-32506-y Figure 5 . 5The absorption Re(ε T ) as a function of x/κ 4 . (a-d) Illustrate the cases of two Rydberg atoms trapped in 1st cavity coupled with the mechanical oscillator, and correspond to the DDI with V(R)/g mj = (0, 2, 4, 6, 10, 30), respectively.ScieNtific RepoRTs \| (2018) 8:14367 \| DOI:10.1038/s41598-018-32506-y Figure 6 . 6The absorption Re(ε T ) as a function of x/κ 4 . (a-d) Describe the cases of two Rydberg atoms trapped in 2nd cavity coupled with a mechanical oscillator, and correspond to DDI with different strengthes V(R)/g mj = (0, 2, 4, 6, 10, 30), respectively.ScieNtific RepoRTs \| (2018) 8:14367 \| DOI:10.1038/s41598-018-32506-y respectively. Compared with Figure 7 . 7Real part Re(ε T ) as a function of x/κ 4 . (a-d) Illustrate the cases of two Rydberg atoms trapped in the 1st cavity and the mechanical oscillators do not couple with cavities, which correspond to DDI with V(R)/g mj = (0, 2, 4, 30), respectively. ScieNtific RepoRTs \| (2018) 8:14367 \| DOI:10.1038/s41598-018-32506-y case [See Fig. 3(a)] without Rydberg atom. One can also find that the positions of the two extra resonances move to the right with the increase of the DDI strength as shown in Fig. 5(b-d). But the position of the left extra resonance moves slowly than the right one. In Fig. 6, one mechanical oscillator and two Rydberg atoms coupled with 2nd cavity have been discussed. The variation tendencies of two extra resonances are the same as the ones in Fig. 5. However, the widths, the positions and the amplitudes of two extra resonances are different. When the Rydberg atoms are trapped in 3rd and 4th cavities, numerical results also show that same variation tendencies of two extra resonances can be obtained in Figs 5 and 6, respectively. But the widths and the amplitudes of the two extra resonances have little difference. In addition, the amplitudes of two extra resonances become smaller and experience Fano resonance with the increase of DDI strength. When DDI strength increases, the left extra resonance gets close to the central absorptive dip and then both extra resonances die out. Compared Figs 5(d), (6(d)) with 2(a), Figure 8 . 8Real part Re(ε T ) as a function of x/κ 4 . (a-d) Describe the cases of two Rydberg atoms trapped in 2nd cavity and the mechanical oscillators do not couple with cavities, which correspond to DDI with V(R)/g mj = (0, 2, 4, 30), respectively. ScieNtific RepoRTs \| (2018) 8:14367 \| DOI:10.1038/s41598-018-32506-y © The Author(s) 2018 AcknowledgementsAuthor ContributionsAdditional InformationCompeting Interests: The authors declare no competing interests.Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. 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[ "Predicting the species abundance distribution using a model food web", "Predicting the species abundance distribution using a model food web" ]	[ "Craig R Powell \nSchool of Physics and Astronomy\nTheoretical Physics Group\nUniversity of Manchester\nM13 9PLManchesterUK\n", "Alan J Mckane \nSchool of Physics and Astronomy\nTheoretical Physics Group\nUniversity of Manchester\nM13 9PLManchesterUK\n" ]	[ "School of Physics and Astronomy\nTheoretical Physics Group\nUniversity of Manchester\nM13 9PLManchesterUK", "School of Physics and Astronomy\nTheoretical Physics Group\nUniversity of Manchester\nM13 9PLManchesterUK" ]	[]	A large number of models of the species abundance distribution (SAD) have been proposed, many of which are generically similar to the log-normal distribution, from which they are often indistinguishable when describing a given data set. Ecological data sets are necessarily incomplete samples of an ecosystem, subject to statistical noise, and cannot readily be combined to yield a closer approximation to the underlying distribution. In this paper we use empirical data obtained from an ecosystem model to study the predicted SAD in detail, resolving features which can distinguish between models but which are poorly seen in field data. We find that the power-law normal distribution is superior to both the log-normal and logit-normal distributions, and that the data can improve on even this at the high-population cut-off.	10.1016/j.jtbi.2008.09.005	[ "https://arxiv.org/pdf/0805.0084v1.pdf" ]	15,319,636	0805.0084	107a1def3df26506651f0a803c3c8f4994f735e6	Predicting the species abundance distribution using a model food web 1 May 2008 (Dated: May 1, 2008) Craig R Powell School of Physics and Astronomy Theoretical Physics Group University of Manchester M13 9PLManchesterUK Alan J Mckane School of Physics and Astronomy Theoretical Physics Group University of Manchester M13 9PLManchesterUK Predicting the species abundance distribution using a model food web 1 May 2008 (Dated: May 1, 2008)numbers: 8723Cc Keywords: ecological diversitytrophic distributionecological community model A large number of models of the species abundance distribution (SAD) have been proposed, many of which are generically similar to the log-normal distribution, from which they are often indistinguishable when describing a given data set. Ecological data sets are necessarily incomplete samples of an ecosystem, subject to statistical noise, and cannot readily be combined to yield a closer approximation to the underlying distribution. In this paper we use empirical data obtained from an ecosystem model to study the predicted SAD in detail, resolving features which can distinguish between models but which are poorly seen in field data. We find that the power-law normal distribution is superior to both the log-normal and logit-normal distributions, and that the data can improve on even this at the high-population cut-off. I. INTRODUCTION The species abundance distribution (SAD) is one of the most widely studied descriptions of an ecological community. To determine it, the number of species in a given community which have n observed individuals is plotted against n. The shape of this plot has been investigated by a great many empiricists and theorists over the years, beginning with the classic work of Fisher et al. [1] and Preston [2]. Reviews of the subject [3,4,5,6,7] reveal the large number of mechanisms that have been proposed to explain the observed SAD. Many of these mechanisms predict the essential aspects of the observations, that is, a few very abundant species and many rare species. As a consequence it has become very difficult to falsify proposed mechanisms from empirical data, which has led to the authors of the most recent multi-author review [6] to contrast the development of the analysis of SADs with "a healthy scientific field" in which "theoretical, empirical and statistical developments [...] advance roughly in parallel". In this paper we suggest a way forward which is in effect intermediate between the theoretical and empirical approaches. We measure the SAD in an established model which constructs an ecological community as a set of predator-prey interactions [8]. The model itself was originally created so that many of its key properties were emergent and not put in by hand. So, for instance, trophic levels emerge from the nature of the predatorprey interactions; species are not labelled as "plants", "herbivores" or "carnivores" a priori. This contrasts with traditional theoretical approaches which either postulate a mechanism, or if a model community is put forward it is usually rather simple, with the form of the SAD fol-lowing from one of the fundamental aspects of the theory. Conversely, measurements taken in the field will of necessity include numerous influences involving climate, terrain, location etc., which are not present in the model we use to measure SADs. Thus the SADs we measure will be free of these external influences, but still be determined by influences which are too complex to easily characterise. This approach will also allow us to measure SADs for a multi-trophic level community whereas, so far as we are aware, all other predictions for SADs have been for communities of trophically similar species. The model we will be using (called the Webworld model) has been developed over a number of years [8,9,10,11]. In it, species are defined in terms of traits (phenotypic and behavioural characteristics), and it is the nature of the interactions between these traits which define the nature of interactions between species. This community is built up from a small number of species through a speciation mechanism which creates a new species with a novel set of features. Resources are distributed through a quite elaborate set of equations governing population dynamics with adaptive foraging. Overviews of the model are given in review articles [12,13,14], and more briefly in section II. In section 3 we outline the method of our analysis, in section 4 we describe the results obtained and we conclude with a review of the results in section 5. II. MODEL DESCRIPTION The Webworld model consists of a set of species, each defined by its unique combination of ten different features. The features are chosen from a set of L possible features determined at the start of the simulation, at which point two species are created. One of these is the environment species, which has a fixed population for all time and is the ultimate source of energy for all species in the food web. The other initial species is the common ancestor of all species encountered during a simulation run. The dynamics of the Webworld model occur on three separated time-scales. The longest of these is the evolutionary time-scale, on which new species are added as mutated versions of extant species. Specifically to the implementation of Webworld, a species is selected at random without regard to its population, except that this must be non-zero. One individual of that species is then used to found a new species identity, sharing all features but one with the parent species. The remaining feature is selected to avoid repetition either of the same feature within one species or the same set of features between species, but is otherwise selected at random. The newly introduced species is then subject to the same population dynamics as all other species, which is the dynamical process that occurs on the intermediate time-scale. Population dynamics occurs by balance of energy; energy is gained through "predation", which in the case of feeding on the environment species we interpret as autotrophy. Each species i changes its population n i according to the balance equatioṅ n i = λ j g ij n j − j g ji n i − dn i ,(1) where g ij is the functional response, the dependence of the rate of energy consumption by species i on the population of species j. The factor of λ = 0.1 introduces an ecological efficiency whereby the energy lost to species j is greater than that gained by its predator i. Thus the first term on the right hand side of Eq. (1) is the energy income of species i summed across all prey species, while the second term is the energy loss summed across all predators. If species a does not feed on species b then g ab = 0, and hence this does not contribute to either sum. The final term in Eq. (1) is the loss of energy from species i due to death of its constituent individuals at rate d per individual; the expected lifespan of an individual is therefore 1/d, which for simplicity we take to be the same across all species. Death appears in our model purely as an energy loss term and cannot be made an evolvable quantity, since it has a preferred value of zero. The shortest time-scale in the Webworld model reflects the choice of foraging strategy by individuals of each species. The functional response for Eq. (1) is given by g ij = f ij S ij bn j + k α ik f kj S kj n k n i ,(2) where f ij is the fraction of its time species i spends feeding on prey species j, which is the quantity to be optimised in order to maximise j g ij n j . S ij and α ij are constants defined by relating the features of species i and j, S indicating the ability of i to feed on j, and α relating to the degree of inter-specific competition. To prohibit mutual predation the matrix S is made anti-symmetric, thus S ij = −S ji , and the shortest possible feeding loop involves three species. Matrix α is symmetric, with maximum competition α ii = 1 between members of the same species, and minimum competition 0.5 between highlydifferent species. By calculating S and α based on a set of features largely conserved during speciation we ensure that each newly-founded species has similar abilities to its parent species, with which it is also in strong competition, and in particular the dynamics of two identical species, were they allowed, would be indistinguishable from the dynamics of pooling them as one species. In Drossel et al. [8] an evolutionarily stable strategy was shown to exist for foraging, which can be found by iteratively solving Eq. (2) with the condition that f ij = g ij k g ik .(3) The result of the repeated application of these dynamics is the gradual construction of a complex food web. Species are removed if their population falls below 1, and the fixed population of the environment species, R, as such determines the expected number of species present in the food web at any time, though there is a continual turnover of species and consequent fluctuation in any given food web measure. After running the model for a large number of evolutionary time steps, there is no systematic change in quantities such as the number of concurrent species, and the food web structure appears to have matured. It is on such webs that we examine the species abundance distribution. III. METHOD Using the Webworld model discussed in the previous section we generate sets of communities for which the ensemble species abundance distribution (SAD) can be examined in detail. Because we use the same set of possible features and the same environment species in each case, we assume that the underlying SAD does not alter between model realisations. In this case it is possible to pool the resultant communities in order to determine the SAD with improved statistical noise. Details of the computational approach are given in section III A. In section III B we discuss the functions which we fit to the data, and the optimisation criteria of the fitting. In section III C we discuss the problems of generalising fits to include communities differing in size or trophic level. A. Comparative models Although the Webworld model can simulate ecological communities in reasonable time, to create large complex communities takes considerable computation, and to generate enough simulations to get good statistics across a broad range of parameter space is difficult. We therefore perform the first examination on a variant of Webworld in which all species feed exclusively on the environment. Because all species are basal, the relative populations are determined by the relative ability, S, and competition, α, terms between existing species, which are selected by evolution in the same way as in the full model. By avoiding a large part of the computational effort we are able to generate large numbers of webs for comparison, and in the results presented here gather statistics from a set of one hundred model runs for each value of resources, R. In section IV we focus on the fitting of food webs with resources 10 3 , 10 4 , 10 5 and 10 6 , but simulations were performed for numerous other values of R within this range to show that interpolation of the results is reasonable. The minimum value of R results in communities with few species, which become correspondingly harder to characterise in terms of an SAD. Larger values of R become increasingly computationally expensive. Rather than attempting to extend the range of R to larger values, we created a total of 900 basal communities at R = 10 6 for more detailed analysis of the tails of the distribution. Because the common theoretical SADs have been selected based on reproduction of the modal peak, and are poorly constrained by observations, the tails offer the largest differences between candidate SADs. Due to the much larger computational complexity of the full Webworld model, we have only a sample of ten comparable food webs for large R from which to deduce trophic SADs. B. Fitting method As can be seen in Figure 1, the probability distribution function (p.d.f.) of species abundance has a rather noisy histogram even for the largest collection of independent communities we were able to assemble with the available computer time. Fitting a distribution function to such histograms is problematic for several reasons. The noise makes it difficult to algorithmically optimise the fitting function, and hence can obscure differences in the strength of different functional forms. More importantly, the apparently optimal parameters and associated fitness will depend on the arbitrary choice of bin width and position, since changing these parameters can significantly alter the distribution of noise between the bins. Furthermore, the distribution function underlying the observed SAD is likely to have a functional form other than our approximations, and in general may be significantly more complicated than we can extract from data so long as the noise remains. Rather than obtaining a function which closely matches the value of the p.d.f. for most population sizes, but which omits important features, we prefer to recover a smoothed version of the distribution function which correctly predicts the total number of species. As a consequence of these considerations we fit the integrated version of the fitting function to the empirical cumulative distribution function (c.d.f.), whose value at a given population N is the measured number of species with n i < N . This definition matches the type of p.d.f. used by [5] whose integral is the expected number of species. P.d.f.s may also be defined such that the area enclosed is unity. To illustrate the fitting procedure we present plots of the measured and fitted c.d.f.s in addition to the p.d.f.s, and indicate the goodness-of-fit by plotting the residuals of the c.d.f., that is, the difference between the integrated fitting function and the measured c.d.f. The strongest condition that we impose on each fitting function is that it should correctly predict the number of species more abundant than the least abundant species observed. Below this population the distribution may be terminated by a veil line, but we do not allow any such consideration for populations above the most abundant species observed. Subject to this condition we optimise the parameters of each theoretical distribution function, f (ln N ), by minimising a quantity analogous to χ 2 . One such statistic is the Cramér-von Mises test [15], defined as CM = 1 12S + 1 S S i=1 i − 0.5 −f (n i ) 2 ,(4) wheref (n i ) is the predicted number of species less abundant than n i , subject to the veil line at n 1 , and S is the number of species observed. Although this is readily generalised to an ensemble of SADs, it attributes most weight to the peak of the distribution at the expense of fitting the tails, and we instead minimise the quantity k 2 = ln Nmax ln n1 C(N ) −f (N ) 2 d ln N,(5) where C(N ) is the observed number of species less abundant than N . For many distributions N max → ∞, but functions such as the logit-normal distribution are parametrised by the total number of individuals observed, J, in which case N max = J. Unlike the Cramérvon Mises statistic, k 2 places equal weight in all intervals of ln N . Given that the theoretical distribution almost certainly differs from the distribution underlying the data, this tends to avoid problematic regions, such as ranges of N in which few species are observed, but where the empirical and theoretical c.d.f.s differ. The tails of the distribution often behave in this manner. Having identified optimal fitting parameters by minimising k 2 , we follow the advice of [6] that "claim[s] of a superior fit must be robust by being superior on multiple measures" by evaluating the Kolmogorov-Smirnov statistic [16] for each theoretical distribution. Defined for a single realisation as d = S −1/2 max i i − 1 −f (n i ) , i −f (n i ) ,(6) d corresponds to the greatest deviation between the empirical and theoretical c.d.f.s. This must occur at one of the observed species, which correspond to steps in the empirical c.d.f. It is necessary to evaluate the difference between the empirical and theoretical c.d.f. both immediately before and after the step, and hence the The fitted species abundance distribution for basal communities with resources R = 1000, 10 000, 100 000 and 1 000 000. The histogram indicates the data in bins of width 0.1 in ln N . The solid curves indicate optimal log-normal fits, the dotted lines optimal logit-normal fits, and the dashed lines optimal power-law normal fits. Distributions to the right correspond to increasing R. 'maximum' operator in Eq. (6) contains two terms for each observation i. Although the values of d obtained imply rejection of the theoretical distributions given the amount of data available, we use d as a measure of the relative goodness-of-fit to distinguish between theoretical distributions. Other measures of goodness-of-fit tend to relate to binned data rather than the c.d.f., and provide correspondingly weaker evidence [17]. Although the log-normal distribution has been criticised as inappropriate for application to SADs [18], it is a commonly examined form of the SAD and we therefore adopt it as one of the theoretical SADs we fit to the data. We also consider the logit-normal distribution preferred by Williamson & Gaston [18]. Whereas the log-normal distribution appears as a normal distribution when plotted against a logarithmic population-axis, the logit-normal has a normal distribution when plotted against a logit population axis. Our analysis will consistently use the logarithmic axis both for plotting and for the integration of k 2 , so while the log-normal distribution has the form P (ln N ) d ln N = A exp − (ln N − ln µ) 2 2σ 2 d ln N,(7) with the fitting parameters A, µ and σ, the logit-normal distribution includes an extra factor, giving P (ln N ) = A J J − N exp      − ln N J−N − ln µ J−µ 2 2σ 2      .(8) We also consider a third fitting function, the powerlaw normal distribution, which appears normal against a power-law population axis. Transformed to a logarith- mic axis, this has the functional form P (ln N ) = AαN α exp − (N α − µ α ) 2 2σ 2 ,(9) where α is the power-law index. We do not consider the log-series distribution since our data are with few exceptions peaked at large N , whereas the p.d.f. of the log-series distribution decreases from N = 1 even when drawn against a logarithmic population-axis. The broken stick distribution [19] was found to be similar in form to the observed distribution, but inferior to the log-normal in all cases. C. Comparison of food webs Since we are applying the same distribution function with different parameters to basal food webs of different sizes, and to the SADs of different trophic levels within a single community, in the ideal case a parametrisation of the fitting coefficients in terms of resources, R, and trophic level, l, would be found. Because small values of R correspond to food webs with fewer species, complications arise in weighting the contribution to goodness-offit from differently sized webs, and we do not in this paper attempt to simultaneously fit webs of different sizes. By examination of the best-fitting parameters for each web we can determine the dependence of parameters on R except in one case; the power-law index α of the power-law normal distribution. For most values of R the goodnessof-fit depends quite weakly on this parameter, and the optimal value of α is poorly constrained for any one web. Since we were unable to identify a systematic trend or strongly constrain the value of α, we chose α = 0.2 as a constant value consistent with the optimised parameters, and fixed this value for all results presented here. Figure 2 shown as residuals; the solid line corresponds to the empirical c.d.f. minus the log-normal distribution, the dotted line to the data minus the logit-normal distribution, and the dashed line to the data minus the power-law normal distribution. Offsets of -0.5, -1.5 and -2.5 have been applied to data for resources R = 10 000, 100 000 and 1 000 000 respectively. IV. RESULTS In section IV A we present the results of the fitting procedure for the basal communities. These should give the least complicated species abundance distributions (SADs), since all species feed on a single resource and are in direct competition with each other. In comparison, the trophic communities examined in section IV B feed on multiple food sources themselves distributed in abundance, and compete with different subsets of the other species. In section IV C we make use of the large number of simulation runs which can be performed to make a detailed examination of the low-and high-population tails of the empirical distribution, and compare this to the behaviour of the fitted distributions. A. Basal community The results for this version of the model are the most complete in that one hundred simulations runs were examined for each value of resources R, and a large number of values of the continuous parameter were examined. In Figures 1 to 3 only four of these realisations are plotted, corresponding to R = 10 3 , 10 4 , 10 5 and 10 6 , which include the two most extreme values of R for which webs were calculated. The general features of the SAD for these four values are typical, as is the goodness of fit achieved by each of the three fitting functions examined. It is clear from Figure 1 that the observed distribution is left-skewed (has an over-abundance of rarer species), a characteristic absent from the log-normal distribution. The logit-normal distribution does not have significantly improved skew over the log-normal distribution, since the most abundant species from any run has less than one quarter of the mean number of individuals J, and the logit function is therefore well below its asymptotic cut-off. Williamson & Gaston [18] note that in this limit the logit-normal distribution approaches the log-normal. The power-law normal distribution much more closely captures the smaller high-N tail. The corresponding cumulative distribution functions (c.d.f.s) are plotted in Figure 2, where the logit-normal distribution has been omitted for clarity. It can be seen, especially for R = 10 6 , that the log-normal distribution underestimates the cumulative number of species in both tails, which corresponds to the skew of the p.d.f., and that even for one hundred realisations the empirical c.d.f. is far from smooth. More instructive than the c.d.f. are the residuals of this plot, that is, the difference between the instantaneous value of the empirical c.d.f. and the fitting function. These are shown in Figure 3 for all three fitting functions. The integral of the square of this plot is our goodness-of-fit measure k 2 , and the maximum deviation from zero is the Kolmogorov-Smirnov d-measure. Substantial structure can be seen in the residuals, especially the central peak for each value of R when examining the power-law normal distribution, which most closely mimics the tails. Table I records the values of k 2 and the Kolmogorov-Smirnov d value for each fit, for fitting parameters minimising k 2 . Basal communities are labelled by the value of resources, R, while trophic levels examined in section IV B are labelled according to the trophic level, l. For the basal food webs the power-law normal fit always outperforms both the logit-normal and log-normal distributions in terms of k 2 , and is only in one case inferior to the logit-normal distribution as measured by d. A further comparison of the relative merits of the theoretical distribution functions is given in section IV C. In Figure 4 we plot the dependence of the parameters of the power-law normal fit on R, as well as the two goodness-of-fit indicators used. The solid line, marking the population of the peak of the distribution, indicates the very near linearity of the value of the peak of the distribution with ln R. The standard deviation of the distribution increases more rapidly than linearly, as indicated by the dashed line. The value of k 2 increases with ln R for two reasons. Firstly, it is measured on the full c.d.f. rather than the normalised distribution, and so tends to increase as the square of the expected number of species, S. Secondly, because it is an integrated measure, it tends to increase with the width of the distribution, which we characterise by the standard deviation of the log-normal distribution, σ LN . It is more appropriate to use this measure than the standard deviation of the power-law normal itself since the former corresponds naturally to the width along the logarithmic population axis. In Figure 4 we plot the quantity K = 1000k 2 S 2 σ LN ,(10) which compensates for these effects, and includes a factor of 1000 to scale it appropriately for that plot. It can be seen that intermediate values of R are the best fitted, as measured by either K or the Kolmogorov-Smirnov d, perhaps due to relatively small amounts of additional structure. B. Trophic levels Having established that the power-law normal distribution describes the SAD reasonably well for basal communities, we apply it to individual trophic levels of full Webworld communities to determine the relevant fitting parameters. Due to the small number of food webs available, and the small number of species in each trophic level for any one web, it is inappropriate to seek deviations from this distribution with the data available, although we find that the power-law normal distribution is adequate, and superior in all cases to the log-normal distribution, having smaller values of both k 2 and d. As indicated by the values given in table I, the logit-normal dis- tribution marginally improves upon the power-law normal distribution for trophic levels 1 and 3, but is significantly inferior to the power-law normal for trophic level 2. For trophic level 1, the typical number of species observed per web in the data examined was only 5.9, the most abundant species being nearly half the total population of its trophic level. For trophic level 3 the lower tail of the distribution was truncated, and although here the logit-normal distribution performed better than the power-law normal, it is not clear that the logit-normal is able to adequately reproduce the whole SAD. Although four trophic levels were found in the empirical data, a very small number of species were found in trophic level 4. It can be seen in Figure 5 that the distribution function of this level is little more than the high-population tail of the distribution function, and no reliable results can be obtained by its analysis. For comprehension of the empirical distribution being fitted we reproduce, in Figure 6, the cumulative distribution function constructed from the simulation data along with the optimal log-normal and power-law normal fits. It can be seen clearly from this figure that the distribution of the second trophic level, which has the largest number of species in total, is closest in form the those of the basal communities. The distribution of trophic level three, to its left, passes the veil line before a significant fraction of the low-population tail has been exposed, but is otherwise in good agreement with the basal community distributions. The lowest trophic level, however, seems relatively truncated, resulting in a much sharper cutoff at large N than is reproduced by either the log-normal or power-law normal distributions. The cause of this may relate to the presence of predators, who can be expected to preferentially target the most abundant prey, but additional data are required to investigate this hypothesis. The residuals of the c.d.f. fits are shown in Figure 7; it is possible that similar structure in these is present to that seen for the basal communities in Figure 3, but the Figure 6 shown as residuals; the solid line corresponds to the empirical c.d.f. minus the log-normal distribution, the dotted line to the data minus the logit-normal distribution, and the dashed line to the data minus the power-law normal distribution. Offsets of -1.0, -2.0 and -2.5 have been applied to data for trophic levels 2, 3 and 4 respectively. No logit-normal fit was obtained for trophic level 4 due to the absence of a positive optimal mean. degree of noise is greater. In Figure 8 the mean and standard deviation of the power-law normal distribution are plotted as a function of trophic level. While the standard deviation appears to decline linearly with trophic level, the distribution mean may decrease more slowly. However, if the results for trophic level four are misleading due to the extremely high position of the veil line, and the distribution of basal species is possibly altered through predation as discussed, the reliability of these results is limited. The quantity K, defined in Eq. (10), is much better for trophic levels two and three than for either the basal or fourth trophic level, although only marginal improvements in the Kolmogorov-Smirnov d value can be seen. C. Distribution tails An advantage of examining computationally derived communities of species is that extremely large data sets can be constructed with relative ease, subject only to the availability of computer time. In addition, the Webworld model produces complete ecological communities, and the sampling effects associated with field data are avoidable. As such it is much more feasible to examine the form taken by the tails of the distribution function, which McGill et al. [6] note are subject to noisy data, but which often contain the main differences between theoretical distributions. To construct a high-quality empirical SAD whose tails could be examined, nine hundred simulation runs were performed for the basal community with R = 10 6 . The low-population tail of this distribution is plotted in Figure 9, where the logarithm of the binned species abundance has been taken to expose the tail. The fact that a linear regression to this data (not shown) produces a good fit for ln N < 7 implies that in this regime a powerlaw fit, P (ln N ) ∝ N a ,(11) with a ∼ 4/3, is obeyed. The power-law normal distribution is able to reproduce this form reasonably well, while both the log-normal and logit-normal distributions significantly underestimate the number of species present. The distribution tail for large populations is shown in Figure 10. Here bins have been chosen to be uniform in width in population, rather than uniform in ln N , in order to resolve the tail. The result is that a different version of the distribution is shown, P (N ) dN = P (ln N ) N dN,(12) which, when integrated with respect to N , gives the c.d.f. Note that in order to highlight the form of the decay, the population axis has been stretched to a power-law. The regression line, plotted as a dash-dot line, indicates that the high-population tail has the form P (N ) dN ∝ exp − N 7140 1.4116 dN.(13) As can be seen in Figure 10, this form of the decay declines more rapidly with N than any of the log-normal, logit-normal or power-law normal distributions examined. Having established probable forms for the low-and high-population tails by regression to Figures 9 and 10, we combine these into a distribution which has the minimum value of the two tail-fitting functions for all N . In addition to the dashed line marking the empirical c.d.f., identical to that shown in Figure 2, this fit is shown in Figure 11 in two forms. The lower plot is the c.d.f. integrated from zero species at N = 0, and the upper curve is integrated down from the observed number of species so as to converge with the empirical distribution at large N . The fact that the latter curve is above the former indicates that the combined distribution underestimates the total number of species, implying that it under-predicts the p.d.f. near the peak, to which it was not fitted. Figure 11 therefore also plots the residuals of the tail-fitting distribution as a histogram. There appear to be at least three peaks in the residuals, making it difficult to identify a plausible general form. Since we do not have unrelated basal food webs to examine, in particular to establish what parameters of the tail distributions are generic and whether the residuals show a common pattern, it is not appropriate to draw further conclusions about the central part of the distribution. We are also unable to ascribe a goodness-of-fit to the tail-based distribution due to its inability to reproduce the peak of the distribution. V. CONCLUSIONS We have investigated the form of the species abundance distribution empirically derived from simulation results of the Webworld food web model. This model was created to examine patterns of food web assembly, and the form of the species abundance distribution (SAD) was not a factor in its construction. Rather, the use of population dynamics to establish the success of particular species and feeding strategies within the community lead naturally to variation in the abundance of species which appears similar to the SADs identified from real ecosystems. By investigating the empirical SAD from the simulations in the same manner as data from real ecosystems we are able to characterise not only the peak of the distribution, which is frequently observed to have a form similar to the log-normal distribution, but to examine in detail parts of the distribution difficult to obtain data on from field studies. We agree with the conclusion of Williamson & Gaston [18] that the logit-normal dis- tribution fits better, but with particular reference to the tails of the distribution find that the power-law normal distribution function is better still. In particular, the lognormal and logit-normal distributions predict that the number of species with population N falls more rapidly with decreasing N than we obtain from our simulation results, which the power-law normal distribution matches very well in this tail. The presence of structure in Figure 3 suggests that a more complicated function is needed to properly reproduce the observed SAD, but we have not been able to examine the reproducibility of this remaining structure. All the food webs examined were created for the same set of possible features and the same environment species. To fully explore the results even for a single value of R would require the use of food webs constructed for 'worlds' with different environment species and feature sets. In undertaking such a programme it would first be necessary to establish whether such parameters as the mean and variance of the fitted distribution changed, or more generally to construct the meta-distribution of a large number of Webworld 'worlds' and test, using the Kolmogorov-Smirnov d value, whether the empirical distribution constructed from webs of a single family was consistent with the meta-distribution. We find that the power-law normal distribution identified as well describing the SAD of a basal community is also successful in describing individual trophic levels of a food web. It is particularly descriptive of the second trophic level, which can be seen in Figures 5 and 6 to be the most completely realised by our empirical data. The higher trophic levels can also be expected to be well-fitted by the power-law normal distribution, although the truncation of the distribution at low populations results in the log-normal and logit-normal descriptions also being ad- equate. The empirical distribution of the lowest trophic level is more sharply truncated at high populations than seen for other communities, the reason for which would require substantial additional investigation. Unlike the case of examining basal communities at different values of R, only a small number of trophic levels are ever possible, and hence the relation between them is harder to quantify. While it would be possible to construct metadistributions from larger numbers of food webs, it is more feasible to first examine the agreement between the metadistributions of basal communities and the constituent distributions. If there is good agreement, the agreement between the meta-distribution and the trophic distributions should be examined. If not, then a very large number of communities need to be evolved in the same environment in order to study the relation between trophic levels, potentially also examining the effect of different values of R. The main problem in investigating the SAD of numerically modelled ecosystems is the extensive computer time required to provide data. The SADs constructed for this paper are complete not only in the sense that they contain all individuals present in the sample area, but also in that they do not feature immigrant or transient species, which can contribute to the low-population tail without representing a viable population. While features such as immigration from surrounding communities can easily be incorporated into our model, as can finite population effects, their exclusion demonstrates the existence of an extensive low-population tail to the distribution even for a closed ecosystem. This contrasts with the proposal by Magurran [20] that the low-population tail is a log-series distribution of "occasional" species added to a core log-normal distribution. Although we do not agree with McGill [21] that left-skew is purely an effect of sampling, it may be the case that the left-skew of incomplete samples does not reflect the underlying distribution. McGill et al. [6] observe that most proposed SADs are similar to one another except in the tails, which is precisely the region which field observations are least able to address due to paucity of data. This issue can be addressed by the use of any model which can produce multiple independent realisations of its dynamics from which a composite SAD can be constructed, but this process can only be used to inform the analyses which should be performed on ecological data, since it is not known a priori that any given model accurately reproduces the real SAD. A virtue of the Webworld model is that is produces a plausible SAD without any such consideration having been used during the model design, being based rather on plausible ecological rules. FIG. 1: The fitted species abundance distribution for basal communities with resources R = 1000, 10 000, 100 000 and 1 000 000. The histogram indicates the data in bins of width 0.1 in ln N . The solid curves indicate optimal log-normal fits, the dotted lines optimal logit-normal fits, and the dashed lines optimal power-law normal fits. Distributions to the right correspond to increasing R. FIG. 2 : 2The fitted cumulative species abundance distribution for basal communities with resources R = 1000, 10 000, 100 000 and 1 000 000. The solid line shows the data, the dotted line marks the log-normal distribution, and the dashed line the power-law normal distribution. Distributions to the right correspond to increasing R. FIG. 3 : 3The same data plotted in FIG. 4 : 4Parameters of the power-law normal fit to the basal community SAD for all values of resources examined. The solid line passes through points marking the mean population, µ in Eq. (9); the dashed line marks the standard deviation, σ. Squares mark the Kolmogorov-Smirnov d value, and stars mark the quantity K described in the text. FIG. 5 : 5Histograms mark the observed species abundance distribution for the four trophic levels found in the ten Webworld communities examined. Trophic levels two and four are marked by dotted and dashed lines respectively. Solid curves mark the optimal log-normal fits to each trophic level, and dashed lines the optimal power-law normal fits. FIG. 6 : 6The cumulative species abundance distribution for Webworld communities corresponding to the four observed trophic levels. Higher trophic levels are to the left of lower levels, having smaller typical populations. Optimal log-normal fits are marked by dotted lines, and optimal power-law normal fits by dashed lines. FIG. 7 : 7The same data plotted in FIG. 8 : 8Parameters of the power-law normal fit to the trophic community SAD for all values of resources examined. The solid line passes through points marking the mean value of N . The lower dashed line marks the standard deviation, while the upper dashed line multiplies this quantity by 10 for clarity. Squares mark the Kolmogorov-Smirnov d value. Stars mark the quantity K defined in the text. FIG. 9 : 9The small population tail of the basal community p.d.f. for resources R = 10 6 . The p.d.f. is described in the text. The solid, dotted and dashed curves mark the lognormal, logit-normal and power-law normal fits toFigure 2respectively. FIG. 10 : 10The high population tail of the basal community p.d.f. for resources R = 10 6 . The x-axis is linear in N 1.4116 , which was found to be the power-law index minimising the χ 2 of the regression line, but has been marked with corresponding values of N for clarity. The histogram marks the value of P (N ), the population density in bins of equal width in N . The solid, dotted and dashed curves mark the log-normal, logitnormal and power-law normal fits toFigure 2respectively. The dash-dotted line indicates the best-fitting regression for N > 2000. FIG. 11 : 11The c.d.f. for the basal community with resources R = 10 6 , as shown inFigure 2, is shown as a dashed line. The solid lines mark the c.d.f. constructed from fits to the tails as described in the text. The histogram marks the residuals of the p.d.f. of this fit. TABLE AcknowledgementsThe authors thank Carlos A. Lugo for providing additional simulation data. This work was supported by EPSRC under grant GR/T11784. . R A Fisher, A S Corbet, C Williams, Fisher, R. A., Corbet, A. 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[ "Metal Artifact Reduction with Intra-Oral Scan Data for 3D Low Dose Maxillofacial CBCT Modeling", "Metal Artifact Reduction with Intra-Oral Scan Data for 3D Low Dose Maxillofacial CBCT Modeling" ]	[ "Chang Min Hyun˚ ", "Taigyntuya Bayaraa˚ ", "Hye Sun Yun˚ ", "Tae-Jun Jang˚ ", "Hyoung Suk ", "Park : ", "Jin Keun Seos ", "\nchool of Mathematics and Computing (Computational Science and Engineering)\nYonsei University\n03722SeoulSouth Korea\n", "\nNational Institute for Mathematical Sciences\n34047DaejeonSouth Korea\n" ]	[ "chool of Mathematics and Computing (Computational Science and Engineering)\nYonsei University\n03722SeoulSouth Korea", "National Institute for Mathematical Sciences\n34047DaejeonSouth Korea" ]	[]	Low-dose dental cone beam computed tomography (CBCT) has been increasingly used for maxillofacial modeling. However, the presence of metallic inserts, such as implants, crowns, and dental filling, causes severe streaking and shading artifacts in a CBCT image and loss of the morphological structures of the teeth, which consequently prevents accurate segmentation of bones. A two-stage metal artifact reduction method is proposed for accurate 3D low-dose maxillofacial CBCT modeling, where a key idea is to utilize explicit tooth shape prior information from intra-oral scan data whose acquisition does not require any extra radiation exposure. In the first stage, an image-to-image deep learning network is employed to mitigate metal-related artifacts. To improve the learning ability, the proposed network is designed to take advantage of the intraoral scan data as side-inputs and perform multi-task learning of auxiliary tooth segmentation. In the second stage, a 3D maxillofacial model is constructed by segmenting the bones from the dental CBCT image corrected in the first stage. For accurate bone segmentation, weighted thresholding is applied, wherein the weighting region is determined depending on the geometry of the intra-oral scan data. Because acquiring a paired training dataset of metal-artifact-free and metal artifact-affected dental CBCT images is challenging in clinical practice, an automatic method of generating a realistic dataset according to the CBCT physics model is introduced. Numerical simulations and clinical experiments show the feasibility of the proposed method, which takes advantage of tooth surface information from intra-oral scan data in 3D low dose maxillofacial CBCT modeling.Index Terms-cone beam computed tomography, metal artifact reduction, deep learning, digital dentistry, intra-oral scan arXiv:2202.03571v1 [eess.IV]	null	[ "https://arxiv.org/pdf/2202.03571v1.pdf" ]	246,652,207	2202.03571	e1a0983e43cbf82a77d1aa0f3cc33258280e0d04	Metal Artifact Reduction with Intra-Oral Scan Data for 3D Low Dose Maxillofacial CBCT Modeling Chang Min Hyun˚ Taigyntuya Bayaraa˚ Hye Sun Yun˚ Tae-Jun Jang˚ Hyoung Suk Park : Jin Keun Seos chool of Mathematics and Computing (Computational Science and Engineering) Yonsei University 03722SeoulSouth Korea National Institute for Mathematical Sciences 34047DaejeonSouth Korea Metal Artifact Reduction with Intra-Oral Scan Data for 3D Low Dose Maxillofacial CBCT Modeling 1 Low-dose dental cone beam computed tomography (CBCT) has been increasingly used for maxillofacial modeling. However, the presence of metallic inserts, such as implants, crowns, and dental filling, causes severe streaking and shading artifacts in a CBCT image and loss of the morphological structures of the teeth, which consequently prevents accurate segmentation of bones. A two-stage metal artifact reduction method is proposed for accurate 3D low-dose maxillofacial CBCT modeling, where a key idea is to utilize explicit tooth shape prior information from intra-oral scan data whose acquisition does not require any extra radiation exposure. In the first stage, an image-to-image deep learning network is employed to mitigate metal-related artifacts. To improve the learning ability, the proposed network is designed to take advantage of the intraoral scan data as side-inputs and perform multi-task learning of auxiliary tooth segmentation. In the second stage, a 3D maxillofacial model is constructed by segmenting the bones from the dental CBCT image corrected in the first stage. For accurate bone segmentation, weighted thresholding is applied, wherein the weighting region is determined depending on the geometry of the intra-oral scan data. Because acquiring a paired training dataset of metal-artifact-free and metal artifact-affected dental CBCT images is challenging in clinical practice, an automatic method of generating a realistic dataset according to the CBCT physics model is introduced. Numerical simulations and clinical experiments show the feasibility of the proposed method, which takes advantage of tooth surface information from intra-oral scan data in 3D low dose maxillofacial CBCT modeling.Index Terms-cone beam computed tomography, metal artifact reduction, deep learning, digital dentistry, intra-oral scan arXiv:2202.03571v1 [eess.IV] I. INTRODUCTION In the field of dentistry, dental cone beam computed tomography (CBCT)-based maxillofacial modeling has been widely utilized to understand the complicated anatomical structures of the mandible, maxilla, or skeleton for various clinical tasks [19], [33], [34], [53], [55], [56], [61], [66]. Dental CBCT provides a three-dimensional (3D) maxillofacial model with a reasonably high resolution at a low radiation dose and cost. However, in the presence of metallic inserts, such as implants, crowns, and dental filling, metal artifacts cause the reconstructed CBCT image to deteriorate, making it difficult to segment tooth structures accurately [18], [40], [52], [58]. The metal artifacts are caused by several physical factors, such as beam hardening, scattering, noise, nonlinear partial volume, and photon starvation. Reducing metal-related artifacts in lowdose dental CBCT has drawn increased attention in digital Manuscript received XXX; revised XXX. Corresponding author: Hyoung Suk Park ([email protected]). dentistry workflows because the number of people with oral metallic appliances is rapidly and steadily growing [11], [14], [43], [48], [54], [57], [60]. There have been numerous studies on metal artifact reduction (MAR) in computed tomography (CT) imaging, which include sinogram inpainting-based correction [1], [4], [27], [38], [44], statistical iterative correction [10], [13], [37], [41], [67], and dual-energy reconstruction approaches [3], [32], [68]. However, existing MAR methods are not fully satisfactory for clinical use. The inpainting-based correction approach can generate secondary artifacts owing to inaccurate interpolation along the metal trace in the sinogram. The statistical iterative correction and dual-energy approaches require a large computational cost and an additional radiation dose, respectively. Recent advances in deep learning technology are progressing in MAR. [70] used deep learning to generate an artifact-reduced prior image, then used the projection of the prior image to replace the metal-affected projection, and then performs the final MAR CT reconstruction. [20] performed a residual learning to correct metal streaking artifacts after the first pass by normalized metal artifact reduction (NMAR). Metal artifacts are non-local and highly associated with various factors, including the geometry of metallic inserts and the energy spectrum of the incident X-ray beam, making it difficult to learn their complicated structures in the image domain. [46] proposed a deep learning-based sinogram correction method to reduce the primary metal-induced beam-hardening factors along the metal trace in the sinogram. This method was applied in the restricted situation of a patient-implantspecific model in which two simple metallic objects were placed in the hip joint. [29] proposed a dual-domain network (DuDoNet) to restore sinogram consistency and enhance CT images simultaneously. DuDoNet pursues MAR enhancement by leveraging dual-domain learning networks: sinogram and image enhancement networks. Here, the dual networks operate separately on the sinogram and image domains, but are trained in an end-to-end fashion. The sinogram enhancement network learns how to correct a metal-affected sinogram by using an inpainting loss on the metal trace and sinogram consistency loss. Since the final image is the output of the image enhancement network, the data fidelity may be compromised, resulting in anatomical structure changes. [69] proposed a dual-domain joint learning network that first generates a good prior image with fewer metal artifacts; then, the forward projection of the prior image is utilized for sinogram enhancement. The final Fig. 1. Low-dose dental CBCT uses a small detector with offset array. The small detector size leads to a small area of the scanner FOV, which causes the patient's head to be cut off the sinogram data in the transversal direction. This incomplete sinogram data can be combined with beam hardening of the teeth, creating streaked artifacts. Photon starvation is very common in dental low-dose X-ray CBCT, especially when the patient has many implants. output is the reconstructed image from the sinogram modified only in the metal trace area. Although the above sinogram-domain learning methods, including dual-domain learning, have shown potential to improve the overall image quality, it is difficult to apply such approaches to a practical dental CBCT environment in terms of offset detection, field of view (FOV) truncation, low radiation dose, and 3D characteristics in image reconstruction. First, dental CBCT sinogram data are severely contaminated under the influence of complex factors related to complex geometries of teeth and bones, FOV truncation, offset detection, scattering, etc. See Fig. 1. Hence, sinogram inpainting using deep learning is much more difficult than that in a standard CT environment. Sinogram inpainting is very different from image inpainting, because sinogram inpainting requires sophisticated utilization of global information outside the inpainting area, as well as surrounding local information, while maintaining sinogram consistency. Second, in 3D CT, sinogram learning or even simultaneous learning of image and sinogram domain networks is challenging because of several factors, such as high dimensionality, and huge computation and memory burdens. The aforementioned MAR methods, including sinogramdomain learning, have focused on a 2D fan-beam CT environment. Most importantly, dental CBCT scans are obtained with a significantly lower radiation dose. Therefore, when multiple and strong metallic inserts occupy a significant area, the corresponding sinogram is frequently missing along the metal trace, resulting in the loss of tooth structures around the metallic inserts in the reconstructed image. Unfortunately, it is still arduous to restore the missing morphological structures effectively, regardless of the remarkable capability deep learning has shown for estimating expected values by exploring prior information from the training dataset [23], [62]. With the development of intra-oral scanning technology, intra-oral scanners are rapidly being adopted in workflows 3D dental CBCT and intra-oral scan data. The intra-oral scan data can provide 3D surface information of teeth. We assume that intra-oral scanning provides exact tooth boundary information. of digital dentistry [45]. As shown in Fig. 2, the intra-oral scanner makes it possible to acquire 3D surface information of teeth directly from the oral cavity, and a joint utilization of intra-oral scan with CBCT has been being considered for several clinical applications (e.g., orthognathic surgery planning, implant guiding, and occlusion analysis) that require sophisticated understanding of both maxillofacial structure and dentition [2], [16], [17], [31], [42], [50], [51], [64]. In this study, a novel MAR method is developed to construct a 3D maxillofacial model from low-dose dental CBCT, where a key idea is to utilize explicit tooth shape prior information from intra-oral scan data whose acquisition does not require any extra radiation exposure. The proposed method comprises two core stages; (i) deep learning-based MAR and (ii) maxillofacial modeling. In the first stage, image-to-image deep learning is employed to mitigate metal-related artifacts from CBCT images, where metal-artifacted and images are used as input and ground-truth, respectively. To improve the learning ability, the proposed network is designed to take advantage of the intra-oral scan data as side-inputs and perform the multi-task learning of auxiliary tooth segmentation [9], [63]. The suitable incorporation with explicit shape-prior of intra- Fig. 3. Overall process of the proposed metal artifact reduction method with explicit shape-prior of intra-oral scan data for 3D low dose maxillofacial CBCT imaging. oral scan data can bring significant positive effect in terms of learnability and feature extraction [30]. In the second stage, a 3D maxillofacial model is constructed by segmenting the bones from the CBCT images corrected in the previous stage. For more accurate modeling, the weighted thresholding is applied, where the weight region is determined depending on the geometry of the intra-oral scan data. In clinical practice, it is challenging to obtain a paired training dataset of and metal-artifacted CBCT images and corresponding intra-oral scan data, simultaneously. To circumvent this difficulty, a realistic paired training dataset for MAR is generated through the following procedures, which do not involve any time-consuming and labor-intensive manual process. Metallic inserts, such as dental crowns and tooth implants, are automatically generated in basis of individual tooth segmentation from a metal-artifact-free CBCT image. Then, according to the CBCT physics model, realistic metalaffected sinogram data are generated. For intra-oral scan data generation, it is assumed that intra-oral scanning provides exact tooth boundary information in CT images. This study, for the first time, investigates the potential impact of using intra-oral scan data in MAR and maxillofacial modeling. Numerical simulations and clinical experiments demonstrated the feasibility of the proposed method and the benefit of using intra-oral scan data in 3D low dose maxillofacial CBCT imaging. II. METHOD The proposed method is developed for 3D low-dose maxillofacial CBCT imaging, where the sinogram data P can be expressed as P " S ub p´lnˆE ηpEq expp´R˛µ E qdE`nq(1) Here, µ E is the attenuation coefficient distribution of a 3D human body to be scanned at an energy E, η is the normalized energy distribution of the X-ray source, n is the CT noise, and S ub is a subsampling operator determined by the size and arrangement of the detector (typically, small and offset). In the presence of metallic objects inside the FOV, the standard FDK algorithm [15], denoted by R : , produces severe streaking and shadowing artifacts that cause the image quality of maxillofacial structures to deteriorate. Hence, high-quality 3D maxillofacial imaging is arduous only with the image R : P. The goal of the proposed method is to provide a high-quality 3D maxillofacial image from metal-affected sinogram data P by leveraging intra-oral scan data O. The output should be competitive with a "gold-standard" maxillofacial image y mf acquired from an artifact-free CT image y " R : P ‹ , where P ‹ represents the artifact-free sinogram data corresponding to P. The intra-oral scan data O provide a 3D tooth surfaces that can be useful as prior information about tooth geometry. It is assumed that intra-oral scan data O provides exact tooth boundary information. The proposed method is based on the image-to-image learning approach and weighted thresholding that leverages intra-oral scan data as explicit shape prior information of tooth geometry for MAR. As illustrated in Fig. 3, the reconstruction map f can be expressed as f " f α-WT˝fIE˝R : (2) where ‚ R : is the weighted FDK algorithm involving the sinogram extrapolation method for addressing offset detector arrangement and FOV truncation. ‚ f IE is the tooth geometry prior information-based-imageenhancing network f IE , which mitigates metal-related artifacts. ‚ f α-WT is a weighted thresholding, wherein the weighting region is determined in basis of the α-shape from intraoral scan data. This procedure is used for further removing the remaining streaking artifacts around the teeth in constructing a maxillofacial model. Here, the input of f is a pair of metal-affected data P and intra-oral scan data O (i.e., f : pP, Oq Þ Ñ f pP, Oq « y mf ). The FDK reconstruction R : is as follows. For given P, an image R : Ppx, zq is defined bŷ 2π 0 ωpu β,x qˆR PpP)pβ, u, v β,x,z qR 3 pu β,x´u q 4πU 2 β,x b R 2`u2`v2 β,x,z dudβ (3) where ω is a weighting function satisfying ωpuq`ωp´uq " 1 and P is a constant padding operator that fills the truncated regions by boundary values [6], [59], [65]. Here, px, zq with x " px 1 , x 2 q is the 3D Cartesian coordinate system, β is the projection angle of the X-ray source rotated along the circular trajectory, pu, vq is the coordinate system of the 2D flatpanel detector, R is the distance from the X-ray source to the isocenter, is the 1D ramp filter, U β,x " R`x¨θ K β , v β,x,z " zR{U β,x , u β,x " Rpx¨θ β q{U β,x , θ β " pcos β, sin βq, and θ K β " p´sin β, cos βq. See the blue box in Fig. 1. Stage 1. Image-enhancing network f IE The image-enhancing network f IE is designed to utilize intra-oral scan data as the tooth shape prior while mitigating metal-related artifacts. In our experience, an image domainlearning-based approach can mitigate metal-related artifacts effectively, whereas it tends to have weakness in recovering tooth shape. especially when being destroyed by severe artifacts or when being missed. To compensate for this weakness, we attempt to take advantage of supplemental shape information from intra-oral scan data. We emphasize that data acquisition by the intra-oral scanner does not increase the total amount of radiation exposure to a patient. Let x be a 3D CBCT image reconstructed using the FDK algorithm (i.e., x " R´1 pPq). The image-enhancing network f IE aims to provide f IE px j , O j q « y j(4) where y j is the j-th slice of a metal=artifact-free image (i.e., y = R´1 P ‹ ). It is also desirable that the output satisfies Bf IE px j , O j q\| teeth « O j(5) where Bf IE px j , O j ; w 2 q\| teeth is a binary mask of the tooth surface region on the output image f IE px j , O j ; w 2 q. To accomplish these goals, two strategies are adopted; sideinput layer and multi-task learning. First, additional information of intra-oral scan data is repeatedly enriched during feature extraction in an encoding path. These side inputs can help the network extract tooth shape while compensating for missing or severely distrusted structures through high quality shape information provided by intra-oral scan data. Second, multi-task learning is applied, which learns image reconstruction and auxiliary tooth segmentation in a parallel fashion. In the medical imaging field, it has been reported that deep learning-based image reconstruction ability can be boosted by learning other image-related tasks, such as segmentation and registration [30], [63]. In terms of image recovery, the auxiliary tooth segmentation is expected to reveal the shapes of the teeth in the decoding path and the interference of tooth features, which are joint domain information of the interrelated tasks, through the shared parameters. Fig. 3 shows the overall procedure of the proposed imageenhancing network f IE . Inspired by M-net [36], the proposed network has side-input and side-output layers. In the side-input layers, intra-oral scan data O with suitable resizing is repeatedly added to the encoding path after 3ˆ3 convolution. In the side-output layers, tooth segmentation masks are obtained during the decoding path. The detailed backbone structure can be found in [36]. When ps 0 q piq j is the final network output of i-th training data and j-th slice (i.e., ps 0 q where pps 0 q piq j q p1q denotes the first channel output of ps 0 q piq j , S piq j denotes a binary segmentation mask of the reference tooth region in the metal-artifact-free image y piq j , tps k q piq j u 2 k"1 is a set of side outputs in the decoding path, L 2 is the standard 2 loss, and L ce is the cross-entropy loss. For convenience, the notation f IE px, Oq is used to represent the output image (i.e., the first channel output). Fig. 4. α-shape-based region determination for weighted thresholding f α-WT Stage 2. α-shape-based weighted thresholding f α-WT The next step is maxillofacial imaging from the metalartifact-reduced CBCT image obtained in the previous stage. A final 3D maxillofacial image is obtained by weighted thresholding, which can further reduce the remaining streaking artifacts around teeth. The weighting region is determined depending on the geometry of the intra-oral scan data O. To extract the geometry, the α-shape technique [12] is used. It provides a family of piece-wise linear lines associated with the shape of the teeth. Fig. 4 shows the overall process. When y dl is y dl " f IE px, Oq, the weight thresholding f α-WT can be expressed as f τ α-WT py dl , Oq " p y mf(7) where p y mf " " p y mf ppq " 0 if p P A O or y mf ppq ă τ p y mf ppq " 1 otherwise(8) Here, p is a point in a grid of y mf , τ is a thresholding constant, and A O is a thresholding region obtained using the α-shape from O. The region A O is obtained as follows. For given intra-oral scan data O, O is a point cloud corresponding to O. Denoted by α O , an α-shape of O is given by a polytope with a boundary Bα O , which is defined by Bα O " t ∆ T \| T Ă O, \|T\| ď 3, ∆ T is α-exposed u(9) where ∆ T denotes a simplex for T, and ∆ T is α-exposed if and only if there exists an open ball B α with radius α such that B α XO " H and BB α XO " T. Here, BB α is a boundary of B α . After the α-shape is obtained, an extension direction on each vertex of α O is defined by taking the average of the normal vectors on the faces that contain the vertex. Along the direction, α O is extended while preserving its shape and converted into a binary mask α O , where the inner regions of the shape boundary are filled with one. Finally, the region A O is determined by A O " α O´O(10) where O is the binary mask where the inner part of tooth surfaces in O is filled with one. III. EXPERIMENT AND RESULT A. Experiment Setting The sinogram data of a real patient were obtained from a commercial CBCT machine (Q-FACE, HDXWILL). The voxel size was 1200ˆ654ˆ658 with real scale of 0.2 mm for each axis, where 1200 is the number of uniformly sampled projection views in r0, 2πq, and 654ˆ658 is the number of samples measured by the 2D flat detector for each projection view. CBCT images were reconstructed in a voxel size of 800ˆ800ˆ400 with a real scale of 0.2mm. For cone beam projection and back-projection (FDK reconstruction), an opensource code, TIGRE [5], was used after modification to make it suitable for the experimental setting. All simulated data were consistently generated to have same scale as the real data. A self-developed fully-automated paired data generation tool was used. The detailed process is described in Section III-B. Metal-free CBCT sinogram data were collected from 20 normal patients. They were used for training data generation. Metal-affected CBCT data were collected from nine patients. They were used for test purposes. Among the metal-affected data, real intra-oral scan data for one patient was provided. The intra-oral scan data was acquired from a scanner (i500, MEDIT), where the file format was provided by the Standard Triangle Language (STL). A set of its vertices is a point cloud in millimeters, where the maxilla and mandible are represented by approximately 100,000 and 70,000 points, respectively. For registration into the dental CBCT system, the method described by [26] was applied. In PyTorch environment [47], all deep learning experiments were conducted with a computer system equipped with two Intel Xeon CPUs E5-2630 v4, 128GB DDR4 RAM, and four NVIDIA GeForce GTX 2080ti GPUs. The optimization was conducted using Adam optimizer [28] and multi-GPUs. Batch normalization was applied to achieve fast convergence and minimization [24]. The network capacity (i.e., feature and network depths) was minimized as much as possible while maintaining the backbone structure because of the huge computational cost associated with the CBCT image size of 800ˆ800ˆ400. For α-shape implementation, open source packages, Visualization ToolKit (VTK) and Alpha Shape Toolbox (AST), were used. The adaptive values α and τ were selected empirically. B. Fully-Automated Paired Data Generation Fully automated realistic paired data generation for supervised learning was employed. For given metal-affected data, it is a huge challenge to find the corresponding metal-free data owing to the nonlinear nature of the artifacts. Hence, collecting a clinical paired CBCT dataset of metal-affected and metal-free data from many patients is almost impossible. Instead, a paired dataset can be generated by artificially producing metal artifacts using the Beer-Lambert law (1) with simulated surgery on many normal patient data. However, the generation of realistic data requires time-consuming and laborintensive manual processes to suitably place dental metallic prostheses within the oral structure. To address this difficulty, a fully-automated tool is proposed that provides realistic shape generation and placement of dental metallic prostheses. The tool is designed for dental crown and tooth implants, which are commonly encountered in clinical dentistry. The overall workflow is shown in Fig. 5. As a first step, fully-automated individual tooth segmentation was performed on normal patient data by using the technique reported by [25]. Several tooth positions were chosen randomly in which virtual metal implants could be placed. For the crown case, a crown mask was constructed by cutting the roots of chosen teeth based on crown height information for each tooth [39], and then by the erosion process. The crown thickness was randomly set from 0. 6 tooth that passed through two points of the tooth center in the lowest and middle slices, except those containing a tooth root. Then, the root parts were filled with circles whose center was located at the line, and the radius was empirically set. Using the generated dental crown or tooth implant mask, metal-affected sinogram data was artificially synthesized using the Beer-Lambert law (1) and combined with normal patient sinogram. Simulated intra-oral scan data (i.e., a binary voxel mask of the surface of the teeth) were synthesized by erosion on the combined mask of tooth segmentation and the inserted metal masks. In the experiment, the same settings were used as those of the real dental CBCT machine: an X-ray source with a tube voltage of 85keV and a tube current of 8mA. A metal attenuation coefficient was randomly assigned from {Au, Pd, Ni, Cr, Zr, Al}. For the numerical simulation, the energy distribution of the X-ray source and attenuation coefficient values were those described elsewhere [21], [35]. Poisson and Gaussian noise were added to take account of the CT noise. Seventy 3D CBCT data pairs were generated from 20 normal patient data, where the number of inserted metal implants was randomly set from two to five. Fig. 6 shows several samples of the simulated data using the data generation tool. Fig. 7 shows 3D segmented maxillofacial models by uncorrected image + image thresholding, the proposed network + image thresholding, and the proposed method (the proposed network + the α-shape-based weighted thresholding). The result was obtained using clinical CBCT data and real intraoral scan data. The proposed method clearly enhanced the quality of a 3D maxillofacial model so that it precisely depicted the tooth and bone structures. The α-shape-based weighted thresholding was found to be powerful in real intraoral scan data for high quality maxillofacial modeling. C. Experimental Results To investigate the advantages of the proposed network, performance comparisons were conducted with various MAR methods. The experiments were based on three test sets: synthesized CBCT data + simulated intra-oral scan data, clinical CBCT data + simulated intra-oral scan data, and clinical CBCT data + real intra-oral scan data. Qualitative and quantitative evaluations were conducted on the synthesized CBCT dataset in which the corresponding ground-truth images are given. For clinical CBCT data, qualitative evaluations were performed. It should be mentioned that comparison with other methods is unfair, because the proposed method takes advantage of additional information from intra-oral scan data. 1) Test on synthesized CBCT and simulated intra-oral scan data: Fig. 8 and Table I show qualitative and quantitative performance comparisons of the proposed network with linear interpolation, an image domain network, a sinogram domain network, and a sinogram inpainting network. For the linear interpolation, the sinogram reflection technique reported by [6] was applied to deal with metal trace truncation. Image thresholding was used to extract metal traces. For the image domain network, U-net [49] was trained, which directly maps from an uncorrected image to the corresponding ground truth image. For the sinogram domain network, U-net was trained, which directly maps from an uncorrected sinogram to the corresponding ground truth sinogram. For the sinogram inpainting network, U-net was trained such that only the metal traces in the sinogram were corrected by a network output. In the experiments, the proposed network exhibited the best performance, significantly improving the shape quality of teeth and bone associated with maxillofacial imaging. In particular, the proposed network appears to have an outstanding ability to recover the tooth shape, even if it is fairly disrupted or missed because of metal-related artifacts. As shown in Fig. 9 and Table I, an ablation study for multitask learning (MT) and side input layer (SI) in the proposed network was conducted qualitatively and quantitatively. The single use of MT did not provide any advantage in the sense of improving the reconstruction ability in the quantitative and qualitative sense. Either SI or a combination of SI and MT enhances the reconstruction performance both qualitatively and quantitatively. The combination of SI and MT appears to provide an optimal result owing to the synergistic effect. 2) Test on clinical CBCT and simulated intra-oral scan data: Fig. 10 shows a qualitative comparison of the test set of real clinical CBCT data and simulated intra-oral scan data, where the intra-oral scan data were obtained by tooth segmentation from the clinical CBCT data. Here, the method of [25] was utilized, which provides considerably accurate tooth segmentation, even in the presence of metal-related artifacts. Several simulated intra-oral scan data are listed in the first column of Fig. 10. In three cases from different patients, the proposed network successfully reduced metal artifacts while recovering the boundary of the teeth effectively, whereas the image domain network tended to suffer from loss, blurring, or disruption of the tooth boundary around metal objects. 3) Test on clinical CBCT and real intra-oral scan data: Fig. 11 shows reconstructed results using clinical CBCT and real intra-oral scan data. The proposed method consistently preserves or recovers the boundary of the teeth around metal objects compared with the image domain network. See regions highlighted by yellow arrows in Fig. 11. The performance of the proposed method was compared as well when using simulated and real intra-oral scan data for the same clinical CBCT data. There was some performance degradation in the case of real intra-oral scan data relative to the simulated intra-oral scan case. See the region indicated by the orange arrows in Fig. 11. IV. CONCLUSION AND DISCUSSION In response to concerns about radiation, dental CBCT has been being developed toward the direction of minimizing radiation exposure while maintaining diagnostic image quality. Scanning using the lowest possible tube current setting and the shortest possible exposure time is recommended. Moreover, most dental CBCTs use a detector with a limited size and an offset arrangement to maximize the axial FOV while reducing the radiation dose or manufacturing cost [7], [8]. In the presence of strong and multiple metallic implants in the FOV of dental CBCT, the measured sinogram is highly contaminated under the influence of complex factors related to low radiation. In particular, the sinograms are frequently missing along the metal trace, resulting in loss of tooth structures around the metallic implants. Moreover, the limited size of a detector and offset arrangement result in significant truncation of the sinogram, as shown in Fig. 1. This limited CBCT environment makes it more difficult to apply existing MAR methods. In this article, an MAR method for high-quality 3D maxillofacial modeling in a low-dose dental CBCT environment was proposed. With the rapid development of digital dentistry technology, it has become possible to obtain 3D surface information about teeth using an intra-oral scanner, which has a huge potential to improve the MAR process. This study propose for the first time a MAR method utilizing an intraoral scanner for both MAR and maxillofacial modeling in low-dose dental CBCT, which does not require additional radiation exposure for data acquisition. In the experiments, the tremendous potential of intra-oral scan data to have a significant positive effect on the restoration of tooth shape loss caused by low-dose scans was demonstrated. Although there is still much room for improvement, this study is meaningful as a first attempt to pave the way toward MAR utilizing the shape prior from intra-oral scan data. To train the proposed network, a paired dataset of metalartifacted CBCT, metal-artifact-free CBCT, and intra-oral scan data is required, but data accessibility is limited in clinical practice. Hence, the data generation tool was utilized to provide realistic metal-artifacted CBCT and intra-oral scan data from metal-free CBCT data, where the intra-oral scan data for training are simulated as a set of boundaries of individual teeth segmented in an artifact-free CBCT image. However, the simulation does not fully reflect the real scanning environment, such as scanning protocol, condition, and performance. The difference between the training and test domains results in the performance degradation of the proposed MAR network, as shown in Fig. 11. The performance of the proposed network on real intra-oral scan data can be improved if more realistic simulated oral scan data or a sufficient number of real oral scan data for training can be obtained. Moreover, the ability of the learning-based MAR method can be further improved through complex network architectures (e.g., deep layers or large feature depths) and a largescale paired training dataset. However, there is a trade-off with the total computational cost for learning that can be critical, especially in high-dimensional data applications [22]. Even for the simple M-net architecture shown in Fig. 3, at least 10 days are required for training of 300 epochs with a dataset of 60 Fig. 11. Qualitative comparison of metal artifact reduction with clinical CBCT data; image domain learning (Img DL), the proposed method with real intra-oral scan data, the proposed method with simulated intra-oral scan data. In the second row, we provides an overlapped image of a reconstructed image with the corresponding intra-oral scan data (solid line with apricot color). image voxels under the computational resources used in this study. Even though the use of sophisticated networks or large training datasets can potentially enhance MAR capability, associated hurdles involving high data dimensionality should be addressed for practical dental CBCT applications. Fig. 2 . 2Fig. 2. 3D dental CBCT and intra-oral scan data. The intra-oral scan data can provide 3D surface information of teeth. We assume that intra-oral scanning provides exact tooth boundary information. Fig. 5 . 5Overall process of fully-automated paired training data generationFig. 6. Real and simulated CBCT data. The simulated data is generated by the fully-automated data generation tool represented in Section III-B. Fig. 7 . 7CBCT-based 3D maxillofacial modelling via the proposed method with clinical CBCT and real intra-oral scan data. Fig. 8 . 8Qualitative comparison of metal artifact reduction over simulated data with various MAR methods; linear interpolation (LI), image domain learning (Img DL), sinogram domain learning (Sino DL), sinogram inpainting learning (Sino Inpaint DL), and the proposed network. Case 1 is the best MAR case and Case 2 is the worst MAR case. Fig. 9 . 9Qualitative ablation study for the proposed method; multitask learning (MT), side input layer (SI). Fig. 10 . 10Qualitative comparison of metal artifact reduction with clinical CBCT data and simulated intra-oral scan data; image domain learning (Img DL) and the proposed method. to 1.4 mm. For an implant case, instead of erosion, another process was applied to create an implant screw bar. A line was defined for eachNormal patient CBCT data Individual tooth segmentation Tooth selection [Crown case] [Implant case] Tooth root Cut and Erosion Tooth root Inferred center Cut and Center inference Filling Crown shape Implant shape Normal patient sinogram R −1 R −1 Fully-automated generation of metal shape and location Metal artifact generation Beer Lambert Law P = −ln E η(E) exp(−R µ E )dE + n Erosion and Binarization Intra-oral scan data Metal artifacted image Metal artifacted image QUANTITATIVE COMPARISON OF DEEP LEARNING-BASED MAR RESULTS FOR SIMULATED PATIENT DATA IN TERMS OF NORMALIZED MEAN SQUARE ERROR (NMSE), STRUCTURAL SIMILARITY INDEX (SSIM), AND PEAK SIGNAL-TO-NOISE RATIO (PSNR).Metric Uncorrected LI Sino DL Sino Inp DL Img DL Img DL+MT Img DL+SI Proposed Network NMSE 0.6298 0.4158 0.5445 0.7017 0.3577 0.3885 0.3458 0.3421 SSIM 0.9908 0.9948 0.9882 0.9846 0.9954 0.9924 0.9963 0.9965 PSNR 52.32 55.74 53.40 51.20 57.06 56.36 57.35 57.44 TABLE I Reduction of dental filling metallic artifacts in CT-based attenuation correction of PET data using weighted virtual sinograms optimized by a genetic algorithm. 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[ "Automated Parking Space Detection Using Convolutional Neural Networks", "Automated Parking Space Detection Using Convolutional Neural Networks" ]	[ "Julien Nyambal [email protected] \nSchool of Computer Science and Applied Mathematics\nSchool of Computer Science and Applied Mathematics\nUniversity of the Witwatersrand Johannesburg\nSouth Africa\n", "Richard Klein [email protected] \nUniversity of the Witwatersrand Johannesburg\nSouth Africa\n" ]	[ "School of Computer Science and Applied Mathematics\nSchool of Computer Science and Applied Mathematics\nUniversity of the Witwatersrand Johannesburg\nSouth Africa", "University of the Witwatersrand Johannesburg\nSouth Africa" ]	[]	Finding a parking space nowadays becomes an issue that is not to be neglected, it consumes time and energy. We have used computer vision techniques to infer the state of the parking lot given the data collected from the University of The Witwatersrand. This paper presents an approach for a realtime parking space classification based on Convolutional Neural Networks (CNN) using Caffe and Nvidia DiGITS framework. The training process has been done using DiGITS and the output is a caffemodel used for predictions to detect vacant and occupied parking spots. The system checks a defined area whether a parking spot (bounding boxes defined at initialization of the system) is containing a car or not (occupied or vacant). Those bounding box coordinates are saved from a frame of the video of the parking lot in a JSON format, to be later used by the system for sequential prediction on each parking spot. The system has been trained using the LeNet network with the Nesterov Accelerated Gradient as solver and the AlexNet network with the Stochastic Gradient Descent as solver. We were able to get an accuracy on the validation set of 99% for both networks. The accuracy on a foreign dataset(PKLot) returned as well 99%. Those are experimental results based on the training set shows how robust the system can be when the prediction has to take place in a different parking space.	10.1109/robomech.2017.8261114	[ "https://arxiv.org/pdf/2106.07228v1.pdf" ]	38,367,937	2106.07228	f330c9be0232e6cf40e42a89a62657655241ead2	Automated Parking Space Detection Using Convolutional Neural Networks Julien Nyambal [email protected] School of Computer Science and Applied Mathematics School of Computer Science and Applied Mathematics University of the Witwatersrand Johannesburg South Africa Richard Klein [email protected] University of the Witwatersrand Johannesburg South Africa Automated Parking Space Detection Using Convolutional Neural Networks Finding a parking space nowadays becomes an issue that is not to be neglected, it consumes time and energy. We have used computer vision techniques to infer the state of the parking lot given the data collected from the University of The Witwatersrand. This paper presents an approach for a realtime parking space classification based on Convolutional Neural Networks (CNN) using Caffe and Nvidia DiGITS framework. The training process has been done using DiGITS and the output is a caffemodel used for predictions to detect vacant and occupied parking spots. The system checks a defined area whether a parking spot (bounding boxes defined at initialization of the system) is containing a car or not (occupied or vacant). Those bounding box coordinates are saved from a frame of the video of the parking lot in a JSON format, to be later used by the system for sequential prediction on each parking spot. The system has been trained using the LeNet network with the Nesterov Accelerated Gradient as solver and the AlexNet network with the Stochastic Gradient Descent as solver. We were able to get an accuracy on the validation set of 99% for both networks. The accuracy on a foreign dataset(PKLot) returned as well 99%. Those are experimental results based on the training set shows how robust the system can be when the prediction has to take place in a different parking space. I. INTRODUCTION As the population grows, the number of private vehicles increases as well. But the number of parking spaces most of the time remains. Sometimes, and most of the time, there are vacant spots, but the drivers do not have any information about them. It could be, either the free spot is far from them, or it is hidden by some other cars or any other objects big enough to hide the spot. In the past, and maybe at some places now, parking spaces are managed by some persons in the parking lot who might not have the total view of the next available parking space. Sometimes the driver him/herself has to check for a vacant space by circling in the parking lot, and another driver will come and many losses are generated: time, fuel, and maybe temper. Some researchers came up with different parking space detection algorithms using gadgets like video cameras or sensors to detect a vacant spot when needed. Those algorithms combined gave birth to automated parking detection. The target of the system is mostly open areas like shopping mall parking lots or university parking lots. Implementing a sensor-based approach to automating the parking spot detection will require paperwork, time will be wasted and more trouble to the users when trying to park their vehicles while installing those devices on the ground. On the other hand, those parking lots are monitored by video cameras for security purposes, which is the case in most shopping malls as well. Video-based detection for detection is being used in many areas like obstacle detection, human detection [1]. Convolutional Neural Networks architecture (CNN) is similar to the human neural network build with synapses (weights) and neurons. From this point of view, complex tasks can be learned through the network. This uses CNN with pre-existing architectures to detect in real-time the vacancy of a parking spot. II. RELATED WORK A sensor-based approach can be used to detect a moving object entering the parking lot. The size of the object and the time it takes to cross the gate where the sensor is located infers the occupation of a spot by the car [2]. Due to the cost linked to the installation and maintenance of sensors, their use is broad but restricted when budget is an issue. Another big issue of the sensors, they cannot be implemented outdoors, because there is no roof to install a sensor. The approach in [3] presents a video-based system for parking space classification that has been tested and resulted in a success when it has been also working on another parking lot without re-training the model. To solve the problem of occupancy of a parking spot, color histograms and Differenceof-Gaussian (DoG) have been used for feature extraction, and three machine learning algorithms have used for classifying between a free or occupied parking spot. Those three learning algorithms are: K-Nearest Neighbor (k-NN), Linear Discriminant Analysis (LDA) and the Support Vector Machines (SVM). To avoid errors during the classification process, temporal integration exponential smoothing has been applied to the classification results. DoG and color histograms (features extractors) have been combined to the three machine learning algorithms in a one-to-one relationship, to produce the best performing combination of features extractors and machine learning algorithms used. The system has been tested with a success rate of more than 92% on an unseen parking lot [3]. Wu et al. [4] proposed a method for detecting available parking space using both Support Vector Machines and a Markov Random Field framework. The input is a video, and from the images, the system distinguishes between empty spaces from occupied spaces in a parking lot. The system is trained to recognize empty parking spaces by applying a machine learning algorithm rather than segmenting the image of the vehicle from the parking space. The system is composed of 4 parts: pre-processing, ground model feature extraction, multi-class Support Vector Machines recognition, and Markov Random Field based correction. Features are extracted from the color histogram, 50 features are selected after dimension reduction. Support vector machines are used to classify the occupancy of a parking spot based on the patches of the parking lot manually determined. Markov Random Field (MRF) is used to improve the performance done by the Support Vector Machines. The results of the experiments show that Support Vector Machines used without Markov Random Field produce 84.35% using a single space for detection and 85.57% using 3 spots. When using Markov Random Field, the accuracy climbs to 93.53%. The accuracy increases with the training sample, which shows the strength of the algorithm used. A robust dataset of images of parking spaces under different weather conditions can be used to better generalize classification of the state of a spot [5]. The authors use the Local Binary Pattern (LPB) and the Local Phase Quantization (LPQ) textual descriptors for feature extraction. For their classifier, they combined the Support Vector Machines with some variants of the LPQ and the LPB which are the LBP Rotation Invariant, LPQ with Gaussian window, and the LPQ gaussian derivative quadrature filter pair to achieve a classification rate of 99.64%. Amato et al. [6] present an approach to detecting parking spots using deep learning techniques. The authors trained their classifier on two datasets: PKLot [5], and their own dataset from the National Research Council in Pira (CNRPark). Two network configurations have been used: mAlexNet: is a mini version of the AlexNet network and mLeNet: is the mini version of the LeNet network. Those mini-networks are modified well-known CNN architectures (AlexNet and LeNet respectively) to accelerate the process of classification, as the original ones are very expensive in time when resources are not equivalent to the big size of the network. This method produces an accuracy greater than 90% inter and intra-datasets. III. THE DATASET The dataset is made of 782 images of parking lots. Those parking spots have been extracted from 2 parking lots of the University of the Witwatersrand. A. Data Collection The data collection, using a GoPro Hero 4 Black camera, took a working week (5 days), from 6 am to approximately 8 am at the rate of one frame per minute. During that period (a) Original image with radial distortion of type Barrel. (b) Image with radial distortion corrected. of time, the parking gets filled actively and rapid change of illumination is captured. The camera was set to take pictures at 5 MegaPixels which is in a resolution of 2560 x 1920 pixels. Due to the "Wide" setting of the GoPro, severe radial distortion appears on all the frames. It is of type Barrel due to the fisheye lens of the camera [7], which affects the straight lines of the parking spots as they tend to be curved. The position of the camera and the choice of the buildings have been chosen in such a way that a car from a parking spot does not hide more than 60% of the next car around. The radial distortion (Barel) has been manually fixed using distort tools and the size of the images manually fixed using convert. Distort and convert are both Unix utilities [8]. Each frame has been resized from 2560 x 1920 pixels to 1000 x 1000 pixels. Fig.1a is the original frame captured by the camera that is distorted due to the fisheye lens of the camera. Fig.1b is the result of all the manipulations on the original images coming directly from the camera to correct the distortion. B. Labeling The process used is to crop from each image of a parking lot each individual parking spot so that a label (occupied or vacant) can be manually assigned to it. To do so, the cartesian coordinates of each spot stored in an XML file generated by a labeling tool will help to extract them from the image. That tool extracts the patches of all the manually selected parking spots, using the XML file generated from a single image. Since the parking lot does not move, the same XML file has been reused on the other images to get the multiple patches [9], [10]. Fig.2a and Fig.2b represent respectively the cropped image of an occupied and vacant parking spot. After the labeling, vacant and occupied parking spots patches have been separated into different folders, vacant and occupied folders respectively. The raw dataset contains 782 images of parking lots. IV. DEEP LEARNING WITH CAFFE AND NVIDIA DIGITS A. The System The system built works on the fact that the images of the parking spot will represent a fixed parking lot. The system analyzes the parking lot after every fifth frame. A frame represents a state of the parking spot at a precise moment. The system works on live feed or on a recorded video. Fig.3 represents the flow of the data from raw images from video frames to take a decision on the vacancy of a parking spot. On Fig.3, first frame and video on frame-by-frame represent a state of the parking lot and "1", "2", "3", "4" represent the parking spots on each frame. A-The System launches the frame required to define the parking spots, B-The coordinates of the parking spots are stored in a JSON file (.json), will get generated if it does not exist, C-The System loads the video and the JSON file (.json), D- Fig.3 corresponds to one iteration of the loop in the number of parking spots. The first iteration will crop the spot one from the frame n of the loaded video, E-The System loads the labels file, the caffemodel, and the deploy file corresponding to the caffemodel, F-Those three arguments on the system are applied to the cropped image of the parking lot, save on the disk. In Fig.3, the cropped image is the parking spot "1", G-The result of the prediction from the System with the arguments applied on the frame n. A red or green rectangle corresponding respectively to either an occupied or vacant parking spot will surround the cropped image after the classification. B. Configuration Nvidia Deep Learning package DIGITS, which a web server that manages the activities of the GPU (Nvidia only) to efficiently perform the classification tasks. The background heavy tasks are done with Caffe. was used for both the training and testing phases. It has been used for training and testing. The CPU could also be used for training and testing but the performances will be relatively slower if a single CPU is to be used. 1) Hardware and software: The implementation of the system has been done on a desktop with an Intel i5 processor at 3.20GHz with 4 cores. The GPU used is a Nvidia GeForce GTX 750 Ti with 2GB video RAM. The system is trained under Ubuntu 14.04 with Nvidia DIGITS 3, using cuDNN [11] which has been shown to accelerate CNN training and testing processes [12]. 2) Different Convolutional Neural Networks Configurations Used: Two CNNs architectures were used and compared, AlexNet [13] and LeNet [14]. AlexNet is very good for image classification, first due to the depth of the architecture, and other factors like dropouts. Another important factor to note, AlexNet has been trained on 1000 classes with more than 1 million images, which implies that the weights are the depth of the network could be reused for fine-tuning. with some tweaking LeNet in its parameters performs better. LeNet architecture takes as input grayscale images of size 28x28, whereas AlexNet takes as input color images of size 256x256. All images are normalized by subtracting the training mean from all the images before the training begins. AlexNet has been slightly modified as follows: five convolutional layers, followed by max-pooling, norm, ReLU layers, and three fully-connected layers. Fig. 4: Definition of x-min, x-max, y-min, y-max of a single spot. Those coordinates are relative to the current image of the parking lot and will not change since the setup of the frame will remain static. LeNet as well has been slightly modified as follows: two convolutional layers, followed by max-pooling layers and two inner products layers. C. Processing and Prediction We show some details of the processing that happens in the system, from the image acquisition to the prediction of the specific status of the selected parking spot. 1) Image Acquisition: The images are taken from a video of a parking lot stored on a local workstation. Using OpenCV libraries, the video is read and generates frames to perform the predictions. 2) Definition of Parking Spots: The first frame of the footage (or a frame that contains mostly occupied parking spots) is used to define what a parking spot is in that current frame when the program starts. This technique of defining a parking spot on one frame relies on the fact that the camera will not move during the whole process of prediction and logically the parking lot does not move, only the cars are moving in or out of the different defined parking spots. Therefore this definition applies to the subsequent frames. The user defines the spot by clicking on two corners of the parking spot (the upper corner left and the lower corner right). After that set of clicks, the parking spot is defined and classified on the go. 3) Storage of Coordinates on JSON file: A JSON file is created by the system to hold the coordinates of the different parking spots defined by the user. The JSON coordinates (terminologies used in the JSON format are explicitly explained in Fig.4) are stored in the following format: { "spot_1": { / * Spot identifier * / "name": "spot_1", / * Spot identifier * / "xmax": 448, / * Spot maximum x-coordinate * / "xmin": 345, / * Spot minimum x-coordinate * / "ymax": 542, / * Spot maximum y-coordinate * / "ymin": 478 / * Spot minimum y-coordinate * / }, ... Algorithm 1 Classification Input: model, deploy file, cropped image, label file Output: The actual class of the cropped image (string) 1 4) Prediction and classification: The JSON file previously generated is used to perform the classification of each spot. The following set of operations will take place: 1) Cropping: OpenCV allows processing the spots on a frame-by-frame basis. The system takes as one of its parameter the JSON file that contains the set of coordinates and the video. Parking spots will be cropped from each frame of the video based on the JSON file. The cropping is done sequentially and the corresponding image is stored on the disk, in other words, only one cropped image is stored and the next picture overrides it later on. 2) Prediction and classification: After the training phase, the generated Convolutional Neural Network model comes along with two other files: label.txt containing the different label of a spot (vacant or occupied) and the deploy.prototxt, which is the network configuration used by the model. This file defines the type of input allowed by the model and the protocol followed to produce the output. The system takes as arguments the model created at the training phase, the corresponding labels, the network configuration (deploy.prototxt), and the cropped image of the parking (one at a time). Those arguments are processed through a borrowed function written by Nvidia, classify [15], which is a part of DiGITS. Modifications were made only in the classify function as shown in Algorithm 1. Those modifications allow to visually determine whether a spot is either vacant (green rectangle) or occupied (red rectangle), based on the score or confidence level (result of the probability model computed) produced by the classification method. Those results are then streamed to the main system to produce the output. (a) Prediction on some spots 3. The system does get triggered with a human passing through the classification area, which is classified as vacant. The spot 1 and 3 (first two rectangles to the left) are correctly classified are occupied. (b) Prediction on some spots 2. The street sign does not trigger the classifier as it is too small to change the color distribution of the area covered by the green rectangle. For the experiments, the comparison of LeNet and AlexNet allowed us to run the classification on a recorded video 1 containing a parking lot, with some modifications on the hardware and the weight of the inputs (images) that each network has to process. The next section presents the different sets of CNN and parameters employed and the corresponding performances on the earlier stated video. A. Dataset: Input to the Convolutional Neural Networks From Section III, a set of images of both statuses a parking spot: occupied or vacant. As shown in Table I, we have a very small number of images in the dataset for training compared to the number of images in the dataset from Section III. Many instances of identical pictures were removed to avoid any type of overfitting. Based on the two selected CNNs to develop the system, we shaped the data according to the input requirements of the concerned CNN. For the AlexNet, all the images are resized to 256x256. Those images are in RGB. For the LeNet, all the images are resized to 28x28. Those images are in grayscale. From Fig.6, the image mean is giving a clear approximation of all the status of a parking spot at any given time, including the parking lines. Table I shows the split of the dataset used for training and validation processes with the ratio of 75% for training and 25% for testing. B. Configurations of Convolutional Neural Networks Given the datasets generated from Section V-A, we built some CNN configurations, based on LeNet and AlexNet, to use them in the system for classification. For all the network configurations, the Base Learning Rate or Base lr is initialized at 0.01. The Learning Rate Policy or Policy lr is the way the learning rate will change with time. The learning rate is be divided by 10 after each 33% portion of the training set. Three solvers for the classifier are mainly used to compare the results. The three solvers used are: 1) Stochastic Gradient Descent (SGD) 2) Adaptive Gradient Descent (AdaGrad) 3) Nesterov's Accelerated Gradient (NAG) We have made 30 passes through the training set (epochs). Table II is the summary of all the custom configurations to get the best accuracy with the validation set along with the speed performance. The validation set is made out of 25% of the entire set. The validation set is entirely independent of the training set. C. Results After the training phase, very high accuracy values are noted at validation level based on Table II (accuracy calculated during training with images that are closely similar to the training set, but not in the training set). For the accuracy of the system, PKLot dataset has been used [5] to assess the strength of the system to classify on a foreign parking lot. Table II shows the testing results of the classifier after the training and validation process. The test data has been generated from unseen images of parking spots to the classifier. The performance of all different CNN configurations has been tested against the test images. All those configurations received the same batch of test images at once to simulate a close to real-time heavy flow. In the same table, during testing AlexNet with the Nesterov Accelerated Gradient, the batch of images took very long to terminate. Giving that the system is meant for a real-time purpose, time is an important factor to monitor and delays are considered as failure. Fig.5a and Fig.5b represent the results produced by the system with AlexNet as the classifier. The change of lighting, which is a big factor misbehavior of many classifiers, does not affect too much the classification done by the system trained on this relatively small dataset. The classification runs on a single frame rather than a stream of frames, where instead of 25 frames per second (fps), it classifies on 5 fps, which takes into account the 2GB of the Nvidia graphic memory. VI. CONCLUSION In this paper, we have shown with success that computer vision with a single camera, using Convolutional Neural Networks is having a success rate close to traditional methods used in the past. The classifier made of LeNet architecture with the Nesterov's Accelerated Gradient as solver produced an accuracy of 93.64%, and the AlexNet architecture with the Stochastic Gradient Descent as solver produced an accuracy of 95.49% which are better than both Amato et al. [6] and Wu et al. [4] who produced respectively 90.4% and 93.52%. Given the hardware used for the research, the LeNet architecture with the Nesterov Accelerated Gradient is faster than the AlexNet architecture with the Stochastic Gradient Descent when tested. The use of Convolutional Neural Networks facilitates the process of building a classifier as they automatically extract and use the features from the dataset. As shown in Fig.5, the system is able to produce an accurate output based on the situation of the current parking spot the camera is facing, given that spot is well defined by the user when launching the system. We have noticed during the research the presence of noise due to either the light, some elements stuck on the ground (oil leaks, cracks ...), or humans passing across parking spots. Those events have been corrected by improving the dataset based on the event of false classification due to noise. We have obtained the best combination in terms of parameters of the classifier by having the Nesterov Accelerated Gradient as solver and LeNet as network architecture, especially for the speed of execution compared to the better accuracy produced by the AlexNet architecture with the Stochastic Gradient, but with poor results when it comes to speed (speed is entirely relative to the hardware used). The classification on video is faster and accurate enough to be deployed in a real-life situation, on a parking that does not have any image appearing in the training set. It implies that the system can be used to a new parking without retraining of the classifier (given that the amount of noise is under control. E.g.: lighting, oil, or any type of leaks on the ground...). For future work, we are planning on first improving the positioning of the camera so that we get a clear view of each car in the parking lot without a car hiding any part of the next car. Given that the availability of the parking spots is deduced from a well-defined parking lot, the system could direct the driver to the nearest available spot using some artificial intelligence techniques. Fig. 1 : 1Correction of radial distortion (Chamber of Mines,University of the Witwatersrand). Fig. 2 : 2Samples of vacant and occupied parking spot. Fig. 3 : 3System Processing Flow. Fig. 5 : 5System in action on predicting vacancy of parking spot V. EXPERIMENTS AND RESULTS 1 https://www.youtube.com/watch?v=U7HRKjlXK-Y Fig. 6: Image Mean of the dataset: Vacant and Occupied spots. TABLE I : IConfiguration of the 2 datasets used for to train the system Dataset Dimensions Size Entries(train & val) Parking(Colour) 256x256x3 84.3MB 2026 & 676 Parking(Grayscale) 28x28x1 2.16MB 2026 & 676 TABLE II : IIAll configurations for the system training using AlexNet and LeNet using different types of solversNetwork Solver Accuracy(Train) Accuracy(Test) AlexNet SGD 0.99 0.95 AlexNet NAG 0.99 Never Ended AlexNet AdaGrad 0.98 0.89 LeNet SGD 0.99 0.92 LeNet NAG 0.99 0.93 LeNet AdaGrad 0.99 0.89 ACKNOWLEDGEMENT I would like to thank the Council for Scientific and Industrial Research (CSIR), Adrian[16], Prof. Turgay Ç elik, and the PIMD (Property and Infrastructure Management Division) at the University of the Witwatersrand for their contribution in this research project. 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[ "Surface structures of ZrO 2 films on Rh(111): From two layers to bulk termination", "Surface structures of ZrO 2 films on Rh(111): From two layers to bulk termination" ]	[ "Peter Lackner \nInstitute of Applied Physics\nTU Wien\nViennaAustriâ\n", "Zhiyu Zou \nInstitute of Applied Physics\nTU Wien\nViennaAustriâ\n", "Sabrina Mayr \nInstitute of Applied Physics\nTU Wien\nViennaAustriâ\n", "Joong-Il Jake Choi \nInstitute of Applied Physics\nTU Wien\nViennaAustriâ\n", "Ulrike Diebold \nInstitute of Applied Physics\nTU Wien\nViennaAustriâ\n", "Michael Schmid [email protected] \nInstitute of Applied Physics\nTU Wien\nViennaAustriâ\n" ]	[ "Institute of Applied Physics\nTU Wien\nViennaAustriâ", "Institute of Applied Physics\nTU Wien\nViennaAustriâ", "Institute of Applied Physics\nTU Wien\nViennaAustriâ", "Institute of Applied Physics\nTU Wien\nViennaAustriâ", "Institute of Applied Physics\nTU Wien\nViennaAustriâ", "Institute of Applied Physics\nTU Wien\nViennaAustriâ" ]	[]	We have studied zirconia films on a Rh(111) substrate with thicknesses in the range of 2-10 monolayers (ML) using scanning tunneling microscopy (STM) and lowenergy electron diffraction (LEED). Zirconia was deposited using a UHV-compatible sputter source, resulting in layer-by-layer growth and good uniformity of the films. For thicknesses of 2-4 ML, a layer-dependent influence of the substrate on the structure of the thin films is observed. Beyond this thickness, films show a (2 × 1) or a distorted (2 × 2) surface structure with respect to cubic ZrO 2 (111); these structures correspond to tetragonal and monoclinic zirconia, respectively. The tetragonal phase occurs for annealing temperatures of up to 730 °C; transformation to the thermodynamically stable monoclinic phase occurs after annealing at 850 °C or above. High-temperature annealing also breaks up the films and exposes the Rh(111) substrate. We argue that the tetragonal films are stabilized by oxygen deficiency, while the monoclinic films are only weakly defective and show band bending at defects and grain boundaries. This observation is in agreement with positive charge being responsible for the grain-boundary blocking effect in zirconia-based solid electrolytes. Our work introduces the tetragonal and monoclinic 5 ML-thick ZrO 2 films on Rh(111) as well-suited model system for surface-science studies on ZrO 2 as they do not exhibit the charging problems of thicker films or the bulk material and show better homogeneity and stability than the previously-studied ZrO 2 /Pt(111) system.	10.1016/j.susc.2018.09.004	[ "https://arxiv.org/pdf/1808.08301v1.pdf" ]	102,488,935	1808.08301	916afeba6bcbefd1d3686f414458ac830579f514	Surface structures of ZrO 2 films on Rh(111): From two layers to bulk termination Peter Lackner Institute of Applied Physics TU Wien ViennaAustriâ Zhiyu Zou Institute of Applied Physics TU Wien ViennaAustriâ Sabrina Mayr Institute of Applied Physics TU Wien ViennaAustriâ Joong-Il Jake Choi Institute of Applied Physics TU Wien ViennaAustriâ Ulrike Diebold Institute of Applied Physics TU Wien ViennaAustriâ Michael Schmid [email protected] Institute of Applied Physics TU Wien ViennaAustriâ Surface structures of ZrO 2 films on Rh(111): From two layers to bulk termination Present address: Center for Nanomaterials and Chemical Reactions, Institute for Basic Science (IBS), Daejeon 305-701, South Korea. *email address: We have studied zirconia films on a Rh(111) substrate with thicknesses in the range of 2-10 monolayers (ML) using scanning tunneling microscopy (STM) and lowenergy electron diffraction (LEED). Zirconia was deposited using a UHV-compatible sputter source, resulting in layer-by-layer growth and good uniformity of the films. For thicknesses of 2-4 ML, a layer-dependent influence of the substrate on the structure of the thin films is observed. Beyond this thickness, films show a (2 × 1) or a distorted (2 × 2) surface structure with respect to cubic ZrO 2 (111); these structures correspond to tetragonal and monoclinic zirconia, respectively. The tetragonal phase occurs for annealing temperatures of up to 730 °C; transformation to the thermodynamically stable monoclinic phase occurs after annealing at 850 °C or above. High-temperature annealing also breaks up the films and exposes the Rh(111) substrate. We argue that the tetragonal films are stabilized by oxygen deficiency, while the monoclinic films are only weakly defective and show band bending at defects and grain boundaries. This observation is in agreement with positive charge being responsible for the grain-boundary blocking effect in zirconia-based solid electrolytes. Our work introduces the tetragonal and monoclinic 5 ML-thick ZrO 2 films on Rh(111) as well-suited model system for surface-science studies on ZrO 2 as they do not exhibit the charging problems of thicker films or the bulk material and show better homogeneity and stability than the previously-studied ZrO 2 /Pt(111) system. Introduction The search for a detailed understanding of a material is often driven by the technological applications that rely on it. This is also true for zirconia (ZrO 2 ), which is used as catalyst support [1] and catalyst [2], as a refractory ceramic due to its high thermal stability and strength [3][4][5], and as dental implant material due to its good biocompatibility [6]. Chemically doped zirconia is heavily used as a solid-state electrolyte in solid oxide fuel cells [7] and gas sensors [8]. While the material is an electronic insulator up to high temperatures, it can conduct oxygen (and, thereby, electric charge) via vacancy diffusion, which forms the basis for using zirconia as an electrolyte. As the intrinsic concentration of oxygen vacancies (V O s) in ZrO 2 is very low even at high temperatures and reducing atmosphere, V O s are introduced by chemical doping with trivalent elements such as yttrium. Depending on temperature or dopant concentration, zirconia exhibits three stable bulk structures at atmospheric pressure: For pure, stoichiometric ZrO 2 , the cubic structure (c-ZrO 2 , fluorite lattice) is found above 2377 °C; at lower temperatures the tetragonal phase (t-ZrO 2 , above 1205 °C), and finally monoclinic ZrO 2 (m-ZrO 2 , also known as baddeleyite) are stable [9], see Figure 1a. While all these phases are related to the cubic fluorite structure, pure c-ZrO 2 does not exist at room temperature due to the small O-O distance imposed by the short (strong) Zr-O bonds (d O-O ≈ 256 pm for hypothetical room-temperature c-ZrO 2 [10]). The average O-O distance can be increased by shifting the O atoms alternatingly up or down in [001] direction, leading to the tetragonal phase (d O-O ≈ 260 pm), which, however, is still unstable for pure, stoichiometric ZrO 2 at room temperature. Upon transformation to m-ZrO 2 , Zr-O bonds are broken, the coordination of Zr changes from 8 to 7 and for half of the O atoms the coordination is reduced from fourfold (tetrahedral) to threefold (planar); the volume increases by ≈ 5%. These changes substantially increase the average O-O distance, while the Zr-O bonds remain short. The tetragonal and cubic phases can be also stabilized at room temperature by introducing V O s and by increasing the lattice parameter, which occurs when doping with yttria [9,[11][12][13]. Y 2 O 3 concentrations above ≈ 8 mol% (corresponding to Zr 0.85 Y 0.15 O 1.93 ) stabilize the cubic phase (yttria-stabilized zirconia, YSZ). At lower dopant concentrations one finds mixtures of cubic and tetragonal, or cubic and monoclinic zirconia. The lower doping limit is 1.5-2 mol% Y 2 O 3 , where the monoclinic phase becomes stable [14]. The tetragonal phase is also found in nanoscale ZrO 2 at room temperature; while this was initially attributed to its favorable surface energy [15], newer works rather point towards a stabilization by V O s instead [12,13]. Figure 1b shows the surface termination of c-ZrO 2 (111) and the corresponding lowest-energy terminations of the other phases. Despite the distortions with respect to c-ZrO 2 , all these surfaces are non-polar. The cubic phase exhibits a hexagonal (1 × 1) structure. The shifted O rows in the tetragonal phase lead to a (2 × 1) unit cell w.r.t. the cubic phase. The monoclinic phase features a distorted (2 × 2) surface unit cell (again, w.r.t. c-ZrO 2 ). For this latter phase, density functional theory (DFT) calculations [16] predict that the surface has the lowest surface energy; in (111) contrast to (111), it has only one (instead of two) surface O with twofold coordination per unit cell (marked by an asterisk in Figure 1b). Apart from the above-mentioned structures, several orthorhombic high-pressure phases of zirconia exist; some of these are metastable at ambient conditions [17]. Recently, the orthorhombic phases of ZrO 2 and mixed ZrO 2 /HfO 2 have received increased attention as candidate materials for ferroelectric memory devices [18]. Similar to monoclinic ZrO 2 , these orthorhombic phases are based on distortions of the cubic fluorite structure, again having 7-fold coordinated Zr and O with 3-fold and 4fold coordination [19,20]. When cut along a direction equivalent to c-ZrO 2 (111), the most common orthorhombic I and II phases would exhibit a (2 × 2) or (2 × 4) surface unit cell with respect to the cubic phase, respectively. In spite of its technological importance, zirconia has received surprisingly little attention from the surface-science community. This is partly due to its insulating nature, as most surface-science methods rely on electronic conductance, e.g., scanning tunneling microscopy (STM), low-energy electron diffraction (LEED), or x-ray photoelectron spectroscopy (XPS). Furthermore, the phase transitions make it impossible to grow a ZrO 2 single crystal from the melt. This second limitation does not exist for cubic YSZ, where single crystals are readily available and inexpensive. Morrow et al. [21] used a YSZ single crystal for high-temperature STM studies; this work was conducted at ~300 °C to ensure sufficient conductivity. While atomic resolution was achieved, this approach is limited to high temperatures, and due to Y segregation the surfaces had a rather high Y concentration. Several groups have followed a different approach and used several-monolayer-thick films of pure zirconia on Pt(111) as model systems. Meinel et al. [22][23][24] performed STM studies on up to 10 ML-thick zirconia films that were deposited onto Pt(111) via physical vapor deposition (PVD) of Zr in an O 2 atmosphere. Their work had been built on a previous LEED study by Maurice et al. [25], but involved annealing at higher temperatures. Depending on film thickness and annealing temperature, Meinel et al. found a large number of superstructures in LEED and STM. It must be noted that films as thick as 10 ML broke up upon annealing and eventually dissolved in the Pt substrate, so it is not straightforward to decide which structures should be assigned to the multilayer films and which ones to the Pt-Zr or Pt-Zr-O structures. Nevertheless, it is clear that the initial structures were based on ZrO 2 (111), with a ZrO 2 (111)-(2 × 2) LEED pattern. Also spots interpreted as ZrO 2 (111)-(1 × 1) rotated by ±6.6° w.r.t. Pt(111) after 3 min of annealing at 680 °C can be attributed to the ZrO 2 films. Submonolayer films exhibited (5 × 5) and (√19 × √19)R36.6° superstructures with respect to Pt(111) [26]; the latter structure also appeared when thicker films were annealed at high temperatures. Possibly those formed at areas where the thicker ZrO 2 film has disappeared. STM indicates a band gap at least for films ≥ 2 ML and density functional theory (DFT) indicates that bulk-like band gaps are reached at 5 ML [23]. A different approach to zirconia model systems is the growth of ultra-thin zirconia films by oxidation of alloy single crystals as first shown by Antlanger et al. [27]. Two substrates were used: Pt 3 Zr(0001) [27,28] and Pd 3 Zr(0001) [29]. By annealing these crystals at 400 °C in O 2 , disordered zirconia formed, which consumed Zr from the top layers of the alloy. In the case of Pt 3 Zr, Zr diffusion is slow, so the interface was essentially pure Pt(111). By annealing at 900 °C in UHV, the disordered zirconia transformed to an ordered monolayer corresponding to one O-Zr-O trilayer repeat unit of ZrO 2 (111). For ZrO 2 /Pt 3 Zr, the film exhibited the same (√19 × √19)R23.4° superstructure (this rotation angle is equivalent to 36.6°) as found in previous studies of zirconia on Pt(111) [24,26], confirming that this structure on Pt(111) likely corresponded to an ultrathin film. ZrO 2 /Pd 3 Zr formed an O-Zr-O trilayer with an almost identical in-plane lattice constant (0.35 nm) and a large (√217 × √217)R10.16° superstructure cell. Both, STM measurements and DFT calculations indicated a substantial buckling of the films. These alloy-based, ultrathin zirconia films were successfully used as model systems for surfaces of bulk ZrO 2 in metal growth [30] and water adsorption [31] studies. Thin zirconia films can also be grown by atomic layer deposition using the precursor zirconium (IV) tert-butoxide (ZTB). While this technique is typically used in industry [32] and not in UHV studies, it was successfully applied to deposit submonolayer coverages of ZrO 2 on Pd(111) [33,34] and Cu(111) [35] . However, it remains to be seen whether this method can be used to grow atomically flat zirconia films with a thickness of a few monolayers. In the current work, we present results obtained from zirconia films deposited with a UHV-compatible sputter source [36]. This sputter source features a highly reproducible deposition rate, high purity of the films and higher deposition rates than typically achievable by PVD. We found varying structures with increasing film thickness; for thicknesses below 5 ML the films are strongly influenced by the underlying Rh(111) substrate. For thicknesses ≥ 5 ML, bulk-terminated tetragonal or monoclinic zirconia can be stabilized, depending on the annealing temperature. [10,37] for cubic (extrapolated from doped to undoped ZrO 2 ), [38] for tetragonal, and [39] for monoclinic ZrO 2 . The surface cell of t-ZrO 2 deviates only slightly from two unit cells of c-ZrO 2 . Experimental The UHV system used in this work comprises two-chambers, one for sample preparation and one for analysis. The preparation chamber (base pressure below 10 −10 mbar) contains an ion source for sputtering, and an electron-beam heater for preparation of the substrate Rh(111) single crystal, as well as a home-built, UHVcompatible sputter source for deposition of Zr [36]. The thermocouple for temperature measurement is attached to the fixed part of the sample holder; temperatures above 700 °C are therefore corrected by using a disappearing-filament pyrometer and corrections for lower temperatures are extrapolated from the high temperature values. We estimate the temperatures to be accurate within ±30 °C. The analysis chamber (p base < 7 × 10 -11 mbar) houses a room-temperature STM (Omicron micro STM) and LEED optics (ErLEED). The whole system is suspended on springs for vibration damping. Etched W tips cleaned by Ar + sputtering were used in all STM measurements and conditioned by voltage pulses on a Au(110) crystal. All STM images showing atomic lattices or well-ordered superlattices were corrected for piezo drift as described in Ref. [29]. We chose Rh(111) as a substrate. Compared with Pt(111), it has the advantage of lower solubility of Zr in the bulk, and the 4:3 ratio of lattice constants between ZrO 2 (111) and Rh favors the growth of unrotated zirconia films. A Rh(111) single crystal (diameter 9 mm, height 2 mm, from MaTecK, Germany) was cleaned by cycles of sputtering (2 keV Ar + , 3.6 μA/cm 2 , 10 min) and annealing (T = 920 °C, 10 min). Zirconium was sputter-deposited on the clean Rh substrate at RT in a mixed Ar/O 2 atmosphere (p Ar = 8 × 10 -6 mbar, p O2 = 1 × 10 -6 mbar); these films exhibit excellent purity [36]. The sputter deposition source also leads to some Ar + bombardment of the sample (ion-beam assisted deposition, IBAD); we chose rather gentle operating conditions with Ar + energies below 150 eV (grid voltages of 150 V and 100 V for the front and rear grid, respectively, unless noted otherwise) [36]. The amount of deposited material was calibrated by deposition of metallic Zr and measuring island areas with STM; the coverage was reproducible within 0.1 ML. We give the thickness in ZrO 2 monolayers (ML), with one O-Zr-O repeat unit of c-ZrO 2 (111) defined as one monolayer, which corresponds to ≈ 9 × 10 18 Zr atoms/m 2 or ≈ 0.3 nm thickness. The as-deposited films were not fully oxidized and were therefore post-annealed for 10 min in O 2 (p O2 = 5 × 10 −7 mbar) at temperatures of at least 550 °C. In most experiments the post-annealing temperatures were such that a continuous but well-ordered film was obtained at the given film thickness; at higher temperatures and low film thickness (≤3 ML), holes down to the Rh substrate appeared. Results Zirconia layers of increasing thickness When employing ZrO 2 films as a model system for the surface of bulk ZrO 2 , a compromise between bulk-like properties (requiring thick films) and easy imaging by STM (requiring thin films due to their insulating nature) must be sought. Therefore, we have studied the structure of the films as a function of their thickness, starting from 1.5 ML. After annealing 1.5 ML of zirconia at 550 °C in O 2 (p O2 = 5 × 10 -8 mbar), the film partially de-wetted the surface and a 2 ML-thick film with holes to the Rh(111) substrate formed, see Figure 2a. A single monolayer was found to be unstable under these annealing conditions. At first glance, the LEED pattern suggests a structure with (3 × 3) oxide units per (4 × 4) Rh cells (marked by red lines in Figure 2a), which would require an oxide lattice of 358 pm. This would be an expected structure, as three c-ZrO 2 unit cells (3 × 0.36 nm = 1.08 nm) nearly coincide with 4 Rh unit cells (4 × 0.269 nm = 1.076 nm). However, when using the Rh spots as a gauge to measure the true value of the oxide lattice constant, we find a value of approximately 0.34 nm -far shorter than the 358 pm required for a true (3 × 3)/(4 × 4) structure. The oxide structure can therefore not be explained by this superstructure. STM shows that the film is not perfectly ordered, as can be seen from the variations in the surface structure. In ordered areas, the most common feature resembles a rosette. The rosettes are hexagonally ordered with a periodicity of 1.2 nm, marked in the inset of Figure 2a. Usually, the domains of well-ordered rosettes are much smaller than in the inset of Figure 2a, however. From comparison with atomically resolved images of the Rh(111) surface (not shown), we find that the rosette lattice corresponds to a (√21 × √21)R10.9° superstructure with respect to Rh(111). We can explain the ideal rosette The in-plane lattice constant of 341 pm for ZrO 2 is surprisingly short: For metastable tetragonal or cubic ZrO 2 the corresponding value would be about 359-362 pm [38], and even 1 ML (single-trilayer) ZrO 2 films have a larger in-plane lattice constant of ≈350 pm [27,29] . As decreasing the lattice constant is constrained by O-O repulsion (see above), we consider it likely that these films are substantially oxygen-deficient. Based on the LEED image, however, the rotation of the oxide is less than 1° in most areas of the surface and therefore smaller than the 3° expected from the epitaxial relationship. This deviation can be explained, as we observe only small patches of well-ordered rosettes by STM, so the superstructure measured above is only an approximation. Thus, also the in-plane lattice constant may be slightly different from the one calculated assuming a perfectly commensurate superstructure. We also observe LEED spots from a (2 × 1)-O structure on Rh(111) in the holes of the film. This structure is common when annealing Rh(111) in oxygen [40]; the corresponding periodicity can be also detected by STM in the holes (not shown). A 3 ML-thick film annealed at 610 °C appears quite different in STM, see Figure 2b. Apart from some disordered regions in the upper half of the image, the predominant structure shows a (4 × 4) cell with respect to the substrate, which now nicely corresponds to (3 × 3) cells of the oxide (see the LEED pattern). This corresponds to an in-plane lattice constant of 358 pm, which is already close to the value for cubic zirconia (≈362 pm). There is no sign of the rosette structure that was found at 2 ML. It would be tempting to anneal to a higher temperature in order to improve the ordering and eliminate the disordered patches. Unfortunately, these thin films break up easily, forming thicker films with holes. These then have the structures of the respective thicker films. The 3 ML-thick film shown here already has a small number of holes down to the Rh substrate, which explains the bright Rh spots in LEED. In addition to the ZrO 2 (3 × 3) superstructure, LEED shows a ZrO 2 (2 × 1) structure that originates from 4 ML-high terraces (according to STM). This structure is discussed below. In Figure 2c, a 4 ML-thick film is shown, with a few terraces having a total height of 5 ML. The LEED image again indicates a (3 × 3) oxide lattice per (4 × 4) Rh units. The 4 ML-thick areas are covered by isolated protrusions with a typical height of 60 pm, see the contrast-enhanced inset and the line profile in the inset. These protrusions can form a honeycomb-type short range order with a (3√3 × 3√3)R30° superstructure with respect to the oxide or (4√3 × 4√3)R30° with respect to Rh (hexagon in inset). We consider it likely that these protrusions are adatoms or molecules, but not impurities, as these features are solely present on the 4 ML films. The protrusions nicely mark the lattice periodicity: A fast Fourier transform (FFT) of their positions extracted from the STM image (bottom right of Figure 2c) shows the Rh and ZrO 2 periodicities, as well as weaker spots for the (3√3 × 3√3)R30° lattice. Note that the circles in the FFT are exactly equidistant, marking the exact positions for a (4 × 4) superstructure. The oxide lattice is rotated by ≈0.5° with respect to the Rh lattice, which causes slight deviations of the maxima in the FFT from the center of the circles. Between the protrusions, rows can be made out in the STM image in Figure 2c (marked by green lines in the top left of the inset). These rows have the same periodicity as the rows on the 5 ML-thick structure, see Figure 3a and the islands in Figure 2c. The distance between the rows is ≈0.6 nm, which corresponds to a (2 × 1) structure with respect to the oxide, the expected unit cell of tetragonal zirconia. The 4 ML-high islands on the 3 ML film also show these rows (Figure 2b), thus the (2 × 1) already weakly appears in LEED at this coverage. This row structure persists also for thicker films, see below. Tetragonal zirconia films Zirconia films with a thickness of 5 ML and annealed at temperatures of up to 730 °C in 5 × 10 −7 mbar of O 2 are dominated by the row structure mentioned above. This structure has a (2 × 1) periodicity with respect to a ≈360 pm c-ZrO 2 (111) lattice, see Figure 3a. This is the structure expected for a tetragonal ZrO 2 film. As expected for the ABC stacking of t-ZrO 2 , the rows of adjacent layers are laterally shifted by 1/3 of their spacing. This can be seen in panel a1 of Figure 3 (green lines). Domain sizes of ≈30 nm can be reached upon annealing at 730 °C in O 2 (Figure 3a). The apparent corrugation of the tetragonal rows is surprisingly high (typically 30 pm; up to 100 pm at 2.4 V sample bias). This cannot be explained by the geometric heights of the surface atoms in the tetragonal structure (∆z = 35 pm for the O atoms in a bulkterminated structure, less for a relaxed surface [16]; the Zr atoms have roughly equal heights). Thus, the high corrugation stems from either a surface reconstruction or an electronic effect. As neighboring domains of the tetragonal surface appear to blend into each other in some places, then appearing like a (2 × 2) structure (yellow circle in Fig. 3a), we consider the latter explanation more likely. At a thickness of 5 ML, the surface structure can be atomically resolved with STM. With increasing film thickness, the bias voltage has to be increased for stable imaging, and the resolution of the images decreases accordingly (see the image of the 7.5 ML film in Figure 3b). It is difficult to obtain stable tunneling at a thickness of 10 ML; a minimum bias of 7.2 V is required. Nevertheless, the row structure of t-ZrO 2 remains visible at 7.5 ML (frame b1 of Figure 3) and at 10 ML (not shown), and the LEED pattern always shows a (2 × 1) pattern w.r.t. c-ZrO 2 (111). . Green lines mark the (2 × 1) surface structure with respect to cubic ZrO 2 (111). When going from the 5 th to the 6 th ML, the rows shift by 1/3 of a unit cell as is expected for t-ZrO 2 (a1). Orange lines indicate the moiré structure visible in 5 ML films and some of its domain boundaries. The moiré superstructure cells are shown superimposed on the Fourier-filtered STM image in frame (a2), and a point defect is visible in frame (a3). The STM images have been processed to increase the contrast on the terraces. Although an almost perfect 4:3 lattice match between tetragonal ZrO 2 and Rh(111) would be possible, the oxide is not exactly commensurate with the underlying Rh substrate. Upon careful inspection of the STM images, we find more than the three directions of the rows (in 120° increments) expected from the rotational symmetry of the substrate: The rows do not run exactly along the Rh directions, but deviate 〈 110 〉 from the close-packed directions of the substrate by up to ≈3°. This is accompanied by a moiré pattern, which is clearly visible in the STM images of the 5 th ML (orange lines in Figure 3a). The moiré pattern becomes almost invisible in regions with 6 ML thickness and cannot be discerned in STM images of thicker films. There are several similar moiré patterns, however, and each type of moiré has six possible orientations of the oxide (three rotational domains, plus mirror symmetry). The different rotations of the zirconia film in different domains, which lead to the different moirés, cannot be resolved in LEED; rather than split into separate spots, the diffraction maxima of the hexagonal pattern in Figure 3 are only slightly elongated in the azimuthal direction. For one of the domains with a nearly commensurate lattice, we could determine the moiré structure with respect to the substrate below (Figure 3, frame a2). This moiré cell corresponds to a (7 × 7)R21.8° superstructure with respect to Rh(111), which corresponds to (2√7 × 2√7)R19.1° cells of cubic ZrO 2 (111), or half that number of tetragonal cells. This yields a rotation of 2.7° between the oxide lattice and the Rh(111) substrate; the average in-plane nearest-neighbor distance in the zirconia lattice is calculated as 355 pm and the in-plane angles between the nearest-neighbor directions would be multiples of exactly 60°. This moiré cell is only approximate, however. The moiré changes phase on a length scale of 10 nm; this can be seen at the orange lines in Figure 3a. The phase change probably happens because the interatomic distance of 355 pm would be too short for t-ZrO 2 . In addition, this deviation from a perfectly commensurate cell also leads to a deviation from angles of exactly 60° (as expected for t-ZrO 2 , see Figure 1b). In other parts of the surface, we find roughly a 4:3 lattice match with the substrate in one direction, but nevertheless a moiré structure indicating a different (shorter) lattice constant in the other directions and deviations from 60° angles. Our best estimate for the average in-plane interatomic distances in the t-ZrO 2 films is around 357 pm, about 0.5% smaller than the room-temperature values from the literature [38], see Figure 1b. Monoclinic zirconia films Upon annealing a 5 ML-thick ZrO 2 film at 850 °C in 5 × 10 -7 mbar O 2 , a phase transformation from t-ZrO 2 to m-ZrO 2 occurs. (Between 730 °C and 850 °C, the film is partially transformed.) Figure 4a shows a high-resolution STM image of the surface; the surface lattice appears hexagonal at first glance and no signs of the tetragonal row structure are visible. However, the monoclinic phase of zirconia is distorted with respect to the cubic and tetragonal phases, see Figure 1b. Due to this distortion, in order to compare the unit cell of our film with the cell size of bulk m-ZrO 2 , we have to compare three different in-plane distances (or two distances plus one angle); approximate values for these three distances are shown in the inset of Figure 4a. In contrast to t-ZrO 2 , the monoclinic lattice does not have an approximate 6-fold symmetry, which would help us correct for distortions of the STM images and thereby make an accurate determination of the lattice constants possible. As a way out, we took three sets of images with the fast scanning direction aligned with each of the ZrO 2 directions. We then measured the distances along the fast scanning 〈110〉 direction, which is almost unaffected by thermal drift or piezo creep. For calibration, we used atomically resolved images of the Rh(111) lattice recorded with the same scanning angle and scan speed (after removal of the oxide by sputtering). In our experience, this procedure should be accurate within ≈1-2%. The side lengths of the unit cell measured by this procedure are 727, 708, and 664 pm, which compares reasonably well with the values for m-ZrO 2 : 745, 733, and 678 pm [39]. The (111) deviations from the expected values may hint at monoclinic distortions in the thin films being slightly different from the bulk. For comparison, the cell side lengths for the energetically less-favorable termination m-ZrO 2 (111) [16] would be 797, 745, and 733 pm [39]. Thus, we can rule out the (111) orientation, which would be the only other symmetry-inequivalent type of m-ZrO 2 {111} surfaces. The measured distances are also far from those expected for the orthorhombic structures; their unit cells have much less distortion with respect to c-ZrO 2 . Thus, these films can be safely identified as m-ZrO 2 . In the FFT of an STM image with four different domains (Figure 4d), spots from the different domains can be seen in each direction. In LEED, these spots are smeared out, indicating slight variations in azimuthal orientation (Figure 4b). Nevertheless, the splitting of the monoclinic spots makes it easy to distinguish monoclinic and tetragonal films by LEED. Figure 4b also shows the expected diffraction pattern from six domains of m-ZrO 2 ; these show a good agreement with experiment except (111) for the right edge of the LEED screen, where the experimental image is distorted. Apart from the ZrO 2 spots, only Rh(111)-(1 × 1) and (2 × 1) spots are visible; the Rh(111)-(2 × 1) again stems from the (2 × 1)-O overlayer that is formed on bare Rh in the holes upon annealing in O 2 [40]. Together with the tetragonal-to-monoclinic transformation, the film usually breaks up, which leads to holes down to the Rh(111) substrate, see Figure 4c. The material from the holes spreads over the remaining zirconia areas, and locally increases the thickness (e.g., from 5 to 6 ML). The formation of holes and the phase transformation do not always go hand in hand, however: By changing the deposition parameters, we can prepare a tetragonal film that breaks up at temperatures below the phase transition point. In this case, we have lowered the front grid voltage of the sputter source from 150 V to 60 V, which reduced the energy of the Ar + ions that are hitting the surface during deposition [36]. Thus, the growth conditions become more comparable to thermal deposition. Such a weakly-sputtered film breaks up already at an annealing temperature of 670 °C, while it remains in the tetragonal structure. The phase transition then happens after annealing the film at 820 °C. The phase transformation from a tetragonal to a monoclinic film can be reversed by annealing at very high temperatures of 920 °C in UHV (p base < 1 × 10 -10 mbar). This preparation leads to a tetragonal film with holes down to Rh(111). Discussion Thin-film zirconia model systems have been studied since 1990 [25], yet the relation of these films to the ZrO 2 bulk structures could not be determined unambiguously. In hindsight, the LEED patterns originally interpreted as (2 × 2) with respect to c-ZrO 2 (111) by Maurice et al. and Meinel et al. [24,25] most likely correspond to three domains of the tetragonal structure, i.e., the three (2 × 1) domains in Figure 3. On Pt(111), these structures were not stable when annealing for more than one minute at 680 °C and transformed into (√19 × √19)R36.6° w.r.t. Pt(111) [24]; this is the same structure as the ultrathin zirconia films on Pt 3 Zr(0001) [27]. It is possible that this low stability is due to easy dissolution of Zr in the Pt bulk (the dissolution enthalpy of Zr in Pt is exceptionally high [41,42]). On the other hand, the lower stability compared to our films might also be a consequence of thermal deposition in Ref. [22][23][24] vs. sputter deposition with additional ion bombardment in our case (remember that our films grown with softer ion bombardment than usual are less stable). The gentle Ar + ion bombardment by the sputter deposition source may help stabilizing the films by creating defects or slight intermixing; especially reactive Zr atoms embedded in Rh at the interface may help stabilizing the films by providing strong Zr-O bonds with O in the bottommost oxide layer (cf. the stabilization of zirconia films on Pd 3 Zr(0001) [29]). The transformation of the films to monoclinic zirconia was not reported in literature previously. It occurs at temperatures of 850 °C, so the higher thermal stability of our sputter-deposited films on Rh(111) compared to films created by thermal deposition on Pt(111) is certainly beneficial. Our attempts to obtain m-ZrO 2 by sputter deposition on Pt(111) were not successful when annealing in the same p O2 = 5 × 10 -7 mbar as on Rh(111). The films broke up, but remained tetragonal up to 900 °C (not shown). However, at this temperature a significant part of the t-ZrO 2 film was reduced and Zr diffused into the Pt substrate. This does not happen on the Rh substrate. Starting from this partially reduced state on Pt(111), the transformation to m-ZrO 2 can be induced by annealing at 610 °C and higher p O2 = 2 × 10 -6 mbar (not shown). Whether reoxidation of dissolved Zr or the higher p O2 is the reason for the stabilization of the monoclinic phase is a question for further studies. Let us consider the stability of the bulk phases (Figure 1a), where m-ZrO 2 is stable at room temperature and t-ZrO 2 is the high-temperature phase. It is then surprising that the tetragonal phase in the thin films is stable at lower annealing temperatures (T ≤ 730 °C) and transforms to the monoclinic phase when annealed at 850 °C. Assuming a lower surface energy for t-ZrO 2 than m-ZrO 2 , it has been suggested that the monoclinic-to-tetragonal transition temperature decreases with decreasing film thickness (below 1 μm) and should reach room temperature in the range of 20 nm [43]. As mentioned in the introduction, the role of the surface energy stabilizing the tetragonal phase has to be questioned [12,13,15]. In any case, this is not the behavior encountered in our case, as the 5 ML (1.5 nm) tetragonal films can still be transformed to the monoclinic phase, which then stays stable upon cooling to RT. The tetragonal film is therefore in a metastable state, stabilized by the interface to the Rh substrate and, possibly, oxygen deficiency. Considering that Zr-O bonds get broken and the lattice gets distorted upon the tetragonal-to-monoclinic transformation (Fig. 1a), it is clear that a substantial activation energy is required to transform the films. The area density changes from 8.74 to 8.99 × 10 18 Zr atoms m −2 per layer (based on room-temperature bulk lattice constants [38,39]), and the transformation also involves in-plane shear. In contrast to the expansion of the interlayer spacing (295 to 317 pm), changing the area density and in-plane shear clearly require thermal activation. When comparing the tetragonal and the monoclinic films on Rh(111), the tetragonal films appear rather flat, while the monoclinic films show long-distance modulations in their apparent height, with bright halos around both point defects and grain boundaries (Figure 4c). At positive STM sample bias, such an increase of the apparent height (increased tunneling current at constant height) is typical for downwards band bending [44]. These observations are important for the use of (chemically doped) zirconia as a solid-state electrolyte: Grain boundaries (GBs) in YSZ impede oxygen ion transport ("grain boundary blocking"); this is attributed to the positive charge at GBs [45], probably caused by oxygen vacancies at GBs (oxygen vacancies carry 2+ charge with respect to the undisturbed lattice with O 2− at the respective site; V O •• in Kröger-Vink notation [46]). Our STM images are consistent with this model. The flat appearance of the tetragonal films points at a fixed position of the bands. Downwards band bending cannot occur if the conduction band minimum is close to the Fermi level, as is expected for a strongly n-doped oxide. This nicely fits the notion that oxygen vacancies (providing n doping) are responsible for the stabilization of the tetragonal phase in nanoscale ZrO 2 , and could therefore be expected as a stabilizing factor in our thin films. Let us finally discuss the films of lower thickness. While LEED indicates that all of these films are based on c-ZrO 2 (111), the influence of the substrate does not allow these films to develop the surface structures expected for bulk ZrO 2 . The in-plane lattice constants of the 2 ML film are clearly below those of the thicker films and the bulk phases. Again, this indicates sub-stoichiometric (oxygen-deficient) films; for example, it is possible that the rosette structure is related to ordering of oxygen vacancies. Starting from 3 ML, the in-plane lattice constants are already close to the bulk values (≈3:4 lattice match with Rh). For 4 ML films, the row structure of the tetragonal films is already locally visible by STM. It is unclear whether the bright features in Figure 2c are topographic (e.g. adatoms, small molecules, or clusters on top of the t-ZrO 2 surface) or electronic features. The high corrugation of these features (line scan in Figure 2c) points to a topographic feature. It is unlikely that these features are due to impurities, as they were not observed on thicker films. Conclusion Few-layer zirconia films can be reliably prepared by UHV-compatible sputter deposition. They show layer-by layer growth and good homogeneity. Up to a thickness of 4 ML, the surface structure of the zirconia films is influenced by the underlying Rh(111) substrate. For each thickness below 5 ML, a different superstructure is found; apart from the 2 ML films, all structures are close to a commensurate lattice with (3 × 3) c-ZrO 2 (111) cells on (4 × 4) Rh(111) unit cells. Zirconia films of 5 ML or larger thickness show the surface structures of either tetragonal or monoclinic zirconia, depending on the annealing temperature. Both structures can be prepared with large, atomically flat terraces; their surface lattices were resolved by STM, confirming their crystallographic structure. Preparation of a completely monoclinic film needs annealing at temperatures of at least 850 °C; at these temperatures, the film breaks up and holes reaching down to the substrate appear. Thus, the films show some instability towards dewetting. Due to the insulating nature of ZrO 2 , imaging the surface with STM becomes increasingly difficult with increasing film thickness, thus the thinnest films showing the structures of the ZrO 2 bulk phases (5 ML) are the best choice for an STM-accessible ZrO 2 model system. In comparison with the previous ZrO 2 /Pt(111) model system, we believe that there are three reasons for the superior film homogeneity and stability on Rh(111): Firstly, the lower solubility of Zr in Rh than in Pt. Secondly, the 3:4 lattice matching, which leads to low rotation angles between different domains, and thirdly the use of a UHV-compatible sputter source providing additional slight ion bombardment. Figure 1 : 1The three stable ambient-pressure bulk phases of zirconia. (a) Bulk unit cells and (b) bulk-terminated surfaces equivalent to the (111) surface of cubic ZrO 2 , with the surface unit cells marked in yellow. Oxygen (2−) ions are depicted in red, zirconium (4+) ions in green. Room-temperature cell sizes in (b) are given in pm and based on Ref. Figure 2 : 2Surface structures of zirconia films with 2-4 ML thickness, as seen with STM (left) and LEED (right). Each thickness has its own surface reconstruction: Rosettes at 2 ML (a, unit cell marked in pink), (3 × 3) at 3 ML (b), and a (3√3 × 3√3) superstructure of small protrusions at 4 ML (c). For 4 ML, the line scan shows the height of the protrusions, and an FFT of the protrusions shows weak spots of their superstructure. Superstructures are given with respect to c-ZrO 2 (111). lattice using the lattice constant from LEED if it is a (√13 × √13)R13.9° superstructure w.r.t. a cubic ZrO 2 (111) lattice. This results in an oxide lattice constant of 341 pm and a small 3° rotation of the oxide w.r.t. the Rh lattice. Figure 3 : 3Tetragonal zirconia films with (a) 5 ML and (b) 7.5 ML, as seen with STM (large and zoom-in frames) and LEED (top right) Figure 4 : 4Monoclinic zirconia films: Upon annealing a 5 ML-thick film at 850 °C in 5 × 10 -7 mbar O 2 , the film breaks up and transforms into the thermodynamically stable monoclinic structure. (a) High-resolution STM image of the structure. The inset shows a zoom to one unit cell, with approximate lattice parameters given in nm. (b) LEED image. The calculated LEED pattern of the surface of monoclinic (111) zirconia is shown by red dots: spots originating from Rh(111) or Rh(111) (2 × 1)-O are blue. (c) STM overview image showing the holes down to Rh(111) and different domains (a few grain boundaries are marked by broken lines): In the Fourier transform, these domains result in a splitting of the spots (d). 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[ "Evidence for a developing gap in a 10 Myr old protoplanetary disk", "Evidence for a developing gap in a 10 Myr old protoplanetary disk" ]	[ "Nuria Calvet [email protected] \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA. Electronic\n", "Paola D'alessio \nInstituto de Astronomia\nUNAM\nAp.P. 70-26404510MéxicoD.F., México\n\nAmerican Museum of National History\nCentral Park West at 79th Street, New YorkNY10024-5192\n", "Lee Hartmann [email protected] \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA. Electronic\n", "David Wilner [email protected] \nHarvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA. Electronic\n", "Andrew Walsh [email protected] \nMax Planck Institut für Radioastronomie\nauf dem Hügel 6953121BonnGermany\n", "Michael Sitko [email protected] \nDepartment of Physics\nUniversity of Cincinnati\n45221Cincinnati, -0011 -2OH\n" ]	[ "Harvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA. Electronic", "Instituto de Astronomia\nUNAM\nAp.P. 70-26404510MéxicoD.F., México", "American Museum of National History\nCentral Park West at 79th Street, New YorkNY10024-5192", "Harvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA. Electronic", "Harvard-Smithsonian Center for Astrophysics\n60 Garden St02138CambridgeMAUSA. Electronic", "Max Planck Institut für Radioastronomie\nauf dem Hügel 6953121BonnGermany", "Department of Physics\nUniversity of Cincinnati\n45221Cincinnati, -0011 -2OH" ]	[]	We have developed a physically self-consistent model of the disk around the nearby 10 Myr old star TW Hya which matches the observed spectral energy distribution and 7mm images of the disk. The model requires both significant dust size evolution and a partially-evacuated inner disk region, as predicted by theories of planet formation. The outer disk, which extends to at least 140 AU in radius, is very optically thick at infrared wavelengths and quite massive (∼ 0.06M ⊙ ) for the relatively advanced age of this T Tauri star. This implies long viscous and dust evolution timescales, although dust must have grown to sizes of order ∼ 1 cm to explain the sub-mm and mm spectral slopes. In contrast, the negligible near-infrared excess emission of this system requires that the disk be optically thin inside ∼ < 4 AU. This inner region cannot be completely evacuated; we need ∼ 0.5 lunar mass of ∼ 1 µm particles remaining to produce the observed 10µm silicate emission. Our model requires a distinct transition in disk properties at ∼ 4 AU, separating the inner and outer disk. The inner edge of the optically-thick outer disk must be heated almost frontally by the star to account for the excess flux at mid-infrared wavelengths. We speculate that this truncation of the outer disk may be the signpost of a developing gap due to the effects of a growing protoplanet; the gap is still presumably evolving because material still resides in it, as indicated by the silicate emission, the molecular hydrogen emission, and by the continued accretion onto the central star (albeit at a much lower rate than typical of younger T Tauri stars). TW Hya thus may become the Rosetta stone for our understanding of the evolution and dissipation of protoplanetary disks.	10.1086/339061	[ "https://arxiv.org/pdf/astro-ph/0201425v1.pdf" ]	8,706,944	astro-ph/0201425	37424ab5711bc26428c2ac4643da593f20b15f96	Evidence for a developing gap in a 10 Myr old protoplanetary disk 25 Jan 2002 Nuria Calvet [email protected] Harvard-Smithsonian Center for Astrophysics 60 Garden St02138CambridgeMAUSA. Electronic Paola D'alessio Instituto de Astronomia UNAM Ap.P. 70-26404510MéxicoD.F., México American Museum of National History Central Park West at 79th Street, New YorkNY10024-5192 Lee Hartmann [email protected] Harvard-Smithsonian Center for Astrophysics 60 Garden St02138CambridgeMAUSA. Electronic David Wilner [email protected] Harvard-Smithsonian Center for Astrophysics 60 Garden St02138CambridgeMAUSA. Electronic Andrew Walsh [email protected] Max Planck Institut für Radioastronomie auf dem Hügel 6953121BonnGermany Michael Sitko [email protected] Department of Physics University of Cincinnati 45221Cincinnati, -0011 -2OH Evidence for a developing gap in a 10 Myr old protoplanetary disk 25 Jan 2002Subject headings: Accretionaccretion disksStars: Circumstellar MatterStars: FormationStars: Pre-Main Sequence We have developed a physically self-consistent model of the disk around the nearby 10 Myr old star TW Hya which matches the observed spectral energy distribution and 7mm images of the disk. The model requires both significant dust size evolution and a partially-evacuated inner disk region, as predicted by theories of planet formation. The outer disk, which extends to at least 140 AU in radius, is very optically thick at infrared wavelengths and quite massive (∼ 0.06M ⊙ ) for the relatively advanced age of this T Tauri star. This implies long viscous and dust evolution timescales, although dust must have grown to sizes of order ∼ 1 cm to explain the sub-mm and mm spectral slopes. In contrast, the negligible near-infrared excess emission of this system requires that the disk be optically thin inside ∼ < 4 AU. This inner region cannot be completely evacuated; we need ∼ 0.5 lunar mass of ∼ 1 µm particles remaining to produce the observed 10µm silicate emission. Our model requires a distinct transition in disk properties at ∼ 4 AU, separating the inner and outer disk. The inner edge of the optically-thick outer disk must be heated almost frontally by the star to account for the excess flux at mid-infrared wavelengths. We speculate that this truncation of the outer disk may be the signpost of a developing gap due to the effects of a growing protoplanet; the gap is still presumably evolving because material still resides in it, as indicated by the silicate emission, the molecular hydrogen emission, and by the continued accretion onto the central star (albeit at a much lower rate than typical of younger T Tauri stars). TW Hya thus may become the Rosetta stone for our understanding of the evolution and dissipation of protoplanetary disks. Introduction The discovery of extrasolar planets (Marcy & Butler 1998 and references therein) has opened up a new era in the study of planetary systems. While many important clues to the processes of planet formation can be obtained from studies of older systems, the best tests of formation scenarios will require the direct detection of actively planet-forming systems. It is thought that the formation of giant planets involves the sweeping up of material in a wide annulus in the circumstellar disk, resulting in the development of a gap (Lin & Papaloizou 1986Bryden et al. 1999). Material inside the planet-driven gap can continue to accrete onto the central star; if the planet can prevent material from accreting across the gap into the inner disk, the eventual result would be the evacuation of the region interior to the planet. In the case of the Solar System, the formation of Jupiter might have prevented outer disk gas from reaching the inner solar system; the inner gas disk accreted into the Sun, while solid planetesimals remaining behind eventually formed the terrestrial planets. The above scenario suggests that the signature of a forming giant planet would be the presence of a gap that is not entirely evacuated. In this case dusty emission from the inner disk might still be observable. In addition, giant planet formation requires the prior consolidation of large solid bodies to serve as cores for subsequent gas accretion; while such bodies would be invisible with current techniques, one would expect to see evidence for substantial growth in dust particles. Finally, if the inner disk has not been completely evacuated by the forming planet, one might expect to observe continued accretion onto the central star, as all T Tauri systems with inner disks as detected from near-infrared disk emission are also accreting (Hartigan et al. 1990). In this article we propose that the relatively young low-mass star TW Hya has a developing gap in its inner disk qualitatively similar to that expected from planet formation. The evidence supporting this this proposal is: (1) reduced (optically-thin) emission from the inner disk; (2) mm-wave spectra which seem to require grain growth; (3) extra emission from the edge of the outer disk; and (4) continued accretion onto the central star, albeit at a rate substantially lower than that observed from most T Tauri stars. TW Hya has an age of 10 Myr and so according to current theories it is quite likely to be close to the epoch of planet formation. It is also part of an association of young stars of similar age, but it stands out in that it is the only system still accreting at a substantial rate (Muzerolle et al. 2000). The (outer) disk of TW Hya is quite massive, so that there is likely to be more than enough material available to form giant planet(s). TW Hya is also the closest known such system, and so it will be a prime target for following studies to confirm our model. Observations The observations used to constrain our disk model are taken from the literature. In addition, we have obtained a narrow-band L (3.55-3.63 microns) measurement of TW Hya on June 14th, 2000 with the Australian National University 2.3-m telescope at Siding Spring, using the near infrared camera CASPIR (Cryogenic Array Spectrometer Imager; McGregor et al. 1994). CASPIR contains a 256x256 InSb array. TW Hya was imaged at L with a pixel scale of 0.25 arcseconds and an on-source integration time of 236 seconds. The standard star BS4638 (magnitude 4.50 at L) was observed at a similar airmass to TW Hya and was used to calibrate the magnitude of TW Hya. The L magnitude we obtained was 7.12, equivalent to 0.39Jy. This measurement is important in constraining the amount of near-infrared excess in the system. Using K=7.37 (Webb et al. 1999), K-L = 0.25. The K-L color for a K7V star is 0.11 (Kenyon & Hartmann 1995), which would yield an infrared excess of 0.14. However, such small infrared excess does not necessarily imply emission from disk material (Wolk & Walter 1996). For example, the non-accreting stars in Taurus (spectral types ∼ K7 to M2) have K-L between -0.05 and 0.25, while 95% of the accreting stars in Taurus have K-L between 0.35 and 1.2 (Meyer, Calvet, & Hillenbrand 1997). This small value of K-L is consistent with the simultaneous flux measurements of Sitko et al. (2000) indicating little if any hot dust emission at wavelengths < 5µm, though it is difficult to rule out completely any excess. Model assumptions We assume that the heating of the disk is due to stellar irradiation and viscous dissipation, and calculate self-consistently the disk heights and temperatures following the methods we have applied to interpret young ∼ 1 Myr old disks = Paper I, 1999= Paper II, 2000 We adopt M * = 0.6 M ⊙ , R * = 1 R ⊙ and T * = 4000 K for the stellar mass, radius, and effective temperature (Webb et al. 1999), and a mass accretion rate ofṀ = 5 × 10 −10 M ⊙ yr −1 as derived by Muzerolle et al. (2000), which we take as constant through the whole disk. The inclination of the disk axis to the line of sight is i ∼ 0 • , in agreement with the nearly symmetric HST images (Krist et al. 2000), and we adopt the Hipparcos distance of 55 pc (Wichmann et al. 1998). With the assumption of steady accretion, the disk surface density scales as Σ ∝Ṁ α −1 T −1 , with T is the midplane temperature (see Paper II). The temperature does not vary much between acceptable models, partly because it is mostly determined by irradiation heating, not viscous energy dissipation. The disk mass therefore is roughly constant for constantṀ α −1 . Moreover, for long wavelengths where much of the disk is optically thin, the emergent fluxes also tend to scale in the same way. For the purposes of fitting, we vary the parameter α as the dust opacities are varied, but note that varying α is equivalent to varying the disk mass, becauseṀ is fixed. To first order, however, choices withṀ α −1 ≈ constant are acceptable. We use a dust mixture consisting of silicates, refractory organics, troilite and water ice, following Pollack et al. (1994 = P94). The dust size distribution is taken to be n(a) ∼ a −p , with p = 3.5, between given a min and a max . Optical properties for the compounds are taken from Jäger et al. (1994), P94, Begemann et al. (1994;also see Henning et al. 1999), andWarren (1984). We consider the grains to be compact segregated spheres, and calculate the opacity using a Mie scattering code (Wiscombe 1979). Sublimation temperatures for the different grain types are taken from P94. Disk model The spectral energy distribution (SED) of TW Hya is shown in Figure 1. References for the observational data are given in the figure caption. Note that our narrow-band L observation overlaps with Sitko et al. (2000) fluxes. In Figure 1, we compare the SED with the median SED for ∼ 1 Myr-old T Tauri stars in the Taurus molecular cloud (Paper II), normalized to the TW Hya stellar photosphere at H (1.6µm), thus compensating for the differing distances and stellar luminosities. Even though the fluxes of TW Hya are relatively high compared with the median Taurus SED at λ ∼ > 20µm, there is a large flux deficit in TW Hya below 10 microns (Jayawardhana et al. 1999); in particular, fluxes are essentially photospheric below ∼ 6µm (Sitko et al. 2000). The flux deficit below 10 µm has lead to inferences of disk clearing inside a few AU from the central star (Jayawardhana et al. 1999), but since the disk is still accreting mass onto the star (Muzerolle et al. 2000), it has to extend all the way into the corotation radius at least, so the inner disk radius has to be ∼ < 0.03 AU (inferred from the 2 day photometric period [Mekkaden 1998]). We show below that models with a uniform dust well-mixed with the gas throughout the disk extending all the way onto the inner radius cannot explain these features of the SED. The observations are much better understood if the disk is divided into two regions: the outer disk, which is more nearly comparable to the structure inferred for typical T Tauri disks (Paper III), and the inner disk, which is much more optically thin than in typical T Tauri disk models. Outer disk As discussed in Paper II, ISM dust mixtures (with small a max ) cannot explain the far-IR and mm-wave fluxes of T Tauri stars. The similarity between the median Taurus SED and that of TW Hya at wavelengths λ ∼ > 100 µm suggests disk models of the type explored in Paper III, in which we allow for growth to large particles, can in principle explain the observations. Figure 2 shows results for flared, irradiated disk models calculated with the methods of Paper III. The model SEDs shown are for a max = 1 mm (which fits the median SED in Taurus; Paper III), 1 cm, and 10 cm, calculated with abundances in the dust mixture usually assumed for protoplanetary disks (P94), which yield a dust-to-gas ratio of 0.013. It can be seen that models where grains have grown to a max ∼ 1 − 10 cm provide a much better fit to the long-wavelength SED than the a max = 1 mm model, requiring no or very little additional emission from a wind or non-thermal sources. Models with a max ≪ 1 mm fail to reproduce either the sub-mm and mm spectral slopes or the total flux levels for reasonable disk masses. Assuming an outer disk radius of ∼ 140 AU, comparable to the radius at which Krist et al. (2000) find a rapid decline in disk density, the disk mass is ∼ 0.03, 0.06, and 0.11M ⊙ for the a max = 1 mm, 1 cm, and 10 cm models, or α = 5 × 10 −4 , 3 × 10 −4 , and 1 × 10 −3 , respectively. The a max = 1 cm model has a Toomre parameter Q ∼ 1 at its outer edge and so is near the limit expected for gravitational stability. The high mass values are due to the opacity at 7 mm, which decreases as a −1/2 max (for p = 3.5, see Paper III), so higher masses are needed to account for the flux for larger grain mixtures. For example, for the a max = 10 cm mixture, the opacity at 7 mm is a factor of 3 lower than the frequently assumed law κ BS = 0.1(λ/250 µm) −1 (Beckwith & Sargent 1991) and the slope is slightly flatter, ∝ λ 0.8 (cf. Paper III, Figure 2). Our total disk mass estimates, using the dust-to-gas ratio of 0.013, are much higher than the gas mass obtained by Kastner et al. (1997) from 12 CO emission, 3.5 × 10 −5 M ⊙ . This discrepancy may be attributed to a large degree of molecular depletion or to the fact that optical depth effects may not have been properly included (Beckwith & Sargent 1993); it could also be due to molecules existing in the gas phase only in the hot upper atmospheric layers of the disk where only a small amount of mass resides (Willacy & Langer 2000). An alternative to high disk masses, not considered in this work, is to have larger opacities at 7 mm than we obtain; porous aggregates, specially for amorphous carbon particles, may result in such larger opacities (Stognienko, Henning & Ossenkopf 1995). Our results for grain growth are not unique. However, within the assumptions we have made concerning dust opacities, we cannot reproduce the mm-wave fluxes and spectral slope without including dust particles much larger than those of a "standard ISM" mixture (see Paper III). Beckwith & Sargent (1991) noted that optical depth effects could make the mm-wave spectral slope flatter, and thus reduce or eliminate the need for grain growth. However, we are unable to make TW Hya disk sufficiently optically thick over its large radial extent. Edge of the outer disk The well-mixed grain-growth models of Figure 2, with self-consistently calculated temperature structures, exhibit too little emission in the 20 − 60µm wavelength region in comparison with observations. In principle, disk models with more small particles could have higher fluxes. However, it is apparent that these models also predict far too much flux at wavelengths ∼ < 10µm, because they are too optically thick in the inner regions; this suggest that some clearing has occurred in the inner disk. Moreover, it suggest that the outer disk should have an optically-thick edge in which extra heating could be important. In the case of irradiation of an optically thick disk with a smoothly-varying thickness, the stellar flux captured by the disk (and thus the disk heating per unit area) depends on the cosine of the angle between the direction of incidence of the stellar beam and the normal to the disk surface µ 0 (Kenyon & Hartmann 1987). In general, stellar radiation penetrates the disk very obliquely, so µ 0 is fairly small, ∼ 10 −3 − 10 −2 (cf. D'Alessio 1996). However, if the disk had an inner edge, this portion of the disk would be illuminated by the star more directly, increasing µ 0 dramatically and thus increasing the amount of irradiation heating. We propose that the outer regions of the TW Hya disk can be described as in §4.1, but that it is truncated at a few AU by a steep, optically thick region, where most of the mid-IR flux excess arises ( Figure 3). Inside this region lies the inner optically thin disk, which produces negligible continuum flux. To calculate the emission of the disk edge, we have assumed that its irradiation surface, that is the surface where most of the stellar energy is deposited, has a shape given by z s = z o (R o )exp[(R − R o )/∆R], where z o is the height of irradiation surface of the outer disk at radius R o and ∆R is a characteristic width of the edge. The edge continuum emission is produced in the photosphere, which for simplicity we take as the irradiation surface; above this layer, there is a hotter optically thin region, the atmosphere, which we take as isothermal (cf. Chiang & Goldreich 1997). We think that a more detailed treatment of the structure of this region is not necessary given the exploratory nature of this study. The temperature of the edge photosphere, assuming that half of the intercepted flux reaches the photosphere, is given by T phot (R) ≈ T * R * R 1/2 µ 0 2 1/4 . (1) where µ 0 = cos(θ o ) is obtained from θ o = π/2 − tan −1 (dz o /dR). The flux is evaluated as F ν = Ro R th I ν 2πRdR,(2) where R th is a the radius where the disk becomes optically thin, and I ν = B ν (T phot ). The location of the outer disk edge and its width, characterized by R o and ∼ ∆R, are fairly well restricted by the mid-infrared excess, although the actual shape is not so well constrained as long as most of it faces the star. In general we find ∆R ∼ 0.5 AU, because the shape of the excess is fairly narrow and thus cannot be produced by a region with a large range of temperatures. In addition, if R o is much smaller than ∼4 AU, then the edge is too hot and there is too much excess below 10 µm. Similarly, if R o is much larger, there is too little flux. With R o ∼ 4 AU and ∆R ∼ 0.5 AU, µ o varies from ∼ 0.2 at R = 3 AU to ∼ 0.7 at R = 4 AU, and T phot varies from ∼ 80 to 100 in this range. The resulting continuum flux is shown in detail in the lower panel of Figure 4, while the upper panel shows the fit to the SED of the composite disk model. The temperature of the optically thin atmosphere of the edge is given by W (R)κ * P T 4 * = κ P (T up )T 4 up(3) where W (R) is the geometrical dilution factor, W (R) = Ω * /4π ≈ (R * /2R) 2 , for R >> R * , and κ * P and κ P are Planck mean opacities calculated at the stellar and local radiation fields, respectively (cf. Paper I, Paper II, Paper II). The contribution to the flux from this region is given by eq. (2), with I ν = B ν (T up )τ ν and τ ν ≈ κ ν τ * /χ * , where τ * and χ * are the optical depth and extinction coefficient at the characteristic wavelength of the stellar radiation. We follow Natta et al. (2000) taking τ * ≈ µ 0 , but in this work we include the effect of scattering at the wavelengths at which the stellar radiation is absorbed. The neglect of scattering in the calculation of χ * leads to artificially large values of τ ν and thus of I ν , and would be only appropriate if scattering was completely forward. However, the asymmetry parameter g = cos Θ , with Θ the scattering angle, is in general < 1, and this approximation is not valid. We have included the effect of scattering using χ * = κ * + (1 − g * )σ * , where κ * , σ * , are the absorption and scattering coefficients and g * the asymmetry parameter for the assumed dust mixture, all evaluated at the characteristic wavelength of the stellar radiation. We assume that the atmosphere of the edge has small grains so it can produce emission in the silicate feature. We considered different glassy and crystalline pyroxenes and olivines (with optical properties from Laor & Draine 1993, Jäger et al. 1994, and Dorschner et al. 1995. We find that glassy pyroxene Mg 0.5 F e 0.43 Ca 0.03 Al 0.04 SiO 3 has one of the highest ratios κ ν /χ * as required for producing a strong band (see also Natta et al. 2000). However, even with the increasing heating at the disk edge, the conspicuous silicate emission feature seen in the SED of TW Hya cannot be explained by emission from the atmospheric layers of the edge. This is due in part to the inclusion of scattering of stellar light in the calculation of χ * , as discussed above, and in part to the adopted dust mixture; organics dominate the opacity at the stellar wavelength (Paper III, Figure 2), yielding κ 10µm /χ * ∼ 1. Note that the atmosphere of the optically thick outer disk, which is irradiated less frontally by the star and is thus cooler than the atmosphere of the edge, cannot produce significant silicate emission either. Inner disk Since the region inside the outer disk edge is not empty because material is still accreting onto the star (Muzerolle et al. 2000), we have explored the possibility that a small amount of particles coexists with the accreting gas of the inner disk, giving rise to silicate feature emission, while on the other hand still resulting in negligible continuum flux. We assume that the temperature of the dust in this region is given by equation (3). The emergent intensity is calculated as I ν = B ν κ ν τ 10 /κ(10µm), where τ 10 , the optical depth at 10µm, is a parameter. This region extends to ∼ 0.02 AU, the radius where grains sublimate. We find that we can fit the silicate feature with τ 10 ∼ 0.05, see Figure 4, lower panel; this optical depth is small enough for the disk continuum emission in the near infrared to be negligible. The best fit to the profile is achieved with glassy pyroxene, in agreement with the fact that the profile is closer to that in young stars than in comets (Sitko et al. 2000). The grain sizes are in the range a min ∼ 0.9µm and a max ∼ 2µm. Smaller grains have much higher temperatures and produce too much emission at short wavelengths, resulting in a narrower silicate profile than observed. Bigger grains produce too little emission. From τ 10 ∼ 0.05, we get a column density for the ∼ 1µm dust of Σ d ∼ 4 × 10 −3 g cm −2 , which implies a mass inside 4 AU of ∼ 4 × 10 25 g ∼ 0.5 lunar masses. A strong lower limit to the mass of gas in the inner disk can be obtained from the observed rate of gas accretion onto the star assuming that the material in the inner disk is in free-fall towards the star, M gas >>ṀR 3/2 /(2GM * ) 1/2 . Inside 4 AU, we obtain M gas >> 6 × 10 −10 M ⊙ . Another lower limit can be obtained from Σ d , assuming a normal dust-to-gas ratio, yielding Σ g ∼ 0.4 g cm −2 and a mass M gas > 2 × 10 −6 M ⊙ ∼ 0.6 earth masses. If solid material is hidden into large bodies, then the mass in gas could be much higher than this value. From this limit, nonetheless, we can get an upper limit for the radial velocity of the gas at 1 AU, v R <Ṁ /2πRΣ g ∼ 0.008 km s −1 << v K (1AU) ∼ 23 km s −1 , so the gas probably drifts inwards slowly, following nearly Keplerian orbits in the inner disk. Weintraub, Kastner, and Bary (2000) have detected a flux of ∼ 1.0 × 10 −15 erg s −1 cm −2 in the 1-0 S(1) line of H 2 towards TW Hya. Using Tiné et al. (1997) models, we can estimate a flux in this line at 55 pc, F (H 2 ) ∼ 2 × 10 −15 (ǫ/10 −22 s −1 )(M H 2 /10 −8 M ⊙ )erg s −1 cm −2 , where ǫ is the emissivity per H 2 molecule. For a gas temperature of ∼ 1000K and a ionization rate of 3 × 10 −10 s −1 , estimated with typical parameters and an X-ray luminosity L X ∼ 10 30 ergcm −3 (Kastner et al. 1997) using the Glassgold, Najita, & Igea (1997) corrected expression, ǫ is 2.3 × 10 −22 erg s −1 for densities n H ≥ 10 7 cm −3 (because the line becomes thermalized; S. Lepp, personal communication). Assuming a disk height of 0.05 AU, we estimate n H >> 4 × 10 7 cm −3 inside 4 AU, so from the mass limit estimated above, we see that the observed H 2 very likely arises in the inner disk. Firmer conclusions require a determination of the gas temperature in the inner disk, which is left for future work. Model tests: comparison with VLA 7 mm Data In principle, observations of disk surface brightness distributions can be used to test our physically self-consistent models based on SED fitting. The optical and near-infrared scattered light distributions presented by Krist et al. (2000) and Trilling et al. (2001) do not have sufficient resolution to probe our inner disk region; however, they do provide constraints on our outer disk structure. We find that the scattered light fluxes predicted by the a max = 1 cm model presented above matches the observations of Krist et al. (2000) reasonably well, although Krist et al. find evidence for structure that cannot be reproduced in detail with disk properties smoothly-varying with radius, as assumed here. This work will be reported in a forthcoming paper (D'Alessio et al. 2001). The high angular resolution data at 7 mm from the Very Large Array (Wilner et al. 2000) begin to approach the resolution needed to probe the inner disk, as well as constrain the thermal dust emission of the outer disk instead of simply the scattering surface, and thus provide an additional test of our model. Moreover, the imaging data provide an independent test, since the model was constructed only to match the SED. The value of τ 10 sets an upper limit to the opacity at λ > 10µm, so the inner disk should not contribute significantly to the emission at 7 mm. We have calculated the intensity profile of the disk, assuming different values for the brightness temperature T b of the edge between 3 and 4 AU, since we do not know the properties of its interior. An upper limit for T b is the surface temperature, ∼ 100K, but it could be lower if it was optically thin at 7 mm, so we have varied T b between 60K and 100K. Figure 5a shows the 7 mm images at two angular resolutions. The "low" resolution image (∼ 0. ′′ 6, about 35 AU) emphasizes extended low brightness emission, and the "high" resolution (∼ 0. ′′ 1, about 6 AU) shows the smallest size scales where emission remains detectable with the available sensitivity. Figure 5b shows the result of imaging the model brightness distribution at these two angular resolutions using the same visibility sampling as the observations. For a more quantitative comparison, Figure 5c shows the residual images obtained by subtracting the model visibilities from the 7 mm data, and then imaging the difference with the standard algorithms. The low resolution model image, with the centrally peaked dust emission (T b ≥ 60K), compares very favorably with the 7 mm images of Figure 5a; unfortunately, the higher resolution is still not adequate to resolve directly the presence of the inner hole. But in both cases, the residual images show only noise. The agreement between the model and observations confirms the insightful parametric modeling of Wilner et al. (2000), who fitted the observations with radial power laws for the temperature and surface density. This was to be expected, since Wilner et al. adopted powers that were consistent with the predictions of irradiated accretion disk models at large radii. However, since our models are constructed from first principles, by solving the disk structure equations in the presence of stellar irradiation subject to the constraint of steady accretion, we are able to predict and confirm not only the radial dependences of the physical quantities (which are not exactly power laws), but also their absolute values. The free parameters of our model are essentially the disk mass (or equivalently, the parameter α, see §3) and the dust mixture. The models were constructed primarily to fit the SED, but they can also fit the radial brightness distribution, as the good agreement between the observations and the model indicate. On the other hand, although there is no guarantee that the outer disk is completely steady, or has a constant α, or ours is the correct prescription for the dust mixture, these assumptions seem to provide a good description of the physical situation in the disk of TW Hya. Discussion The lack of infrared flux excess at wavelengths below 10 microns could be easily understood if the inner disk was evacuated inside a few AU. However, the fact that material is still being transfered onto the star, as evidenced by the broad emission line profiles and the ultraviolet excess emission (Muzerolle et al. 2000), indicates that gas still exists in the inner disk. However, this material must be optically thin to explain the flux deficit in the near infrared. If the outer disk extended inwards to the magnetospheric radius, then the surface density Σ at 2 AU would be ∼ 320g cm −2 , implying a surface density in solids of Σ dust ∼ 3g cm −2 (for a dust to gas mass ratio of 0.01). To obtain an optical depth at the near-infrared of < 0.05 with this amount of dust, the opacity should be of order κ dust < 0.05/Σ dust ∼ 0.02cm 2 g −1 . If we assume that the grains are large enough that the opacity per gram of grain is ∼ Q ext πa 2 / (4/3)πρ g a 3 , with Q ext the extinction cross section and ρ g the grain density, both or order 1, then κ dust ∼ 1/a. Using this estimate, we find that the original solids should have grown to sizes a > 50 cm to account for the low optical depth in the near infrared. While this estimate is very rough, it suggests that very significant grain growth has probably already taken place in the inner disk of TW Hya, especially if the solid to gas ratio has increased significantly from solid matter left behind as gas flows onto the star. This inside-out coagulation and settling is consistent with the evolution of solid particles predicted in solar nebula theories. Dust grains are expected to coagulate by sticking collisions at a rate ∝ 1/Ω 2 ∝ P 2 and reach a maximum size at the midplane in a few ×1000P , where Ω is the Keplerian angular velocity and P the orbital period (Safronov 1972;Goldreich & Ward 1973;Weidenschilling 1997;Nakagawa 1981). Since P ∼ 1.4 × 10 3 (R/100AU) 3/2 yrs, coagulation and settling are expected to occur first in the inner disk. In the outer disk, a dust mixture where grains have reached ∼ 1 cm and dust and gas are well mixed can explain the observations, within the framework of our assumptions. This interpretation is consistent with the Weidenschilling's (2000) argument that grain growth to at least 1 cm is necessary before particles start settling to the midplane, because smaller grains get stirred around by turbulence, and we suggest that turbulence plays an important role in the evolution of dust in protoplanetary disks. Grains in the upper layers can absorb stellar radiation and efficiently heat the disk and make it flare, resulting in the large far infrared and millimeter fluxes observed. Nonetheless, grains in the outer disk have grown significantly from ∼ 1 mm, the size that characterizes the median SED of Taurus (Paper III). To get high mm fluxes for such large grains, the disk mass has to be high, ∼ 0.06M ⊙ , since the opacity at 7 mm decreases as grains grow (cf. Paper III, §2). This high value for the mass is surprising; it is similar or higher than values typically found for disks around 1 Myr old T Tauri stars (Beckwith et al. 1990), and comparable to that of objects emerging from the embedded phase (D'Alessio, Calvet, & Hartmann 1997). On the other hand, as we discussed in Paper III, disk masses may have been underestimated if the standard power law opacity is applied indiscriminately to disks where significant grain growth may have taken place. It may be possible that the 3.6 cm data point contains a significant contribution from hot (ionized) stellar plasma. This contribution is unlikely to come from a wind. If the flux emitted by the wind is proportional to the mass loss rate, then scaling down from the wind/jet contribution in the case of HL Tau, which hasṀ ∼ 10 −6 M ⊙ yr −1 to the case of TW Hya, and assuming a ratio of mass loss rate to mass accretion rate of ∼ 0.1 (Calvet 1998), we find that this contribution at 3.6 cm should be ∼ 2 × 10 −4 mJy, too low to be important. A non-thermal contribution due to stellar surface activity is still possible, given the high X-ray flux of TW Hya (Kastner et al. 1997). Second epoch observations should resolve this question, because any non-thermal contribution should be highly time-dependent. In any event, the high flux at 7 mm, at which a non-thermal plasma contribution is much less likely, indicates that the grain mixture must contain grains that have grown at least to ∼ 1 cm, otherwise the opacity would already turn to the optical limit and drop, as is apparent in Figure 2 (cf. Paper III, Figure 2). The peculiarities of the disk of TW Hya suggest an advanced state of evolution of the dust. In addition, the presence of a steep transition between the inner and outer disk suggest the action of an agent other than dust settling to the midplane, which in first approximation would be expected to produce effects that vary monotonically with radius. We speculate that we may be seeing the outer edge of a gap opened by the tidal action of a growing protoplanet. Numerical models of gap formation in disks result in a sharp drop of the surface density at the gap and enhancements near the edges, as disk material is pulled away from the protoplanet and into the outer disk (Bryden et al. 2000;Nelson et al. 2000). The edge of the gap would be facing the star, and if it was optically thick, would result in excess emission similar to that modeled schematically by our truncation region. The situation could be comparable to the one considered by Syer & Clarke (1995), who studied the evolution of the disk spectrum under the presence of a planet formed in the disk, in the case in which the mass of the protoplanet was larger than the mass of the disk at the planet location. In this case, the disk cannot push the planet inwards at the local viscous time-scale and material forms a reservoir upstream of the gap (Lin & Papaloizou 1986). The surface density enhancement at the edge could make it optically thick, so the midplane temperature would not increase as much as the surface temperature; the scale height, in this case, would not increase significantly, justifying our assumption of similar heights for the edge region and outer disk and the neglect of the effects of shadowing. The details of the structure of this region, in any event, are of necessity very schematic. The extrapolated mass of the disk inside ∼ 4 AU is ∼ 10 −3 M ⊙ ∼ 1 Jupiter mass, so this mechanism would work if a giant planet were forming in this region of the disk. Finally, we address the question of what TW Hya tells us about protoplanetary disk evolution. Although we do not have direct measurements of the gas mass of the outer disk, it would be very surprising if the gas were strongly depleted there, as suggested by Zuckerman et al. (1995). Both the SED and the scattered light emission as a function of radius follow the predictions of models in which the dust is well-suspended in the gas, producing finite scale heights of the dust disk. Based on this inference, we suggest that the existence of the (fairly massive) TW Hya disk is evidence that neither UV radiation nor stellar winds can efficiently remove gas at radii ∼ > 5 AU over timescales of 10 Myr for stars formed in dispersed, isolated environments. The survival of disk material at 10 Myr also places constraints on viscous timescales. The surface density of an accretion disk evolves with time, as viscous dissipation makes the disk transfer mass onto the star and expand to conserve angular momentum in a viscous time-scale given by R 2 c /ν, where R c is a characteristic radius and ν the viscosity (cf. Pringle 1981). Here we use the similarity solutions for disk evolution of Hartmann et al. (1998), which assume temperature and surface density distributions comparable to our outer disk models. The viscous time-scale in these solutions is t s ∼ R 2 1 /ν ∼ 8 × 10 4 (10 −2 /α)(R 1 /10AU) yrs, where R 1 is the radius that contains ∼ 60 % of the initial mass (at t = 0), and the viscosity ν is expressed in terms of the standard Shakura-Sunyaev (1973) α parameterization. Adopting α ∼ 5 × 10 −4 and R 1 ∼ 100 AU, then t s ∼ 1.6 × 10 7 yrs. In the similarity model, the mass of the disk falls in time as M d ∝ T −1/2 , where T = 1 + t/ts. Thus, if the total disk mass at 10 Myr is 0.06M ⊙ , then the initial disk mass must have been 0.08M ⊙ , comparable to that of objects still surrounded by infalling envelopes, like HL Tau (D'Alessio, Calvet, & Hartmann 1997). The surface density drops below a Σ ∝ R −1 power-law by a factor of 1/e at a distance R 1 (T ) = R 1 T ∼ 160 AU, close to the ∼ 140 AU transition in midplane density found in Krist et al. (2000). With these values of disk mass and radius, the value of the Toomre parameter Q remains close to unity and so gravitational instabilities (which could transfer angular momentum rapidly and thus invalidate the purely viscous evolution model) need not arise. The mass accretion rate for this constant-α model 6 at an age of 1 Myr iṡ M = M d (0)/(2t s T 3/2 ) ∼ 2 × 10 −9 M ⊙ yr −1 , near the lowest values observed in accreting T Tauri stars of this age Hartmann et al. 1998). The mass accretion rate at 10 Myr is then ∼ 1 × 10 −9 M ⊙ yr −1 , which is slightly higher than estimated by Muzerolle et al. (2000). It would not be surprising if accretion is being halted in the inner disk, and that inner and outer disk are becoming decoupled. If material is prevented from reaching the inner disk, the material inside the gap would drain onto the central star on a viscous time at ∼ 4AU, which with α ∼ 10 −3 , would be ∼ 3 × 10 5 yrs. Such a short timescale suggests that the probability of observing a system in this stage is very low, but still possible. Instead of the α ∼ 10 −3 adopted above, Hartmann et al. (1998) estimated a typical value of α = 10 −2 for T Tauri disks based on mm-wave disk mass estimates and assuming that these disks expand to ∼ 100 AU in ∼ 1 Myr. If the TW Hya disk (the region containing most of the mass) is not much larger than 200 AU in radius, then α cannot be this large. A smaller viscosity could be consistent with the typical T Tauri data if disk masses are a few times larger than typically estimated, as suggested by D' Alessio et al. (2000). One remaining question is why TW Hya's disk evolution has been so much slower than that of other members of the association with presumably similar ages. In part, this must be due to the presence of binary (or multiple) stellar companions in several other systems (e.g., HD 98800; Soderblom et al. 1998) which can disrupt disks. In addition, we conjecture that TW Hya may simply have had a larger initial disk radius (equivalent to a large R 1 in terms of the similarity solution above), and thus a lower average surface density, than other systems. A larger radius would lead to slower viscous evolution for a given α. Also, in contrast to timescales for dust coagulation and sedimentation, which depend mostly on orbital period (see above), theoretical models of the runaway growth of giant planets indicate that the timescale for such growth is a very sensitive function of surface density (Pollack et at. 1996). The lower surface density of the disk around TW Hya would imply a slower growth of giant planets. This would imply, in turn, that the final clearing of disks is due to sweeping-up of material by large bodies. Conclusions The disk of TW Hya seems to be in advanced state of dust evolution. A planet or other large perturbing body may have already formed, opening a gap with outer edge around ∼ 4 AU, and although there is still material in the inner disk, it is rapidly dissipating. In the outer disk, grains may be reaching the size necessary to start settling towards the midplane. This interpretation can consistently explain the large degree of activity of TW Hya, the lack of disk emission at near-IR wavelengths and the large fluxes beyond ∼ 10 µm, as well as the 7 mm images. Evacuated regions in the disks are also known to result from the effects of companion stars (GG Tau), and in some cases disk accretion can still occur, especially if the binary has an eccentric orbit (DQ Tau). Indeed, HD98800, another member of the TW Hya association, shows evidence for an inner disk hole; because this system is quadruple, it is quite likely that the inner disk regions have been evacuated by the companion stars. Although we cannot rule out such a possibility in TW Hya, the fact that accretion still occurs onto the central star, which is not the case in HD 98800, suggests that the companion body responsible for the gap is much less massive than a typical companion star, and thus much less effective in clearing out the disk. Interferometric imaging should be attempted to constrain the mass of any companion object, expected to have a separation of order 2-3 AU (0.04 -0.06 arcsec at 55 pc). -SED of TW Hya and disk models with gas and dust well mixed and extending to the magnetospheric radius, for a max = 1 mm (solid line), 1 cm (dotted line), and 10 cm (long dashed line). Larger maximum grain sizes produce a better match to the sub-mm and mm-wave spectrum. The SED of the stellar photosphere is also shown for comparison (short dashed line.) Fig. 3.-Model disk adopted for TW Hya. The outer disk, where grains have grown to ∼ 1 cm, has a edge at R ∼ 3 − 4 AU, and surrounds the inner optically thin disk, which has a vertical optical depth at 10µm τ 10 ∼ 0.05. Gas still exists in the inner disk accreting onto the star through a magnetosphere. A minute amount of ∼ 1 µm dust permeates this gas. The dotted line is the surface of the outer disk if it extended inward; the resulting SED for this model is shown in Figure 4. The emission from each region is indicated: outer disk (dot-dash), edge of outer disk (dashed), inner disk (dotted), star (light solid), total (heavy solid). The stellar spectrum is taken from the Allard & Hauschildt (1995) M dwarf library, where a model with appropriate effective temperature and gravity (log g =4) could be found. The model has been scaled to the observations at K (2.2 µm). The excess near ∼ 4.5µm could be CO fundamental emission; the amount of excess (if any) at λ < 5µm is very uncertain, as it depends critically on the effective temperature (and model) for the stellar photosphere. Fig. 1 . 1-Spectral energy distribution of TW Hya. Observations are from Rucinski & Krautter (1983; average of UBVRI measurements) (open circles), Webb et al. (1999) (triangles), Sitko et al. (2000) (small open circles), this paper (open square), Jayawardhana et al. (1999) (squares), IRAS (de la Reza et al. 1989; Gregorio-Hatem et al. 1987; crosses), Weintraub et al. (1989) (hexagons), Wilner (2001) (open circle), and Wilner et al. (2000) (filled circles) The solid line is the median SED of classical T Tauri stars inTaurus (Paper II). Note the excess flux at mid-infrared wavelengths and the flux deficit below 10 µm (see text). 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W J Wiscombe, Boulder, ColoradoNational Center for Atmospheric ResearchWiscombe, W.J., 1979, Mie scattering calculations: advances in technique and fast, vector- speed computer codes, NCAR/TN-140 + STR, National Center for Atmospheric Research, Boulder, Colorado . S J Wolk, F M Walter, AJ. 1112066Wolk, S. J. & Walter, F. M. 1996, AJ, 111, 2066 . B Zuckerman, T Forveille, J H Kastner, Nature. 373494Zuckerman, B., Forveille, T., & Kastner, J. H. 1995, Nature, 373, 494 The synthesized beam sizes are 0. ′′ 82 × 0. ′′ 61 p.a. 20 (left) and 0. ′′ 13 × 0. ′′ 10Fig. 5.-(a) VLA 7 mm images of TW Hya at two resolutions, from Wilner et. 39right). The contour levels are ±(2, 3, 4, 5, 6)×Fig. 5.-(a) VLA 7 mm images of TW Hya at two resolutions, from Wilner et al. (2000). The synthesized beam sizes are 0. ′′ 82 × 0. ′′ 61 p.a. 20 (left) and 0. ′′ 13 × 0. ′′ 10, p.a. 39 (right). The contour levels are ±(2, 3, 4, 5, 6)×	[]
[ "SOLVING PARAMETERIZED POLYNOMIAL SYSTEMS WITH DECOMPOSABLE PROJECTIONS", "SOLVING PARAMETERIZED POLYNOMIAL SYSTEMS WITH DECOMPOSABLE PROJECTIONS" ]	[ "Carlos Améndola ", "ANDJulia Lindberg ", "Jose Israel Rodriguez " ]	[]	[]	The Galois group of a parameterized polynomial system of equations encodes the structure of the solutions. This monodromy group acts on the set of solutions for a general set of parameters, that is, on the fiber of a projection from the incidence variety of parameters and solutions onto the space of parameters. When this projection is decomposable, the Galois group is imprimitive, and we show that the structure can be exploited for computational improvements. Furthermore, we develop a new algorithm for solving these systems based on a suitable trace test. We illustrate our method on examples in statistics, kinematics, and benchmark problems in computational algebra. In particular, we resolve a conjecture on the number of solutions of the moment system associated to a mixture of Gaussian distributions.	null	[ "https://arxiv.org/pdf/1612.08807v2.pdf" ]	26,189,993	1612.08807	7fa910efab588f01d8b57f2537dd6a91f1455a34	SOLVING PARAMETERIZED POLYNOMIAL SYSTEMS WITH DECOMPOSABLE PROJECTIONS Carlos Améndola ANDJulia Lindberg Jose Israel Rodriguez SOLVING PARAMETERIZED POLYNOMIAL SYSTEMS WITH DECOMPOSABLE PROJECTIONS The Galois group of a parameterized polynomial system of equations encodes the structure of the solutions. This monodromy group acts on the set of solutions for a general set of parameters, that is, on the fiber of a projection from the incidence variety of parameters and solutions onto the space of parameters. When this projection is decomposable, the Galois group is imprimitive, and we show that the structure can be exploited for computational improvements. Furthermore, we develop a new algorithm for solving these systems based on a suitable trace test. We illustrate our method on examples in statistics, kinematics, and benchmark problems in computational algebra. In particular, we resolve a conjecture on the number of solutions of the moment system associated to a mixture of Gaussian distributions. Introduction A parameterized system of polynomial equations F = 0 arises from a polynomial map F : C k × C n → C N where C k is the space of parameters, C n is the space of solutions and N is the number of equations. The polynomial map F gives rise to the incidence variety: V (F ) := {(u, z) ∈ C k × C n : F (u, z) = 0}. The projection of the incidence variety to C k has a fiber over a general point in the image. When this fiber is zero dimensional, its cardinality gives a general root count to the system of equations. The Galois group or monodromy group is an invariant of a general fiber, that is, an invariant of the solutions to the parameterized polynomial system. This group acts on the solutions by permuting the elements of the fiber. When considering an irreducible component of the incidence variety, the Galois group is known to be transitive. With the transitivity property, one is able to use numerical homotopy continuation to collect solutions of the system if given a starting point. This powerful technique has been used in many instances [4,6,7,9,12,22,23]. In addition, numerical algorithms for computing Galois groups have been developed in [14,20], and examples from applications include formation shape control and maximum likelihood estimation in algebraic statistics. Many of these instances have a Galois group with special block symmetries, and we say the group is imprimitive (see Definition 2.4). The main theoretical connection is that the Galois group of a parameterized polynomial system is imprimitive if and only if the system has a decomposable projection (Proposition 2.6). We exploit this structure by generalizing witness sets of projections in Section 2, leading to Algorithm 2. An illustrative simple example is the following. of this curve to the u-coordinate is 2000 to one for all u in C \ {0, 1}. The Galois group of the cover associated to π is not the full symmetric group S 2000 . This is seen by decomposing the projection as the following sequence of maps: Z α → V β → Y (u, z) → (u, z 1000 ) → u. The degree of the map α : Z → V is 1000. By setting y = z 1000 , the defining equation of V is seen to be y 2 − 2y + u = 0. See Figure 1. The map β : V → Y is a projection with degree two. Thus, we have (non-trivially) decomposed the projection π into a composition of maps α • β. One way to describe the the fiber π −1 (u) is by listing all 2000 points over a general point. We prefer to list only 2 = deg β points that map to distinct points under α. Often, this description is sufficient as the other solutions are equivalent up to an easily described action. In this example, the action is given by multiplying the z-coordinate by a primitive 1000th root of unity. In the above, eliminating the z-coordinate to compute the defining equation of V is easily performed via substitution. However, in examples of Section 4, this elimination is a bottleneck that we avoid by using the numerical homotopy continuation method of monodromy. The rest of the paper is structured as follows. In Section 2.1, we review monodromy groups and the notion for a group to be imprimitive. In Section 2.2 we define decomposability for a projection and explain its connection to imprimitivity of the Galois group. In Section 2.3, we define witness sets, coming from numerical algebraic geometry, in the decomposable context. We put things together in a decomposable monodromy algorithm in Section 3.1, and illustrate with elementary examples in Section 3.2. In order to have a stopping criterion for the first algorithm, we introduce a trace test algorithm in Section 3.3. Finally, Section 4 is devoted to applications: we disprove a conjecture on Gaussian mixtures models, explore a kinematics problem, and provide some computations related to the benchmark cyclic n-roots problem. Monodromy, decompositions, and invariants In this section, we recall basic facts about monodromy groups of parameterized polynomial systems, we define decomposability and witness sets. 2.1. Monodromy and Galois groups. As defined in the introduction, we will consider the incidence variety V (F ) := {(u, z) ∈ C k × C n : F (u, z) = 0} of a square system of polynomial equations F = 0 parameterized by C k and their sets of solutions in C n . We consider the fiber of the projection π : V (F ) → C k . We denote the fiber over a point u ∈ π(V (F )) as π −1 (u ) := {z ∈ C n : F (u , z) = 0}. In the cases we are interested in, we can assume this fiber is finite over a generic point in C k . In other words, we are assuming the dimension of V (F ) is k and π : V (F ) → C k is dominant. As a consequence, we see that π : V (F ) → C k is a branched cover, with a branch locus denoted by B. Over C k \ B, the projection π admits a covering space, which has a monodromy group. This monodromy group is equivalent to the Galois group, see [11] for a modern reference. Definition 2.1. Let γ ⊂ C k \ B denote a loop in C k based at u * . Then, γ induces an action on π −1 (u ). We denote the permutation of the fiber induced by γ as σ γ . The group of such permutations is the monodromy group G π:V (F )→C k . When it is clear, we denote this group by G F . Remark 2.2. When V (F ) is a curve (k = 1), the branch locus is a finite set of points. The monodromy group is generated by fixing a base point and taking simple loops around each of these branch points. Proposition 2.3. If Z 1 , Z 2 are distinct irreducible components of V (F ) such that π : Z i → C k is dominant and the fiber is finite, then the monodromy group of π : Z i → C k is transitive for each i, and the monodromy group of π : Z 1 ∪ Z 2 → C k is not transitive. Proof. See Proposition 2.5 of [14]. The main consequence of this proposition is that we can use homotopy continuation to populate the fiber if given a starting point. This is described as a special case of Algorithm 1 and has been exploited in numerous instances as mentioned in the introduction. We make the assumption that one solution to a system is easy to find; this is typically done by fixing some of the variables and solving for the parameters. Using the terminology of [24], if the monodromy group of π is the full symmetric group S d , then we say that G π is uniform. Our methods focus on monodromy groups that are imprimitive; in particular these are not uniform. Decomposing a projection. We have the following definition from [24]. Definition 2.5. Let π : Z → Y be a generically finite dominant map of degree d between complex algebraic varieties. We say that π is (nontrivially) decomposable if there exists an open dense subset U ⊆ Y over which π factors as (2.1) π −1 (U ) α → V β → U where α and β are finite morphisms of degree at least two. If either deg α = 1 or deg β = 1, then we say the decomposition π = β • α is trivial. We will be interested in the case where Z is an irreducible component and curve in V (F ) with F : C × C n → C n . It follows that if π is decomposable then there is an intermediate cover. This leads directly to the following proposition. Proposition 2.6. The projection π : Z → C is decomposable as in (2.1) if and only if the Galois group G π is imprimitive. Moreover, if π : Z → C is decomposable, then G π is a subgroup of a wreath product S a ∝ S b where a = deg α, b = deg β, and the Galois group G β:V →U is a transitive subgroup of S b . Proof. The first part is immediate with Galois theory by using the one-to-one correspondence between the intermediate subfields of the field extension induced by π and the subgroups of G π . The second part follows by our assumption that Z is an irreducible curve and that the projection to C is dense. We will use Proposition 2.6 in Algorithm 1. Definition 2.7. For a parametric polynomial system F : C k × C n → C n define the projection map π : V (F ) → C k , (u, z) → u. We say F is decomposable w.r.t g : C n → C if there exists a dense Zariski open subset U of C k such that π factors as π −1 (U ) α → α(π −1 (U )) β → U where α(u, z) = (u, g(z)), β(u, y) = u. In the definition, it is important that the first coordinates are consistently u, otherwise the composition β • α does not necessarily decompose π. For most choices of α, we have deg α is one and will not yield a nontrivial decomposition of the projection. To find a nontrivial α (when they exist), one can employ algorithms in invariant theory or decomposition of polynomials. Remark 2.8 (Fundamental Invariants). Given a parametric polynomial system F , one can ask how to search for a suitable polynomial function g that makes F decomposable. When F is invariant under a finite group action G, this can be answered using fundamental invariants. We can take the polynomial map g to be a random linear combination of a finite set of fundamental invariants. In practice, it is often enough to take g to be a single low degree invariant, as illustrated in Subsection 4.1. To obtain fundamental invariants, one can use for instance the Reynolds operator from computational invariant theory [8,28]. The Reynolds operator takes a polynomial and maps it to an invariant: f → 1 \|G\| σ∈G σ(f ). When the system F is invariant under G, which is usually easy to check, we can often find a g to decompose F by applying the Reynolds operator to a generic polynomial of suitable degree. For instance, in Example 4.3 we see that the system is invariant under the dihedral group D n , so we can use this technique. 2.3. Homotopy continuation. Homotopy continuation is one of the central tools in numerical algebraic geometry. A homotopy uses a numerical predictor corrector method to deform a solution to one set of equations to another. We want to prescribe homotopies that take advantage of the structure of the system to improve computational performance. This can be done in a number of ways. For example, polyhedral methods use the Newton polytope structure of the system and regeneration uses the equation by equation structure. These methods have led to off the shelf software [29,18,5], and [4] respectively. A homotopy with path-parameter t, is given by H T : C × C n → C n , (t, z) → H t (z). Witness sets are a fundamental data structure in numerical algebraic geometry to describe varieties. The standard witness set consists of a witness point set, a linear space, and equations [27]. When the variety has more structure, additional information can be included in the witness set. This information can include multiplicity like when using deflation [17,21] or multiprojective structure [13]. In many instances, one does not have defining equations for the ideal of a variety. One such case is when a variety is given by the image of a projection; these are described by pseudo witness sets [16] or witness sets of projections [15]. For irreducible curves, which is the case we reduced to in Section 2, we recall the concept of witness sets of projections. Then, we introduce witness set factors for decomposable projections. These are much in the same vein as pseudo witness sets. They describe fibrations and sections of fiber bundles. Definition 2.9. Let F : C k ×C n → C n be a parametrized polynomial system, Z ⊆ C k ×C n an irreducible subvariety and π : C k × C n → C k be the projection given by π(u, z) = u. The witness set W π (Z) of π restricted to Z, consists of the following three pieces of information: {F, q , W π (q )} where W π (q ) := {z ∈ C n : F (q , z) = 0 and (q , z) ∈ Z} where q is a general point in the image π(Z) ⊂ C k . When the context is clear, we denote it simply by W π . The set W π (q ) is said to be a witness point set and its elements witness points. With W π , we are able to easily describe the fiber over another point q ∈ C k . From the witness set W π , we use a homotopy to deform q to q which deforms the witness point set π −1 (q ). Doing so, every nonsingular isolated point of the fiber π −1 (q ) will be a limit point of one of the deformed witness points [27]. When a projection π = β • α is decomposable, the witness set W π has extra structure that we capture with two witness point subsets: • W π (α, q ) ⊂ W π (q ) consists of deg α points that map to a single point under α. • W π (β, q ) ⊂ W π (q ) consists of deg β points that map to distinct points under α. We call such witness points subsets an α-factor and a β-factor of W π (q ), respectively. Remark 2.10. Using the notation in Definition 2.9, suppose W π has a witness point set with ab witness points and imprimitive Galois group (a, b > 1). Consider subsets A, B of W π consisting of a and b distinct witness points of W π respectively. Then, A, B are an α-factor and a β-factor respectively for W π if and only if the following occur: (1) A is a block, i.e. for each γ ∈ G π , the intersection γ · A ∩ A is empty or A. (2) B is a set of representatives for the partition by the blocks {γA} γ∈Gπ , i.e. for each γ ∈ G π , the intersection γ · A ∩ B is precisely one point. W π (α, −3) = iζ, iζ 2 , . . . , iζ 1000 and W π (β, −3) = iζ, 1000 √ −3 , where ζ is a primitive 1000th root of unity. These sets consist of a thousand and two witness points, respectively. Remark 2.10 leads to computational improvements in the following sense. Suppose we were given W π (α, −3) and wish to compute a β-factor W π (β, −3). In this example, this means finding a point in W π (α, −3) and W π (β, −3) \ W π (α, −3). Let γ denote a path in C \ {0, 1} . It would be a waste of resources to use homotopy continuation to track every point of W π (α, −3) along γ. According to Remark 2.10, if tracking W π (α, −3) along γ produces an end point in W π \ W π (α, −3), then tracking any single representative of W π (α, −3) along γ produces an end point in W π \ W π (α, −3). The notion of witness set factors generalizes naturally to projections with more than two factors in their decomposition. Suppose π decomposes as α • · · · • α 2 • α 1 with witness set W π . An α i th-factor of W π is a set W π (α i ) of deg α i distinct witness points of W π satisfying the following properties: (1) the map α i−1 • · · · • α 1 on W π (α i ) is injective, (2) the image of W π (α i ) under α i • α i−1 · · · • α 1 is one point. Example 2.12. We illustrate the decomposition of a projection into multiple factors. Let α i (u, z) = (u, z 2 ) for i = 1, 2, 3, α j (u, z) = (u, z 5 ) for j = 4, 5, 6, and α l (u, z) = u for = 7. Then, the projection π : Z → C from Example 1. 1 decomposes into π = α 7 • α 6 • α 5 • α 4 • α 3 • α 2 • α 1 . The witness point sets for each of these factors corresponding to α 1 , . . . , α 7 consist of 2, 2, 2, 2, 5, 5, 5 points respectively. In this article, we use monodromy to compute the witness set factors. Computing a fiber of a decomposable projection In this section we will give a general monodromy algorithm to populate a fiber and a trace test algorithm as a stopping criterion. 3.1. Decomposable monodromy algorithm. Our aim is to compute a subset S of the solution set to F (u , x) = 0 for generic u . We also want to make use of the information that the system is decomposable. To this end, we present Algorithm 1, which computes solutions via monodromy loops, but only keeps track of solutions that map to different images under a polynomial g : C n → C. Algorithm 1: Decomposable monodromy algorithm Input: Parametric polynomial system: F : C k × C n → C n . General parameters: u ∈ C k . Start solutions: A nonempty finite subset S of {x ∈ C n : F (u , z) = 0} ⊂ C n . Polynomial map: g : C n → C. Stopping criteria: C. Output: A finite subset of {z ∈ C n : F (u , z) = 0} ⊂ C n . 1 while the criterion C is False 2 do 3 Let B ⊂ C k be the branch locus. 4 Set γ to be a loop in C k \ B beginning at u . 5 Do a parameter homotopy along γ with start points S to obtain endpoints E. 6 for each point p in E 7 do 8 if g(p) ∈ g(S )) 9 then 10 S ← S ∪ {p}. 11 return S . We show that this algorithm is consistent with the setting of a parametrized polynomial system F with respect to g in Theorem 3.2. Remark 3.1 (Trivially decomposable). When g is the identity map, i.e., g(z) = z this is the standard monodromy as seen in [9,25]. We call this the Classical Monodromy Method. Theorem 3.2. Using the notation in Algorithm 1, if u is general and S is a single point, then the output is contained in a unique irreducible component Z of V (F ). Moreover, there exists a sequence of loops such that the output is W π (u ). Proof. We have that the monodromy group of F acts transitively on W π (u ) for a general point u ∈ π(Z) by Proposition 2.3. Hence, if g is the identity map, there exists a sequence of loops where the output is the entire set of solutions {z ∈ C n : F (u , z) = 0 and (u , z) ∈ Z}. For an arbitrary g, the output need not return this entire set of solutions: if the distinct solutions p, q are such that g(p) = g(q), then Algorithm 1 only returns one of them according to step 8. Indeed, the algorithm returns a set of solutions that have distinct images under g. This is precisely what is needed to have a witness set W π (u ) = {z ∈ C n : F (u , z) = 0 and (u , z) ∈ Z}, by definition of decomposability with respect to g. Example 3.3. Consider the parametric polynomial system given by the equation F = z 6 + z 4 + z 2 + u = 0. This is decomposable with respect to g(z) = z 2 . The image under α(u, z) = (u, z 2 ) is given by the curve in C 2 defined by y 3 + y 2 + y + u = 0. The steps of Algorithm 1 are illustrated in Figure 2. The loop γ is given by the unit circle in u-space beginning at the red point of Figure 2(c). This loop avoids the branch locus V (u(27u 2 − 14u + 3)). Tracing around γ once lifts to a path in the z-complex plane and y-complex plane connecting two red points. So tracing around γ six times gives one revolution around the "square" in Figure 2(a) but two revolutions around the "triangle" in Figure 2(b). If g(z) = z is the identity, the algorithm will return all six solutions. If g(z) = z 2 , the algorithm returns three solutions which have distinct y-coordinates. From this set of generators, one sees the Galois group G π is imprimitive: the odd and even numbered solutions form two nontrivial blocks. Indeed, note that the odd (even) numbered solutions are permuted amongst themselves or are taken to even (odd) numbered solutions. Example 3.7. Consider the parameterized system of equations F : C 6 ×C 2 → C 2 given by (3.1) (u 1 + u 2 (z 1 z 2 2 + z 2 1 z 2 ))z 1 z 2 = 0 u 4 + u 5 (z 1 + z 2 ) + u 6 z 1 z 2 = 0. We are only interested in the component Z of V (F ) not contained in a coordinate hyperplane. The projection π of Z to C has a fiber of four solutions corresponding to the mixed volume of the system. Let α(u, z 1 , z 2 ) = (u, z 1 + z 2 ). Using decomposable monodromy, we find W π (β, u) has two points, which is less than the mixed volume. is what can we use as a stopping criterion. To answer this, we have our next algorithm, which involves a trace test [19,13,26]. However, since we cannot use a trace test directly on the parametric system F : C k × C n → C n , we will modify it to get a new system. LetF be the system of n polynomial equations in n + 1 unknowns (v, z) ∈ C × C n given by Clearly,F is underdetermined. To get a parametric square system of equations, we introduce a new constraint and a single parameter t. For generic linear functions 1 , 2 : C → C we construct the following nonlinear constraint that depends on t and a polynomial g(z): (3.3) A(t, v, z) = 1 (v) 2 (g(z)) − t. The one parameter system of equations (3.4)F (v, z) = 0, A(t, v, z) = 0 is a square system. When t = 0, the system factors as (3.5)F (v, z) = 0, 1 (v) = 0 andF (v, z) = 0, 2 (g(z)) = 0. The former system corresponds to the solutions of F (u, z) = 0 where u = L(v) and v is fixed. The latter corresponds to solutions of F (u, z) = 0 with the parameters restricted to the line u = L(v) and g(z) fixed to be a generic value. Algorithm 1 (or some other black box polynomial system solving method) can be used to find subsets of the set of solutions for each of the systems by introducing the parameters t 1 , t 2 : (3.6)F (v, z) = 0, 1 (v) = t 1 andF (v, z) = 0, 2 (g(z)) = t 2 . Since i is a generic affine linear function, the parameters t 1 = 0 and t 2 = 0 are generic. Denote by S a subset of solutions W toF (v, z) = 0, A(t, v, z) = 0 with t = 0. For t = ±1 Let S(±1) denote the set of endpoints of a homotopy with t varying from 0 to ±1 with start points S. With the sets of points S(0) := S, S(−1), and S(1) we can do a trace test to verify we have found all of the solutions. We call this a Pseudo-Segre trace test because we use the image coordinate g(z) as opposed to just z, and we use the traces of a curve in a Segre embedding. Remark 3.8 (Degree of an affine curve). A special case of the previous algorithm is a classic technique in numerical algebraic geometry to verify the degree of an irreducible affine curve. Let X be a curve in C n that is an irreducible component of V (h 1 , . . . , h m ) and let z denote a generic point of X. For generic t ∈ C, and generic (b 1 , . . . , b n ) ∈ C n apply Algorithm 1 to the parameterized polynomial system F (t, z) = [h 1 (z) = 0, . . . , h m (z) = 0, n i=1 b i (z i − (z ) i ) = t − t ] with g(z) = z. It has been shown in [26,19] that Algorithm 2 returns = 0 if and only if S(0) = W . We recall that in our general situation for decomposable parametric polynomial systems, the standard trace test cannot be applied directly. Theorem 3.9. We use the preceding notation in this subsection and assume V (F ) is irreducible. Let W ⊂ C n+1 be the set of solutions to the system F (v, z) = 0, A(0, v, z) = 0. Let ψ : C n+1 → C 3 be the map ψ(v, z) = (v, g(z), vg(z)), and suppose ψ restricted to S(0) ⊆ W is one to one. Then ψ(S(0)) = ψ(W ) if and only if the output of Algorithm 2 is = 0. Proof. The main idea of the proof is to reduce to the the case where we are using a trace test to verify the degree of an affine curve in C 3 like in Remark 3.8. The polynomial system F (u, z) = 0 defines an irreducible variety in C k × C n . In (3.4) we restrict the parameter space C k to a general line using L : C → C k parameterized by v. By treating v as an unknown, we haveF (v, z) = 0 defining an algebraic variety in C n+1 ∼ = C × C n , which is in fact an irreducible curve by Bertini's theorem. Recall 1 (v) and 2 (g(z)) from (3.3). Define Y to be the graph of the map π : V (F ) → C, (v, z) → t = 1 (v) 2 (g(z)). The variety Y is irreducible because V (F ) is irreducible. Moreover, V (F ) is defined by the system (3.4). Thus, W is precisely π −1 (0). On the other hand, using g, we map V (F ) into C 2 ∼ = C × C byᾱ(v, z) := (v, g(z)). The varietyᾱ(V (F )) ⊂ C 2 has coordinate projections π 1 (v, g) = g, π 2 (v, g) = v. The degrees of these coordinate projections are say d 1 and d 2 respectively. So we can assume the curvē α(V (F )) ⊂ C 2 is defined by a bivariate polynomial in (v, g) with bidegree (d 1 , d 2 ). Since 1 , 2 : C → C are general affine linear functions, it follows α(V (F )) ∩ V ( 1 (v) 2 (g)) ⊂ C 2 consists of d 1 + d 2 points. More importantly, by taking an affine chart of a Segre embedding, we map C 2 → C 3 by σ(v, g) = (v, g, vg). With this embedding the bi-degree (d 1 , d 2 ) curveᾱ(V (F )) is a degree d 1 + d 2 curve in C 3 . Moreover, in the C 3 coordinates and for t ∈ C, the bilinear constraint 1 (v) 2 (g) = t defines a general hyperplane H t in C 3 . In summary the intersection points of the curve σ(V (f )) with hyperplane H t in C 3 is ψ(W ) = σ(ᾱ(W )), which is the intersection points of an affine curve with a hyperplane. We can use a standard trace test from Remark 3.8 to verify we have found all points of intersection in C 3 , i.e., that ψ(S(0)) = ψ(W ). The (exact) trace test is successful if and only if = 0 as mentioned in Remark 3.8. This completes the proof because we assume ψ restricted to S(0) is one to one. Remark 3.10 (Separable Solve Method). Since the polynomial systemF in equation 3.6 factors when t = 0, we exploit this fact to solve these two systems independently using Algorithm 1, followed by Algorithm 2. Empirically, this speeds up performance by a factor of two for difficult problems and we call this the Separable Method in our computational results. Applications In the first subsection we have a case study on a moment system. In the second subsection we present computational results motivated by kinematics. In the last subsections, we have a case study on the cyclic n-roots problem up to n = 9. Case Study: Gaussian Mixtures. An example from statistics where polynomial systems with symmetry arise naturally is the moment equations of Gaussian mixture distributions. For history and context of this problem, see [2]. The first non-trivial instance of this problem involves the five moment equations corresponding to a mixture of two univariate Gaussians: (4.1) m 0 = λ 1 + λ 2 m 1 = λ 1 µ 1 + λ 2 µ 2 m 2 = λ 1 (µ 2 1 + σ 2 1 ) + λ 2 (µ 2 2 + σ 2 2 ) m 3 = λ 1 (µ 3 1 + 3µ 1 σ 2 1 ) + λ 2 (µ 3 2 + 3µ 2 σ 2 2 ) m 4 = λ 1 (µ 4 1 + 6µ 2 1 σ 2 1 + 3σ 4 1 ) + λ 2 (µ 4 2 + 6µ 2 2 σ 2 2 + 3σ 4 2 ) m 5 = λ 1 (µ 5 1 + 10µ 3 1 σ 2 1 + 15µ 1 σ 4 1 ) + λ 2 (µ 5 2 + 10µ 3 2 σ 2 2 + 15µ 2 σ 4 2 ) . The indeterminates are λ 1 , λ 2 , µ 1 , µ 2 , σ 2 1 , σ 2 2 , and m 0 , m 1 , m 2 , m 3 , m 4 , m 5 are the parameters, which correspond to given numerical moments. Note that if we have a solution (λ 1 , λ 2 , µ 1 , µ 2 , σ 2 1 , σ 2 2 ), then (λ 2 , λ 1 , µ 2 , µ 1 , σ 2 2 , σ 2 1 ) is also a solution. This phenomenon is known in statistics as 'label-swapping'. We claim that this symmetry corresponds to a map decomposition of the projection of the incidence variety defined by the system to the moment space. In general, for a k mixture model, the th moment equation is given by m = λ 1 M (µ 1 , σ 1 ) + · · · + λ k M (µ k , σ k ) (4.2) where M (µ i , σ i ) can be calculated recursively as M 0 (µ i , σ i ) = 1, M 1 (µ i , σ i ) = µ i and M (µ i , σ i ) = µ i M −1 + ( − 1)σ 2 i M −2 for ≥ 2. The k mixture moment problem is to find all isolated solutions defined by the system of polynomials in equations m for = 0, . . . , 3k − 1. Due to the label-swapping symmetry discussed above, S k acts on the solution set given by a k mixture system, partitioning the solutions into equivalence classes of size k!. Beginning with the computation for a mixture of k = 2 univariate Gaussians and restricting the parameters to a general line yields the curve C. Setting y = µ 1 + µ 2 and eliminating the coordinates µ 1 , µ 2 , σ 2 1 , σ 2 2 , λ 1 , λ 2 is nontrivial. Using a combination of substitutions and resultants, after three hours we found the defining equation for α(C). This polynomial is dense in bidegree (9,9) consisting of 100 terms. On the other hand, with standard monodromy we tracked 66 paths and 18 complex solutions are obtained. Moreover, if we use Algorithm 1 instead with α(u, λ, µ, σ) = (u, µ 1 + µ 2 ) to decompose the map, we obtain 9 solution classes (of size 2) tracking only 24 paths in a particular instance. We also run the analogous computation for a mixture of k = 3 univariate Gaussians. This includes the variables λ 3 , µ 3 , σ 2 3 to the six equations in system (4.1), and we need to include three more moment equations m 6 , m 7 , m 8 to make the system zero-dimensional. This yields 225 equivalence classes of size 6 = 3! when using the general coordinate α(λ, µ, σ) = µ 1 + µ 2 + µ 3 . This number coincides with the one found via Gröbner bases in [2]. For general k, one has 3k variables and a corresponding system of 3k moment equations. The fact that this yields a finite number of solutions for generic moments was proved in [3]. For k = 4, the conjectured structure of the solutions to the system of twelve variables and twelve equations according to [2] consists of 264600 complex solutions arranged in 11025 equivalence classes of size 4! = 24. Combining Algorithms 1 and 2 we are able to disprove this conjecture. Method. LetF : C × C 13 → C 13 be the parametric system in the unknowns (v, µ, σ 2 ) given byF (t; v, µ, σ 2 ) = m − (λ 1 M (µ 1 , σ 1 ) + · · · + λ 4 M (µ 4 , σ 4 ) = 0, . . . 11 We give computational results for Gaussian 2, 3 and 4 mixture models in Table 1. All computations were performed using HomotopyContinuation.jl [5] on a 2018 Macbook Pro with 2.3 GHz Quad-Core Intel Core i5 processor. The timings and number of monodromy loops are an average of 5 trials. We initiate the trace test in Algorithm 2 once there are 10 loops with no new solutions. In Table 1, the row Monodromy (degree) corresponds to applying Algorithm 1 to the system F with stopping criterion once the number of solutions reaches the degree. The row Monodromy (trace) corresponds to applying Algorithm 1 to the systemF and then doing a trace test with Algorithm 2. The row Monodromy (separable) follows the algorithm as explained in Remark 3.10. Overall, we observe that Monodromy (degree) majorly outperforms the standard total degree and polyhedral homotopy continuation algorithms. We also see that polyhedral homotopy initially outperforms Monodromy (trace test), but once k = 4 polyhedral homotopy becomes computationally untenable. 4.2. Algebraic kinematics. There will be two ideas illustrated in this example. First, decompositions of projections can have physical meaning in kinematics. Second, even with partial information, we are able to construct an α for decomposing the projection. In this subsection we consider four-bar linkages and Alt's nine-point problem [1,30]. The first linkage is grounded in the plane at the endpoints a 1 := (a 1 ,ā 1 ) and a 2 := (a 2 ,ā 2 ); these endpoints are called the ground pivots. Two links with lengths 1 and 2 will be attached to the respective ground pivots a 1 and a 2 ; the position of the endpoints of these two links are denoted by b 1 := (b 1 ,b 1 ) and b 2 := (b 2 ,b 2 ). The middle linkage is a (coupler) triangle b 1 b 2 p with p := (p,p) called the coupler point of the four bar mechanism. The motion of p is coupled with the motion of the other two linkages. The angle of motion of links a 1 b 1 , a 2 b 2 , and b 1 b 2 p are given by motion indeterminants (θ 1 ,θ 1 ), (θ 2 ,θ 2 ) and (φ,φ) respectively. The motion indeterminants satisfy the angle relations θ iθi = 1 and φφ = 1 and vector loop relations (4.3) 1 θ 1 = p + φb 1 − a 1 , 1θ1 =p +φb 1 −ā 1 , 2 θ 2 = p + φb 2 − a 2 , 2θ2 =p +φb 2 −ā 2 . Thus, the family of four bar linkages (with coupler point p and motion) has twelve configuration indeterminants K := (p, a 1 , a 2 , b 1 , b 2 , 1 , 2 ) and six motion indeterminants M := (θ 1 ,θ 1 , θ 2 ,θ 2 , φ,φ) satisfying (4.3) and the angle relations. Projecting the family of four bar linkages to the configuration space yields a hypersurface defined by the polynomial f cc (p, a 1 , a 2 , b 1 , b 2 , 1 , 2 ) found in Eq. (3.20) in [31,Section 3.2]. The degree of this polynomial with respect to p is six. This means, when the indeterminants a 1 , a 2 , b 1 , b 2 , 1 , 2 are fixed, the polynomial defines a degree six (coupler) curve in the p plane. This curve is the set of points through which the coupler point passes through over the range of motions. If we restrict p,p to a line parameterized by v, then we have a monic univariate polynomial in v whose coefficients are rational functions in the configuration indeterminants. We identify these coefficients with Y 0 , Y 1 , Y 2 , Y 3 , Y 4 , Y 5 in the equation (4.4). (4.4) f cc (L(v), a 1 , a 2 , b 1 , b 2 , 1 , 2 ) = S 6 + Y 5 S 5 + Y 4 S 4 + Y 3 S 3 + Y 2 S 2 + Y 1 S + Y 0 where L : C → C 2 is a general affine linear function. A general coupler curve is determined by the values of these six Y -coordinates. Since the polynomial is of degree six, there is a degree six map from the family of four bar linkages to the coupler curve space given by Y -coordinates. We denote this map by α (K, M). Alt's problem is to find the number of coupler curves that pass through a specified nine general points in the plane d i := (d i ,d i ) for i = 1, 2, . . . , 9. One formulation of the problem is to solve the nine equations f cc, i (d i , a 1 , a 2 , b 1 , b 2 , 1 , 2 ), where p is set to random points d i for i = 1, 2, . . . , 9 in the plane, along with the vector loop relations and angle relations. The number of solutions has been found numerically to be 3! × 1442. The 3! comes from the Robert's cognates and label swapping symmetry. Thus, we can consider the 3! × 1442 as the degree of the fiber of the projection π of the incidence variety of four bar linkages going through nine points to the space of nine points; the incidence variety is in the configuration indeterminants K, motion indeterminants M, and indeterminants d i for i = 1, 2, . . . , 9. What we have discussed shows the projection π decomposes into α • β where Thus, we can use decomposable monodromy to determine a β witness set. Indeed, simplifying the decomposition by restricting d-space to a line and taking α (K, M) = Y 5 we use Algorithm 1 and Algorithm 2 to recover the 1442 different coupler curves. In our computation, we only tracked 5028 paths, which is even less than 3! × 1442. Table 2. Number of paths tracked and average time to compute all solutions to cyclic-n roots using different homotopy continuation algorithms. 4.3. Benchmarks with cyclic n-roots. One of the benchmark systems in polynomial system solving is the cyclic n-roots problem. The system has variables x 0 , x 1 , . . . , x n−1 and parameters u 0 , u 1 , . . . , u n−1 : (4.5) f 0 := x 0 + x 1 + · · · + x n−1 + u 0 = 0 f 1 := x 0 x 1 + x 1 x 2 + . . . + x n−1 x 0 + u 1 = 0 . . . f n−2 := x 0 x 1 · · · x n−2 + . . . + x n−1 x 0 · · · x n−3 + u n−2 = 0 f n−1 := x 0 x 1 · · · x n−1 + u n−1 = 0 The standard cyclic n-roots problem is to solve the system for a special choice of parameters u 0 = . . . = u n−2 = 0 and u n−1 = −1. We will consider a variant of this problem where we solve the system for a general choice of parameters. For n = 5, 6, 7 we solve the parameterized system of equations, which can be deformed to the special choice of parameters and find all isolated nonsingular solutions. In the three cases we considered, the root count for the generic case agrees with the special case (this is no longer true for n = 4, 8, 9 [10]). The system (4.5) is known to have 70 solutions when n = 5. These solutions split into 7 groups of 10 elements via the dihedral action on the coordinates x 0 , x 1 , x 2 , x 3 , x 4 where rotations act cyclicly on the labels and a reflection reverses the ordering of the labels. Defining equations for the irreducible curve C are found by restricting u 0 , u 1 , . . . , u 4 to a line parameterized by v. The projection π : C → C decomposes into α • β where α(v, x) = (v, x 3 x 0 + x 4 x 1 + x 0 x 2 + x 1 x 3 + x 2 x 4 ) and β(v, y) = v. To find this decomposition, we note that the system is invariant under the dihedral group D 5 which acts by label swapping the x-coordinates. We use the Reynolds operator on the monomial x 0 x 2 , i.e., 1 \|D n \| σ∈Dn σ(x 0 x 2 ) = x n−2 x 0 + x n−1 x 1 + x 0 x 2 + x 1 x 3 + · · · + x n−3 x n−1 (n = 5). One might be tempted to take α(v, x) = v, σ∈D 5 σ(x 0 ) or α(v, x) = v, σ∈D 5 σ(x 0 x 1 ) . However, such choices lead to α having degree 70 as the entire fiber is mapped to a single point under α; this means β has degree 1, and we fail to nontrivially decompose the projection. We summarize our computations for this subsection in Table 2. The different homotopy continuation methods are as in Table 1. We see again that Monodromy (degree) is much faster than total degree and polyhedral homotopies. Example 1. 1 . 1Let k = n = N = 1. Consider the curve Z defined by z 2000 −2z 1000 +u = 0. The projection map π : Z → Y (u, z) → u . Figure 1 . 1Curve y 2 − 2y + u = 0 from Example 1.1. When u = −3 we obtain the deg β = 2 intersection points (−3, −1), (−3, 3) of V . Example 2 . 11 . 211Recall the curve Z from Example 1.1. The witness set of π : Z → C consists of 2000 points. For u = −3, the following are factors of the witness set W π (−3): Remark 3. 4 ( 4Choosing loops). While taking a loop given by concatenating random line segments avoids the branch locus with probability one, a systematic choice of loops can lead to computational savings during implementation. Remark 3. 5 ( 5Stopping criteria implementation). The framework that we have provided is flexible and allows for numerous types of stopping criteria. Some typical criteria count the number of times the while loop is entered, an upper bound on the number of solutions, or the wall time of the computation. In our implementation, the default stopping criteria are: (1) the while loop did not find any new solutions ten times in a row or (2) the number of solutions found equals or exceeds D which is chosen accordingly.3.2. Elementary examples. Example 3.6. Following up on Example 1.1, consider the curve C ⊂ C 2 defined by z 2000 − 2z 1000 + u. The branch locus for the projection π : C → C consists of two points: u = 0 and u = 1. The critical locus can be written as the intersection of two ideals: (z 2000 − 2z 1000 + u, 2000z 1999 − 2000z 999 ) = (u, z 999 ) ∩ (u − 1, z 1000 + 1).Let γ 0 , γ 1 denote loops based at a general point of C that encircle u = 0 and u = 1 respectively. These loops induce the following two permutations:(1, 2)(3, 4)(5, 6) · · · (1999, 2000) and (1, 3, 5, . . . , 1999)(2, 4, 6, . . . , 2000). Figure 2 . 2Illustration of Algorithm 1 3.3. A trace test stopping criterion. An immediate question in regards to Algorithm 1 (3.2)F (v, z) := F (L(v), z) where L : C → C k is a general affine linear function. Algorithm 2 : 2Pseudo-Segre Trace test Input: Nonempty subsets S(0), S(−1), and S(1) in C n+1 ∼ = C × C n A polynomial map g : C n → C Output: A nonnegative real number 1 Denote by Tr(j) ∈ C 3 the coordinate-wise average of the set of points {(v, g(z), vg(z)) : (v, z) ∈ S(j)}.2 Set to \|\|(Tr(1) − Tr(0)) − (Tr(0) − Tr(−1))\|\|. 3 return Result 4 . 1 . 41The number of solutions for a k = 4 mixture model is 248400 = 10350 · 4! for generic moments (m 0 , m 1 , . . . , m 11 ). α(d 1 1. . . , d 9 ; K, M) := (d 1 , . . . d 9 ; α (K, M)) and β(d 1 , . . . d 9 ; Y 0 , . . . , Y 5 , ) = (d 1 , . . . d 9 ). Definition 2.4. Let G be a group acting transitively on a finite set [k] = {1, . . . , k}. A subset B ⊆ [k] is a block if gB = B or gB ∩ B = ∅ for every g ∈ G. We say that G is primitive if its only blocks are ∅, [k] and {j} for j ∈ [k]. Otherwise, G is imprimitive. (v) · 2 (g(µ, σ)) + t with L(v) = (m 0 , m 1 , . . . , m 11 ) where L : C → C 12 is a general affine linear function and 1 , 2 : C → C are general affine linear functions.For t = 0, we find 31815 solutions and verify this is a complete set of solutions up to symmetry using Algorithm 2 with < 10 −12 . Of the 31815 solutions, 10350 solutions satisfy 1 (v) = 0. Since 1 is a general affine linear function, all of these solutions have the same v-coordinate, say v . The 10350 are solutions for the moment system chosen as L(v ). AcknowledgementsWe would like to thank Bernd Sturmfels, Jonathan Hauenstein, Anton Leykin, Gunter Malle, and Botong Wang for their helpful comments and suggestions. Über die Erzeugung gegebener ebener Kurven mit Hilfe des Gelenkvierecks. H Alt, Zeitschrift für Angewandte Mathematik und Mechanik. 3114H. Alt.Über die Erzeugung gegebener ebener Kurven mit Hilfe des Gelenkvierecks. 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Verschelde, and C. Wampler. Using monodromy to decompose solution sets of polynomial systems into irreducible components. In Applications of algebraic geometry to coding theory, physics and computation (Eilat, 2001), volume 36 of NATO Sci. Ser. II Math. Phys. Chem., pages 297-315. Kluwer Acad. Publ., Dordrecht, 2001. 7 Symmetric functions applied to decomposing solution sets of polynomial systems. A Sommese, J Verschelde, C Wampler, SIAM J. Numer. Anal. 40611A. Sommese, J. Verschelde, and C. Wampler. Symmetric functions applied to decomposing solution sets of polynomial systems. SIAM J. Numer. Anal., 40(6):2026-2046, 2002. 9, 11 The numerical solution of systems of polynomials. A J Sommese, C W Wampler, I I , World Scientific Publishing Co. Pte. LtdHackensack, NJArising in engineering and scienceA. J. Sommese and C. W. Wampler, II. The numerical solution of systems of polynomials. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. 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[ "Holographic inflation", "Holographic inflation" ]	[ "Shin'ichi Nojiri \nDepartment of Physics\nNagoya University\n464-8602NagoyaJapan\n\nKobayashi-Maskawa Institute for the Origin of Particles and the Universe\nNagoya University\n464-8602NagoyaJapan\n", "Sergei D Odintsov \nInstitut de Ciencies de lEspai (IEEC-CSIC)\nCampus UAB, Carrer de Can Magranss/n 08193 Cerdanyola del Valles, BarcelonaSpain\n\nInstitució Catalana de Recerca i Estudis Avançats (ICREA)\nPasseig Lluís Companys\n23 08010BarcelonaSpain\n", "Emmanuel N Saridakis \nDepartment of Physics\nNational Technical University of Athens\nZografou Campus GR 157 73AthensGreece\n\nDepartment of Astronomy\nSchool of Physical Sciences\nUniversity of Science and Technology of China\n230026HefeiP.R. China\n" ]	[ "Department of Physics\nNagoya University\n464-8602NagoyaJapan", "Kobayashi-Maskawa Institute for the Origin of Particles and the Universe\nNagoya University\n464-8602NagoyaJapan", "Institut de Ciencies de lEspai (IEEC-CSIC)\nCampus UAB, Carrer de Can Magranss/n 08193 Cerdanyola del Valles, BarcelonaSpain", "Institució Catalana de Recerca i Estudis Avançats (ICREA)\nPasseig Lluís Companys\n23 08010BarcelonaSpain", "Department of Physics\nNational Technical University of Athens\nZografou Campus GR 157 73AthensGreece", "Department of Astronomy\nSchool of Physical Sciences\nUniversity of Science and Technology of China\n230026HefeiP.R. China" ]	[]	We apply the holographic principle at the early universe, obtaining an inflation realization of holographic origin. Such a consideration has equal footing with its well-studied late-time application, and moreover the decrease of the horizons at early times naturally increases holographic energy density at inflationary scales. Taking as Infrared cutoff the particle or future event horizons, and adding a simple correction due to the Ultraviolet cutoff, whose role is non-negligible at the high energy scales of inflation, we result in a holographic inflation scenario that is very efficient in incorporating inflationary requirements and predictions. We first extract analytically the solution of the Hubble function in an implicit form, which gives a scale factor evolution of the desired e-foldings. Furthermore, we analytically calculate the Hubble slow-roll parameters and then the inflation-related observables, such as the scalar spectral index and its running, the tensor-to-scalar ratio, and the tensor spectral index. Confronting the predictions with Planck 2018 observations we show that the agreement is perfect and in particular deep inside the 1σ region.	10.1016/j.physletb.2019.134829	null	91,184,461	1904.01345	997638d7ea45ce1da7ef4b0b93396b2002f2089a	Holographic inflation 2 Apr 2019 Shin'ichi Nojiri Department of Physics Nagoya University 464-8602NagoyaJapan Kobayashi-Maskawa Institute for the Origin of Particles and the Universe Nagoya University 464-8602NagoyaJapan Sergei D Odintsov Institut de Ciencies de lEspai (IEEC-CSIC) Campus UAB, Carrer de Can Magranss/n 08193 Cerdanyola del Valles, BarcelonaSpain Institució Catalana de Recerca i Estudis Avançats (ICREA) Passeig Lluís Companys 23 08010BarcelonaSpain Emmanuel N Saridakis Department of Physics National Technical University of Athens Zografou Campus GR 157 73AthensGreece Department of Astronomy School of Physical Sciences University of Science and Technology of China 230026HefeiP.R. China Holographic inflation 2 Apr 2019 We apply the holographic principle at the early universe, obtaining an inflation realization of holographic origin. Such a consideration has equal footing with its well-studied late-time application, and moreover the decrease of the horizons at early times naturally increases holographic energy density at inflationary scales. Taking as Infrared cutoff the particle or future event horizons, and adding a simple correction due to the Ultraviolet cutoff, whose role is non-negligible at the high energy scales of inflation, we result in a holographic inflation scenario that is very efficient in incorporating inflationary requirements and predictions. We first extract analytically the solution of the Hubble function in an implicit form, which gives a scale factor evolution of the desired e-foldings. Furthermore, we analytically calculate the Hubble slow-roll parameters and then the inflation-related observables, such as the scalar spectral index and its running, the tensor-to-scalar ratio, and the tensor spectral index. Confronting the predictions with Planck 2018 observations we show that the agreement is perfect and in particular deep inside the 1σ region. Introduction -The holographic principle originates from black hole thermodynamics and string theory [1][2][3][4], and establishes a connection of the Ultraviolet cutoff of a quantum field theory, which is related to the vacuum energy, with the largest distance of this theory, which is related to causality and the quantum field theory applicability at large distances [5]. This consideration has been applied extensively at a cosmological framework at late times, in which case the obtained vacuum energy constitutes a dark energy sector of holographic origin, called holographic dark energy [6] (for a review see [7]). In particular, the holographic energy density is proportional to the inverse squared Infrared cutoff L IR , which since is related to causality it must be a form of horizon, namely ρ = 3c 2 κ 2 L 2 IR ,(1) with κ 2 the gravitational constant and c a parameter. Holographic dark energy proves to have interesting phenomenology, both in its basic [6][7][8][9][10][11][12][13][14] as well as in its various extensions [15][16][17][18][19][20][21][22], and it can fit observations [23][24][25][26][27][28][29]. Despite the extended research on the application of holographic principle in late-time cosmology and darkenergy epoch, there has not been any attempt in applying it at early universe, namely to obtain an inflationary realization of holographic origin. Nevertheless, such consideration has equal footing with its late-time application, and moreover, observing the form of (1), we deduce that since at early times the largest distance is small, the holographic energy density is naturally suitably large in order to lie in the inflationary scale. In the present Letter we are interested in investigating holographic inflation, namely to acquire a successful inflation triggered by the energy density of holographic origin. Since the involved energy scales at this epoch are high, we should additionally consider a correction coming from the Ultraviolet cutoff. As we show, although the basic scenario is very simple and natural, it can be very efficient and results to inflationary observables in perfect agreement with observations. Furthermore, as we show, one can extend the basic scenario to more subtle constructions. Holographic inflation -In this section we will construct the basic model of holographic inflation. We consider a homogeneous and isotropic Friedmann-Robertson-Walker (FRW) geometry with metric ds 2 = −dt 2 + a 2 (t) dr 2 1 − kr 2 + r 2 dΩ 2 ,(2) where a(t) is the scale factor and with k = 0, +1, −1 corresponding respectively to flat, close and open spatial geometry. In this work for convenience we focus on the flat case, nevertheless the generalization to non-flat geometry is straightforward. In a general inflation scenario the first Friedmann equations is written as H 2 = κ 2 3 ρ inf ,(3) where ρ inf is the energy density of the (effective) fluid that drives inflation, which can originate from a scalar field, from modified-gravity, or from other sources and mechanisms. Note that as usual we have neglected the contributions from other components, such as the matter and radiations sectors. In this work we consider that inflation has a holographic origin, namely that its source is the holographic energy density. Hence, imposing that ρ inf is ρ of (1), the Friedmann equation (3) for an expanding universe becomes simply H = c L IR .(4) As we mentioned earlier, since the Infrared cutoff is related to causality it must be a form of horizon. The simplest choice is the Hubble radius, which however cannot be used at late-times application since it cannot lead to an accelerating universe [30], and this is the case for the next guess, namely the particle horizon. Hence, one can use the future event horizon [6], the age of the universe or the conformal time [31,32], the inverse square root of the Ricci curvature [33], a combination of Ricci, Gauss-Bonnet [34] or other curvature invariants, or even consider a general Infrared cutoff as an arbitrary function of the above and their derivatives [35]. Contrary to the case of late-time application of holographic principle, namely the holographic dark energy, in the present inflationary application almost all the above choices can be successful in driving inflation, due to the absence of matter sector, apart from the simplest case of the Hubble radius which leads to a trivial result. In particular, we may consider the particle horizon L p or the future event horizon L f , which are given as L p ≡ a t 0 dt a , L f ≡ a ∞ t dt a .(5) Inserting these into (4) we obtain d dt c aH = m a ,(6) where m = 1 corresponds to the particle horizon and m = −1 to the future event horizon. In the second case, and for c = 1, we immediately extract the de Sitter solution a = a 0 e H0t , with a 0 ,H 0 the two integration constants. Hence, as we observe, the basic inflationary feature can be straightforwardly obtained. Since we investigate the application of holographic principle at early times, and thus at high energy scales, apart from the Infrared cutoff we should consider the effects of the Ultraviolet cutoff Λ UV too. In particular, at this regime the quantum effects become important, and thus the Infrared cutoff acquires a correction by the Ultraviolet one, which as was shown in [36] takes the simple form L ≡ L 2 IR + 1 Λ 2 UV .(8) Therefore, inserting this corrected expression into (4), with L IR being either the particle horizon L p or the future event horizon L f , we obtain m a = d dt 1 a c 2 H 2 − 1 Λ 2 UV ,(9) or equivalentlẏ H = − H 3 c 2 m c 2 H 2 − 1 Λ 2 UV + H c 2 H 2 − 1 Λ 2 UV . (10) As we will soon see, the above simple modification caused by Λ UV has a crucial effect in the inflation realization, namely it can cause a successful exit from the de-Sitter solution (7) obtained for Λ UV → ∞, and most importantly it can lead to inflationary predictions in perfect agreement with observations. The general solution of (10), for Λ UV not being equal to 0 or infinity, can be written in an implicit form as (11) whose integration provides the scale factor evolution. Note that the above solution is real and well behaved for all c > 0. Hence, since the evolution of H(t) is known, we can straightforwardly obtain the Hubble slow-roll parameters ǫ n (with n positive integer), defined as [37][38][39][40] c 2 − H 2 Λ 2 UV (c 2 − 1) 3 2 Λ 2 UV H 2 − c 2 Λ 2 UV tan −1 mH √ c 2 − 1 H 2 − c 2 Λ 2 UV + m c 2 − H 2 Λ 2 UV −c 2 c 2 (c 2 −1)HΛ 2 UV + tanh −1 H √ c 2 −1Λ UV (c 2 − 1) 3 2 Λ 3 UV = − t c 2 Λ 2 UV +C0,ǫ n+1 ≡ d ln \|ǫ n \| dN ,(12) with ǫ 0 ≡ H ini /H and N ≡ ln(a/a ini ) the e-folding number, and where a ini is the scale factor at the beginning of inflation and H ini the corresponding Hubble parameter (inflation ends when ǫ 1 = 1). Thus, we can calculate the values of the inflationary observables, namely the scalar spectral index of the curvature perturbations n s , its running α s ≡ dn s /d ln k with k the absolute value of the wave number k, the tensor spectral index n T and the tensor-to-scalar ratio r, as [39] r ≈ 16ǫ 1 , n s ≈ 1 − 2ǫ 1 − 2ǫ 2 ,(13)α s ≈ −2ǫ 1 ǫ 2 − ǫ 2 ǫ 3 ,(14)n T ≈ −2ǫ 1 ,(15) where the first three ǫ n are straightforwardly extracted from (12) to be ǫ 1 ≡ −Ḣ H 2 ,(17)ǫ 2 ≡Ḧ HḢ − 2Ḣ H 2 ,(18)ǫ 3 ≡ Ḧ H − 2Ḣ 2 −1 · HḢ ... H −Ḧ(Ḣ 2 + HḦ) HḢ − 2Ḣ H 2 (HḦ − 2Ḣ 2 ) .(19) Relations (13)- (16) are very useful, since they allow for a comparison of the predictions of holographic inflation with observations. In Fig. 1 we present the estimated tensor-to-scalar ratio of the specific scenario for four parameter choices and for e-folding numbers varying between N = 50 and N = 60, on top of the 1σ and 2σ contours of the Planck 2018 results [41]. As we observe, the agreement with observations is very efficient, and in particular well inside the 1σ region. In the scenario of holographic inflation examined above, the differential equation (10) allows to eliminate the time-derivatives of H in terms of H in (17)- (19), and therefore extract analytical and quite simple expressions for the inflationary observables. In particular, doing so we find that r = 16   1 − H 2 c 2 Λ 2 UV + m c 2 − H 2 Λ 2 UV c 2   , (20) n s = −1 − 2m c 2 − H 2 Λ 2 UV − 2H 2 c 2 Λ 2 UV ,(21)α s = mH 4 m + c 2 − H 2 Λ 2 UV c 4 Λ 2 UV (c 2 Λ 2 UV − H 2 ) ,(22)n T = − r 8 .(23) Hence, we can now eliminate H 2 Λ 2 UV between (20), (21) and between (20), (22), and obtain the relation of n s and α s in terms of r, namely n s = −3 + r 8 + 2 − l √ c 2 r + 4 2c 2 − 8m c 2 r + 8 − 4l √ c 2 r + 4 ,(24) and α s = m 8 + c 2 (r − 16) − 4q √ c 2 r + 4 2 64c 4 8 + c 2 r − 4q √ c 2 r + 4 · 4m + 8 + c 2 r − 4q c 2 r + 4 ,(25) where l = ±1 corresponds to two solutions branches. Interestingly enough, we observe that as long as Λ UV is finite and non-zero it does not appear in the above relations (24) and (25), although its effect was crucial in generating a new solution branch that did not exist in the case where Λ UV in (8) was absent (namely when Λ UV → ∞, i.e. when there is no Ultraviolet cutoff). Of course it does appear in the solution for H (relation (11)), and hence along with c it determines the duration of inflation and the e-folding number, and thus the bounds of the above parametric curves. However, the most significant feature is that relation (24) leads to r and n s values in perfect agreement with observations, as long as one chooses suitably the holographic parameter c. In particular, one can deduce that the most interesting case is when m = −1 (i.e. the future event horizon is used) and when q = −1. In this case, as one can see from Fig. 2, with c slightly larger than 1 we can obtain a n s inside its observational bounds and r adequately small. Finally, calculations of α s using (25) result to typical values of the order of 10 −7 , and thus well inside the observational bounds [41]. These are the main result of the present work and reveal the capabilities of holographic inflation, since it is well-known that the majority of inflationary models cannot result to adequately small r. Generalized scenarios -In this section we apply the holographic principle at early times, however considering extended Infrared cutoffs. Such extensions have been applied in the late-time universe, resulting in generalized holographic dark energy [35]. In particular, in these holographic constructions one considers a general Infrared cut-off L IR , which could be a function of both L p and L f [14] and their derivatives, or additionally of the Hubble horizon and its derivatives as well as of the scale factor [35], namely L IR = L IR L p ,L p ,L p , · · · , L f ,L f ,L f , · · · , a, H,Ḣ,Ḧ, · · · . (26) Hence, applying the above general Infrared cutoff at the early universe gives as enhanced freedom to obtain a successful inflationary realization. Without loss of generality we consider the following specific example. We start by considering an Infrared cutoff of the form L IR = − 1 6αḢ 2 a 6 t dta 6Ḣ ,(27) with α the model parameter. Inserting this expression into the inflationary Friedmann equation (4) and taking c = 1 for simplicity, leads to the differential equation 3H 2 = α −108H 2Ḣ + 18Ḣ 2 − 36HḦ .(28) Having in mind that the Ricci scalar in FRW geometry is just R = 6(2H 2 +Ḣ), the above differential equation can be re-written in the form F (R) 2 = 3 H 2 +Ḣ F ′ (R) − 18 4H 2Ḣ + HḦ F ′′ (R),(29) with F (R) = R + αR 2 . Equation (29) is just the first Friedmann equation of the R 2 -gravity [43][44][45][46] (see [47,48] for reviews in F (R) gravity) in the absence of matter. Therefore, the scenario of holographic inflation, under the generalized Infrared cutoff (27) can reproduce Starobinsky R 2 inflation [43], which is known to lead to inflationary observables in a very good agreement with observations [41]. In similar lines, one can consider other generalized Infrared cutoffs in order to obtain a correspondence with other geometrical inflationary models, such as Gauss-Bonnet and f (G) inflation [49], f (T ) inflation, etc. These capabilities act as an additional advantage in favour of generalized holographic inflation. Conclusions -In this work we applied the holographic principle at the early universe, obtaining an inflation realization of holographic origin. Although holographic energy density has been well studied at late times, giving rise to holographic dark energy, up to now it had not been incorporated at early times, although such consideration has equal footing, and moreover despite the fact that the decrease of the horizons at early times naturally increases holographic energy density at inflationary scales. Taking as Infrared cutoff the particle or future event horizons, and adding a simple correction due to the Ultraviolet cutoff, whose role is non-negligible at the high energy scales of inflation, we resulted in a holographic inflation scenario that is very efficient in incorporating inflationary requirements and predictions. In particular, we first extracted analytically the solution of the Hubble function in an implicit form, which can give a scale factor evolution of the desired e-foldings. Furthermore, we analytically calculated the Hubble slow-roll parameters and then the inflation-related observables, such as the scalar spectral index and its running, the tensor-to-scalar ratio, and the tensor spectral index, which were found to follow simple expressions. Confronting the predictions with Planck 2018 observations, we showed that the agreement is perfect, and in particular deep inside the 1σ region. Additionally, we found that with n s being inside its observational bounds r can be adequately small. These are the main result of the present work and reveal the capabilities of holographic inflation, since it is well-known that the majority of inflationary models cannot lead to adequately small r. Finally, we constructed generalizations of holographic inflation which are based on extended Infrared cutoffs. Under these considerations we showed that we can reconstruct Starobinsky inflation, which is also known to be in a very good agreement with the data, as well as obtain a correspondence with other inflationary scenarios of geometrical origin. In summary, holographic inflation proves to have a very interesting phenomenology, and thus it is a good candidate for the description of the early universe. FIG. 1 : 11σ (yellow) and 2σ (light yellow) contours for Planck 2018 results (Planck +T T + lowP ) [41], on ns − r plane. 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[ "Novelty Search in Representational Space for Sample Efficient Exploration", "Novelty Search in Representational Space for Sample Efficient Exploration" ]	[ "Ruo Yu Tao *[email protected] \nMcGill University\n\n\nQuebec Artificial Intelligence Institute\nMila\n", "Vincent François-Lavet \nMcGill University\n\n\nQuebec Artificial Intelligence Institute\nMila\n", "Joelle Pineau \nMcGill University\n\n\nQuebec Artificial Intelligence Institute\nMila\n" ]	[ "McGill University\n", "Quebec Artificial Intelligence Institute\nMila", "McGill University\n", "Quebec Artificial Intelligence Institute\nMila", "McGill University\n", "Quebec Artificial Intelligence Institute\nMila" ]	[]	We present a new approach for efficient exploration which leverages a lowdimensional encoding of the environment learned with a combination of modelbased and model-free objectives. Our approach uses intrinsic rewards that are based on the distance of nearest neighbors in the low dimensional representational space to gauge novelty. We then leverage these intrinsic rewards for sampleefficient exploration with planning routines in representational space for hard exploration tasks with sparse rewards. One key element of our approach is the use of information theoretic principles to shape our representations in a way so that our novelty reward goes beyond pixel similarity. We test our approach on a number of maze tasks, as well as a control problem and show that our exploration approach is more sample-efficient compared to strong baselines.	null	[ "https://arxiv.org/pdf/2009.13579v3.pdf" ]	213,793,858	2009.13579	930428503ea019465a856098f4b9079a8ffd9b62	Novelty Search in Representational Space for Sample Efficient Exploration Ruo Yu Tao [email protected] McGill University Quebec Artificial Intelligence Institute Mila Vincent François-Lavet McGill University Quebec Artificial Intelligence Institute Mila Joelle Pineau McGill University Quebec Artificial Intelligence Institute Mila Novelty Search in Representational Space for Sample Efficient Exploration We present a new approach for efficient exploration which leverages a lowdimensional encoding of the environment learned with a combination of modelbased and model-free objectives. Our approach uses intrinsic rewards that are based on the distance of nearest neighbors in the low dimensional representational space to gauge novelty. We then leverage these intrinsic rewards for sampleefficient exploration with planning routines in representational space for hard exploration tasks with sparse rewards. One key element of our approach is the use of information theoretic principles to shape our representations in a way so that our novelty reward goes beyond pixel similarity. We test our approach on a number of maze tasks, as well as a control problem and show that our exploration approach is more sample-efficient compared to strong baselines. Introduction In order to solve a task efficiently in reinforcement learning (RL), one of the main challenges is to gather informative experiences via an efficient exploration of the state space. A common approach to exploration is to leverage intrinsic rewards correlated with some metric or score for novelty (Schmidhuber, 2010;Houthooft et al., 2016). With intrinsic rewards, an agent can be incentivized to efficiently explore its state space. A direct approach to calculating these novelty scores is to derive a reward based on the observations, such as a count-based reward (Bellemare et al., 2016;Ostrovski et al., 2017) or a prediction-error based reward (Burda et al., 2018b). However, an issue occurs when measuring novelty directly from the raw observations, as some information in pixel space (such as randomness or backgrounds) may be irrelevant. In this case, if an agent wants to efficiently explore its state space it should only focus on meaningful and novel information. In this work, we propose a method of sample-efficient exploration by leveraging intrinsic rewards in a meaningful latent state space. To build a meaningful state abstraction, we view Model-based RL (MBRL) from an information theoretic perspective -we optimize our dynamics learning through the Information Bottleneck (Tishby et al., 2000) principle. We also combine both model-based and model-free components through a joint representation. This method encodes high-dimensional observations into lower-dimensional representations such that states that are close in dynamics are brought close together in representation space (François-Lavet et al., 2018). We also add additional constraints to ensure that a measure of distance between abstract states is meaningful. We leverage these properties of our representation to formulate a novelty score based on Euclidean distance in low-dimensional representation space and we then use this score to generate intrinsic rewards that we can exploit for efficient exploration. One important element of our exploration algorithm is that we take a Model Predictive Control (MPC) approach (Garcia et al., 1989) and perform actions only after our model is sufficiently accurate (and hence ensure an accurate novelty heuristic). Through this training scheme, our agent is 34th Conference on Neural Information Processing Systems (NeurIPS 2020), Vancouver, Canada. arXiv:2009.13579v3 [cs.LG] 15 Apr 2022 also able to learn a meaningful representation of its state space in a sample-efficient manner. The code with all experiments is available 1 . Problem setting An agent interacts with its environment over discrete timesteps, modeled as a Markov Decision Process (MDP), defined by the 6-tuple (S, S 0 , A, τ, R, G) (Puterman, 1994). In this setting, S is the state space, S 0 is the initial state distribution, A is the discrete action space, τ : S × A → S is the transition function that is assumed deterministic (with the possibility of extension to stochastic environments with generative methods), R : S × A → R is the reward function (R = [−1, 1]), G : S × A → [0, 1) is the per timestep discount factor. At timestep t in state s t ∈ S, the agent chooses an action a t ∈ A based on policy π : S × A → [0, 1], such that a t ∼ π(s t , ·). After taking a t , the agent is in state s t+1 = τ (s t , a t ) and receives reward r t ∼ R(s t , a t ) and a discount factor γ t ∼ G(s t , a t ). Over n environment steps, we define the buffer of previously visited states as B = (s 1 , . . . , s n ), where s i ∈ S ∀i ∈ N. In RL, the usual objective is to maximize the sum of expected future rewards V π (s) = E π r t + ∞ i=1 i−1 j=0 γ t+j r t+i \|s = s t . To learn a policy π that maximizes the expected return, an RL agent has to efficiently explore its environment (reach novel states in as few steps as possible). In this paper, we consider tasks with sparse rewards or even no rewards, and are interested in exploration strategies that require as few steps as possible to explore the state space. Abstract state representations We focus on learning a lower-dimensional representation of state when our state (or observations in the partially observable case (Kaelbling et al., 1998)) is high-dimensional (Dayan, 1993;Tamar et al., 2016;Silver et al., 2016;Oh et al., 2017;de Bruin et al., 2018;Ha and Schmidhuber, 2018;François-Lavet et al., 2018;Hafner et al., 2018;Gelada et al., 2019). Information Bottleneck We first motivate our methods for model learning. To do so, we consider the Information Bottleneck (IB) (Tishby et al., 2000) principle. Let Z denote the original source message space andZ denote its compressed representation. As opposed to traditional lossless compression where we seek to find corresponding encodingsZ that compresses all aspects of Z, in IB we seek to preserve only relevant information inZ with regards to another relevance variable, Y . For example when looking to compress speech waveforms (Z) if our task at hand is speech recognition, then our relevance variable Y would be a transcript of the speech. Our representationZ would only need to maximize relevant information about the transcript Y instead of its full form including tone, pitch, background noise etc. We can formulate this objective by minimizing the following functional with respect to p(z \| z): L(p(z \| z)) = I[Z;Z] − βI[Z; Y ] where I[·; ·] is the Mutual Information (MI) between two random variables. β is the Lagrange multiplier for the amount of information our encodingZ is allowed to quantify about Y . This corresponds to a trade-off between minimizing the encoding rate I[Z;Z] and maximizing the mutual information between the encoding and our random variable Y . We now apply this principle to representation learning of state in MBRL. If our source message space is our state S and our encoded message is X , then to distill the most relevant information with regards to the dynamics of our environment one choice of relevance variable is {X, A}, i.e. our encoded state in the previous timestep together with the presence of an action. This gives us the functional L(p(x \| s )) = I[S ; X ] − βI[X ; {X, A}]. (1) In our work, we look to find methods to minimize this functional for an encoding that maximizes the predictive ability of our dynamics model. We first aim to minimize our encoding rate I[S ; X ]. Since encoding rate is a measure of the amount of bits transmitted per message S , representation dimension is analogous to number of bits per message. This principle of minimizing encoding rate guides our selection of representation dimension -for every environment, we try to choose the smallest representation dimension possible such that the representation can still encapsulate model dynamics as we understand them. For example, in a simple Gridworld example, we look to only encode agent position in the grid-world. Now let us consider the second term in Equation 1. Our goal is to learn an optimally predictive model of our environment. To do so we first consider the MI between the random variable denoting our state representation X, in the presence of the random variable representing actions A and the random variable denoting the state representation in the next timestep X (Still, 2009). Note that MI is a metric and is symmetric: I[{X, A} ; X ] = E p(x ,x,a) log p(x \| x, a) p(x ) = H[X ] − H[X \| X, A](2) This quantity is a measure of our dynamics model's predictive ability. If we consider the two entropy terms (denoted H[·]), we see that H[X ] constitutes the entropy of our state representation and H[X \| X, A] as the entropy of the next state X given our current state X and an action A. Recall that we are trying to minimize I[X ; S ] and maximize I[X ; {X, A}] with respect to some encoding function X = e(S). In the next section, we describe our approach for this encoding function as well as dynamics learning in MBRL. Encoding and dynamics learning For our purposes, we use a neural encoderê : S → X parameterized by θê to map our highdimensional state space into lower-dimensional abstract representations, where X ⊆ R n X . The dynamics are learned via the following functions: a transition functionτ : X × A → X parameterized by θτ , a reward functionr : X × A → [−1, 1] parameterized by θr, and a per timestep discount factor functionγ : X × A → [0, 1) parameterized by θγ. This discount factor is only learned to predict terminal states, where γ = 0. In order to leverage all past experiences, we use an off-policy learning algorithm that samples transition tuples (s, a, r, γ, s ) from a replay buffer. We first encode our current and next states with our encoder to get x ←ê(s; θê), x ←ê(s ; θê). The Q-function is learned using the DDQN algorithm (van Hasselt et al., 2015), which uses the target: Y = r + γQ(ê(s ; θê−), argmax a ∈A Q(x , a ; θ Q ); θ Q − ), where θ Q − and θê− are parameters of an earlier buffered Q-function (or our target Q-function) and encoder respectively. The agent then minimizes the following loss: L Q (θ Q ) = (Q(x, a; θ Q ) − Y ) 2 . We learn the dynamics of our environment through the following losses: L R (θê, θr) = \|r −r(x, a; θr)\| 2 , L G (θê, θγ) = \|γ −γ(x, a; θγ)\| 2 and our transition loss L τ (θê, θτ ) = \|\|[x +τ (x, a; θτ )] − x \|\| 2 2 . (3) Note that our transition function learns the difference (given an action) between previous state x and current state x . By jointly learning the weights of the encoder and the different components, the abstract representation is shaped in a meaningful way according to the dynamics of the environment. In particular, by minimizing the loss given in Equation 3 with respect to the encoder parameters θê (or p(x \| s)), we minimize our entropy H[X \|X, A]. In order to maximize the entropy of our learnt abstracted state representations H[X ], we minimize the expected pairwise Gaussian potential (Borodachov et al., 2019) between states: L d1 (θê) = E s1,s2∼p(s) exp(−C d1 \|\|ê(s 1 ; θê) −ê(s 2 ; θê)\|\| 2 2 )(4) with C d1 as a hyperparameter. Losses in Equation 3 and Equation 4 are reminiscent of the modelbased losses in François-Lavet et al. (2018) and correspond respectively to the alignment and uniformity contrastive loss formulation in Wang and Isola (2020), where alignment ensures that similar states are close together (in encoded representation space) and uniformity ensures that all states are spread uniformly throughout this low-dimensional representation space. The losses L τ (θê) and L d1 (θê) maximizes the I[{X, A}; X ] term and selecting smaller dimension for our representation minimizes I[X , S ]. Put together, our method is trying to minimize L(p(x \|s )) as per Equation 1. Distance measures in representational space For practical purposes, since we are looking to use a distance metric within X to leverage as a score for novelty, we ensure well-defined distances between states by constraining the 2 distance between two consecutive states: L csc (θê) = max( ê(s 1 ; θ e ) −ê(s 2 ; θ e ) 2 − ω, 0) (5) where L csc is a soft constraint between consecutive states s 1 and s 2 that tends to enforce two consecutive encoded representations to be at a distance ω apart. We add L csc to ensure a well-defined 2 distance between abstract states for use in our intrinsic reward calculation (a discussion of this loss is provided in Appendix B). We discuss how we use ω to evaluate model accuracy for our MPC updates in Appendix A. Finally, we minimize the sum of all the aforementioned losses through gradient descent: L = L R (θê, θr) + L G (θê, θγ) + L τ (θê, θτ ) + L Q (θ Q ) + L d1 (θê) + L csc (θê).(6) Through these losses, the agent learns a low-dimensional representation of the environment that is meaningful in terms of the 2 norm in representation space. We then employ a planning technique that combines the knowledge of the model and the value function which we use to maximize intrinsic rewards, as detailed in the next section and Section 4.3. Novelty Search in abstract representational space Our approach for exploration uses intrinsic motivation (Schmidhuber, 1990;Chentanez et al., 2005;Achiam and Sastry, 2017) where an agent rewards itself based on the fact that it gathers interesting experiences. In a large state space setting, states are rarely visited and the count for any state after n steps is almost always 0. While Bellemare et al. (2016) solves this issue with density estimation using pseudo-counts directly from the high-dimensional observations, we aim to estimate some function of novelty in our learnt lower-dimensional representation space. Sparsity in representation space as a measure for novelty Through the minimization of Equation 1, states that are close together in dynamics are pushed close together in our abstract state space X . Ideally, we want an agent that efficiently explores the dynamics of its environment. To do so, we reward our agent for exploring areas in lower-dimensional representation space that are less visited and ideally as far apart from the dynamics that we currently know. Given a point x in representation space, we define a reward function that considers the sparsity of states around x -we do so with the average distance between x and its k-nearest-neighbors in its visitation history buffer B:ρ X (x) = 1 k k i=1 d(x, x i ),(7) where x=ê(s; θê) is a given encoded state, k ∈ Z + , d(·, ·) is some distance metric in R n X and x i=ê (s i ; θê), where s i ∈ B for i = 1 . . . k are the k nearest neighbors (by encoding states in B to representational space) of x according to the distance metric d(·, ·). Implicit in this measure is the reliance on the agent's visitation history buffer B. An important factor in this score is which distance metric to use. With the losses used in Section 3, we use 2 distance because of the structure imposed on the abstract state space with Equations 4 and 5. As we show in Appendix D, this novelty reward is reminiscent of recoding probabilities (Bellemare et al., 2016;Cover and Thomas, 2012) and is in fact inversely proportional to these probabilities, suggesting that our novelty heuristic estimates visitation count. This is also the same score used to gauge "sparseness" in behavior space in Lehman and Stanley (2011). With this reward function, we present the pseudo-code for our exploration algorithm in Algorithm 1. Algorithm 1: The Novelty Search algorithm in abstract representational space. 1 Initialization: transition buffer B, agent policy π; 2 Sample n init initial random transitions, let t = n init ; 3 while t ≤ n max do // We update our dynamics model and Q-function every n f req steps 4 if t mod n f req == 0 then 5 while j ≤ n iters or L τ ≤ ω δ 2 do 6 Sample batch of transitions (s, a, r extr , r intr , γ, s ) ∈ B; 7 Train dynamics model with (s, a, r extr , γ, s ); 8 Train Q-function with (s, a, r extr + r intr , γ, s ); 9 end 10 ∀(s, a, r extr , r intr , γ, s ) ∈ B, set r intr ←ρ X (ê(s ; θê)); 11 end 12 a t ∼ π(s t ); 13 Take action in environment: s t+1 ← τ (s t , a t ), r t,extr ← R(s t , a t ), γ t ← G(s t , a t ); 14 Calculate intrinsic reward: r t,intr ←ρ X (ê(s t+1 ; θê)) 15 B ← B ∪ {(s t , a t , r t,extr , r t,intr , γ t , s t+1 )}; 16 end Asymptotic behavior This reward function also exhibits favorable asymptotic behavior, as it decreases to 0 as most of the state space is visited. We show this in Theorem 1. Theorem 1. Assume we have a finite state space S ⊆ R d , history of states B = (s 1 , . . . , s N ), encoded state space X ⊆ R n X , deterministic mapping f : R d → R n X and a novelty reward defined asρ X (x). With an optimal policy with respect to the rewards of the novelty heuristic, our agent will tend towards states with higher intrinsic rewards. If we assume a communicating MDP setting (Puterman, 1994), we have that lim N →∞ρ X (f (s)) = 0, ∀s ∈ S. Proof. We prove this theorem in Appendix E. Combining model-free and model-based components for exploration policies Similarly to previous works (e.g. Oh et al., 2017;Chebotar et al., 2017), we use a combination of model-based planning with model-free Q-learning to obtain a good policy. We calculate rollout estimates of next states based on our transition modelτ and sum up the corresponding rewards, which we denote as r : X × A → [0, R max ] and can be a combination of both intrinsic and extrinsic rewards. We calculate expected returns based on the discounted rewards of our d-depth rollouts:Q d (x, a) =      r(x, a) +γ(x, a; θγ)× max a ∈AQ d−1 (τ (x, a; θτ ), a ), if d > 0 Q(x, a; θ Q ), if d = 0(8) Note that we simulate only b-best options at each expansion step based on Q(x, a; θ Q ), where b ≤ \|A\|. In this work, we only use full expansions. The estimated optimal action is given by a = argmax a∈AQ d (x, a). The actual action chosen at each step follows an -greedy strategy ( ∈ [0, 1]), where the agent follows the estimated optimal action with probability 1 − and a random action with probability . : Two views of the same full history of learned abstract 3-dimensional representation of our multi-step maze after 300 steps. Orange and blue points denote states without and with keys respectively. Our agent is able to disentangle states where the agent has a key and when it doesn't as seen in the distance between the two groups of states. Meaningful information about the agent position is also maintained in the relative positions of states in abstract state space. Experiments We conduct experiments on environments of varying difficulty. All experiments use a training scheme where we first train parameters to converge on an accurate representation of the already experienced transitions before taking an environment step. We optimize the losses (over multiple training iterations) given in Section 3. We discuss all environment-specific hyperparameters in Appendix J. Labyrinth exploration We consider two 21 × 21 versions of the grid-world environment (Figure 7 in Appendix). The first is an open labyrinth grid-world, with no walls except for bordering walls. The second is a similar sized grid-world split into four connected rooms. In these environments the action space A is the set of four cardinal directions. These environments have no rewards or terminal states and the goal is to explore, agnostic of any task. We use two metrics to gauge exploration for this environment: the first is the ratio of states visited only once, the second is the proportion of total states visited. Open labyrinth In the open labyrinth experiments (Figure 2a), we compare a number of variations of our approach with a random baseline and a count-based baseline (Bellemare et al., 2016) (as we can count states in this tabular setting). Variations of the policy include an argmax over state values (d = 0) and planning depths of d ∈ {1, 5}. All variations of our method outperform the two baselines in this task, with a slight increase in performance as planning depth d increases. In the open labyrinth, our agent is able to reach 100% of possible states (a total of 19 × 19 = 361 unique states) in approximately 800 steps, and 80% of possible states (≈ 290 states) in approximately 500 steps. These counts also include the n init number of random steps taken preceding training. Our agent is also able to learn highly interpretable abstract representations in very few environment steps (as shown in Figure 1a) as it explores its state space. In addition, after visiting most unseen states in its environment, our agent tends to uniformly explore its state space due to the nature of our novelty heuristic. A visualisation of this effect is available in Appendix H. 4-room labyrinth We now consider the 4-room labyrinth environment, a more challenging version of the open labyrinth environment (Figure 1a). As before, our encoderê is able to take a high-dimensional input and compress it to a low-dimensional representation. In the case of both labyrinth environments, the representation incorporates knowledge related to the position of the agent in 2-dimensions that we call primary features. In the 4-room labyrinth environment, it also has to learn other information such as agent surroundings (walls, open space) etc., but it does so only via the transition function learned through experience. We call this extraneous but necessary information secondary features. As most of these secondary features are encoded only in the dynamics modelτ , our agent has to experience a transition in order to accurately represent both primary and secondary features. In this environment specifically, our dynamics model might over-generalize for walls between rooms and can sometimes fail at first to try out transitions in the passageways between rooms. However, because our agent tends to visit uniformly all the states that are reachable within the known rooms, the -greedy policy of our approach still ensures that the agent explores passageways efficiently even in the cases where it has over-generalized to the surrounding walls. We run the same experiments on the 4-room labyrinth domain as we do on the open labyrinth and report results in Figure 2b. In both cases, our method outperforms the two baselines in this domain (random and count-based). Control and sub-goal exploration In order to test the efficacy of our method beyond fixed mazes, we conduct experiments on the control-based environment Acrobot (Brockman et al., 2016) and a multi-step maze environment. Our method (with planning depth d = 5) is compared to strong exploration baselines with different archetypes: 1. Prediction error incentivized exploration 2. Hash count-based exploration 3. Random Network Distillation (Osband et al., 2017) 4. Bootstrap DQN (BDQN, Osband et al. (2016)) In order to maintain consistency in our results, we use the same deep learning architectures throughout. Since we experiment in the deterministic setting, we exclude baselines that require some form of stochasticity or density estimation as baselines (for example, Shyam et al. (2018) and Osband et al. (2017)). A specificity of our approach is that we run multiple training iterations in between each environment step for all experiments, which allows the agent to use orders of magnitude less samples as compared to most model-free RL algorithms (all within the same episode). Acrobot We now test our approach on Acrobot (Brockman et al., 2016), which has a continuous state space unlike the labyrinth environment. We specifically choose this control task because the nature of this environment makes exploration inherently difficult. The agent only has control of the actuator for the inner joint and has to transfer enough energy into the second joint in order to swing it to its goal state. We modify this environment so that each episode is at most 3000 environment steps. While this environment does admit an extrinsic reward, we ignore these rewards entirely. To measure the performance of our exploration approach, we measure the average number of steps per episode that the agent takes to move its second joint above a given line as per Figure 3a. To demonstrate the ability of our method to learn a low dimensional abstract representation from pixel inputs, we use 4 consecutive pixel frames as input instead of the 6-dimensional full state Best results are in bold. We provide p-values indicative of the null hypothesis H 0 : ∆µ = µ 1 − µ 2 = 0, calculated using Welch's t-test, all as per (Colas et al., 2019). In this case, we do a pair-wise comparison between the central tendencies of our algorithm (Novelty) and our baselines. Normalized and combined results are also shown -results here are first normalized with respect to the average number of steps taken for our algorithm and then combined on both environments. vector. We use a 4-dimensional abstract representation of our state and results from experiments are shown in Table 1. Our method reaches the goal state more efficiently than the baselines. The passageway to the west portion of the environment is blocked before the key (black) is collected. b) Right: The passageway is traversable after collecting the key, and the reward (red) is then available. The environment terminates after collecting the reward. Multi-step goal maze We also test our method on a more complex maze with the sub-task of picking up a key that opens the door to an area with a reward. We build our environment with the Pycolab game engine (Stepleton, 2017). The environment can be seen in Figure 3b, where the input to our agent is a top-down view of the environment. While this environment does admit an extrinsic reward (1 for picking up the key, 10 for reaching the final state), we ignore these rewards and only focus on intrinsic rewards. In our experiments, we show that our agent is able to learn an interpretable representation of the environment in a sample-efficient manner. Figure 1c shows an example of learnt representations in this domain after reaching the goal -we observe that positions in the maze correspond to a nearly identical structure in the lower-dimensional representation. Our representation also nicely captures internal state information (whether the key has been picked up) by separating the two sets of states (states when the key has been picked up and states when the key has not been picked up). Similar positions in both sets of states are also mapped closely together in lower-dimensional space (ie. (1, 1, with key) is close in 2 to (1, 1, without key)), suggesting good generalization between similar states. Related work The proposed exploration strategy falls under the category of directed exploration (Thrun, 1992) that makes use of the past interactions with the environment to guide the discovery of new states. This work is inspired by the Novelty Search algorithm (Lehman and Stanley, 2011) that uses a nearest-neighbor scoring approach to gauge novelty in policy space. Our approach leverages this scoring to traverse dynamics space, which we motivate theoretically. Exploration strategies have been investigated with both model-free and model-based approaches. In Bellemare et al. (2016) and Ostrovski et al. (2017), a model-free algorithm provides the notion of novelty through a pseudocount from an arbitrary density model that provides an estimate of how many times an action has been taken in similar states. Recently, Taiga et al. (2020) do a thorough comparison between bonusbased exploration methods in model-free RL and show that architectural changes may be more important to agent performance (based on extrinsic rewards) as opposed to differing exploration strategies. Several exploration strategies have also used a model of the environment along with planning. Hester and Stone (2012) employ a two-part strategy to calculate intrinsic rewards, combining model uncertainty (from a random-forest based model) and a novelty reward based on L 1 distance in feature space. A strategy investigated in Salge et al. (2014); Mohamed and Rezende (2015); Gregor et al. (2016); Chiappa et al. (2017) is to have the agent choose a sequence of actions by planning that leads to a representation of state as different as possible to the current state. In Pathak et al. (2017); Haber et al. (2018), the agent optimizes both a model of its environment and a separate model that predicts the error/uncertainty of its own model. Burda et al. (2018a) similarly uses an intrinsic reward based on the uncertainty of its dynamics model. In Shyam et al. (2018), forward models of the environment are used to measure novelty derived from disagreement between future states. Still and Precup (2012) take an information theoretic approach to exploration, that chooses a policy which maximizes the predictive power of the agent's own behavior and environment rewards. In Badia et al. (2020), an intrinsic reward from the k-NN over the agent's experience is also employed for exploration. They instead employ a self-supervised inverse dynamics model to learn the embeddings as opposed to our approach. Beyond improved efficiency in exploration, the interpretability of our approach could also lead to human-in-the-loop techniques (Mandel et al., 2017;Abel et al., 2017) for exploration, with the possibility for the agent to better utilize feedback from interpretability of the agent in representation space. Discussion In this paper, we formulate the task of dynamics learning in MBRL through the Information Bottleneck principle. We present methods to optimize the IB equation through low-dimensional abstract representations of state. We further develop a novelty score based on these learnt representations that we leverage as an intrinsic reward that enables efficient exploration. By using this novelty score with a combination of model-based and model-free approaches for planning, we show more efficient exploration across multiple environments with our learnt representations and novelty rewards. As with most methods, our approach also has limitations. One limitation we may have is the scalability of non-parametric methods such as k-NN density estimation since our method scales linearly with the number of environment steps. A possible solution to this problem would be to use some sampling scheme to sample a fixed number of observations for calculation of our novelty heuristic. Another issue that has arisen from using very low-dimensional space to represent state is generalization. In some cases, the model can over-generalize with the consequence that the low-dimensional representation loses information that is crucial for the exploration of the entire state space. An interesting direction for future work would be to find ways of incorporating secondary features such as those mentioned in Section 5.1.2. An interesting possibility would be to use a similar IB method, but using a full history of states as the conditioning variable. Beyond these points, we discuss limitations and potential improvements to this work in Appendix K. Finally, we show preliminary results of our method on a more complex task -Montezuma's Revenge -in Appendix G. With the theory and methods developed in this paper, we hope to see future work done on larger tasks with more complex environment dynamics. Broader Impact Algorithms for exploring an environment are a central piece of learning efficient policies for unknown sequential decision-making tasks. In this section, we discuss the wider impacts of our research both in the Machine Learning (ML) field and beyond. We first consider the benefits and risks of our method on ML applications. Efficient exploration in unknown environments has the possibility to improve methods for tasks that require accurate knowledge of its environment. By exploring states that are more novel, agents have a more robust dataset. For control tasks, our method improves the sample efficiency of its learning by finding more novel states in terms of dynamics for use in training. Our learnt low-dimensional representation also helps the interpretability of our decision making agents (as seen in Figure 1). More interpretable agents have potential benefits for many areas of ML, including allowing human understandability and intervention in human-in-the-loop approaches. With such applications in mind, we consider societal impacts of our method, along with potential future work that could be done to improve these societal impacts. One specific instance of how efficient exploration and environment modeling might help is in disaster relief settings. With the incipience of robotic systems for disaster area exploration, autonomous agents need to efficiently explore their unknown surroundings. Further research into scaling these MBRL approaches could allow for these robotic agents to find points of interest (survivors, etc.) efficiently. One potential risk of our application is safe exploration. Our method finds and learns from states that are novel in terms of its dynamics. Without safety mechanisms, our agent could view potentially harmful scenarios as novel due to the rarity of such a situation. For example, a car crash might be seen as a highly novel state. To mitigate this safety concern we look to literature on Safety in RL (García and Fernández, 2015). In particular, developing a risk metric based on the interpretability of our approach may be an area of research worth developing. A Using ω to gauge model accuracy The hyperparameter ω can be used to estimate the accuracy of our transition loss, hence of our novelty estimates. In order to gauge when our representation is accurate enough to use our novelty heuristic, we use a function of this hyperparameter and transition loss to set a cutoff point for accuracy to know when to take the next environment step. If ω is the minimum distance between successive states, then when L τ ≤ ω δ 2 , the transitions are on average within a ball of radius ω δ of the target state. Here δ > 1 is a hyperparameter that we call the slack ratio. Before taking a new step in the environment, we keep training all the parameters with all these losses until this threshold is reached and our novelty heuristic becomes useful. The abstract representation dimensionality is also another hyperparameter that requires tuning, depending on the complexity of the environment. Details on the slack ratios, abstract representation dimensionality and other hyperparameters are given in Appendix J. B Discussion on the distance between successive encoded states As for our soft constraints on representation magnitude, we use a local constraint instead of a global constraint on magnitude such that it is more suited for our novelty metric. If we are to calculate some form of intrinsic reward based on distance between neighboring states, then this distance needs to be non-zero and ideally consistent as the number of unique states in our history increases. In the global constraint case, if the intrinsic rewards decrease with an increase in number of states in the agent's history, then the agent will fail to be motivated to explore further. Even though the entropy maximization losses ensures the maximization of distances between random states, if we have \|B\| number of states in the history of the agent, then a global constraint on representation magnitude might lead to lim \|B\|→∞ E (s,s )∼(B,B) [ s − s 2 ] = 0.(9) We also test the necessity of this loss in Appendix F.1 and see that without this loss, we incur a high variance in exploration performance. C Motivation for 2 distance We consider the consecutive distance loss L csc . Minimization of this loss ensures that the distance between two consecutive states is ≤ ω. This along with our alignment and uniformity losses, Lτ and L d1 ensures that temporally close states are close in representational space and states are uniformly spread throughout this space. This implies that the minima of L csc between two consecutive states s and s will occur when: L * consec = min θê L csc (θê) = min θê max( ê(s; θ e ) −ê(s ; θ e ) 2 − ω, 0) = min θê [ ê(s; θ e ) −ê(s ; θ e ) 2 − ω] The minimum of this loss is obtained when the 2 distance between s and s is ω. When this loss is minimized, 2 distance is well-defined in our representation space, which implies that our novelty heuristic will also be well-defined. These losses shape abstract state space so that 2 norm as a distance measure encodes a notion of closeness in state space that we leverage in our novelty heuristic. D Novelty heuristic as an inverse recoding probability score Consider P (X n+1 = x \| X 1:n = x 1:n ), the recoding probability of state x at timestep n + 1. We try to show that our novelty heuristic of a state is inversely proportional to its recoding probability, or that: ρ X (x) = c P (X n+1 = x \| X 1:n = x 1:n ) where c is some constant. If we were to try and estimate our recoding probability first using a non-parametric method then using its inverse, we might consider the K-nearest neighbor approach (Loftsgaarden and Quesenberry, 1965): P (X n+1 = x \| X 1:n = x 1:n ) ≈ k nV x,x k where k < n is an integer and V x,x k is the volume around x of radius d(x, x k ), where x k ∈ X is the kth nearest neighbor to x. The issue with this approach is that our score is only dependent on it's kth nearest neighbor (as this score only depends on V x,x k ), and doesn't take into account the other k − 1 nearest neighbors. In practice, we instead try to find something proportional to the inverse directly: we average each of the 1-nearest neighbor density inverses over the k-nearest neighbors: ρ X (x) = c P (X n+1 = x \| X 1:n = x 1:n ) ≈ nV x,x1 ≈ n k k i=1 V x,xi Since we're only worried about proportionality, we can remove the constant n and replace our volume of radius between two points with a distance metric d: ρ X (x) ∝ρ X (x) = 1 k k i=1 d(x, x i ). Which is our novelty heuristic. E Limiting behavior for novelty heuristic Proof (Theorem 1). Let n s be the visitation count for a state s ∈ S. We assume that our agent's policy will tend towards states with higher rewards. Given the episodic nature of MDPs, we have that over multiple episodes all states communicate. Since our state space is finite, we have that lim n→∞ n s = ∞, ∀s ∈ S. which means that ∃n such that k < n s as n → ∞, and implies that the k nearest neighbors of s are indiscernible from s. Since f is a deterministic function, x i = x for all i. We also assume that our agent's policy will tend towards states with higher rewards. As x and x i are indiscernible and dist is a properly defined metric, dist(x, x i ) = 0 for all i, and we have lim n→∞ρ (x) = 1 k k i=1 dist(x, x i ) (10) = 0.(11) F Ablation study Here we perform ablation studies to test the affects of our losses and hyperparameters. F.1 Consecutive distance loss To further justify our use of the L csc loss, we observe the results of running the same trials in the simple maze environment (with no walls) with no L csc loss in Figure 4. As we increase the number of forward prediction steps the exploration is less effective and variance of our results increases. Without the relative distance constraints of our representation, we observe an increase of forward prediction errors, which is the likely cause of the decrease in performance. These forward prediction errors are further compounded as we increase the number of forward prediction steps (as can be seen when comparing the standard error between d = 0, 1, 5). F.2 Pure model-based/model-free We test the importance of using a combination of both model-based and model-free components on the multi-step maze environment introduced in Section 5.2.2. We train with the same hyperparameters but in the model free (d = 0) and model-based (d = 5, no Q-value tails) settings. We show results in Table 2. G Montezuma's Revenge visualizations We show preliminary results for learning abstract representations for a more complex task, Montezuma's Revenge. Comparing the two figures above, we observe how temporally closer states are closer together in lower-dimensional learnt representational space as compared to pixel space. Transitions are not shown for raw observations. H Labyrinth state count visualization J Experimental setup and hyperparameters For all of our experiments, we use a batch size of 64 and take 64 random steps transitions before beginning training. We also use the same discount factor for all experiments (γ = 0.8) and the same freeze interval for target parameters 1000. The reason behind our low discount factor is due to the high density of non-stationary intrinsic rewards in our state. We also use a replay buffer size corresponding to the maximum number of steps in each environment for all experiments. For all model-based abstract representation training, the following hyperparameters were all kept constant: minimum distance between consecutive states ω = 0.5, slack ratio δ = 12 and transition model dropout of 0.1. For all experiments run with our novelty metric, we use k = 5 for our k-NN calculations. For all experiments that allows for forward planning and not explicitly mention depth d, we set planning depth d = 5. For abstract representation size (n X ), we use a dimensionality of 2 for both labyrinth exploration tasks, a dimensionality of 4 for Acrobot, and finally a dimensionality of 3 for the multi-step maze. J.1 Neural Network Architectures For reference, 'Dense' implies a full-connected layer. 'Conv2D' refers to a 2D convolutional layer with stride 1. 'MaxPooling2D' refers to a max pooling operation. All networks were trained with the RMSProp optimizer. Throughout all experiments, we use the following neural network architectures: J.1.1 Encoder For all our non-control task inputs, we flatten our input and use the following feed-forward neural network architecture forê: • Dense(200, activation='tanh') • Dense(100, activation='tanh') • Dense(50, activation='tanh') • Dense(10, activation='tanh') • Dense(abstract representation dimension). For our control task, we use a convolution-based encoder: • Conv2D(channels=8, kernel=(3,3), activation='tanh') • Conv2D(channels=16, kernel=(3,3), activation='tanh') • MaxPool2D(pool size=(4,4)) • Conv2D(channels=32, kernel=(3,3), activation='tanh') • MaxPool2D(pool size=(3,3)) • Dense(abstract state representation dimension). J.2 Labyrinth environments Both environments used the same hyperparameters except for two: we add an -greedy ( = 0.2) policy for the 4-room maze, and increased n f req from 1 to 3 in the 4-room case due to unnecessary over-training. We have the following hyperparameters for our two labyrinth environments: • n iters = 30000 • α = 0.00025 J.3 Control environment In our Acrobot environment, the input to our agent is 4 stacked consecutive pixel frames, where we reduce each frame down to a 32 × 32 pixel frame. Our abstract representation dimension is 4. We use a learning rate of α = 0.00025 for all experiments. We train for n iters = 50000 for all experiments with the exception of RND and transition loss -this discrepancy is due to time constraints for the latter two experiments which used n iters = 3000, as both these experiments used prohibitively more time to run due to the increased number of steps used to reach the goal state of the environment. J.4 Multi-step maze environment In our multistep maze environment, the input to our agent is a single 15 × 15 frame of an overview of the environment. Our abstract representation dimension is 3. We use an -greedy ( = 0.1) policy for this environment. We use α = 0.00025, n iters = 30000 for our model-free algorithms and α = 0.000025, n iters = 50000 for experiments that include a model-based component. Each episode is at most 4000 environment steps. K Potential improvements and future work K.1 Incorporating agent history for better generalization As mentioned in Section 5.1.2, generalization across states while only conditioning on primary features (X, A in our case) restricts the generalization ability of our agent. An interesting potential for future work would be to somehow incorporate this information into the learnt representation (potentially by using the same IB method, but using a full history of states as the conditioning variable). K.2 Increasing efficiency of learning the abstract state representations Currently, learning our low-dimensional state representation takes many iterations per timestep, and is also sensitive to hyperparameter tuning. Our method requires an accurate state representation and dynamics model according to our losses for our method to be effective -the sample-efficiency from our model-learning methods comes at a cost of more time and compute. This is due to the careful balance our model needs to maintain between its losses for good representations. Another interesting potential for future work would be to find ways to incorporate our model-learning losses using less time and computation. K.3 Extension to stochastic environments One avenue of future work would be to extend this work for stochastic environments. While there has been recent work on using expectation models for planning (Wan et al., 2020) that we could use to extend our algorithm, this still comes with its own limitations. K.4 Empirical validation for representation size Another avenue of investigation is to find a more principled approach to finding the right representation size for a given environment. While we currently simply pick the lowest representation size from prior knowledge about an environment, it may be worthwhile to somehow allow the algorithm to decide this. Figure 1 : 1(a), (b): Plotting the full history of learned abstract representations of both open and 4room labyrinth environments from Figures 7a and 7b after 500 environment steps. Colors denote which side of the maze the agent was in, grid coordinates and transitions are shown. (c) (a ) )Results for open labyrinth and different variations on policies compared to baselines.(b) Results for the 4-room labyrinth and different variations on policies compared to baselines. Figure 2 : 2Labyrinth results for both open labyrinth and 4-room labyrinth over 10 trials, showing mean and standard deviations. ( a ) aLeft: Acrobot start state. right: Acrobot end state (b) Left: Start of our multi-step maze. right: After the agent has collected the key. Figure 3 : 3Illustrations of the Acrobot and multi-step goal maze environments. a) Left: The Acrobot environment in one configuration of its start state. a) Right: One configuration of the ending state of the Acrobot environment. The environment finishes when the second arm passes the solid black line. b) Left: Figure 4 : 4Simple maze (with no walls) experiment with no L csc loss. Figure 5 : 5a) Visualization for 100 observations (4 frames per observation) of Montezuma's Revenge game play. Representation learnt was n X = 5 and visualized with t-SNE (van der Maaten and Hinton, 2008) in 2 dimensions. Labels on top-left of game frames correspond to labels of states in lower-dimensional space. Transitions are shown by shaded lines. b) Original resized game frames visualized using t-SNE with the same parameters. Figure 6 :Figure 7 : 67An example of the state counts of our agent in the open labyrinth with d = 5 step planning. Titles of each subplot denotes the number of steps taken. The brightness of the points are proportional to the state visitation count. The bright spots that begins after 200 counts is the agent requiring a few trials for learning the dynamics of labyrinth walls. Left: Open labyrinth -A 21 × 21 empty labyrinth environment. Right: 4-room labyrinth -A 21 × 21 4-room labyrinth environment inspired by the 4-room domain inSutton et al. (1999). Table 2 : 2A further ablation study on the multi-step maze environment. The MF (model-free) ablation does not employ any forward intrinsic reward planning (d = 0), while the MB (model-based) ablation only uses forward intrinsic reward planning without using or learning Q-values. https://github.com/taodav/nsrs AcknowledgementsWe would like to thank Emmanuel Bengio for the helpful discussions and feedback on early drafts of this work. We would also like to thank all the reviewers for their constructive and helpful comments. Agent-agnostic human-in-the-loop reinforcement learning. D Abel, J Salvatier, A Stuhlmüller, O Evans, arXiv:1701.04079arXiv preprintAbel, D., Salvatier, J., Stuhlmüller, A., and Evans, O. (2017). Agent-agnostic human-in-the-loop reinforcement learning. arXiv preprint arXiv:1701.04079. 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J. 12We use the following architecture • Dense(10, activation='tanh', dropout=0.1)J.1.2 Transition model The input to our transition model is a concatenation of an abstract representation and an action. We use the following architecture • Dense(10, activation='tanh', dropout=0.1) • Dense(30, activation='tanh', dropout=0.1). • Dense(30, activation='tanh', dropout=0.1) • Dense(30, activation='tanh', dropout=0.1). • Dense(30, activation='tanh', dropout=0.1) . • Dense, activation='tanh', dropout=0.1)• Dense(10, activation='tanh', dropout=0.1) . • Dense, abstract representation dimension• Dense(abstract representation dimension) Reward and discount factor models For both reward and discount factor estimators, we use the following architecture: • Dense. J.1.3 Reward and discount factor models For both reward and discount factor estimators, we use the following architecture: • Dense(10, activation='tanh') . • Dense, 50• Dense(50, activation='tanh') . • Dense, 20• Dense(20, activation='tanh') . • Dense, • Dense(1). Q function approximator We use two different architecture based on the type of input. If we use the concatenation of abstract representation and action. we use the following architecture: • Dense(20, activation='relu'J.1.4 Q function approximator We use two different architecture based on the type of input. If we use the concatenation of abstract representation and action, we use the following architecture: • Dense(20, activation='relu') . • Dense, 50• Dense(50, activation='relu') . • Dense, 20• Dense(20, activation='relu') . • Dense, n actions• Dense(n actions ) For the pixel frame inputs for our control environments. we use: • Conv2D(channels=8, kernel=(3,3), activation='tanh'For the pixel frame inputs for our control environments, we use: • Conv2D(channels=8, kernel=(3,3), activation='tanh') . • Conv2d, 16• Conv2D(channels=16, kernel=(3,3), activation='tanh') . • Maxpool2d, pool size=(4,4• MaxPool2D(pool size=(4,4)) . • Conv2d, 32• Conv2D(channels=32, kernel=(3,3), activation='tanh') . • Maxpool2d, 3pool size=• MaxPool2D(pool size=(3,3)) . • Dense, n actions• Dense(n actions ). for our (purely model free) gridworld environments we use: • Dense(500, activation='tanh. Finally, Finally, for our (purely model free) gridworld environments we use: • Dense(500, activation='tanh') . • Dense, • Dense(200, activation='tanh') . • Dense, 50• Dense(50, activation='tanh') . • Dense, • Dense(10, activation='tanh') . • Dense, n actions• Dense(n actions ) As for our Bootstrap DQN implementation, we use the same architecture as above, except we replace the final Dense layer with 10 separate heads. each a Dense layer with n actions nodesAs for our Bootstrap DQN implementation, we use the same architecture as above, except we replace the final Dense layer with 10 separate heads (each a Dense layer with n actions nodes).	[ "https://github.com/taodav/nsrs", "https://github.com/deepmind/pycolab." ]
[ "Mechanism of the insulator-to-metal transition and superconductivity in the spin liquid candidate NaYbSe 2 under pressure", "Mechanism of the insulator-to-metal transition and superconductivity in the spin liquid candidate NaYbSe 2 under pressure" ]	[ "Yuanji Xu \nBeijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n", "Yutao Sheng \nBeijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n", "Yi-Feng Yang \nBeijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nSongshan Lake Materials Laboratory\n523808DongguanGuangdongChina\n" ]	[ "Beijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina", "Beijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina", "University of Chinese Academy of Sciences\n100049BeijingChina", "Beijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina", "University of Chinese Academy of Sciences\n100049BeijingChina", "Songshan Lake Materials Laboratory\n523808DongguanGuangdongChina" ]	[]	The quantum spin liquid candidate NaYbSe2 was recently reported to exhibit a Mott transition under pressure. Superconductivity was observed in the high-pressure metallic phase, raising the question concerning its relation with the low-pressure quantum spin liquid ground state. Here we combine the density functional theory and the dynamical mean-field theory to explore the underlying mechanism of the insulator-to-metal transition and superconductivity and establish an overall picture of its electronic phases under pressure. Our results suggest that NaYbSe2 is a chargetransfer insulator at ambient pressure. Upon increasing pressure, however, the system first enters a semi-metallic state with incoherent Kondo scattering against coexisting localized Yb-4f moments, and then turns into a heavy fermion metal. In between, there may exist a delocalization quantum critical point responsible for the observed non-Fermi liquid region with linear-in-T resistivity. The insulator-to-metal transition is therefore a two-stage process. Superconductivity emerges in the heavy fermion phase with well-nested Yb-4f Fermi surfaces, suggesting that spin fluctuations may play a role in the Cooper pairing. NaYbSe2 might therefore be the 3rd Yb-based heavy-fermion superconductor with a very "high" Tc than most heavy fermion superconductors.	10.1038/s41535-022-00429-7	[ "https://arxiv.org/pdf/2108.03218v1.pdf" ]	236,950,449	2108.03218	e923588420e957d882e178bc57eb39f526ff807d	Mechanism of the insulator-to-metal transition and superconductivity in the spin liquid candidate NaYbSe 2 under pressure 6 Aug 2021 Yuanji Xu Beijing National Laboratory for Condensed Matter Physics Institute of Physics Chinese Academy of Sciences 100190BeijingChina Yutao Sheng Beijing National Laboratory for Condensed Matter Physics Institute of Physics Chinese Academy of Sciences 100190BeijingChina University of Chinese Academy of Sciences 100049BeijingChina Yi-Feng Yang Beijing National Laboratory for Condensed Matter Physics Institute of Physics Chinese Academy of Sciences 100190BeijingChina University of Chinese Academy of Sciences 100049BeijingChina Songshan Lake Materials Laboratory 523808DongguanGuangdongChina Mechanism of the insulator-to-metal transition and superconductivity in the spin liquid candidate NaYbSe 2 under pressure 6 Aug 2021(Dated: August 9, 2021)arXiv:2108.03218v1 [cond-mat.str-el] The quantum spin liquid candidate NaYbSe2 was recently reported to exhibit a Mott transition under pressure. Superconductivity was observed in the high-pressure metallic phase, raising the question concerning its relation with the low-pressure quantum spin liquid ground state. Here we combine the density functional theory and the dynamical mean-field theory to explore the underlying mechanism of the insulator-to-metal transition and superconductivity and establish an overall picture of its electronic phases under pressure. Our results suggest that NaYbSe2 is a chargetransfer insulator at ambient pressure. Upon increasing pressure, however, the system first enters a semi-metallic state with incoherent Kondo scattering against coexisting localized Yb-4f moments, and then turns into a heavy fermion metal. In between, there may exist a delocalization quantum critical point responsible for the observed non-Fermi liquid region with linear-in-T resistivity. The insulator-to-metal transition is therefore a two-stage process. Superconductivity emerges in the heavy fermion phase with well-nested Yb-4f Fermi surfaces, suggesting that spin fluctuations may play a role in the Cooper pairing. NaYbSe2 might therefore be the 3rd Yb-based heavy-fermion superconductor with a very "high" Tc than most heavy fermion superconductors. INTRODUCTION Quantum spin liquid (QSL) is a highly entangled state with nontrivial topological excitations 1-4 . It has attracted intensive interest in modern condensed matter physics and is believed to describe the ground state of the Mott insulator and, upon doping, can induce high-temperature superconductivity as in cuprates [5][6][7][8][9] . Tremendous efforts have thus been devoted to exploring candidate spin liquid compounds [10][11][12][13][14][15] and searching for potential sign of superconductivity via chemical substitution 16,17 . However, doping typically introduces disorder that may obscure the investigation of such delicate intrinsic properties. Recently, superconductivity has been reported by pressurizing the QSL candidate NaYbSe 2 18,19 . In contrast to YbMgGaO 4 which has the same structure (space group R-3mH) but with Mg/Ga disorders [20][21][22][23][24][25][26][27] , NaYbSe 2 contains no intrinsic site disorder and has arguably the simplest crystal structure and chemical formula among all existing QSL candidates 28 . At ambient pressure, analyses of transport and neutron data found an insulating charge gap of 1.9 eV 29 and a crystalline excitation energy of about 15.7 meV [30][31][32] . No magnetic or structural transition was observed down to 50 mK [29][30][31] . Neutron scattering measurement has revealed the signature of magnetic excitation continuum and supported a QSL ground state 31 . The low-energy physics may thus be described by an effective spin-1 2 model on a perfect triangular lattice formed by Yb ions. At 11 GPa, a structural transition was reported 18 . The high-pressure structure has a lower symmetry (space group P -3m1) with two inequivalent Yb ions of different Yb-Se distances 18 . Upon further increasing pressure, an insulator-to-metal transition appears at about 58.9 GPa and superconductivity emerges over a wide pressure range above 103.4 GPa with a maximum T c of about 8 K 18 . The insulator-to-metal transition was claimed to be of Mott type, raising the question concerning the relationship between the low-pressure QSL and the high-pressure superconductivity. To clarify the nature of the "Mott" transition and the superconducting pairing mechanism, we investigate here the electronic structure of NaYbSe 2 and its pressure evolution using the density functional theory (DFT) plus the dynamical mean-field theory (DMFT) calculations [33][34][35][36] . We find that NaYbSe 2 is a charge-transfer insulator at ambient pressure, with the charge gap formed between Se-3p bands and Yb-4f upper-Hubbard bands. The charge gap is closed near about 60 GPa, giving rise to an intermediate semi-metallic phase with small electron and hole pockets from Yb-5d and Se-3p bands, respectively. The Yb-4f electrons remain localized at this pressure and the system is a typical low carrier density Kondo system, which explains the logarithmic temperature dependence of the resistivity observed in transport measurements. Above 73 GPa, the Yb-4f moments are fully screened on one (Yb1) of the two inequivalent Yb-ions, giving rise to flat heavy electron bands near the Fermi energy with sharp quasiparticle peaks in the density of states. The system turns into a heavy fermion metal. But the 4f electrons on the other Yb-ions (Yb2) remain localized and only weakly hybridized with conduction electrons. Superconductivity emerges in the heavy fermion phase when the Yb1-4f Fermi surfaces develop a well nested quasi-two-dimensional structure. The nesting wave vector coincides with the antiferromagnetic wave vector of the localized spins, indicating possible involvement of heavy electrons and antiferromagnetic spin fluctuations. NaYbSe 2 might therefore be the 3rd Yb-based heavy fermion superconductor with a rather "high" transition temperature (T c ∼ 8 K) among all heavy fermion superconductors. RESULTS AND DISCUSSION Optimization of crystal structures The crystal structures of NaYbSe 2 at low and high pressures are compared in Fig. 1a, both composed of edge-shared YbSe 6 and NaSe 6 octahedra. Yb ions form a layer structure of flat trianglular lattices with the shortest Yb-Yb distance in the ab plane 18 . The lattice parameters have been measured below 32 GPa in experiment 18,19 . For numerical calculations, we have used DFT+U to optimize the lattice parameters for both structures 37-39 and the results are compared with existing experimental data in Fig. 1b. Their good agreement over the whole low-pressure range validates the starting point of our numerical calculations. Electronic structures at low pressures The electronic structures of NaYbSe 2 were obtained using DFT+DMFT with the hybridization expansion continuous-time quantum Monte Carlo (CT-HYB) as the impurity solver 40,41 , taking into account both the spin-orbit coupling and electronic correlations. A Coulomb interaction U = 6 eV was chosen to reproduce the correct charge gap of about 1.9 eV at ambient pressure estimated from absorption spectra measurements 29 . Figure 1c plots the calculated density of states and spectral function at T = 10 K and ambient pressure. The Yb-4f Hubbard bands are located at about -3.0 and 2.0 eV, respectively, with a clear charge gap between the Se-3p bands and the Yb-4f upper Hubbard bands. Hence NaYbSe 2 should be identified as a charge-transfer insulator at ambient pressure. Without DMFT, DFT+U calculations alone yield a much larger Yb occupation number and predict incorrectly a metallic ground state, in disagreement with experiment. The band structures at 20 GPa are compared in Fig. 1d. We have now two inequivalent Yb-ions. Their 4f Hubbard bands become blurred, and the Yb-5d bands move down closer to the Fermi energy to form a smaller normal band insulating gap with the slightly upshifted Se-3p bands. 0 -1 1 Enery (eV) A(k,ω) Min Max M H Γ DOS (states / eV f.u.) 60 GPa 75 GPa 100 GPa (a) (b) (c) (d) (e) (f) 0 -1 -0.5 0 0.5 1 Energy (eV) -1 -0.5 0 0.5 1 Energy (eV) -1 -0.5 0 0.5 1 Energy (eV) The intermediate semi-metallic state Upon further increasing pressure, our calculations reveal a sequence of electronic phases, in good correspondence with the experimental observation. This is summarized in the upmost plot of Fig. 2. The first transition occurs near about 60 GPa, where the charge gap is closed and the spectral function reveals a semi-metallic state with a small electron pocket at the M point from Yb-5d bands and a hole pocket at Γ from Se-3p bands, as shown in Fig. 2a. However, the Yb-4f electrons are still localized at this pressure. In Fig. 2d, we see a tiny pseudogap structure near the Fermi energy in the 4f partial density of states, indicating a weak hybridization with conduction bands. Thus NaYbSe 2 should be better identified as a low carrier density Kondo system at this pressure. Consequently, there are no enough conduction electrons to screen the Yb-4f local moments so that the resistivity should be dominated by incoherent Kondo scattering and exhibit typical logarithmic temperature dependence 42,43 . This has indeed been observed in experiment between 58.9 and 74.8 GPa, but was initially attributed to weak localization 18 . We argue that similar properties, including the logarithmic-in-T resistivity and a pseudogap in the quasiparticle density of states, have previously been discussed in other low carrier density Kondo systems [42][43][44][45] . Heavy fermion metal at high pressure The second transition occurs close to 75 GPa beyond which flat heavy electron bands are seen to develop near the Fermi energy in Figs. 2b and 2c. Correspondingly, a sharp quasiparticle peak appears at the Fermi energy in the Yb1-4f density of states shown in Figs. 2e and 2f due to the many-body Kondo effect. The system now turns into a good heavy fermion metal. To determine the exact transition point, we plot in the inset of Fig. 2e the full width at half maximum (FWHM) of the Yb1-4f quasiparticle peak. Its variation with pressure reflects tentatively the strength of the Kondo hybridization 46 . A linear extrapolation to zero suggests that the transition occurs at about 73 GPa, which separates the low carrier density Kondo phase and the heavy fermion metal. Thus, it marks a delocalization transition of the Yb1-4f electrons. Interestingly, experiment did observe a non-Fermi liquid state with linear-in-T resistivity near 74.8 GPa and beyond that a gradual crossover to the Fermi liquid at 126 GPa 18 . This agrees with our theory and strongly suggests the possible existence of a delocalization quantum critical point at zero temperature. However, the broad crossover region is quite unusual and might be associated with the special layer structure of NaYbSe 2 to be discussed later. Nevertheless, the insulator-to-metal transition is not a simple Mott transition, but a two-stage process first from a normal band insulator to a semimetal with incoherent Kondo scattering and then to a heavy fermion metal. Similar transition and non-Fermi liquid property have also been reported in the low carrier density Kondo system CeNi 2−δ (As 1−x P x ) 2 43 . It may be interesting to mention that pressure seems to favor the heavy fermion phase in NaYbSe 2 , while in many other Yb-based heavy fermion compounds, the Kondo effect is typically weakened under pressure and gives in to long-range magnetic orders because of the upshift of the Yb-4f hole (upper-Hubbard) bands further away from the Fermi energy, making Yb-4f electrons more localized. In NaYbSe 2 , our results show that the localized Yb-4f levels are already away from the Fermi energy and the major effect of pressure is to increase the carrier density and enhance the hybridization as will be discussed below. Such anomalous increase of the Kondo effect with pressure has previously been observed in Yb 2 Ni 2 Al and YbCu 2 Si 2 47,48 but attributed to the electron-lattice coupling 49 . Here NaYbSe 2 provides an interesting alternative for further investigation. Alternating heavy fermion and QSL layers To have a better understanding of the peculiarity of NaYbSe 2 , we compare in Figs. 3a and 3b the imaginary part of the Yb-4f hybridization function at four typical pressures. Quite remarkably, this reveals an unexpected but substantial difference between the two inequivalent Yb ions. While no hybridization is observed near the Fermi energy on both ions at low pressure (20 GPa), it begins to develop on Yb1 at 60 GPa and becomes large above 75 GPa, but on Yb2 the hybridization remains weak over the whole pressure range and shows no significant enhancement. These suggest that only Yb1 ions are responsible for the flat bands and heavy fermion properties at high pressure, while Yb2-4f electrons remain localized and weakly interacting with conduction electrons. These observations are further confirmed in the local spin susceptibility, χ zz (ω) = S z (τ )S z (0) ω , calculated using the CTQMC impurity solver. Here S z is the local spin operator of the Yb-4f electrons and τ is the imaginary time. As shown in Fig. 3c, the static susceptibility of Yb1 deviates from the Curie-Weiss law at low temperature above 75 GPa due to the Kondo hybridization, but that of Yb2 always follows the Curie-Weiss law and keeps the local moment behavior. This explains the absence of quasiparticle resonance in the Yb2-4f density of states plotted earlier in Fig. 2f. To understand these huge differences between Yb1 and Yb2 ions, we compare in Fig. 3d their local environment, namely their respective Yb-Se bond distances. With increasing pressure from 20 to 100 GPa, the Yb1-Se distance is seen to decrease from 2.67 to 2.37Å, while the Yb2-Se distance decreases from 3.0 to 2.66Å. Hence the Yb2-Se distance at 100 GPa is roughly equal to the Yb1-Se distance at 20 GPa, explaining why the Yb2-4f electrons are still localized at such high pressure. We thus conclude that the high-pressure structure of NaYbSe 2 contains alternating Yb1 heavy fermion layers and Yb2 QSL layers, a very peculiar structure for studying the interplay between the two exotic quantum many-body states. It is wondering if this might be responsible for the broad crossover to the Fermi liquid behavior observed in the resistivity 18 . Superconductivity Experimentally, superconductivity in NaYbSe 2 emerges 1, k 2 x + k 2 y , k 2 z line * A2g kxkz(k 2 x − 3k 2 y ), kykz(k 2 y − 3k 2 x ) line Eg (k 2 x − k 2 y , Summary Our calculations reveal a sequence of electronic phases in pressurized QSL candidate NaYbSe 2 . We find that it is a charge-transfer insulator at ambient pressure and turns into a low carrier density Kondo system with semi-metallic band structures close to 60 GPa and a good heavy fermion metal above 73 GPa. In between, we predict a delocalization quantum critical point of the Yb1-4f electrons that may explain the observed non-Fermi liquid with linear-in-T resistivity, while the Yb2-4f electrons remain localized over the whole pressure range. The lattice therefore has a special structure constructed with alternating Yb1 heavy fermion layers and Yb2 QSL layers. Superconductivity emerges in the heavy fermion metal with well nested quasi-twodimensional Fermi surfaces, making NaYbSe 2 possibly the 3rd Yb-based heavy-fermion superconductor with a record-high T c . This suggests the possible involvement of both spin fluctuations and phonons as the pairing glues. Our work will be a useful guide for future experimental study of NaYbSe 2 and its other family members. METHODS DFT calculations were performed using the full-potential augmented plane-wave plus local orbital method as im-plemented in the WIEN2k package 37 and the Perdew-Burke-Ernzerhof exchange-correlation potential 39 . The lattice parameters were optimized using DFT+U with an effective Coulomb interaction of 6 eV 38 . The plane-wave cutoff K max was chosen to make R MT × K max = 8.0. All calculations were performed on a grid of 1000 k-points in the Brillouin zone. The spin-orbit coupling was included explicitly. DFT+DMFT calculations were employed to treat the electronic correlations of Yb-4f orbitals. The hybridization expansion continuous-time quantum Monte Carlo (CT-HYB) was used as the impurity solver 40,41 . Yb-4f local orbitals were constructed using projectors with an energy window from -10 to 10 eV relative to the Fermi energy. The Coulomb interaction was chosen to be U = 6 eV and J H = 0.7 eV. A nominal double-counting scheme with n f = 13.0 was used. The self-energy in real frequency was obtained by analytic continuation based on the maximum entropy 35,52 . FIG. 1 : 1Lattice structure and low-pressure electronic structures of NaYbSe2. a Comparison of the crystal structures at low (LP) and high pressures (HP). b Optimized lattice parameters as a function of pressure compared with experiment18,19 . c, d Orbital-resolved density of states and the spectral function at ambient pressure and 20 GPa obtained by DFT+DMFT at 10 K. FIG. 2 : 2Electronic phases of NaYbSe2 under pressure. The upper figure illustrates the predicted phases: the charge transfer or band insulator, the Kondo semimetal (Kondo), the heavy fermion metal (HF), and the superconductivity (SC). The pressures denote a rough boundary of these phases from experiment 18 . a-c The DFT+DMFT spectral functions at 10 K for 60, 75 and 100 GPa, respectively. d-f The corresponding density of states on two inequivalent Yb ions, showing the development of a sharp quasiparticle peak of Yb1-4f electrons. The inset in d is an enlarged plot over a smaller energy window to show the pseudogap structure near the Fermi energy at 60 GPa. The inset in e plots the full width at half maximum (FWHM) of the Yb1 quasiparticle peak as a function of pressure, where the solid line is a linear extrapolation giving a transition at about 73 GPa. FIG. 3 : 3Different properties of the two inquivalent Ybions. a, b Pressure evolution of the minus imaginary part of the hybridization function in real frequency on Yb1 and Yb2 ions from 20 to 100 GPa, obtained by DFT+DMFT calculations at 10 K. c Comparison of static local spin susceptibility of Yb1 and Yb2 4f -electrons at 60, 75, and 100 GPa. d Comparison of the Yb-Se distances as a function of pressure from 20 to 100 GPa. FIG. 4 : 4Fermi surface topology and nesting with pressure. a-c Comparison of the DFT+DMFT calculated Fermi surfaces at 60, 75 and 100 GPa obtained at 10 K. d-f The corresponding Lindhard susceptibility (real part), showing a good nesting property only beyond 100 GPa. TABLE I : ISpin singlet pairing states from irreducible representations of the point group D 3d of Yb1 ions. The line * means possible accidental line nodes for certain parameters of the basis functions and otherwise fully gapped. IR singlet gap functions ∆(k) nodes A1g be involved in the electron pairing. At this stage, phonons also cannot be excluded. But in any case, since NaYbSe 2 is in a heavy fermion state at this pressure, the itinerant heavy Yb1-4f quasiparticles must participate in the superconductivity, making it a candidate heavy fermion superconductor. If this is the case, it would be the 3rd Yb-based heavy fermion superconductors, in addition to YbRh 2 Si 2 of T c ∼ 2 mK 50 and β-YbAlB 4 of T c ∼ 80 mK 51 . The transition temperature of about 8 K in NaYbSe 2 is strikingly high among all heavy fermion superconductors except Pu-115, probably suggesting a potential cooperation of the Fermi surface nesting, spin fluctuations and phonons. For such a Fermi surface topology with the antiferromagnetic nesting wave vector, group-theoretical analyses favor a spin singlet pairing state with gap symmetries listed inTable I. We have either a nodeless s-wave pairing (A 1g ) or a two-component d-wave pairing (E g ), with possible accidental line nodes for appropriate combination of the basis functions, or a less probable g-wave gap (A 2g ) with loop nodes in the k z = 0 plane. More elaborate experiments are needed for better clarification but would be extremely difficult under such high pressures. It will be important if superconductivity can be found at lower pressures in other members of the same family through chemical substitution.kxky), (kxkz, kykz) line * above 103 GPa in the heavy Ferm liquid phase 18 . To gain some insight on its pairing mechanism, we compare in Figs. 4a-c the Fermi surface structures at 60, 75 and 100 GPa, respectively. At 60 GPa, the Fermi surfaces only consist of small electron and hole pockets centered at M and Γ points. The small Fermi volume reflects a low carrier density of the semi-metallic phase for incoherent Kondo scattering. At 75 GPa, the Fermi surfaces become larger with three cylinders around Γ and a small electron pocket around the M point. At 100 GPa, the cylindrical Fermi surfaces continue to grow but the electron pocket at M is gapped out and diminishes due to strong hybridization. We find superconductivity emerges as the Fermi surfaces become well nested and quasi-two-dimensional. This is seen in Figs. 4d-f, where we plot the real part of the Lindhard susceptibility derived from DFT+DMFT Fermi surfaces. Only beyond 100 GPa, we find a well-defined nesting wave vector. It is thus speculated that the Fermi surface nesting and its resulting spin density wave fluctuations may play a major role in the Cooper pairing. Moreover, the nesting pattern is in close resemblance to that of the spin excitation spectra of the Heisenberg model on the same lattice as revealed by neutron scattering experiment at ambient pressure 31 . 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[ "A curved-element unstructured discontinuous Galerkin method on GPUs for the Euler equations", "A curved-element unstructured discontinuous Galerkin method on GPUs for the Euler equations" ]	[ "M Siebenborn ", "V Schulz ", "S Schmidt " ]	[]	[]	In this work we consider Runge-Kutta discontinuous Galerkin methods (RKDG) for the solution of hyperbolic equations enabling high order discretization in space and time. We aim at an efficient implementation of DG for Euler equations on GPUs. A mesh curvature approach is presented for the proper resolution of the domain boundary. This approach is based on the linear elasticity equations and enables a boundary approximation with arbitrary, high order. In order to demonstrate the performance of the boundary curvature a massively parallel solver on graphics processors is implemented and utilized for the solution of the Euler equations of gas-dynamics.	10.1007/s00791-013-0197-0	[ "https://arxiv.org/pdf/1208.4772v2.pdf" ]	5,232,383	1208.4772	946ab4b64227f6e43a4d2947ce18150f55d75773	A curved-element unstructured discontinuous Galerkin method on GPUs for the Euler equations M Siebenborn V Schulz S Schmidt A curved-element unstructured discontinuous Galerkin method on GPUs for the Euler equations In this work we consider Runge-Kutta discontinuous Galerkin methods (RKDG) for the solution of hyperbolic equations enabling high order discretization in space and time. We aim at an efficient implementation of DG for Euler equations on GPUs. A mesh curvature approach is presented for the proper resolution of the domain boundary. This approach is based on the linear elasticity equations and enables a boundary approximation with arbitrary, high order. In order to demonstrate the performance of the boundary curvature a massively parallel solver on graphics processors is implemented and utilized for the solution of the Euler equations of gas-dynamics. Introduction The DG method is based on a discontinuous finite element spatial discretization originally introduced by Reed and Hill in the early 1970s for the neutron transport equation [1]. Later, in a series of papers [2,3,4,5,6], Cockburn and Shu combined the discontinuous Galerkin spatial discretization with an explicit Runge-Kutta time stepping (RKDG method) and extended the method to systems of conservation laws. This created the opportunity for highly parallel implementations, since in the RKDG method, one grid cell only needs information from the immediate neighbouring cells to march in time. Based on this, Biswas et al. investigated the potential of RKDG methods for parallelization in [7]. Another outstanding feature of DG methods is that the degree of basis functions can be chosen arbitrarily, thus leading to high order discretization. However, attention has to be payed to the representation of curved boundaries. Bassi and Rebay applied this method to the two dimensional Euler equations and worked out the importance of the boundary approximation with respect to the solution quality [8]. They pointed out that the order of the method is limited by the order of the boundary representation. Thus, it is necessary to deal with curved element discretizations, in order to overcome this issue. While this seems not to be challenging for some test cases in two dimensions it is quite difficult to approximate complex geometries arising, e.g. , from industrial applications. We present a mesh curvature procedure based on linear elasticity deformations which matches a desired boundary shape. Similar techniques are well known for the deformation of discretization meshes and avoid expensive remeshing [9]. In order to deal with discontinuities arising in the solution of hyperbolic equations, slope limiters where introduced to the RKDG method by Cockburn and Shu. Successful in the finite volume community, limiters are yet complicated for higher order methods. This gives rise to the idea of adding artificial viscosity to the equations in order to smear out discontinuities and control the width of shocks. Persson and Peraire proposed a detector to apply artificial viscosity only in the vicinity of shocks [10]. This approach seems very promising because of its flexibility, since there is no dependency on the order of the scheme or the geometry of the discretization elements like for slope limiters. Recently, there has been payed a lot of attention to discontinuous Galerkin methods because of their potential in terms of parallelization and high performance computing (HPC). Especially the nodal DG method proposed by Hesthaven and Warburton [11] was shown by Klöckner, Warburton, Bridge and Hesthaven [12] to perform very efficiently on modern graphics processors (GPUs) gaining high speedups compared to conventional codes. In this work, we present a novel approach combining the following aspects. We implement a high order Runge-Kutta discontinuous Galerkin method for the Euler equations of gasdynamics on GPUs. For that, we propose an approach introducing unstructured, curvedelement DG discretizations into a massively parallel GPU algorithm. Furthermore, we present a mesh curvature approach, which enables high order accurate boundary representations and seamlessly fits into the DG framework. In particular, we cover the whole simulation chain from the generation of curved, body-fitted meshes up to the parallel HPC system solution. Finally, we demonstrate the performance of this algorithm on some challenging transsonic test cases, where discontinuities arise in the solutions. This paper has the following structure. In section 2, the discontinuous Galerkin method is shortly introduced and it is shown how the discrete operators are composed. It is also covered how discontinuities in the solution are handled. In section 3, we describe an approach for dealing with curvature in the DG discretization. Section 4 shows the implementation on GPUs. Finally, in section 5 and 6 numerical results are presented and discussed. The discontinuous Galerkin method In this paper, we study a discontinuous Galerkin method for systems of hyperbolic conservation laws of the form ∂U ∂t + ∂F 1 (U ) ∂x 1 + · · · + ∂F d (U ) ∂x d = 0 in (0, T ] × Ω, U (0, x) = U 0 (x) x ∈ Ω where U : R × R d → R n , U (t, x) = (U 1 (t, x), . . . , U n (t, x)) T is the vector of conserved quantities at a point x in d-dimensional space and at time t. Here Ω ⊂ R d is the domain of interest and [0, T ] a time interval. The vector fields F i : R n → R n in this system are usually referred to as flux vectors. For the purpose of this work we will maintain a more compact and widely used notation. Introducing the tensor F : R n → R n×d , F (U ) = (F 1 (U ) . . . F d (U )) in terms of the fluxes Ψ(x) = r z y x t s r Figure 1: Mapping from physical to reference element we then obtain ∂U ∂t + ∇ · F (U ) = 0.(1) We follow the approaches in [13] and mostly use the notation therein. For the derivation of the discontinuous Galerkin method, we assume that the domain of interest Ω is subdivided into a finite set of K disjoint, conforming elements Ω = K k=1 Ω k . In our approach this is a tetrahedral mesh with curved elements. The solution is then approximated using a space V h of element-wise defined polynomials ψ j up to degree p V h = K k=1 V k h , V k h = span{ψ j (Ω k ) , j = 1, . . . , N p }. Here, N p = (p+d)! d!p! is the number of basis functions depending on the desired degree p and the spatial dimension d. Multiplying with a test function Φ from the same space and integration over the element Ω k leads to Ω k ∂U ∂t + ∇ · F (U ) ΦdΩ = 0. Integration by parts then yields the weak discontinuous Galerkin formulation Ω k ∂U ∂t Φ dΩ = − ∂Ω k (F (U ) * · − → n ) Φ dS + Ω k F (U ) · ∇ΦdΩ.(2) Here, − → n denotes the outward pointing normal vector and F * an approximate Riemann solver, which deals with the double-valued state at the element interfaces, e.g. upwinding. These approximate Riemann solvers are well known from finite volume context and a survey can be found in [14]. For the purpose of this work, we tested both a local Lax-Friedrichs and a HLLC Riemann solver but without noticing major differences. In the following, a discrete version of equation (2) is derived, which can be implemented on a computer. In order to calculate the integrals occurring in (2) we have to establish a link between an arbitrary curved element Ω k in the mesh and the reference tetrahedron T = {−1 ≤ r, s, t ≤ 1 , r + s + t ≤ −1} on which cubature/quadrature rules are known. This is done by mapping functions Ψ k : Ω k → T for each element as illustrated in figure 1 connecting the physical coordinates x = (x, y, z) to the computational ones r = (r, s, t). Moreover, the partial derivatives of Ψ k have to be known since integration for any functions f, g : Ω k → R n in the physical space is evaluated as Ω k f gdΩ = T =Ψ k (Ω k ) f (Ψ k ) g (Ψ k ) \|det (DΨ k )\| dT with the convention DΨ =   r x r y r z s x s y s z t x t y t z   and J = \|det (DΨ)\| .(3) Hence, it is sufficient to derive the local operators for the reference element and transform them into the physical space. Then with nodal values f = (f (r 1 ), . . . , f (r Np )) T . These grid points are chosen according to [15] to ensure good interpolation properties. Introducing the Vandermonde matrix V ij = ψ j (r i ) we obtain that these two formulations are linked as f = Vf . Since there are as many basis polynomials as collocation points and depending on the interpolation quality V is nonsingular. Now, we are prepared to define the local operators for the integration in (2). For that purpose, we distinguish three sets of nodes: the collocation points r i as introduced above and two sets of nodes used for integration. These are on the one hand cubature points {r cub i , i = 1, . . . , N cub } (figure 2b) for volume integrals [16] and on the other Gauss quadrature points {r g i , i = 1, . . . , N g } (figure 2c) for surface integrals [17]. For the derivation of the discrete operators, we closely follow [13]. First, in order to interpolate the solution given at the nodal points r i to quadrature points r cub i and r g i we need the following interpolation matrices I cub = V cub V −1 ∈ R N cub ×Np , I g = V g V −1 ∈ R Ng×Np where (V cub ) ij = ψ j r cub i and (V g ) ij = ψ j (r g i ). Multiplying the vector of nodal unknown values with these matrices first transforms them to modal expansion coefficients and then evaluates the modal basis functions at the cubature points. For the construction of the stiffness matrices we need to evaluate derivatives like ∂l i ∂r at the cubature points. Here we introduce the operator D r = V r V −1 ∈ R N cub ×Np where (V r ) ij = ∂ψ j (r) ∂r r=r cub i . The s and t derivatives are analogous. With this we are now prepared to assemble the local stiffness matrix for one specific element leaving out the element number k for clarity S x = D T r · diag(r x,i ) + D T s · diag(s x,i ) + D T t · diag(t x,i ) · diag(J i W cub i ) ∈ R Np×N cub(4) where W cub i are cubature weights. S y and S z are obtained in the same way. The local mass matrices are then given by M = I T cub · diag(J i W cub i ) · I cub ∈ R Np×Np(5) Finally, we obtain the local face mass matrices M ∂Ω = I T g · diag(J i W g i ) ∈ R Np×Ng(6) where W g i are the weights of a 2d Gauss integration rule for triangles. Plugging these operators into equation (2) yields the semi-discrete system ∂U h ∂t = M −1 k 3 m=1 S k,xm F m U cub h − M −1 k M ∂Ω k F U g h * · − → n .(7) In the equation above, U cub h = I cub U h denote the degrees of freedom associated with the cubature points used for the volume integration (figure 2b) and U g h = I g U h the ones for surface integration (figure 2c) respectively. Since the operators are only local and therefore represented by small matrices, a LU or Cholesky decomposition of M k can be calculated in an initial step of the solver. Finally, the system is discretized in time using a fourth-order accurate Runge-Kutta pseudo time stepping consisting of five stages. We use a low storage version of this scheme as described in [18] which does not require to keep the intermediate stages in memory. Furthermore, the application of an explicit time integration like this RK method does not require to set up a global matrix system. Thus, the method above is a so called matrix-free approach which is especially in three dimensions very attractive with respect to memory requirements. When dealing with fluid dynamics (see section 5), the method described above works reliably for low Mach numbers. Yet, for higher Mach numbers, shocks might appear in the solution leading to strong non-physical oscillations. This phenomenon is well understood and can be overcome by adding a small amount of artificial viscosity to the equations as proposed by Persson and Peraire in [10]. However, in regions where the solution is smooth there is no need for stabilization. Hence, these authors compare the approximated solution of one component u h to the solutionũ h where they drop the modes with the highest frequencies u h = Np j=1û j ψ j ,ũ h = N p−1 j=1û j ψ j with the following smoothness indicator S k = Ω k (u h −ũ h ) · (u h −ũ h ) dΩ Ω k u h · u h dΩ . In smooth regions of the solution, this indicator will be close to zero whereas it increases in regions with high frequencies. The amount of artificial viscosity in element k is then determined as k =      0 if s k < s 0 − κ 0 2 1 + sin π(s k −s 0 ) 2κ if s 0 − κ ≤ s k ≤ s 0 + κ 0 if s k > s 0 + κ with s k = log 10 (S k ), s 0 ∼ log 10 1 p 4 and empirically chosen parameters κ and 0 . Thus, by applying a shock detector it can be decided where shocks arise and where to add viscosity. The modified system of equation is then given by ∂U ∂t + ∇ · F (U ) = ∇ · ( ∇U ) . This can be rewritten as a system of first order equations and discretized using the same DG approach ∂U ∂t + ∇ · F (U ) − ∇ · √ q = 0 q − √ ∇U = 0. Through this procedure the semi-discrete system (7) is expanded by an additional equation and the viscous fluxes yielding (c.f. [19]) ∂U h ∂t = M −1 k 3 m=1 S k,xm F m U cub h − √ q cub h − M −1 k M g F U g h * − √ q g h * · − → n q h = M −1 k 3 m=1 √ S k,xm U cub h − M −1 k M g √ U g h * · − → n . Mesh generation for DG As pointed out in the introduction, a remarkable feature of the DG method is the freedom to choose the degree of basis functions. However, because of that the mesh has to be dealt with caution. It is tempting to reuse the meshes from the finite volume community. Yet, this leads to major problems, since the discretization with straight-sided elements leads to kinks at the boundary walls, which is problematic whenever the fluid is interacting with curved geometries. In this situation the numerical solution might contain small non-physical shocks in each element at the boundary. In principle, this could be overcome by using a very fine discretization. However, this is conceptually in contrast to higher order discretization methods. Thus, it is necessary to introduce curved elements into the discontinuous Galerkin discretization in order to enable a higher order boundary representation. An isoparametric mapping is applied for the realization of the element curvature. Assuming that the deformation of the elements can be represented in the same Lagrange polynomial basis as used for the DG scheme, we only need to know the displacement of the collocation points with respect to the ones in the straight sided element. Thus, it is not enough to have a functional description of the curved boundary. Additionally, the locations of the collocation points have to be corrected such that they do not fall out of the element. Moreover, this displacement procedure should work smoothly, in order not to perturb the integration. In two dimensions this process is simple. Here, only the elements sharing a face with the boundary wall are affected. By ensuring that the collocation points at the two other faces remain on their initial position, the neighboring elements remain untouched, even if they share one vertex with the boundary. Obviously, this does not hold for the three dimensional case. By deforming a face on the boundary the other three faces in general have to be curved as well. This forces a whole layer of cells around the wall to be curved. In the following, we focus on how to transport the curvature information of the boundary into the mesh. As mentioned before, in three dimensions a lot of cells are affected. Hence, we suggest instead of tracking these cells to put a bounding box around the geometry marking an area in which cells get curved. We follow the ideas in [9] in order to model the discretization Figure 4: Leading edge of ONERA M6 airfoil surface -initial and curved geometry mesh as embedded in some flexible material and solve the linear elasticity equations with a finite element method. By this, the curvature information of the boundary is transported into the mesh and can be retrieved at arbitrary points. In order to keep this process as cheap as possible, we extract the cells which should become curved from the CFD-mesh and solve the linear elasticity equation on the resulting sub-mesh. The governing equations of the linear elasticity are given by ∇ · σ = f on Ω c . Here f denotes body forces acting on the solid, which will not be present in our case. Ω c ⊂ Ω is the region of the DG mesh containing the cells to be curved. Finally, σ denotes the stress tensor and is defined by the strain tensor as σ = λTr( )I + 2µ , = 1 2 ∇u + ∇u T . In the equations above, λ and µ are the Lamé parameters, which can be expressed in terms of the Young's modulus E and Poisson's ratio ν λ = νE (1 + ν)(1 − 2ν) , µ = E 2(1 + ν) . The vector-valued function u : Ω c → R 3 describes the displacement which the material would perform under the prescribed forces. In our case, it is not important whether the chosen coefficients represent a physical material. Nevertheless, it is rather crucial to know how they act on the solution. Young's modulus can be seen as a measure for the stiffness and Poisson's ratio describes how much a material expands in two coordinate directions when compressed in the third. Since f = 0 in our case the solution does not depend on the stiffness parameter E. Thus, the mesh deformation is controlled by ν only. This parameter is then chosen depending on the situation. If we want to compute the flow past a sphere, like illustrated in figure 3, we apply a rubber-like material (ν ≈ 0.5) in order to obtain a smooth deformation, which propagates in each coordinate direction. In the NACA0012 case, which is a 3d staggered version of the 2d test case, we choose ν close to 0 in order to avoid deformation into the third dimension. This system is then completed by the boundary conditions Figure 5: Tip of NACA0012 profile -initial and curved grid u = g on Γ D1 ,(8)u = 0 on Γ D2 ,(9)∂u ∂ − → n = 0 on Γ N .(10) We choose the boundaries such that Γ D1 is the objects surface. Γ D2 is the bounding box defining the sub-mesh for the elasticity equation. Finally, Γ N is used to model symmetry walls, where collocation points are allowed to slide but may not leave the plane. One great advantage of this approach is the variable order of boundary representation. Since, we are free to choose the degree of the finite element basis functions, we can fit the boundary representation to the order of the DG scheme. Moreover, in most situations this approach avoids singularities due to the deformation. This means, if the gap between the straight-sided mesh and the curved boundary is not to large, we can expect a feasible deformation of the mesh without overlappings or zero-volume cells. However, this approach requires a parameterized representation of the surface, since the displacements at the boundary have to be evaluated at arbitrary coordinates. While this seems not to be a problem for many geometries existing as CAD models, we are also dealing with geometries only available as a set of vertices. Yet, there is a lot of literature dealing with the problem how to find a NURB surface representation to a given set of points. For a parameterized surface it must be decided how the straight-sided mesh must be mapped to fit the curved geometry. For this purpose a closest point problem has to be solved min (α,β) 1 2 x − S(α, β) 2 s.t. 0 ≤ α, β ≤ 1(11) where S(·, ·) denotes a NURB surface and x is a point on the straight-sided boundary. We solve this problem by the Gauss-Newton algorithm. Let (α * , β * ) be the optimal solution, then g(x) = S(α * , β * ) − x is plugged into the boundary condition on Γ D1 . With these boundary conditions we are now able to apply a high order finite element solver for the linear elasticity equation. For this purpose we use the GETFEM finite element toolbox. Then, we store the solution obtained together with the CFD-mesh and use it later in the discontinuous Galerkin solver in order to retrieve mesh displacement information at arbitrary locations. Application on GPUs Reviewing the DG method described above, it turns out that there is a lot of parallelism in the executions. On the coarsest level one can proceed as in other discretization techniques like finite volume methods and partition the mesh. Due to the nature of DG this does not lead to much data transfer between the compute nodes since only values at the element faces have to be communicated. The same holds on a second level of parallelism. As seen in the derivation of the DG method in section 2, there is only a loose coupling between elements. Most operations can be performed independently. Just the evaluation of the surface integral requires the nodal values on the face of the neighbouring element to be known. Furthermore, we can distinguish a third level of parallelism, the most interesting one. Both on the volume and on the surface quadrature nodes, the nonlinear flux function F has to be computed, which is a very expensive operation. Not only the high order basis functions but also the rational Jacobian of the non-affine mapping force us to apply a very high order quadrature rule leading to a huge number of quadrature points. However, these operations are also independent per node. These features suggest the application of graphics processing units, i.e. GPUs, to the DG method, since the design considerations above perfectly fit into the CUDA hardware model. Originally designed to render many geometrical primitives or pixels in parallel the Nvidia Fermi architecture nowadays is based on a set of 16 so called streaming multiprocessors (SM). Each of them features 32 cores which leads to a total number of 512 CUDA cores enabling hundreds of floating point operations at a time. From the programming point of view, CUDA organizes threads in a hierarchical structure. On the lowest level there are the CUDA threads each having its own registers for temporary variables. These threads are then gathered in thread blocks and identified through a three dimensional index. Each of these blocks comes with 64 kb shared memory in which data can be collected and distributed between the threads. One major advantage of this concept is that shared memory is on chip. Thus, data which is used multiple times or has to be shared between threads has to be requested from the global RAM of the GPU only once. Finally, on the top level blocks are organized in a grid structure and again identified through three dimensional indices. For further details we refer to the CUDA documentation [20]. Yet, one crucial aspect should be addressed. In most cases the performance gain of a massively parallel GPU algorithm strongly depends on memory fetching strategies. Generally, whenever the threads of a block {t 0 , . . . , t n } fetch data from an array A = {a 0 , . . . , a N } out of the global RAM of the GPU, these accesses should be organized in a blockwise fashion, where the blocksize is a multiple of 16. The threads t 0 , . . . , t n should access successive data elements -each thread one element -where the first index is a multiple of 16. Thus, memory accesses into the ith block should be organized as t τ (j) a (16i+j) , j ∈ {0, . . . , n} where τ is a permutation on {0, . . . , n}. In this case, multiple memory accesses can be performed at once leading to so called coalesced accesses. Whereas, when a thread block requests data which is scattered throughout the memory this ends up in serialization of the accesses. This issue is crucial particularly for older GPU hardware where finite volume methods (FVM) have been implemented for fluid dynamics. Since the FVM can be seen as a special case of the DG method for p = 0, there is only one unknown per cell. All neighbouring variables are involved in the update of the value of this unknown, which leads to a high ratio of scattered data accesses in case of unstructured meshes. While GPU acceleration only seemed to work on structured meshes in FVM, the discontinuous Galerkin method overcomes this problem. Due to the higher order of the method, there are plenty of values per cell which can be organized successively in memory allowing high speedups even on unstructured meshes. This hardware and programming layout can now be identified with the basic steps of the DG method. As previously mentioned, operations on one element are mostly independent of the executions in other elements. Thus, it is obvious to assign one CUDA block to each DG element. Then, the arithmetically expensive nonlinear flux evaluations can be performed in parallel. Moreover, the subsequent matrix vector products can also be handled very efficiently within each block since we are dealing with small, dense matrices. Finally, the CUDA execution handler decides how many thread blocks can be executed in parallel depending on the order of the DG method. We have to reorganize the unknown values in order to meet the memory pattern of the GPU. The length of each field should be a multiple of 16 which is achieved by so-called zero padding as demonstrated in (12). Here, U k h denotes the unknown values inside element k. As illustrated in chapter 2 we are dealing with a vector of n unknowns interpolated at N p collocation points. Thus, the padding length is determined such that the block length is enlarged from N p to b p = Np+15 16 and we ensure that the start address of each field is a multiple of 16 U k h = U 1 0 , U 1 1 , . . . , U 1 Np−1 , 0, . . . , 0, U 2 0 , U 2 1 , . . . , U 2 Np−1 , 0, . . . , 0, . . . . . . . . . . . . . . . U n 0 , U n 1 , . . . , U 2 Np−1 , 0, . . . , 0 . padding(12) The same works for the unknowns at the quadrature points. In contrast, the values at the cubature points are only used once and do not need to get shared with other elements. Thus, we do not need to store them in the global RAM. In the following, we will consider the element specific volume integral in equation (2) as an example. For the integrations inside each element, we also need to fetch the element specific local operators from the global GPU RAM. Thus, they are stored in a similar way like the vector of unknowns. For example the S x operator in equation (4) is stored as padding (14) Note that these operators are stored in column-major form, which is important for the matrixvector product. S x = s x (0,0) , s x (1,0) , . . . , s x (Np−1,0) , 0, . . . , 0, s x (0,1) , s x (1,1) , . . . , s x (Np−1,1) , 0, . . . , 0, . . . . . . . . . . . . . . . s x (0,N cub −1) , s x (1,N cub −1) , . . . , s x (Np−1,N cub −1) , 0, . . . , 0 . padding(13) With this data structures in global GPU RAM we implement the CUDA grid to have K thread blocks, which will be associated with the DG elements. Each of these blocks should contain N cub threads {t 0 , . . . , t N cub −1 } and a matrix U [n][N cub ] in shared memory to cache the unknown values. At this point, we can assume that the matrix U is large enough to store both U h and U cub since N cub > N p . This holds, because for a proper integration of the nonlinear flux function there have to be more cubature points than collocation points. The initial step is to cache the nodal values U h into the shared matrix. For that purpose the first N p threads load and store the n fields of U h into U . Then the product I cub U k h is evaluated column-wise. The threads t 0 , . . . , t N cub −1 load one column of I cub , each thread one value. Since I cub is stored in column-major form, these loads are coalescent. Then the n values in U corresponding to this column are broadcasted to all threads and the multiplication is performed. The broadcast operation is very efficient, since the matrix U was fetched to the shared memory which is on chip in contrast to the global GPU RAM. This is done successively for all columns and each thread is tracking the sum over all columns in its registers. Finally, by this procedure, we obtain the nodal values interpolated to the cubature points U cub h , which are again stored in U . Through this procedure the dot products between the N cub rows of I cub and the fields of U are performed in parallel each by one CUDA thread. In order to calculate the volume integral in (2) which is approximated in the discrete system (7) by 3 m=1 S k,xm F U cub h the threads t 0 , . . . , t N cub −1 evaluate the flux function F on U in parallel. Finally, the multiplication with the three stiffness matrices S x 1 , S x 2 , S x 3 is treated as described above. However, this matrix vector product can not be handled as one thread per output value, since S x i has more columns than rows in general. Thus, the threads are distributed over several matrix columns, which are then handled in parallel to maintain the occupancy of the streaming processors high. For this implementation, we were inspired by the MIDG code [21]. MIDG is a very lightweight discontinuous Galerkin solver for Maxwell's equations designed to run on multiple GPUs. A mesh partitioning with ParMetis is involved which minimizes the cuts between processes and thus reduces communication. The time discretization is realized by a low storage Runge-Kutta scheme as described in section 2. For the spatial discretization, the nodal DG scheme as described in [13] is applied. We mostly adopted the parallel framework of this code including the mesh partitioning and the MPI communications. However, the quadrature free approach therein does not support curved elements and leads to errors when dealing with nonlinear PDEs like the Euler equations (cf. section 5). We thus had to apply a computationally more expensive, quadrature based scheme as introduced in section 2. An overview of our algorithm is given in figure 6. We work on a compute cluster -shared or distributed memory -on which each node is equipped with one GPU. Here, the executions on compute node j are described and it can be seen that after the interpolation to the quadrature nodes on the element faces the communication with other processes takes place. For that purpose, the values on processor cuts are downloaded from the GPU and distributed to other compute nodes. In order to save time, this is done asynchronously, while the volume integration is executed. In our test setting, we have a shared memory system equipped with 8 Intel Xeon E5620 CPU cores and 8 Nvidia Tesla M2050 GPUs. For measurements of the GPU accelerated code against a conventional CPU code, we run a test on one GPU and compare it to a CPU code where the matrix vector multiplication is implemented using BLAS routines. Since the underlying CPU has four cores, we use an OPENMP parallelization to achieve full utilization of its capacity. For that purpose we hint the compiler to apply a parallelization of the k-loop over all elements (cf. equation (2)). In the test setting of Numerical Results As a special case of equation (1) we consider the Euler equations of gas-dynamics in three spatial dimensions. These equations describe the motion of an inviscid fluid without heat conduction       ρ ρu ρv ρw ρE       t +       ρu ρu 2 + p ρuv ρuw u(ρE + p)       x +       ρv ρuv ρv 2 + p ρvw v(ρE + p)       y +       ρw ρuw ρvw ρw 2 + p w(ρE + p)       z =       0 0 0 0 0       . Here, the unknown function reads as U : R × R 3 → R 5 , U (t, x) = (ρ(t, x), ρu(t, x), ρv(t, x), ρw(t, x), ρE(t, x)) T where ρ denotes the density, u, v, w the fluid velocities in the three space directions and E the total energy. Finally, the system is completed by the perfect gas law for the pressure p = (γ − 1) ρE − u 2 + v 2 + w 2 2ρ where γ is the adiabatic index of the fluid [22]. Our first test case is a subsonic flow past a sphere at M ∞ = 0.38 which was examined together with the discontinuous Galerkin method in [8]. This test case is very attractive, since the surface curvature is straightforward. Nevertheless, we apply our curvature approach to obtain a smooth volume mesh. Let r denote the radius and x 0 the centroid of the sphere. Then the boundary conditions in equation (8) for the mesh deformation are given by Thus, we obtain a smoothly curved volume discretization, which we then use as input for the DG Euler solver also based on p = 4 basis functions. We note that best results are obtained when the order of the linear elasticity solution matches the order of the DG solver. In this case the deformation lies within the range of the DG basis functions and can be resolved properly. Figure 7 shows the effect of curved boundaries on the solution quality. On the left hand side the solution is disturbed due to straight sided elements in the discretization mesh. This leads to small yet problematic kinks between the elements on the surface and results in a non-physical solution. In contrast, the computations on the right hand side were performed on a curved grid, which is in addition coarser. In this case the linear elasticity deformation was calculated using polynomials of order four. The next test case is the NACA0012 symmetric airfoil. Although the geometry is only two dimensional, we stretch the profile into the third coordinate direction in order to apply the same solver as for the other test cases. Here, the effect of the boundary deformation on the volume mesh can be visualized as figure 5 shows. g(x) = x 0 + r x − x 0 x − x. In the NACA0012 situation we consider two flow conditions. First, a pure subsonic flow with no angle of attack and Mach number M ∞ = 0.4 (figure 8a). Figure 8b shows a M ∞ = 0.8 transonic flow with angle of attack α = 1.25 • . In both pictures, the density distribution is plotted. Since we are dealing with discontinuities in this situation, we have to add artificial viscosity to the system of equations in order to ensure stability of the method. For that goal, we proceed as described in section 2 and apply a shock detector to select the troubled cells. In the following we consider the Onera M6 wing. This test case is well known and examined with a variety of numerical methods. The original experiment was defined in [23] at Mach number M ∞ = 0.8395 and angle of attack α = 3.06 • . Again, we start with a straight-sided, tetrahedral mesh and a NURB representation of the ONERA M6 geometry and use the deformation approach to obtain the curved mesh (figure 4). Figure 9 shows the density distribution on the upper surface of the airfoil. Also in this case, artificial viscosity is applied to the system to deal with the shocks on the upper surface of the airfoil. In both the NACA0012 and ONERA For this computations we utilize a so-called p-refinement as a acceleration technique. This means that we start the time stepping with low order polynomials of degree p 1 , e.g. p 1 = 1, iterate until the Runge-Kutta update is below a specified tolerance and then switch to a richer polynomial space of degree p 2 = p 1 + 1. This process is repeated until the desired order is reached. The transfer operation between the coarser and the finer polynomial space is straightforward, since we are dealing with a hierarchical set of basis functions as introduced in section 2. Thus, the solution on level p i can be directly embedded into the p i+1 space, where the higher order polynomials are initially weighted with zeros. This procedure offers two advantages in terms of acceleration. Firstly, computations are much cheaper on a coarser level of p, since the number of unknown values is smaller. Furthermore, the cubature and quadrature formulas do not have to be as precise as for higher order polynomials, which leads to smaller local operators. Secondly, the timestep for the Runge-Kutta time integration scales with O(p −2 ), c.f. [13]. This enables a much larger timestep for smaller values of p, which reduces the number of iterations needed to reach the steady state solution. For the computations shown in this work, we started with p = 2 basis functions and subsequently refined to p = 3 and p = 4. We found out that this is a convenient choice, since we are dealing with relatively coarse grids, where computations with p = 1 or even p = 0 do not lead to suitable results. Table 2: Algorithm execution for the sphere -with and without p-refinement Figure 9: Flow over Onera M6 airfoil with p = 4 elements discretization mesh consists of 25956 tetrahedra and 4457 of those are curved elements. Moreover, on the finest level, each element contains 35 collocation nodes, 70 cubature nodes and 4 · 16 surface quadrature nodes. On the left hand side, the simulation was started with p = 2 basis functions and then subsequently refined to p = 3 and p = 4 functions. Here, the number of iterations on level 2 and 3 was chosen empirically. In contrast, on the right hand side, the simulation was run using p = 4 basis functions only. Both executions terminated when the infinity norm of the residual for p = 4 was below a tolerance of 1e-9. For this particular case, the execution time is more then halved by the p-refinement procedure. The number of iterations on each level may look surprising, since it is multiple of 1000. This stems from the fact that the calculation of a norm is an expensive operation in parallel. In contrast, a few extra Runge-Kutta evaluations are relatively cheap compared to the parallel reduction for the norm. Thus, we evaluate the norm of the residual only after multiples of 1000 iterations. Conclusion In this work, we have presented a curved mesh generation approach. This approach is based on the linear elasticity deformation of an initial grid with straight sided elements until it meets the desired curved boundary. We demonstrated that the boundary approximation can be of arbitrary high order reducing the error induced into the discontinuous Galerkin discretization due to under-resolved geometries. The solution of the linear elasticity equations with a high order finite element method seems computationally very expensive at a first glance. However, this has to be done only once. We precompute the curvature and the DG solver only needs to evaluate the FEM-solution at the desired nodes. In a second step, we showed how these curved meshes are embedded into a GPU based parallel DG solution of the Euler equations of gas-dynamics. Furthermore, the performance of this massively parallel GPU code was tested where we gained a speedup of up to 18 compared to the serial version of this code. We have chosen some challenging test cases including transonic flows leading to discontinuities which were treated with an artificial viscosity approach. Although the combination of an explicit Runge-Kutta time integration with diffusion terms leads to very small time steps we have seen that the outstanding parallel performance of the GPU overcomes this issue. And we believe that this fact justifies the additional implementation effort for the GPU code. , we introduce a orthonormal, hierarchical set of modal basis functions {ψ i , i = 1, . . . , N p } on the reference element T . An arbitrary function f on T is then interpolated on a given set of collocation points {r i , i = 1, . . . , N p } (figure 2a) as f (r i ) = Np j=0f j ψ j (r i ) , ∀i = 1, . . . , N p , with modal expansion coefficientsf = (f 1 , . . . ,f Np ) T . The interpolation can be reformulated in terms of a multivariate Lagrange polynomial basis f (r i ) = Np j=0 f (r i )l j (r i ) , ∀i = 1, . . . , N p Figure 2 : 2Collocation and quadrature points Figure 3 : 3Deformation to a sphere Figure 6 : 6Algorithm execution on one compute node In order to store the interpolation matrix I cub the block length has to be as b cub = N cub +15 16 leading to I cub = ι (0,0) , ι (1,0) , . . . , ι (N cub −1,0) , 0, . . . , 0, ι (0,1) , ι (1,1) , . . . , ι (N cub −1,1) , 0, . . . , 0, . . . . . . . . . . . . . . . ι (0,Np−1) , ι (1,Np−1) , . . . , ι (N cub −1,Np−1) , 0, . . . , 0 . Figure 7 : 7Density distribution of a subsonic flow past a sphere with p = 4 basis functions As shown in figure 3, we start with a straight sided grid and solve the linear elasticity equation on the surrounding volume grid with p = 4 basis functions. Figure 8 : 8Flow past NACA0012 airfoil with p = 4 elements M6 situation we set the parameter for the viscosity κ = 4 and 0 = 0.3. Table 2 2shows two different algorithm executions for the sphere test case. The underlyingp iterations timestep time consumed 2 8000 6e-5 105.8 s 3 10000 2e-5 196.5 s 4 20000 5e-6 650.0 s 38000 952.3 s p iterations timestep time consumed 2 3 4 61000 5e-6 1982 s 61000 1982 s AcknowledgementsOur research is supported by BMBF (Bundesministerium für Bildung und Forschung) within the collaborative project DGHPOPT. Triangular mesh methods for the neutron transport equation. W H Reed, T R Hill, LA-UR-73-479Los Alamos ReportW.H. Reed and T.R. Hill. Triangular mesh methods for the neutron transport equation. Los Alamos Report LA-UR-73-479, 1973. The Runge-Kutta local projection P1-discontinuous Galerkin finite element method for scalar conservation laws. B Cockburn, C W Shu, Modél. Math. Anal. Numér. 25B. Cockburn and C.W. Shu. 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The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comp, 54:545-581, 1990. The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems. B Cockburn, C W Shu, Journal of Computational Physics. 141B. Cockburn and C.W. Shu. The Runge-Kutta discontinuous Galerkin method for conser- vation laws V: Multidimensional systems. Journal of Computational Physics, 141:199-224, 1998. Parallel, adaptive finite element methods for conservation laws. R Biswas, K D Devine, J E Flaherty, Applied Numerical Mathematics. 14R. Biswas, K. D. Devine, and J. E. Flaherty. Parallel, adaptive finite element methods for conservation laws. Applied Numerical Mathematics, 14:255 -283, 1994. A high-order discontinuous Galerkin finite element method solution of the 2d Euler equations. F Bassi, S Rebay, J. Comput. Phys. 138F.Bassi and S. Rebay. A high-order discontinuous Galerkin finite element method solution of the 2d Euler equations. J. Comput. Phys., 138:251-285, 1997. Robust mesh deformation using the linear elasticity equations. R P Dwight, Computational Fluid Dynamics 2006: Proceedings of the Fourth International Conference on Computational Fluid Dynamics, ICCFD. Ghent, BelgiumSpringer Verlag401R.P. Dwight. Robust mesh deformation using the linear elasticity equations. In Com- putational Fluid Dynamics 2006: Proceedings of the Fourth International Conference on Computational Fluid Dynamics, ICCFD, Ghent, Belgium, 10-14 July 2006, page 401. Springer Verlag, 2009. Sub-cell shock capturing for discontinuous Galerkin methods. P O Persson, J Peraire, AIAA paper112P.O. Persson and J. Peraire. Sub-cell shock capturing for discontinuous Galerkin methods. AIAA paper, 112, 2006. Nodal high-order methods on unstructured grids:: I. time-domain solution of maxwell's equations. 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V Schmitt, F Charpin, Report of the Fluid Dynamics Panel Working Group 04, AGARD AR 138. V. Schmitt and F. Charpin. Pressure distributions on the ONERA-M6-wing at transonic mach numbers. Report of the Fluid Dynamics Panel Working Group 04, AGARD AR 138, 1979.	[]
[ "Correlated Band Structure of a Transition Metal Oxide ZnO Obtained from a Many-Body Wave Function Theory", "Correlated Band Structure of a Transition Metal Oxide ZnO Obtained from a Many-Body Wave Function Theory" ]	[ "Masayuki Ochi \nDepartment of Physics\nOsaka University\nMachikaneyama-cho560-0043ToyonakaOsakaJapan\n\nRIKEN Center for Emergent Matter Science (CEMS)\n351-0198WakoSaitamaJapan\n", "Ryotaro Arita \nRIKEN Center for Emergent Matter Science (CEMS)\n351-0198WakoSaitamaJapan\n", "Shinji Tsuneyuki \nDepartment of Physics\nThe University of Tokyo\nBunkyo-ku113-0033Hongo, TokyoJapan\n\nInstitute for Solid State Physics\nThe University of Tokyo\n277-8581KashiwaChibaJapan\n" ]	[ "Department of Physics\nOsaka University\nMachikaneyama-cho560-0043ToyonakaOsakaJapan", "RIKEN Center for Emergent Matter Science (CEMS)\n351-0198WakoSaitamaJapan", "RIKEN Center for Emergent Matter Science (CEMS)\n351-0198WakoSaitamaJapan", "Department of Physics\nThe University of Tokyo\nBunkyo-ku113-0033Hongo, TokyoJapan", "Institute for Solid State Physics\nThe University of Tokyo\n277-8581KashiwaChibaJapan" ]	[]	Obtaining accurate band structures of correlated solids has been one of the most important and challenging problems in first-principles electronic structure calculation. There have been promising recent active developments of wave function theory for condensed matter, but its application to band-structure calculation remains computationally expensive. In this Letter, we report the first application of the bi-orthogonal transcorrelated (BiTC) method: self-consistent, free from adjustable parameters, and systematically improvable many-body wave function theory, to solid-state calculations with d electrons: wurtzite ZnO. We find that the BiTC band structure better reproduces the experimental values of the gaps between the bands with different characters than several other conventional methods. This study paves the way for reliable first-principles calculations of the properties of strongly correlated materials.	10.1103/physrevlett.118.026402	[ "https://arxiv.org/pdf/1701.02412v2.pdf" ]	13,463,473	1701.02412	6a9bac811b12c8363ad50bb450581e3f9407583f	Correlated Band Structure of a Transition Metal Oxide ZnO Obtained from a Many-Body Wave Function Theory 12 Jan 2017 Masayuki Ochi Department of Physics Osaka University Machikaneyama-cho560-0043ToyonakaOsakaJapan RIKEN Center for Emergent Matter Science (CEMS) 351-0198WakoSaitamaJapan Ryotaro Arita RIKEN Center for Emergent Matter Science (CEMS) 351-0198WakoSaitamaJapan Shinji Tsuneyuki Department of Physics The University of Tokyo Bunkyo-ku113-0033Hongo, TokyoJapan Institute for Solid State Physics The University of Tokyo 277-8581KashiwaChibaJapan Correlated Band Structure of a Transition Metal Oxide ZnO Obtained from a Many-Body Wave Function Theory 12 Jan 2017(Dated: October 15, 2018) Obtaining accurate band structures of correlated solids has been one of the most important and challenging problems in first-principles electronic structure calculation. There have been promising recent active developments of wave function theory for condensed matter, but its application to band-structure calculation remains computationally expensive. In this Letter, we report the first application of the bi-orthogonal transcorrelated (BiTC) method: self-consistent, free from adjustable parameters, and systematically improvable many-body wave function theory, to solid-state calculations with d electrons: wurtzite ZnO. We find that the BiTC band structure better reproduces the experimental values of the gaps between the bands with different characters than several other conventional methods. This study paves the way for reliable first-principles calculations of the properties of strongly correlated materials. To reveal fertile and nontrivial physics in condensed matter, first-principles electronic-structure calculation has established itself as an indispensable tool in recent studies. For this purpose, density functional theory (DFT) [1,2] has played a leading role and has been applied to various materials; however, the limitations of this theoretical framework have come to light. One of the major problems is an inaccurate description of strong electron correlations, e.g., in transition metal oxides. The GW method [3][4][5] is a promising way to ameliorate the inaccuracy of the band structures and has been applied to several solids, including d-electron systems. However, because the GW method is often applied without satisfying self-consistency, a nontrivial dependence on the initial DFT calculations is introduced. It has also been reported that the GW method sometimes exhibits severe difficulty in obtaining converged results [6], owing to its perturbative nature. Another possible choice is to construct effective models from DFT that include correlation terms such as Hubbard U and to solve the models using elaborated methodologies [7] such as dynamical mean-field theory [8][9][10]. However, the exact correspondence between the effective models and the first-principles Hamiltonian is a nontrivial problem. Recently, wave function theory (WFT), which had been mainly applied to molecular systems and established itself as the gold standard in theoretical chemistry [11], has become a promising alternative to DFT for accurate descriptions of electron correlation in solids [12]. Among the most powerful frameworks in WFT are first-principles quantum Monte Carlo (QMC) methods [13,14], such as the variational Monte Carlo (VMC), diffusion Monte Carlo (DMC), auxiliary-field quantum Monte Carlo (AFQMC), and full-configurationinteraction (FCI) QMC methods. Other kinds of WFT, called post-Hartree-Fock (post-HF) methods, have also been applied to condensed matter in recent years [15][16][17][18][19]. However, their targets are in most cases limited to solids with small unit cells, owing to their expensive computational cost. In addition, the correlated band structure, which is quite useful in various kinds of theoretical analyses, is not easily obtained in many WFTs. For example, calculation of the band structure in the framework of VMC or DMC requires a large number of single-point calculations of the excited states, which is a clear difference from a mean-field-like approach such as DFT, whereby the whole band structure is obtained at once. From this viewpoint, the transcorrelated (TC) method [20][21][22][23] is a fascinating WFT that can be applied to solids with reliable accuracy and moderate computational cost [24][25][26][27][28][29]. The TC method adopts the socalled Jastrow ansatz, which is based on a promising strategy often adopted in several WFTs such as QMC methods to describe strong electron correlations; i.e., the electron-electron distance is included into many-body wave functions. Explicitly correlated electronic-structure theory [30] in quantum chemistry also adopts this strategy. In fact, the Gutzwiller-and Jastrow-correlation factors have often been used to describe strong electron correlation, including the Mott physics in systems such as the Hubbard model [31][32][33][34]. It is also important to note that, unlike several WFTs, the whole band structure is obtained at once by solving a one-body self-consistentfield (SCF) equation in the TC method, as described later in this Letter. Moreover, the TC method is deterministic, i.e., free from the statistical error, unlike the QMC methods. Accurate calculations for the Hubbard model [35,36] and molecular systems [37][38][39] were also reported using the TC method or other theories that have a close relationship with the TC method. However, insofar as solid-state calculations are concerned, the TC method has so far been applied only to weakly correlated systems. In this Letter, we present the first application of the TC method to the band-structure calculation of a delectron system: wurtzite ZnO. 3d transition metal oxides have posed theoretical challenges for first-principles band-structure calculations, as it is well known that popular approximations such as the local-density approximation (LDA) fail to provide their band structures accurately, as we shall see later. We find that the TC method with the bi-orthogonal formulation (the BiTC method) [29,40] successfully reproduces the experimental band structure of ZnO. We also clarify how the Jastrow factor improves the first-principles description of the correlated electronic states through comparison of the band structures and electron densities among the BiTC and other methods. The central concept of the TC method is to make use of the similarity transformation of the many-body Hamiltonian with the Jastrow factor F = exp(− i,j u(x i , x j )), HΨ = EΨ ⇔ H TC Φ = EΦ (H TC = F −1 HF ),(1) where the correlated wave function is represented as Ψ = F Φ and H TC is called the TC Hamiltonian. Here, x denotes a pair of space and spin coordinates: x = (r, σ). By adopting the so-called Slater-Jastrow ansatz, Φ becomes a Slater determinant consisting of one-electron or- bitals, φ(x): Φ = det[φ i (x j )] , and Eq. (1) yields an SCF equation for one-electron orbitals that experience the effective interaction described with the TC Hamiltonian: − 1 2 ∇ 2 1 + v ext (x 1 ) φ i (x 1 ) + N j=1 dx 2 φ * j (x 2 )v 2body (x 1 , x 2 )det φ i (x 1 )φ i (x 2 ) φ j (x 1 )φ j (x 2 ) − 1 2 N j=1 N k=1 dx 2 dx 3 φ * j (x 2 )φ * k (x 3 )v 3body (x 1 , x 2 , x 3 ) × det   φ i (x 1 ) φ i (x 2 ) φ i (x 3 ) φ j (x 1 )φ j (x 2 )φ j (x 3 ) φ k (x 1 )φ k (x 2 )φ k (x 3 )   = N j=1 ǫ ij φ j (x 1 ),(2) where v ext (x 1 ), v 2body (x 1 , x 2 ), and v 3body (x 1 , x 2 , x 3 ) are the external potential including the nucleus-electron interaction [41] and the two-and three-body effective interactions in the TC Hamiltonian, defined as v 2body (x 1 , x 2 ) ≡ 1 \|r 1 − r 2 \| + 1 2 2 i=1 [∇ 2 i u(x 1 , x 2 ) − (∇ i u(x 1 , x 2 )) 2 + 2∇ i u(x 1 , x 2 ) · ∇ i ],(3) and v 3body ( x 1 , x 2 , x 3 ) ≡ ∇ 1 u(x 1 , x 2 ) · ∇ 1 u(x 1 , x 3 ) + ∇ 2 u(x 2 , x 1 ) · ∇ 2 u(x 2 , x 3 ) + ∇ 3 u(x 3 , x 1 ) · ∇ 3 u(x 3 , x 2 ),(4) respectively. As is evident, the HF method can be regarded as the TC method with u = 0. Owing to this effective one-body picture, it is possible to treat the manybody correlation with moderate computational cost. In addition, one can obtain the band structure of the quasiparticles by using the real part of the eigenvalues of the ǫ matrix on the right-hand side of Eq. (2). One of the authors proved in prior work [23] that such a use of the ǫ matrix as quasiparticle energies is consistent with Koopmans' theorem. We note that one can systematically improve the accuracy of the TC method by utilizing quantum chemical methodologies such as the coupled-cluster and configuration interaction methods to go beyond a single Slater determinant (e.g. Refs. [22,29,40]). Here, the Jastrow function, u(x, x ′ ), is set to the following simple form without adjustable parameters: [13,24,42,43] u(x, x ′ ) = A \|r − r ′ \| (1 − exp (−\|r − r ′ \|/C σ,σ ′ )) ,(5) where A = V /(4πN ) (N is the number of valence electrons in the simulation cell, V is the volume of the simulation cell) and C σ, σ ′ = √ 2A (spin parallel: σ = σ ′ ), √ A (spin antiparallel: σ = σ ′ ). The long-range asymptotic form of this function describes the screening effect of the electron-electron Coulomb interaction [44]. The short-range behavior of the exact Jastrow function should obey the cusp condition [45][46][47]. The Jastrow ansatz adopted here works well for state-of-the-art QMC methods [13,14]. Although our choice of the Jastrow function is rather simple, we shall see that, nevertheless, it works well not only for weakly correlated systems [24,25] but also for the 3d-electron system. Of course, it is possible to improve a quality of the many-body wave function by using a complicated Jastrow factor, but we adopted this simple trial wave function to realize moderate computational cost. In this study [49], we adopted the BiTC method [29,40], in which the left one-electron orbitals, χ(x) in the left Slater determinant X = det[χ i (x j )], replace the bra orbitals in the SCF equation (2), while the ket orbitals remain φ(x). Because the TC Hamiltonian is non-Hermitian, φ(x) and χ(x) become different. We do not show the band structure calculated using the TC method without the bi-orthogonal extension here, because of the large imaginary part of the eigenvalues [57]. Figure 1 presents the band structures of ZnO calculated with the LDA, all-electron G 0 W 0 starting from LDA using the LAPW [58] method, BiTC, and HF methods. The characteristic energy values in these band structures, as evaluated at the Γ point, are listed in Table I. Table I also lists calculated values with various other methods. By focusing on the gaps between the bands with different characters as listed in Table I, the BiTC band structure exhibits better accuracy than [74]. The bottoms of the Zn-3d and O-2p bands are evaluated at the Γ point where the valence-band top is set to zero for each method. The errors between the calculated and experimental values shown in Ref. [62] are presented in parentheses. For LDA, the O-2p bottom and Zn-3d-averaged levels are not presented here because of the overlap among the O-2p and Zn-3d bands. For G 0 W 0 (HSE03), the Zn-3d bottom position was read from the density of states presented in Ref. [64]. All values are in eV. a Ref. [64]. b Ref. [60]. c Ref. [63]. d Ref. [65]. e Ref. [66]. f Ref. [67]. g Ref. [68]. h Ref. [62]. i Ref. [69]. j Ref. [70]. k Ref. [71]. [62], and the Zn-3d peak position [69], which might correspond to the averaged position of the Zn-3d bands. The valence-band maximum energy is set to zero. many other conventional methods, including the G 0 W 0 method starting from LDA, and accuracy comparable to that for the most accurate varieties of the GW scheme, i.e., the G 0 W 0 method starting from HSE03 and the self-consistent GW methods such as the QSGW method [59]. It is noteworthy that both the GW and BiTC methods yield such successful results despite being based on conceptually different formulations. It is important that the BiTC method is based on the self-consistent formulation and thus is independent of DFT calculations, whereas the G 0 W 0 method strongly depends upon the unperturbed DFT calculations, as seen in Table I. We should also note that, while some methods employ parameters U and V d , which are difficult to determine in the ab initio way, the BiTC method does not use such parameters. (U − J = 6 eV) G 0 W 0 +V d (GGA+U ) c 3.30 [−0.1] −5.5 [−0.2] −7.45 −8.0 (U − J = 6 eV, V d = 1.5 eV) QSGW d 3.87 [+0.47] −5.3 [± 0] − −7.2 scGW (RPA) e 3.8 [+0.4] − − −6.4 − scGW (e-h) e 3.2 [−0.2] − − −6.7 − WFT AFQMC f 3.26(16) [−0.14] − − − − DMC g 3.8(2) [+0.4] − − − − BiTC 3.1 [−0.3] −5.1 [+0.2] −9.1 −9.7 HF 11.4 [+8.0] −5.7 [−0.4] −9.3 −9.9 Expt. 3.4 h −5.3 h , −5.2(3) i −7.5 c , −7.5(2) j , −8.5(4) k , − −8.6(2) k , −8.81(15) i Consistency of the calculated band gaps among WFTs that use similar trial wave functions (AFQMC, DMC, and BiTC) is also remarkable. We again stress that the whole band structure is not easily obtained in QMC simulations and requires many single-point excited-state calculations. We can see that the band structures calculated with the BiTC and HF methods are very similar, except for the band gap. Therefore, the main role of the Jastrow factor used in this study on the band structure seems to be improvement of the size of the band gap through the screening effect of the electron-electron interaction, which is described with the long-ranged asymptotic behavior of the Jastrow factor. To obtain a more accurate band structure, e.g., with respect to the depth of the Zn-3d bands, more elaborated Jastrow factors, as used in QMC studies [13,14], will be necessary. Because the Zn-3d bands are almost flat, a key point might be accurate description of the atomic states, which is an important issue for future investigation. For the GW method, it was pointed out that the shallow Zn-d bands can strengthen the p-d hybridization, and thus can result in underestimation of the band FIG. 2: Calculated electron densities on the blue line shown in the crystal structure are presented for the LDA, BiTC, and HF methods with solid black, broken blue, and broken red lines, respectively. The crystal structure was depicted using the VESTA software [75]. gap [63,66]. A similar situation might also be realized in the BiTC band structure, whereas the Zn-d bands are rather deep [72] and so the band gap can be overestimated. However, the BiTC band gap is also affected by the A parameter in the Jastrow factor, as mentioned in the above comparison between the BiTC and HF band structures. Because the A parameter used in this study was determined by RPA analysis of the uniform electron gas, it can cause overscreening in the insulator [26], thereby decreasing the band gap. One possibility is that these two factors are canceled here, but more detailed investigation on other materials is also an important future issue [73]. Figure 2 presents the electron densities calculated with the LDA, HF, and BiTC methods on the line shown in the crystal structure. The electron density obtained by the BiTC method is defined as n(r) = Re[ N i=1 χ * i (r)φ i (r)] , where the condition dr n(r) = N is satisfied due to the bi-orthonormalization condition χ i \|φ j = δ ij . We can see that the electron densities of these methods are almost the same. However, a slight increase of the electron density at the atomic sites is observed for the BiTC method compared with the others. Such a tendency is consistent with the fact that the strong divergence of the electron-electron Coulomb repulsion is alleviated for the effective two-body interaction in the similarity-transformed TC Hamiltonian, because the Jastrow factor satisfies the cusp condition. More concretely, the ∇ 2 1 u(x 1 , x 2 ) and ∇ 2 2 u(x 1 , x 2 ) terms in Eq. (3) yield 1/\|r 1 − r 2 \| divergence with a different sign than the electron-electron Coulomb repulsion. As can be seen in the proof of the cusp condition [45,46], the true manybody wave function should exhibit deformation described with the two-body degrees of freedom near the electronelectron coalescence point, which cannot be represented solely with one-body degrees of freedom. This is a characteristic advantage of the Slater-Jastrow-type wave function for the description of localized electronic states, such as in strongly correlated systems. It is noteworthy that the atomic calculations of the TC method also exhibit a similar tendency for localization [23]. Finally, we mention the computational effort required for the BiTC calculation. Computation takes place on time scales given by O(N 2 k N 2 b N pw log N pw ), where N k , N b , and N pw are the numbers of k-points, occupied bands, and plane waves, respectively [76]. This is the same order as that for the HF or hybrid DFT calculations with a prefactor about 20 to 40 [77]. The BiTC calculation involves neither the frequency index nor the convergence with respect to the number of conduction bands, unlike some perturbative methodologies such as the GW method. As can be seen from the fact that the hybrid DFT calculations have now been applied to various periodic systems, the computational cost of the BiTC method is reasonable for solid-state calculations. One remaining obstacle for wide application of the BiTC method is that we use the norm-conserving pseudopotential with a very high cutoff energy to handle the semicore states at present. However, this problem is not inherent to the BiTC method and can be overcome in principle by the development of a pseudopotential formalism such as the PAW method [78,79] adapted to the TC method, which is an important future issue [80]. We also note that the deformation of the electron density near atomic sites shown in Fig. 2 also implies the importance of careful treatment for the core states. This can also be a common problem for QMC calculations using the Jastrow correlation factor. To conclude, we apply the bi-orthogonal version of the TC method to wurtzite ZnO and find that it well reproduces the experimental band structure. Our study encourages further investigation of other strongly correlated materials using the BiTC method. A part of the calculations was performed using supercomputers at the Supercomputer Center, Institute of Solid State Physics, The University of Tokyo. This study was supported by Grant-in-Aid for young scientists (B) (No. 15K17724) from the Japan Society for the Promotion of Science, MEXT Element Strategy Initiative to Form Core Research Center, CREST, JST, and Computational Materials Science Initiative, Japan. SUPPLEMENTAL MATERIAL CONVERGENCE WITH RESPECT TO THE NUMBER OF k-POINTS We have performed band-structure calculations using a 2 × 2 × 2 k-mesh (32 atoms in the unit cell) for the HF and BiTC methods to confirm the convergence with respect to the number of k-points. For the BiTC calculations, the band gap and the relative energy of the Zn-d-band bottom to the valence-band top are 2.92 and −9.73 eV, respectively, for a 2 × 2 × 2 k-mesh, whereas they are 3.09 and −9.68 eV, respectively, for a 3 × 3 × 3 k-mesh (108 atoms in the unit cell). For the HF calculations, the corresponding values are 11.53 and 10.07 eV, respectively, for a 2 × 2 × 2 k-mesh, whereas they are 11.42 and −9.90 eV, respectively, for a 3 × 3 × 3 k-mesh. The difference between the values calculated using the two k-meshes is less than 0.2 eV for all of these values. In our previous study on the TC method [25], we saw that the calculated band gap E gap (N k ) with N k k-points obeys an approximate relation: E gap (N k ) ≃ E gap (N k = ∞) + C/N k ,(6) where C is a constant. Using this approximate relation for the band energies, the residual finite-size error in our calculations using a 3 × 3 × 3 k-mesh is estimated to be less than 0.1 eV. Therefore, we can say that the convergence with respect to the number of k-points is sufficiently achieved. Although it is possible that the approximate relation that is used above does not hold for this case [68], we can expect that the convergence will be sufficient for our discussion because the finite-size error is expected to be significantly reduced in a 3 × 3 × 3 k-mesh from its value for a 2 × 2 × 2 one. IMAGINARY PART OF THE TC EIGENVALUES WITHOUT THE BI-ORTHOGONAL FORMULATION We do not show the band structure calculated using the TC method without the bi-orthogonal extension in the main text. This is because we found that the imaginary part of the eigenvalues of the Zn-3d bands obtained by solving the one-body SCF equation became as large as 0.1 eV in the TC method, which is more than 1,000 times larger than that in the BiTC method. Because the non-Hermiticity in the (Bi)TC formulation originates from the similarity-transformation, the presumed equivalency between the TC solution and the true many-body eigenstate is suggested to break when the imaginary parts of the eigenvalues become too large. The difference between the band structures obtained with the TC and BiTC methods was small for weakly correlated systems except LiF that is rather localized-electron system [29]. ELECTRON-DENSITY DIFFERENCES AMONG THE LDA, HF, AND BITC METHODS The differences in the electron densities calculated with the LDA, HF, and BiTC methods are shown in Fig. S1. In Fig. S1, we can verify the tendencies explained in the main text, i.e., that the BiTC electron density tends to be localized near the atomic sites. In addition, we can see an increase of the BiTC electron density along the Zn-O bonds near the oxygen atoms. PACS numbers: 71.10.-w, 71.15.-m, 71.20.-b FIG. 1 : 1Calculated band structures using the LDA (solid lines), all-electron G 0 W 0 [60] (green circles), BiTC (solid line), DMC [68] (orange circles), and HF (solid line) methods. Experimental data taken from Ref. [61] are shown with black dots in the middle figure. Blue broken lines show the other experimental data for the positions of the conductionband minimum, the O-2p bottom TABLE I : ISome characteristic values in the band structures of ZnO, as calculated by several methods . P Hohenberg, W Kohn, Phys. Rev. 136864P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). . W Kohn, L J Sham, Phys. Rev. 1401133W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965). . L Hedin, Phys. 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In our observation, the similar trend is also realized for other semiconductors and insulators with sp electrons, which include both the ionic and covalent crystals. a bit different from ours. However, such difference yields a difference in the band gap and the d-band position for less than 0.1 eV in LDA calculations. Refs. [60] and [63] used the lattice parameters that are. Therefore, we can conclude that the difference of the lattice parameters among several references has a minor effect on the comparison presented in Table 1Refs. [60] and [63] used the lattice parameters that are a bit different from ours. However, such difference yields a difference in the band gap and the d-band position for less than 0.1 eV in LDA calculations. Therefore, we can con- clude that the difference of the lattice parameters among several references has a minor effect on the comparison presented in Table 1. . K Momma, F Izumi, J. Appl. Crystallogr. 441272K. Momma and F. Izumi, J. Appl. Crystallogr. 44, 1272 (2011). This scaling is valid only when a single Slater determinant is used (i.e. multi-determinant methodologies are not used) and the Jastrow function is represented as u(x, x ′ ) = p wp(x)v p;σ,σ ′ (\|r − r ′ \|)wp(x ′ ) (i.e. not a general two-body function). This scaling is valid only when a single Slater determi- nant is used (i.e. multi-determinant methodologies are not used) and the Jastrow function is represented as u(x, x ′ ) = p wp(x)v p;σ,σ ′ (\|r − r ′ \|)wp(x ′ ) (i.e. not a gen- eral two-body function). The ratio of computational time required for the TC and HF calculations is about 15. to 20 [25]. As for the BiTC method, this ratio becomes about 20 to 40 owing to the nonequivalency of the bra and ket orbitalsThe ratio of computational time required for the TC and HF calculations is about 15 to 20 [25]. As for the BiTC method, this ratio becomes about 20 to 40 owing to the nonequivalency of the bra and ket orbitals. . P E Blöchl, Phys. Rev. B. 5017953P. E. Blöchl, Phys. Rev. B 50, 17953 (1994). . G Kresse, D Joubert, Phys. Rev. B. 591758G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). Our present implementation requires about 20,000 core hours for solving the BiTC-SCF equation, which is determined by a very large cutoff energy of plane waves. If we can make the cutoff energy an order of magnitude lower, which is a usual demand for other methods, by development of efficient core treatment. this cost will be reduced also by an order of magnitudeOur present implementation requires about 20,000 core hours for solving the BiTC-SCF equation, which is de- termined by a very large cutoff energy of plane waves. If we can make the cutoff energy an order of magnitude lower, which is a usual demand for other methods, by development of efficient core treatment, this cost will be reduced also by an order of magnitude. 1: The differences between the HF and LDA electron densities and between the BiTC and LDA electron densities are presented in panels (b) and (c), respectively. The electron densities on the plane. shown in panel (a) are depicted hereFIG. 1: The differences between the HF and LDA electron densities and between the BiTC and LDA electron densities are presented in panels (b) and (c), respectively. The electron densities on the plane shown in panel (a) are depicted here.	[]
[ "The environs of the ultracompact HII region G45.45+0.06", "The environs of the ultracompact HII region G45.45+0.06" ]	[ "S Paron [email protected] \nInstituto de Astronomía y Física del Espacio (IAFE)\nCC 67, Suc. 281428Buenos AiresArgentina\n", "S Cichowolski \nInstituto de Astronomía y Física del Espacio (IAFE)\nCC 67, Suc. 281428Buenos AiresArgentina\n", "M E Ortega \nInstituto de Astronomía y Física del Espacio (IAFE)\nCC 67, Suc. 281428Buenos AiresArgentina\n" ]	[ "Instituto de Astronomía y Física del Espacio (IAFE)\nCC 67, Suc. 281428Buenos AiresArgentina", "Instituto de Astronomía y Física del Espacio (IAFE)\nCC 67, Suc. 281428Buenos AiresArgentina", "Instituto de Astronomía y Física del Espacio (IAFE)\nCC 67, Suc. 281428Buenos AiresArgentina" ]	[]	Aims. G45.45+0.06 is an ultra-compact HII (UCHII) region extensively studied. It is known that G45.45+0.06 is embedded in a complex of UCHII regions, but up the date, the surrounding ISM in a larger spatial scale has not been analyzed.Methods. Using data from large-scale surveys: Two Micron All Sky Survey, GLIMPSE, MIPSGAL, MAGPIS and GRS, we performed a multiwavelength study of a region about 7 ′ × 7 ′ in the vicinity of G45.45+0.06.Results. We found that the UCHII complex lies in a border of a larger (diameter of ∼ 3 ′ ) and fainter HII region, which is located at the same distance as G45.45+0.06, ∼ 8 kpc.In this work, this larger HII region was called G45L. A good morphological correlation is observed between the PDRs and the molecular gas mapped in the 13 CO J=1-0 and CS J=2-1 lines, suggesting that G45L may be collecting the molecular material. From a near-and mid-IR photometric study, we found three sources, likely O-type stars, that are possibly responsible for the creation of G45L. Additionally we found several YSO candidates lying preferently in the molecular shell that surrounds G45L. Our results confirm that the region southeastern the UCHII complex where G45.45+0.06 is embedded and eastern G45L is active in star formation. We suggest that G45L has been expanding during about 2 × 10 6 yr and could have triggered the formation of the zero-age main sequence stars that are ionizing the UCHII region G45.45+0.06. However we can not discard that both HII regions are coeval.	10.1051/0004-6361/200912646	[ "https://arxiv.org/pdf/0907.4679v1.pdf" ]	16,296,801	0907.4679	d78d16df59ecafc28e1748abe68acbf7ef96e3ce	The environs of the ultracompact HII region G45.45+0.06 27 Jul 2009 July 27, 2009 S Paron [email protected] Instituto de Astronomía y Física del Espacio (IAFE) CC 67, Suc. 281428Buenos AiresArgentina S Cichowolski Instituto de Astronomía y Física del Espacio (IAFE) CC 67, Suc. 281428Buenos AiresArgentina M E Ortega Instituto de Astronomía y Física del Espacio (IAFE) CC 67, Suc. 281428Buenos AiresArgentina The environs of the ultracompact HII region G45.45+0.06 27 Jul 2009 July 27, 2009Received ¡date¿; Accepted ¡date¿Astronomy & Astrophysics manuscript no. g45-correcc1ISM: HII regions -ISM: clouds -stars: formation Aims. G45.45+0.06 is an ultra-compact HII (UCHII) region extensively studied. It is known that G45.45+0.06 is embedded in a complex of UCHII regions, but up the date, the surrounding ISM in a larger spatial scale has not been analyzed.Methods. Using data from large-scale surveys: Two Micron All Sky Survey, GLIMPSE, MIPSGAL, MAGPIS and GRS, we performed a multiwavelength study of a region about 7 ′ × 7 ′ in the vicinity of G45.45+0.06.Results. We found that the UCHII complex lies in a border of a larger (diameter of ∼ 3 ′ ) and fainter HII region, which is located at the same distance as G45.45+0.06, ∼ 8 kpc.In this work, this larger HII region was called G45L. A good morphological correlation is observed between the PDRs and the molecular gas mapped in the 13 CO J=1-0 and CS J=2-1 lines, suggesting that G45L may be collecting the molecular material. From a near-and mid-IR photometric study, we found three sources, likely O-type stars, that are possibly responsible for the creation of G45L. Additionally we found several YSO candidates lying preferently in the molecular shell that surrounds G45L. Our results confirm that the region southeastern the UCHII complex where G45.45+0.06 is embedded and eastern G45L is active in star formation. We suggest that G45L has been expanding during about 2 × 10 6 yr and could have triggered the formation of the zero-age main sequence stars that are ionizing the UCHII region G45.45+0.06. However we can not discard that both HII regions are coeval. Introduction Nowadays it is well established that the formation of massive stars can be triggered by the action of expanding HII regions through the "collect and collapse" process. During its supersonic expansion, an HII region can collect a dense layer of material between its ionization and shock fronts. This layer can be fragmented in massive condensations that Send offprint requests to: S. Paron then may collapse to lead to the formation of new stars. Thus, it is expectable the presence of protostars, young stars, and ultra-compact HII (UCHII) regions on the borders of HII regions. Several observational evidence have been found supporting this star forming mechanism (see e.g. Pomarès et al. 2009;Zavagno et al. 2007, and reference therein). G45.45+0.06 is a luminous Galactic UCHII region that has been extensively studied in the radio continuum and in molecular lines (e.g. Wood et al. 1988;Garay et al. 1993;Wilner et al. 1996;Hatchell et al. 1998). This UCHII region presents CH 3 OH and H 2 O maser emission (Codella et al. 2004). Recently, G45.45+0.06 was included in the Boston University Catalog of Galactic HII Region Properties (Anderson et al. 2009). Using the 13 CO J=1-0 emission obtained from the GRS 1 , the authors derive a v LSR ∼ 55.6 km s −1 and a kinematical distance of 8.2 kpc for this UCHII region, in agreement, within errors, with previous estimations (e.g. v LSR ∼ 55.9 km s −1 and d ∼ 7.7 kpc, Kolpak et al. 2003). As early proposed by Matthews et al. (1977), G45.45+0.06 is part of a cluster of several UCHII regions. Giveon et al. (2005a,b) generated a catalog matching VLA Galactic plane catalogs at 5 and 1.4 GHz with new radio continuum observations and the MSX6C Galactic plane catalog. According to this catalog, G45.45+0.06 is part of a complex of five radio compact HII regions. Feldt et al. (1998) performed a near-and mid-infrared study of G45.45+0.06 and concluded that this UCHII region, the oldest in the complex, is a young OB cluster around which sequential star formation is taking place. Recently, Blum & McGregor (2008) using NIFS behind ALTAIR on Gemini North identified several massive O-type stars that are ionizing G45.45+0.06. The complexity of this region is evident. Moreover, in the vicinity of this complex (see Figure 1), Cyganowski et al. (2008) discovered an "extended green object" (EGO), a source with extended Spitzer-IRAC 4.5 µm emission which is usually presented in green colour. According to the authors, an EGO is a massive young stellar object (MYSO) driven outflows. The presence of this EGO reveals that star formation is taking place in this region. Indeed, this EGO coincides with the high-mass star forming region G45.47+0.05 studied by Remijan et al. (2004). In this work, we present a multiwavelength study of the spatial environment surrounding the UCHII region G45.45+0.06, with the purpose of exploring the ISM around the complex. We use survey and archival data to show that G45.45+0.06 is located in the environs of a larger HII region whose expansion could have originated the conditions for the formation of the UCHII region G45.45+0.06. Data The data presented and analyzed in this work were extracted from five large-scale surveys: Two Micron All Sky Survey (2MASS) 2 , Galactic Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE), MIPSGAL, MAGPIS and GRS. GLIMPSE is a mid infrared survey of the inner Galaxy performed using the Spitzer Space Telescope. We used the mo-saiced images from GLIMPSE and the GLIMPSE Point-Source Catalog (GPSC) in the 4.5,5.8 and 8 µm). IRAC has an angular resolution between 1. ′′ 5 and 1. ′′ 9 (see Fazio et al. 2004 andWerner et al. 2004). MIPSGAL is a survey of the same region as GLIMPSE, using MIPS instrument (24 and 70 µm) on Spitzer. The MIPSGAL resolution at 24 µm is 6 ′′ . MAGPIS is a radio continuum survey of the Galactic plane at 6 and 20 cm using the VLA in configurations B, C and D combined with the Effelsberg 100 m single-dish telescope (White et al. 2005). The GRS is being performed by the Boston University and the Five College Radio Astronomy Observatory (FCRAO). The survey maps the Galactic Ring in the 13 CO J=1-0 line with an angular and spectral resolution of 46 ′′ and 0.2 km s −1 , respectively (see Jackson et al. 2006). The observations were performed in both position-switching and On-The-Fly mapping modes, achieving an angular sampling of 22 ′′ . We also analyzed the additional data that this survey presents: the CS J=2-1 line, which has similar angular and spectral resolutions as the 13 CO J=1-0 line. Figure 1 shows a Spitzer-IRAC three color image as extracted from GLIMPSE of a region about 7 ′ × 7 ′ in the vicinity of the UCHII complex where G45.45+0.06 lies. The three IR bands are 3.6 µm (in blue), 4.5 µm (in green) and 8 µm (in red). The contours correspond to the 13 CO J=1-0 emission as extracted from the GRS integrated between 50 and 65 km s −1 , a velocity range around the v LSR of G45.45+0.06. The UCHII complex lies on a border of a more extended structure as seen mainly in the Spitzer-IRAC 8 µm band. The morphological correlation between the 8 µm, mainly originated in the policyclic aromatic hydrocarbons (PAHs), and the molecular gas is suggestive of an expansive and "collective" HII region. Results and discussion It is important to note that the PAHs emission delineates the HII region boundaries. This is because these molecules are destroyed inside the ionized region, but are excited in the photodissociation region (PDR) by the radiation leaking from the HII region (Pomarès et al. 2009). Although G45L is not completely bordered by a PDR, its morphology resembles the structure of the IR dust bubbles associated with O and early-B type stars: a PDR visible in the 8 µm band, which encloses ionized gas observed at 20 cm and hot dust observed at 24 µm (see Churchwell et al. 2006Churchwell et al. , 2007Watson et al. 2008). Figure 2 (left) displays a composite two color image, where the red represents the Spitzer-IRAC 8 µm band and the green the above 3σ radio continuum emission at 20 cm extracted from MAGPIS. The high angular resolution of the MAGPIS 20 cm image allows us to distinguish two different radio continuum structures: that associated with the UCHII complex, where both colors are combined and satured (yellow), and other more diffuse towards the West, enclosed by the 8 µm emission that indicates the photodissociation regions (PDRs). Hereafter, this latter structure, an HII region of about 3 ′ in diameter, possible related to the molecular gas shown in Figure 1 as contours, will be called G45L. As indicated in Figure 2 color image where again the red is the Spitzer-IRAC 8 µm emission and the blue represents the MIPSGAL emission at 24 µm. The red structure observed onto the UCHII complex is not real, it is just due to the presence of saturated pixels in the MIPSGAL image. Distance We are studying a region in the first Galactic quadrant, thus we have to take into account the distance ambiguity that exists when using radial velocities and a Galactic rotation curve to assign distances to sources. According to Kolpak et al. (2003) and Anderson et al. (2009), G45.45+0.06 has a radio recombination line velocity of ∼ 56 km s −1 , which gives the possible distances of ∼ 4 or ∼ 8 kpc. As the HI absorption spectrum towards G45.45+0.06 presents an absorption feature at the tangent point velocity (∼ 64 km s −1 ), the authors adopted the farther distance for G45.45+0.06. The molecular gas belonging to the shell associated with G45L (see Figure 1) is at the same velocity range as the gas related to the UCHII complex where G45.45+0.06 is embedded, which may suggest that G45L is at the same distance as the UCHII complex. In order to prove this suggestion, using HI data extracted from the VLA Galactic Plane Survey (VGPS) (Stil et al. 2006), we studied the absorption features towards two regions: the UCHII complex where G45.45+0.06 is embedded (Figure 3-up) and the radio continuum structure enclosed by North and middle PDR (Figure 3-down). As both profiles have the same HI absorption features, we conclude that G45L is at the same distance as G45.45+0.06. Hereafter we adopt the distance of 8 kpc. HI absorption towards the radio continuum structure enclosed by North and middle PDR. Note that both profiles have the same HI absorption features. Molecular Environment As shown in Figure 1, an incomplete 13 CO shell at the velocity range between 50 and 65 km s −1 presents a good morphological correlation with the North and South PDRs visible in the 8 µm band (see Figure 2-left). This suggests that the expansion of the HII region, that we called G45L, is collecting the molecular gas as is observed in other HII regions (e.g. Pomarès et al. 2009;Cichowolski et al. 2009;Deharveng et al. 2008). Figure 4 shows the integrated velocity channel maps of the 13 CO J=1-0 emission every ∼ 1.25 km s −1 . For reference, the circle highlights the position and the size of G45L as suggested by the North and South PDRs. From this figure is clear that the molecular gas is more abundant towards the East. In particular, the v = 58.1 km s −1 channel map shows two molecular clumps spatially coincident with the UCHII complex and the region where the EGO G45.47+0.05 lies, respectively. It is also evident that the molecular gas encompasses the circle, showing the velocity structure of the molecular shell related to G45L. As Deharveng et al. (2005) propose, the presence of a molecular shell surrounding the ionized gas of an HII region, or the presence of massive fragments regularly spaced along the ionization front, suggest that the collect and collapse process is at work in the region. On the other hand, the channels with velocities of 56.8 and 58.1 km s −1 show a smaller molecular shell interior to the circle and in spatial coincidence with the structure delimited by the middle and North PDRs (see Figure 2-left). J=2-1 emission presents a similar structure as the 13 CO. Given that this line requires high densities, 10 4 − 10 5 cm −3 , to be excited (see e.g. Luna et al. 2006), its detection indicates the presence of high density gas in the molecular shell, mainly towards the East. In order to analyze the kinematics of the molecular gas, we study the molecular spectra from three different regions: Region 1 (the North portion of the molecular shell, in coincidence with the North PDR), Region 2 (the South portion of the molecular shell, in coincidence with the South PDR) and Region 3 (where the molecular emission peaks). Besides, taking into account that the 13 CO J=1-0 beam (46 ′′ ) covers completely the middle PDR, we present in Figure The molecular spectra analysis shows that the North PDR has associated two possible molecular structures centered at v ∼ 55 km s −1 and v ∼ 59 km s −1 , respectively. The main velocity component of the molecular gas associated with the middle PDR is centered at v ∼ 55 km s −1 , while the molecular gas related to the South PDR is centered at v ∼ 59 km s −1 . If the molecular gas is indeed associated with G45L, this may suggest that we have to take into account projection effects to understand the three dimensional structure of G45L. This will be discussed in Section 3.5. Using the 13 CO J=1-0 line and assuming local thermodynamic equilibrium (LTE) we estimate the H 2 column density towards the three regions shown in Figure 6. We use: N( 13 CO) = 2.42 × 10 14 T ex τ 13 dv 1 − exp(−5.29/T ex ) to obtain the 13 CO column density. τ 13 is the optical depth of the line and following Anderson et al. (2009) the T ex was assumed to be 13 K. We assume the 13 CO emission is optically thin and use the Gaussian fit line parameters (Table 1) to find the optical depth integral. Finally, we use the relation N(H 2 )/N( 13 CO)∼ 5 × 10 5 (e.g. Simon et al. 2001) to estimate the following values: N(H 2 ) ∼ 1.5 × 10 22 cm −2 for Region 1 and 2, respectively and N(H 2 ) ∼ 2.8 × 10 22 cm −2 for Region 3. In addition we estimate the CS column densities, N(CS), towards these regions. Although the CS J=2-1 line is generally optically thick, a lower limit to the N(CS) can be estimated under the assumption of optically thin CS emission. We use the following equation (Ohashi et al. 1991): N(CS) = 8.5 × 10 11 exp(2.4/T ex ) 1 − exp(−4.7/T ex ) × T mb × ∆v, where T mb is the CS brightness temperature and ∆v is the velocity width of the CS line. Following Goicoechea et al. (2006), who studied the Horsehead PDR using CS J=2-1 between other molecular lines, we assume T ex = 9 K. We obtain N(CS) ∼ (8 and 6) ×10 12 cm −2 for Region 1 and 2, respectively, and N(CS) ∼ 2 × 10 13 cm −2 for Region 3. From the H 2 and CS column densities, we can estimate the CS abundances, X(CS) = N(CS)/N(H 2 ). We obtain X(CS) ∼ (5, 3.5 and 7) ×10 −10 for Region 1, 2 and 3, respectively. Taking into account the LTE approximation and that the estimated CS column densities are lower limits, these values are comparable with those obtained towards the Orion Bar (a warm PDR), X(CS) ∼ 2.9 × 10 −9 (Johnstone et al. 2003) and towards the Horsehead PDR, X(CS) ∼ (7 ± 3) × 10 −9 (Goicoechea et al. 2006). New CS and C 34 S observations in several lines would be very useful to study the sulfur depletion in G45L PDRs. Finally, we estimate the total mass of the whole molecular shell in M ∼ 10 5 M ⊙ . This value was obtained from: M = µ m H D 2 Ω N(H 2 ) , where Ω is the solid angle subtended by the 13 CO J=1-0 beam size, m H is the hydrogen mass, µ, the mean molecular weight, is assumed to be 2.8 by taking into account a relative helium abundance of 25 %, and D is the distance. Summation was performed over all the observed positions within the 27 K km s −1 contour level (see Figures 1 and 6). The exciting star(s) of G45L Given that we do not find any cataloged massive star in the area that may be related to G45L, we have used the GLIMPSE I Spring'07 Catalog to analyze the infrared sources seen in projection onto G45L to identify the star(s) responsible for its creation. Considering only the sources that have been detected in the four Spitzer-IRAC bands, we found 14 sources towards the region located inside the observed PDRs. The main parameters of these sources are shown in Table 2. Figure 8 indicates the location of the sources with respect to the emission distribution at 8 µm (red) and the radio continuum at 20 cm (green). To examine the evolutionary stage of these sources, we analyze their location onto a colorcolor IRAC diagram as shown in Figure 9. Following Allen et al. (2004) color criteria, we found that 8 sources could be classified as main sequence stars (Class III). Among these sources we look for O-type stars as responsible for ionizing the surrounding gas. Using the J, H and K apparent magnitudes as obtained from the 2MASS Point Source Catalog, we estimate the absolute magnitudes for the mentioned 8 sources. To convert the apparent magnitudes in absolute ones, we assume a distance of 8 kpc and a visual absorption between 8 and 12 mag. The extinction values were obtained by inspecting the infrared sources location onto the color-color diagram (H-K) vs (J-H) (not presented here). By comparing the estimated absolute magnitudes with those tabulated by Martins & Plez (2006), we found that sources #1, #9 and #14 are probably O-type stars (likely between O4V and O8V), which is consistent with their position in the color-magnitude diagram K vs (H-K) (not presented here). Source #1 is slightly displaced towards the region of giants stars in the color-magnitude diagram, but it could be a reddened O-type star. An inspection of Figure 8 shows that sources #1 and #9 are located almost onto the central part of G45L, while source #14 lies towards its boundary. This fact suggest that sources #1 and #9 are the most promising candidates for being related to G45L. In particular, source #9 is projected onto a local minimum observed in the brightest part of the G45L radio continuum emission, suggesting that this source could have been blowing its environs creating a small cavity around it. Table 2). In order to investigate if these three sources can provide the energy necessary to ionize the gas, we need to estimate the radio continuum flux of G45L. Using the radio continuum emission data at 1420 MHz, we estimate a flux density of S 1420MHz ∼ 1.0 ± 0.2 Jy for G45L. The number of UV photons necessary to keep the gas ionized is derived using N UV (photons s −1 ) = 0.76 × 10 47 T −0.45 4 ν 0.1 GHz S ν D 2 kpc (Chaisson 1976), where T 4 is the elec- tron temperature in units of 10 4 K, D kpc is the distance in kpc and S ν is the flux density in Jy. Adopting an electron temperature of 10 4 K, we obtain N UV = (5 ± 2) × 10 48 s −1 . On the other hand, the UV photon flux corresponding to the contribution of the three stars is ∼ 3.3 × 10 49 photons per second (Martins et al. 2005). Thus, we conclude that the three stars can maintain the HII region G45L ionized and heat the dust emitting at 24 µm in the ionized gas. It is important to mention that one of the central sources by itself, #1 or #9, can provide the necessary UV photons. On the contrary, source #14 can not have created the HII region G45L alone. Assuming that sources #1, #9, and #14 are the sources responsible for the HII region G45L, we estimate its dynamical age using a simple model described by Dyson & Williams (1980). In this model the radius of the HII region at a given time t is given by R(t) = R s (1 + 7 c s t/ 4 R s ) 4/7 , where c s is the sound velocity in the ionized gas (c s = 15 km s −1 ) and R s is the radius of the Strömgren sphere, given by R s = (3 S * /4 π n 2 o α B ) 1/3 , where α B = 2.6 × 10 −13 cm 3 s −1 is the hydrogen recombination coefficient to all levels above the ground level. S * is the total number of ionizing photons per unit of time emitted by the stars, and n o is the original ambient density. As a rough estimate n o can be obtained by distributing the total molecular mass related to the structure (M ∼ 10 5 M ⊙ ) over a sphere of about 7 pc (3 ′ at 8 kpc) in diameter, which yields n 0 ∼ 10 4 cm −3 . Given that the actual diameter of G45L is about 7 pc, we infer that the HII region has been expanding during about 2 × 10 6 yr. Star formation around G45L To look for primary tracers of star formation activity around G45L, we use the GLIMPSE I Spring'07 Catalog to perform photometry. Considering only sources that have been detected in the four Spitzer-IRAC bands, we found 151 sources in the area delimited by the dashed circle shown in Figure 10. The circular area was chosen in order to cover the molecular gas that surrounds the HII region and where the YSO candidates are expected to be located. This figure displays the spatial distribution of the YSO candidates over the Spitzer-IRAC 8 µm emission. The green crosses indicate Class I sources, the red boxes indicate Class II or intermediate Class I/II sources and the cyan circles indicate the sources that could be reddened Class II objects. The sources classification was performed according the photometric study presented in Figure 11, which shows the IRAC color-color diagram of the sources found. The different regions indicated correspond to different stellar evolutionary stages, as defined by Allen et al. (2004). Seventeen sources lie in the region of the protostars with circumstellar envelopes (Class I, green triangles), only 4 sources lie in the region of young stars with only disk emission (Class II and intermediate Class I/II, red triangles) and 104 sources lie in the region of the main sequence and giant stars (Class III, blue triangles). Sources represented as cyan triangles, located outside the delimited regions, could therefore be reddened Class II objects ). Unfortunately we can not perform an additional NIR photometric study because the 2MASS data from most of these sources are either missing or given as lower limits. In Figure 12 the same sources shown in Figure 10 are displayed over the 13 CO J=1-0 (left) and CS J=2-1 (right) integrated emissions between 54 and 66 km s −1 . Of course we do not know if all the sources seen towards G45L are at the same distance as the HII region. However, the location of some of them, mainly the Class I objects (green crosses in Figure 12), suggest that they are possibly embedded in the molecular gas related to G45L. Figure 10. As in Figure 9, Class I, II and III regions are indicated following Allen et al. (2004). In this case we consider Class I (green), Class II and intermediate Class I/II (red) and reddened Class II (cyan) objects to study star formation around G45L. As mentioned in Section 1, Feldt et al. (1998) and Blum & McGregor (2008) identified several massive O-type stars, that are presumably on, or near, the zero-age main sequence, and are responsible for the ionization of the UCHII region G45.45+0.06. The authors also proposed that this UCHII region triggered the formation of younger UCHII regions in its surroundings, generating an UCHII complex. These studies do not mention any agent responsible for the formation of such stars. Taking into account the estimated age of G45L (about 2 × 10 6 yr), that star formation is taking place around it, probably through the collect and collapse process, and that the UCHII complex lies on one of its border, we propose that G45L could have triggered the formation of the zero-age main sequence stars that are ionizing G45.45+0.06. However we can not discard the possibility that both HII regions, G45.45+0.06 and G45L, could be coeval. HII region G45L spatial structure In this section, based on the IR emission and the molecular environment study, we attempt to describe the possible spatial structure of G45L. As described in previous sections, a view of a larger area in the vicinity of the UCHII region G45.45+0.06 reveals that it actually lies in a border of a larger and fainter HII region that we called G45L. The morphological study of G45L suggests that this HII region is far from being spherical. The observed morphology is quite complex and then projection effects should be considered to analyze the three dimensional structure. In this way, taking into account the location of the PDRs together with the spatial and velocity molecular gas distribution, we suggest that G45L has a pear-like morphology, as it is sketched in Figure 13. This figure presents two sketches of G45L possible structure as seen from two different positions. In Figure 13 (left) can be appreciated that the North PDR is composed by two structures seen superimposed along the line of sight and the middle PDR is the projection of the southern border of the HII region portion closer to us. The UCHII complex that lies towards the East of G45L is represented as a red cloud. Figure 13 (right) shows a longitudinal cut of G45L remarking the molecular gas components related to the PDRs. As the molecular analysis shows, the structure delimited by the North and the middle PDRs has associated molecular gas centered at v ∼ 55 km s −1 , while the structure delimited by the South and the North PDRs has associated molecular gas centered at v ∼ 59 km s −1 . Most ionized nebulae, in particular HII regions, have complex morphologies structures (Morisset et al. 2005). There are many physical causes that may account for such morphologies. Some causes for the G45L spatial structure could be: the presence of density gradients in the ISM where the HII region is evolving, the existence of more than one exciting star, the fact that these exciting stars could have high spatial velocities with respect to the local ISM, and the effects of possible stellar and/or interstellar magnetic fields. Additionally we found several YSO candidates lying preferently in the molecular shell and concentrating towards the East where the CS J=2-1 emission peaks. Our results confirm that the southeastern region of the UCHII complex where G45.45+0.06 is embedded (eastern part of G45L) is an active star formation region. (g) Assuming that three O-type stars are responsible of G45L, we suggest this HII region has been expanding during about 2 × 10 6 yr and could have triggered the formation of the zero-age main sequence stars that are ionizing the UCHII region G45.45+0.06. However we can not discard that both HII regions are coeval. Figure 2 Fig. 1 . 21(left), the brightest part of the radio continuum diffuse structure located at l ∼ 45. • 42, b ∼ 0. • 072 is bordered by two PDRs: the North and the middle PDRs. Towards the South, the HII region is fainter and bordered by what we called South PDR. Spitzer-IRAC three color image (3.5 µm = blue, 4.5 µm = green and 8 µm = red). The contours correspond to the 13 CO J=1-0 emission as extracted from the GRS integrated between 50 and 65 km s −1 , the levels are 27, 37 and 48 K km s −1 . The UCHII complex is remarked with a circle. Also, the EGO discovered byCyganowski et al. (2008) is indicated. The angular resolutions are ∼ 1. ′′ 5 and ∼ 46 ′′ for the Spitzer and molecular data, respectively. Fig. 2 . 2Two color images. Left: the red is the Spitzer-IRAC 8 µm emission and the green is the above 3σ radio continuum emission at 20 cm extracted from MAGPIS. Yellow is the superposition of the IR and radio continuum emissions. Right: the red is the same as the left image and the blue represents the MIPSGAL emission at 24 µm. The red structure onto the UCHII complex is not real, it is just due to the presence of saturated pixels of the MIPSGAL emission. Fig. 3 . 3Up: HI absorption towards the UCHII complex where G45.45+0.06 is embedded. Down: Fig. 4 . 4Integrated velocity channel maps of the 13 CO J=1-0 emission every ∼ 1.25 km s −1 . The grayscale is displayed at the top of the figure and is in K km s −1 , the contour levels are 5, 9.5 and 11 K km s −1 . Note that the molecular gas encompasses the circle, which highlights the position and size of G45L as suggested by the North and South PDRs (seeFigure 2-left). Figure 5 5displays the CS J=2-1 emission integrated between 54 and 66 km s −1 . The CS Fig. 5 . 5CS J=2-1 emission integrated between 54 and 66 km s −1 . The contour levels are 2.4, 3.6, 5 and 8 K km s −1 . Note that the CS emission presents a similar structure as the 13 CO. Figure 6 (Fig. 6 . 66left) shows the Spitzer-IRAC 8 µm emission with contours of the integrated 13 CO J=1-0 as presented inFigure 1. Regions 1, 2 and 3 are indicated with boxes of approximately 2 ′ × 1. ′ 8, 2. ′ 2 × 1. ′ 2 and 1. ′ 5 × 2. ′ 5 in size, respectively. Towards the right, the 13 CO J=1-0 and CS J=2-1 average spectra corresponding to the emission of each region are displayed. Between 40 and 50 spectra were averaged to obtain each average emission spectrum. The parameters determined from Gaussian fitting of these lines are presented inTable 1. T mb represents the peak brightness temperature, V lsr the central velocity, ∆v the line width and I the integrated line intensity. Errors are formal 1σ value for the model of the Gaussian line shape. The 13 CO and CS average lines corresponding to Region 1 were best fitted with two Gaussians. Left: Spitzer-IRAC 8 µm emission with contours of the integrated 13 CO J=1-0 as presented inFigure 1. The boxes represent the regions from which were obtained the average of the molecular emissions. Towards the right the 13 CO J=1-0 and CS J=2-1 average spectra from each region are shown. The rms noise are ∼ 0.03 and ∼ 0.04 K for the 13 CO and CS, respectively. 7 a spectrum obtained from l =45. • 430, b =0. • 055. This spectrum shows the molecular emission probably related to the middle PDR. The velocity of the main component of this profile is v ∼ 56 km s −1 , in coincidence with one of the two molecular components of the North PDR (Region 1 inFigure 6). The spectrum is not symmetrical: it presents another weaker component or a wing towards larger velocities. Fig. 7 . 713 CO J=1-0 spectrum obtained towards the middle PDR, at l =45. • 430, b =0. • 055. The main component velocity is v ∼ 56 km s −1 , which coincides with one of the two molecular components of the North PDR. Fig. 8 . 8Colour composite image, with the radio continuum emission at 1420 MHz in green, and the infrared emission at 8 µm in red. Yellow is the superposition of the IR and radio continuum emissions. The exciting stars candidates are identified (see Fig. 9 . 9GLIMPSE-IRAC color-color diagram [3.6] -[4.5] versus [5.8] -[8.0] for the sources displayed in Figure 8. Class I, II and III regions are indicated following Allen et al. (2004). The ellipse (Class III region: main sequence and giant stars) encloses the sources that we consider to be possible exciting star(s) of G45L. Table 1 . 1Observed parameters of the 13 CO J=1-0 and CS J=2-1 average emissions towards theregions shown in Figure 6. Emission T mb V lsr ∆v I (K) (km s −1 ) (km s −1 ) (K km s −1 ) Region 1 13 CO J=1-0 3.80 ±0.50 58.75 ±1.05 5.05 ±0.25 17.00 ±3.00 4.10 ±0.50 55.25 ±1.05 3.00 ±0.15 15.00 ±2.00 CS J=2-1 0.46 ±0.11 59.60 ±1.10 4.00 ±0.20 2.00 ±0.30 0.42 ±0.15 55.80 ±1.10 3.45 ±0.15 1.90 ±0.30 Region 2 13 CO J=1-0 4.10 ±0.50 59.50 ±0.50 6.75 ±0.85 30.00 ±3.80 CS J=2-1 0.35 ±0.05 59.90 ±0.50 6.25 ±0.95 2.30 ±0.50 Region 3 13 CO J=1-0 6.40 ±0.50 58.65 ±0.25 6.10 ±0.50 41.50 ±3.50 CS J=2-1 1.00 ±0.10 59.25 ±0.55 7.00 ±1.00 7.70 ±1.50 Table 2 . 2Main parameters of the infrared sources found towards G45L.# Glimpse designation Galactic coordinates J H K S 3.6 µm 4.5 µm 5.8 µm 8.0 µm Notes 1 G045.4260+00.0590 45. • 426, 0. • 058 13.93 11.89 11.08 10.47 10.55 10.21 10.18 Class III -(O-type) 2 G045.4061+00.0506 45. • 406, 0. • 050 13.13 11.67 11.08 10.71 10.79 10.58 11.32 3 G045.4255+00.0282 45. • 425, 0. • 028 null null null 13.7 12.3 11.09 10.05 4 G045.4135+00.0528 45. • 413, 0. • 052 11.51 10.45 10.09 9.86 9.8 9.69 9.68 Class III 5 G045.4250+00.0668 45. • 425, 0. • 066 null null 14.43 11.33 10.31 9.77 9.81 6 G045.4420+00.0415 45. • 442, 0. • 041 null null null 12.07 12.03 11.64 11.06 7 G045.4437+00.0458 45. • 443, 0. • 045 null null 12.67 10.68 10.36 9.81 9.84 8 G045.4048+00.0438 45. • 404, 0. • 043 null 13.65 12.17 11.04 10.99 10.58 10.6 Class III 9 G045.4274+00.0672 45. • 427, 0. • 067 13.11 11.64 11.07 10.53 10.49 9.94 9.63 Class III -(O-type) 10 G045.4347+00.0627 45. • 434, 0. • 062 null null null 12.77 12.54 9.40 7.81 11 G045.3969+00.0433 45. • 396, 0. • 043 11.79 10.41 9.85 9.55 9.6 9.41 9.39 Class III 12 G045.4024+00.0412 45. • 402, 0. • 041 10.41 9.21 8.73 8.39 8.51 8.34 8.22 Class III 13 G045.3976+00.0473 45. • 397, 0. • 047 null 13.67 12.42 11.5 11.33 10.82 10.89 Class III 14 G045.4020+00.0361 45. • 402, 0. • 036 13.42 12.28 11.88 11.52 11.55 11.38 11.73 Class III -(O-type) The Class I objects lie preferently in the molecular shell and concentrate towards the East, where the CS J=2-1 emission peaks. They lie in the region where the EGO G45.47+0.05 is embedded(Cyganowski et al. 2008), confirming that in this area, southeastern the UCHII complex and eastern G45L, star formation is taking place.Fig. 10. Spitzer-IRAC 8 µm emission with YSO candidates superimposed. Green crosses indicate Class I sources, red boxes are Class II or intermediate Class I/II sources and the cyan circles are the sources that could be reddened Class II objects. The circle encloses the region where the photometric study was performed.Fig. 11. GLIMPSE-IRAC color-color diagram [3.6] -[4.5] versus [5.8] -[8.0] for the sources observed inside the dashed circle displayed in45.480 45.440 45.400 0.120 0.100 0.080 0.060 0.040 0.020 0.000 Galactic longitude Galactic latitude -0.5 0 0.5 1 1.5 [5.8]-[8.0] 0 1 2 [3.6]-[4.5] Class I Class II Class III Fig. 12. Left: 13 CO J=1-0 integrated emission between 54 and 66 km s −1 with YSO candidates superimposed. The contour levels are 27, 37 and 48 K km s −1 . Right: same, but grays are the CS J=2-1 integrated emission in the same velocity range. The contour levels are 2.4, 3.6, 5 and 8 K km s −1 . Green crosses indicate Class I sources, red boxes are Class II or intermediate Class I/II sources and the cyan circles are the sources that could be reddened Class II objects.45.480 45.440 45.400 0.120 0.100 0.080 0.060 0.040 0.020 0.000 Galactic longitude Galactic latitude CS 2-1 45.480 45.440 45.400 0.120 0.100 0.080 0.060 0.040 0.020 0.000 Galactic longitude Galactic latitude 13CO 1-0 Fig. 13. Left: sketch of the possible shape of the HII region G45L. The PDRs are remarked. The UCHII complex is represented as a red cloud. Right: a longitudinal cut of G45L remarking the molecular gas components related to the PDRs.North PDR South PDR "projection" of Middle PDR To Earth UCHII HII Region CO associated with North PDR at v ~ 55 km/s CO associated with North PDR at v ~ 59 km/s CO associated with Middle PDR at v ~ 55 km/s CO associated with South PDR at v ~ 59 km/s To Earth . SummaryUsing multiwavelength survey and archival data, we studied the ISM towards a region about 7 ′ × 7 ′ in the vicinity of the G45.45+0.06 UCHII complex. The main results can be summarized as follows:(a) We found that the UCHII complex lies in a border of a larger (∼ 3 ′ of diameter) and fainter HII region, here named G45L.(b) Although G45L is not completely border by a PDR, its morphology resembles the structure of the IR dust bubbles associated with O and early-B type stars: a PDR visible in the 8 µm band, which encloses ionized gas observed at 20 cm and hot dust observed at 24 µm.(c) We find a good morphological correlation between the PDRs and the molecular gas, which suggests that the HII region may be collecting the molecular material.(d) Taking into account the velocity (v LSR ∼ 55 − 60 km s −1 ) of the molecular gas related to G45L and the HI absorption study, we conclude that this HII region is at the same distance as the UCHII complex, ∼ 8 kpc.(e) The PDRs position and the spatial and velocity distribution of the associated molecular shell suggest that G45L has a pear-like morphology.(f) From a near-and mid-IR photometric study, we found three sources, likely O-type stars (between O4V and O8V) that are possibly responsible for the creation of G45L. 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[ "Learning and Free Energies for Vector Approximate Message Passing", "Learning and Free Energies for Vector Approximate Message Passing" ]	[ "Alyson K Fletcher ", "Philip Schniter " ]	[]	[]	Vector approximate message passing (VAMP) is a computationally simple approach to the recovery of a signal x from noisy linear measurements y = Ax + w. Like the AMP proposed by Donoho, Maleki, and Montanari in 2009, VAMP is characterized by a rigorous state evolution (SE) that holds under certain large random matrices and that matches the replica prediction of optimality. But while AMP's SE holds only for large i.i.d. sub-Gaussian A, VAMP's SE holds under the much larger class: right-rotationally invariant A. To run VAMP, however, one must specify the statistical parameters of the signal and noise. This work combines VAMP with Expectation-Maximization to yield an algorithm, EM-VAMP, that can jointly recover x while learning those statistical parameters. The fixed points of the proposed EM-VAMP algorithm are shown to be stationary points of a certain constrained free-energy, providing a variational interpretation of the algorithm. Numerical simulations show that EM-VAMP is robust to highly ill-conditioned A with performance nearly matching oracle-parameter VAMP.	10.1109/icassp.2017.7952957	[ "https://arxiv.org/pdf/1602.08207v3.pdf" ]	3,798,279	1602.08207	182fdf6236612d3082e3ecfaf42bfc9d2542b41c	Learning and Free Energies for Vector Approximate Message Passing Alyson K Fletcher Philip Schniter Learning and Free Energies for Vector Approximate Message Passing arXiv:1602.08207v3 [cs.IT] 9 Jan 2017 1 Vector approximate message passing (VAMP) is a computationally simple approach to the recovery of a signal x from noisy linear measurements y = Ax + w. Like the AMP proposed by Donoho, Maleki, and Montanari in 2009, VAMP is characterized by a rigorous state evolution (SE) that holds under certain large random matrices and that matches the replica prediction of optimality. But while AMP's SE holds only for large i.i.d. sub-Gaussian A, VAMP's SE holds under the much larger class: right-rotationally invariant A. To run VAMP, however, one must specify the statistical parameters of the signal and noise. This work combines VAMP with Expectation-Maximization to yield an algorithm, EM-VAMP, that can jointly recover x while learning those statistical parameters. The fixed points of the proposed EM-VAMP algorithm are shown to be stationary points of a certain constrained free-energy, providing a variational interpretation of the algorithm. Numerical simulations show that EM-VAMP is robust to highly ill-conditioned A with performance nearly matching oracle-parameter VAMP. I. INTRODUCTION Consider the problem of estimating a random vector x from linear measurements y of the form y = Ax + w, w ∼ N (0, θ −1 2 I), x ∼ p(x\|θ 1 ), where A ∈ R M×N is a known matrix, p(x\|θ 1 ) is a density on x with parameters θ 1 , w is additive white Gaussian noise (AWGN) independent of x, and θ 2 > 0 is the noise precision (inverse variance). The goal is to estimate x along while simultaneously learning the unknown parameters θ := (θ 1 , θ 2 ) from the data y and A. This problem arises in Bayesian forms of linear inverse problems in signal processing, as well as in linear regression in statistics. Even when the parameters θ are known, exact estimation or inference of the vector x is intractable for general priors p(x\|θ 1 ). The approximate message passing (AMP) algorithm [1] and its generalization [2] are powerful, relatively recent, algorithms that iteratively attempt to recover x. These methods are computationally fast and have been successfully applied to a wide range of problems, e.g., [3]- [11]. Most importantly, for large, i.i.d., sub-Gaussian random matrices A, their performance can be exactly predicted by a scalar state evolution (SE) [12], [13] that provides testable conditions for optimality, even for non-convex priors. When the parameters θ in the model are unknown, AMP can be combined with expectation A maximization (EM) methods [14]- [16] for joint estimation and learning. As it turns out, the AMP methods [1], [2] are fragile with regard to the choice of the matrix A, and can perform poorly outside the special case of zero-mean, i.i.d., sub-Gaussian A. For example, AMP diverges with even mildly non-zero-mean and/or mildly ill-conditioned A [17]. Several techniques have been proposed to improve the robustness of AMP including damping [17], [18], mean-removal [17], and sequential updating [19], but these remedies have limited effect. Recently, the Vector AMP (VAMP) algorithm [20] was established as an alternative to AMP that is much more robust to the choice of matrix A. In particular, VAMP has a rigorous SE that holds under large right-rotationally invariant A, i.e., A whose right singular-vector matrix is uniformly distributed on the group of orthogonal matrices. VAMP can be derived in several ways, such as through expectation propagation (EP) [21] approximations of belief propagation [20] or through expectation consistent (EC) approximation [22]- [24]. But the existence of a rigorous state evolution establishes it firmly in the class of AMP algorithms. However, a shortcoming of the VAMP method [20] is that it requires that the parameters θ in the model (1) are known. In this paper, we extend the VAMP method to enable learning of the parameters θ via Expectation-Maximization (EM) [25], [26]. We call the proposed method EM-VAMP. As described below, exact implementation of EM requires estimating the posterior density p(x\|y, θ) for each parameter estimate θ. This is computationally not possible for the model (1). EM-VAMP is instead derived using a technique from Heskes [27] for combining EM with approximate inference of the posterior. Specifically, it is well-known that EM can be interpreted as a method to minimize a certain energy function. Here, we construct an approximation of the EM cost function that we call the EM-VAMP energy function and derive an algorithm to minimize this function. Our main theoretical result shows that the fixed points of the EM-VAMP method are local minima of the EM-VAMP energy function and thus provide estimates of the parameters θ and posterior density with a precise variational interpretation. By including the parameter learning, this result generalizes the fixed-point energy-function interpretation of EC given in [28], [29] and its variants [24]. Unfortunately, our results do not guarantee the convergence of the method to the fixed point. However, in numerical experiments on sparse regression problems, we show that the proposed method exhibits extremely stable convergence over a large class of matrices that cause AMP to diverge. Moreover, the performance of EM-VAMP is almost identical to that of VAMP with known parameters. In particular, the method is able to obtain close to the theoretically optimal performance predicted by the replica method [30]. II. EM-VAMP A. Review of VAMP To describe the VAMP method in [20], we need to introduce some additional notation. First suppose that we can write the prior on x as p(x\|θ 1 ) = 1 Z 1 (θ 1 ) exp [−f 1 (x\|θ 1 )] ,(2) where f 1 (·) is some penalty function and Z 1 (θ 1 ) is a normalization constant. We assume that f 1 (·) is separable, meaning that f 1 (x\|θ 1 ) = N n=1 f 1n (x n \|θ 1 ),(3) for scalar functions f 1n . This corresponds to the case that, conditional on θ 1 , x has independent components. Also, we write the likelihood for the Gaussian model (1) as p(y\|x, θ 2 ) := 1 Z 2 (θ 2 ) exp [−f 2 (x, y\|θ 2 )] (4) f 2 (x, y\|θ 2 ) := θ 2 2 y − Ax 2 , Z 2 (θ 2 ) = 2π θ 2 N/2 .(5) The joint density of x, y given parameters θ = (θ 1 , θ 2 ) is then p(x, y\|θ) = p(x\|θ 1 )p(y\|x, θ 2 ).(6) The VAMP algorithm [20] considers the case where the parameters θ are known. In this case, VAMP attempts to compute belief estimates of the posterior density p(x\|y, θ) of the form (for i = 1, 2) b i (x\|r i , γ i , θ i ) ∝ exp −f i (x, y\|θ i ) − γ i 2 x − r i 2 ,(7) where the parameters r i , γ i are optimized by the algorithm. To keep the notation symmetric, we have written f 1 (x, y\|θ 1 ) for f 1 (x\|θ 1 ) even though the first penalty function does not depend on y. The steps of VAMP are identical to those shown for proposed EM-VAMP in Algorithm 1, except that VAMP skips the parameter updates in lines 4 and 11. Instead, VAMP fixes θ ik for all iterations k. In Algorithm 1, we have focused on the MMSE version of VAMP since we are interested in approximate inference. There we use E [φ(x)\|r i , γ i , θ i ] := φ(x)b i (x\|r i , γ i , θ i ) dx to denote the expectation with respect to the belief estimate b i (·) in (7). Similarly, Cov(·\|·) is the covariance matrix with respect to the belief estimate and tr Cov(·\|·) is its trace. Hence, the VAMP method reduces the inference problem on the joint density (6) to computing expectations and variances with respect to the belief estimates (7). One of the main motivations of the VAMP method is that, for the penalty functions (3) and (5) considered here, the Algorithm 1 EM-VAMP Require: Matrix A ∈ R M×N , penalty functions f i (x, y\|θ i ), measurement vector y, and number of iterations N it . 1: Select initial r 10 , γ 10 ≥ 0, θ 1,−1 , θ 2,−1 . 2: for k = 0, 1, . . . , N it − 1 do 3: // Input Denoising 4: θ 1k = arg max θ1 E ln p(x\|θ 1 ) r 1k , γ 1k , θ 1,k−1 5: η −1 1k = (1/N ) tr Cov x r 1k , γ 1k , θ 1k 6: x 1k = E x r 1k , γ 1k , θ 1k 7: γ 2k = η 1k − γ 1k 8: r 2k = (η 1k x 1k − γ 1k r 1k )/γθ 2k = arg max θ2 E ln p(x, y\|θ 2 ) r 2k , γ 2k , θ 2,k−1 12: η −1 2k = (1/N ) tr Cov x r 2k , γ 2k , θ 2k 13: x 2k = E x r 2k , γ 2k , θ 2k 14: γ 1,k+1 = η 2k − γ 2k 15: (3), the belief estimate b 1 (·) separates as r 1,k+1 = (η 2k x 2k − γ 2k r 2k )/γ 1,b 1 (x\|r 1 , γ 1 , θ 1 ) ∝ N n=1 exp −f 1 (x n \|θ 1 ) − γ 1 2 (x n − r 1n ) 2 . Thus, the expectation and variance computations in lines 5 and 6 decouple into N scalar computations. Furthermore, for the quadratic penalty (5), the belief estimate b 2 (·) is Gaussian, i.e., b 2 (x\|r 2 , γ 2 , θ 2 ) ∝ exp − θ 2 2 y − Ax 2 − γ 2 2 r − x 2 , with mean and covariance given by E [x\|r 2 , γ 2 , θ 2 ] = Q −1 θ 2 A T y + γ 2 r (8) Cov [x\|r 2 , γ 2 , θ 2 ] = Q −1 (9) Q = θ 2 A T A + γ 2 I.(10) Although (8)-(9) may suggest that VAMP requires an N × N matrix inverse at each iteration, it is shown in [20] that two M ×N matrix-vector multiplications per iteration are sufficient if the SVD of A is precomputed before initialization. Thus, VAMP reduces the intractable posterior inference problem to an iteration of N scalar estimation problems and 2 matrixvector multiplies per iteration, just like AMP. B. Learning the parameters θ To learn the parameters θ, the EM-VAMP methods adds two steps, lines 4 and 11, to update θ ik . These maximizations are similar to those in the EM method, and we formalize this connection in the next section. The updates may be performed once per VAMP iteration, as written, or several times per VAMP iteration, since in practice this seems to speed convergence of EM-VAMP. For now, observe that due to the structure of the prior in (2) and the likelihood in (4), we have that θ i,k+1 = arg min θi E f i (x, y\|θ i ) r ik , γ ik , θ ik + ln Z i (θ i ) .(11) This minimization is often tractable. For example, when the penalty function corresponds to an exponential family (i.e., y)), the minimization in (11) is convex. In particular, for the quadratic loss (5), the minimization is given by f i (x, y\|θ i ) = θ T i φ i (x, y) for sufficient statistic φ i (x,θ −1 2,k+1 = 1 N E y − Ax 2 \|r 2k , γ 2k , θ 2k = 1 N y − Ar 2k 2 + tr(AQ −1 k A T ) ,(12) where Q k = θ 2k A T A + γ 2k I. As mentioned earlier, it is possible to reduce the complexity of evaluating (12) by precomputing the SVD of A [20], since tr(AQ −1 k A T ) = R i=1 s 2 i /(θ 2k s 2 i + γ 2k ) where {s i } R i=1 are the non-zero singular values of A. In this case, the update of θ 2 is very simple, computationally. III. FIXED POINTS OF EM-VAMP We will now show that the parameter updates in EM-VAMP can be understood as an approximation of the EM algorithm. We first briefly review the standard energy-function interpretation of EM [26]. Consider the problem of finding the maximum likelihood (ML) estimate of the parameter θ: θ = arg max θ p(y\|θ) = arg max θ p(x, y\|θ) dx.(13) Due to the integration, this minimization is generally intractable. EM thus considers an auxiliary function, Q(θ, b) = − ln p(y\|θ) + D(b p(·\|y, θ)),(14) defined for an arbitrary density b(x). In (14), D(b p(·\|y, θ)) is the KL divergence between b(x) and the posterior density p(x\|y, θ). Note that, for any parameter estimate θ, min b Q(θ, b) = − ln p(y\|θ), where the minimum occurs at the posterior b(x) = p(x\|y, θ). Hence, the MLE (13) can, in principle, be found from the joint minimization θ = arg min θ min b Q(θ, b).(15) This fact leads to a natural alternating minimization, E-step: b k = arg min b Q( θ k , b) = p(x\|y, θ k )(16) M-step: θ k+1 = arg min θ Q(θ, b k ).(17) This recursion is precisely the EM algorithm, written in a slightly non-standard form. Specifically, (16) is the E-step, which computes the posterior density of x given y and the current parameter estimate θ k . A simple manipulation shows that Q(θ, b) = −E [ln p(x, y\|θ)\|b] − H(b),(18) where the expectation is with respect to the density b(x) and H(b) is the differential entropy of b. Equation (18) shows that the minimization in (17) can equivalently be written as θ k+1 = arg max θ E ln p(x, y\|θ) b k ,(19) which is a familiar expression for the M-step. Unfortunately, the computation of the posterior density required by the E-step (16) is generally intractable for joint density (6) considered here. We thus consider an alternate energy function, similar to that used by Heskes in [27] for understanding EM combined with belief propagation-based inference. First observe that, using (18) and (6), we can write the auxiliary function as Q(θ, b) = 2 i=1 {E [f i (x, y\|θ i )\|b] + ln Z i (θ i )} − H(b) = 2 i=1 D i (b, θ i ) + H(b),(20) where D i (b, θ i ) is the KL divergence, D i (b, θ i ) = D b Z i (θ i ) −1 e −fi(·,y\|θi) .(21) Now, given densities b 1 , b 2 and q, we define the energy function J(b 1 , b 2 , q, θ) := D 1 (b 1 , θ 1 ) + D 2 (b 2 , θ 2 ) + H(q),(22) which matches the original auxiliary function Q(θ, b) under the matching condition b = b 1 = b 2 = q. Hence, we can rewrite the joint minimization (15) as θ = arg min θ min b1,b2 max q J(b 1 , b 2 , q, θ) s.t. b 1 = b 2 = q. (23) We call (22) the EM-VAMP energy function. Now, as mentioned in the Introduction, VAMP-like many algorithms-can be viewed as an example of expectation consistent (EC) approximate inference [22]- [24]. Specifically, following the EC framework, we relax the above GFE optimization by replacing the constraints in (23) with so-called moment matching constraints: E(x n \|b 1 ) = E(x n \|b 2 ) = E(x n \|q), ∀n, E( x 2 \|b 1 ) = E( x 2 \|b 2 ) = E( x 2 \|q).(24) Thus, instead of requiring a perfect match in the densities b 1 , b 2 , q as in (23), we require only a match in their first moments and average second moments. Using the above approximation, we can then attempt to compute parameter estimates via the minimization (25) as L(b 1 , b 2 , q, θ, β, γ) := J(b 1 , b 2 , q, θ)− 2 i=1 β T i [E(x\|b i )−E(x\|q)] + 2 i=1 γ i 2 E( x 2 \|b i )−E( x 2 \|q) ,(26) where β = (β 1 , β 2 ) and γ = (γ 1 , γ 2 ) represent sets of dual parameters for the first-and second-order constraints. We then have the following. Theorem 1. At any fixed point of the EM-VAMP algorithm with γ 1 + γ 2 > 0, we have η 1 = η 2 = η := γ 1 + γ 2 , (27a) x 1 = x 2 = x := (γ 1 r 1 + γ 2 r 2 ) /(γ 1 + γ 2 ). (27b) Also, let β i := γ i r i , let b i be the density b i (x) := b i (x\|r i , γ i , θ i ),(28) where b i (·) is given in (7) and let q(x) be the Gaussian density q(x) ∝ exp − η 2 x − x 2 .(29) Then, b i , θ, and q are critical points of the Lagrangian (26) that satisfy the moment matching constraints (24). The proof is given in Appendix A and is an adaptation of a similar result in [24] with the addition of the parameters θ. The consequence of this result is that, if the algorithm converges, then its limit points are local minima of the EM-VAMP energy minimization. IV. NUMERICAL EXPERIMENTS While the above analysis characterizes the fixed points of EM-VAMP, it does not provide any guarantees on the convergence of the algorithm to the fixed points. To study the convergence and evaluate the algorithm's performance, we conducted a numerical experiment. We considered sparse linear regression, where the goal is to recover the signal x from measurements y from (1) without knowing the signal parameters θ 1 or the noise precision θ 2 > 0. For our experiment, we drew x from an i.i.d. Bernoulli-Gaussian (i.e., spike and slab) prior, p(x n \|θ 1 ) = (1 − β x )δ(x n ) + β x N (x n ; µ x , τ x ),(30) where parameters θ 1 = {β x , µ x , τ x } represent the sparsity rate β x ∈ (0, 1], the active mean µ x ∈ R, and the active variance τ x > 0. Following [17], we constructed A ∈ R M×N from the singular value decomposition (SVD) A = USV T , whose orthogonal matrices U and V were drawn uniformly with respect to the Haar measure and whose singular values s i were constructed as a geometric series, i.e., s i /s i−1 = α ∀i > 1, with α and s 1 chosen to achieve a desired condition number s 1 /s min(M,N ) as well as A 2 F = N . It is shown in [17], [18] that standard AMP (and even damped AMP) diverges when the matrix A has a sufficiently high condition number. Thus, this matrix-generation model provides an excellent test for the stability of AMP methods. Recovery performance was assessed using normalized mean-squared error (NMSE) x − x 2 / x 2 averaged over 100 independent draws of A, x, and w. Figure 1 shows NMSE versus condition number for sparse linear regression under M = 512, N = 1024, β x = 0.1, µ x = 0, and (τ x , θ 2 ) giving a signal-to-noise ratio of 40 dB. EM-VAMP was initialized with β x = (M/2)/N , τ x = y 2 / A 2 F β x , µ x = 0, and θ −1 2 = M −1 y 2 . It is compared with (i) VAMP under perfect knowledge of θ = {τ w , β x , µ x , τ x }; (ii) the EM-AMP algorithm from [15] with damping from [17]; and (iii) the replica prediction for Bayes minimum MSE from [31]. It was recently shown [32], [33] that the replica method gives the correct prediction in sparse linear regression when A is i.i.d. Gaussian. Figure 1 shows that the NMSE of EM-VAMP is nearly indistinguishable from that of VAMP and much more robust than EM-AMP to illconditioning in A. Figure 2(a) shows EM-VAMP and VAMP converging in ∼ 10 iterations (whereas EM-AMP requires > 100 iterations) at condition number 32, and Figure 2(b) shows EM-VAMP converging in ∼ 20 iterations at condition number 3162. These plots suggest that the convergence rate of EM-VAMP is i) nearly identical to that of genie-aided VAMP and ii) relatively insensitive to the condition number of A. We note that, in generating the above figures, we used multiple updates of the noise precision θ 2 per VAMP iteration. In particular, (12) was iterated to convergence. A Matlab implementation of our EM-VAMP method can be found in the GAMPmatlab software package at http://sourceforge.net/projects/gampmatlab/. V. CONCLUSIONS AND FUTURE WORK We presented an approach for recovering the signal x from AWGN-corrupted linear measurements y = Ax + w by posing recovery in the MMSE framework while simultaneously learning the parameters θ governing the signal prior p(x\|θ) and the AWGN variance. The proposed method combines EM and VAMP algorithms for approximate inference of the posterior. We showed that, if the algorithm converges, then its fixed points coincide with stationary points of a certain energy function. Simulations show the proposed method exhibits robustness to the condition number of A and MMSE closely matching that of the replica prediction under known θ. While the algorithm has great potential, one outstanding issue is that its convergence has not been established. One possible solution is to extend the convergence proofs in [24] or the state evolution analysis of VAMP [20]. Another avenue for future work is the application of EM-VAMP to sparse Bayesian learning (SBL) [34]. SBL tackles sparse linear regression using a Gaussian-scale-mixture prior p(x\|θ 1 ) = N (x; 0, Diag(θ 1 )) with a deterministic unknown variance vector θ 1 ∈ R N + learned by the EM algorithm. While the standard SBL implementation uses an N × N matrix inverse at each EM iteration, the EM-VAMP implementation of SBL could avoid matrix inversions by precomputing an SVD. (36) where step (a) uses the facts that γ 1 + γ 2 = η and β 1 + β 2 = γ 1 r 1 + γ 2 r 2 = η x, step (b) follows by completing the square, and step (c) uses the density in (29). Hence, the maximizer of (33) is given by (29). Also, from the updates of x i and η i in Algorithm 1, we have x = E(x\|b i ), η −1 = 1 N tr Cov(x\|b i ). Since q is Gaussian, its mean and average covariance are E(x\|q) = x, 1 N tr Cov(x\|q) = η −1 . This proves that the densities satisfy the moment matching constraints (24). J (b 1 , b 2 , q, θ) s.t. (24) are satisfied.(25) Our main result shows that the fixed points of EM-VAMP are stationary points of the optimization (25). To state the result, we write the Lagrangian of the constrained optimization Fig. 1 . 1For sparse linear regression, recovery NMSE versus condition number of A. Also shown is the replica prediction of the MMSE. Fig. 2 . 2For sparse linear regression, recovery NMSE versus iteration for condition number 32 in (a) and condition number 3162 in (b). . K. Fletcher (email: [email protected]) is with the Departments of Statistics, Mathematics, and Electrical Engineering, University of California, Los Angeles, CA, 90095. The work of A. K. Fletcher was supported by the NSF under grant CCF-1254204. P. Schniter (email: [email protected]) is with the Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH, 43210. The work of P. 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[ "Trees with unique minimum glolal offensive alliance sets", "Trees with unique minimum glolal offensive alliance sets" ]	[ "Mohamed Bouzefrane [email protected] ", "Isma Bouchemakh s:[email protected] ", "Mohamed Zamime [email protected] ", "Noureddine Ikhlef-Eschouf ", "\nFaculty of Technology\nFaculty of Mathematics\nFaculty of Technology\nLaboratory L'IFORCE, University of Sciences and Technology Houari Boumediene (USTHB)\nUniversity of Médéa\nB.P. 32 El-Alia16111Bab-Ezzouar, AlgiersAlgeria, Algeria\n", "\nFaculty of Sciences\nDepartment of Mathematics and Computer Science\nUniversity of Médéa\nAlgeria\n", "\nUniversity of Médéa\nAlgeria\n" ]	[ "Faculty of Technology\nFaculty of Mathematics\nFaculty of Technology\nLaboratory L'IFORCE, University of Sciences and Technology Houari Boumediene (USTHB)\nUniversity of Médéa\nB.P. 32 El-Alia16111Bab-Ezzouar, AlgiersAlgeria, Algeria", "Faculty of Sciences\nDepartment of Mathematics and Computer Science\nUniversity of Médéa\nAlgeria", "University of Médéa\nAlgeria" ]	[]	Let G = (V, E) be a simple graph. A non-empty set S ⊆ V is called a global offensive alliance if S is a dominating set and for every vertex v in V − S, at least half of the vertices from the closed neighborhood of v are in S. The global offensive alliance number is the minimum cardinality of a global offensive alliance in G. In this paper, we give a constructive characterization of trees having a unique minimum global offensive alliance.	null	[ "https://arxiv.org/pdf/1804.07113v1.pdf" ]	119,722,004	1804.07113	b87fcf75bce61cbf9342e5f5f8db517e0076d941	Trees with unique minimum glolal offensive alliance sets 19 Apr 2018 Mohamed Bouzefrane [email protected] Isma Bouchemakh s:[email protected] Mohamed Zamime [email protected] Noureddine Ikhlef-Eschouf Faculty of Technology Faculty of Mathematics Faculty of Technology Laboratory L'IFORCE, University of Sciences and Technology Houari Boumediene (USTHB) University of Médéa B.P. 32 El-Alia16111Bab-Ezzouar, AlgiersAlgeria, Algeria Faculty of Sciences Department of Mathematics and Computer Science University of Médéa Algeria University of Médéa Algeria Trees with unique minimum glolal offensive alliance sets 19 Apr 2018arXiv:1804.07113v1 [math.CO]Domination, global offensive alliance Let G = (V, E) be a simple graph. A non-empty set S ⊆ V is called a global offensive alliance if S is a dominating set and for every vertex v in V − S, at least half of the vertices from the closed neighborhood of v are in S. The global offensive alliance number is the minimum cardinality of a global offensive alliance in G. In this paper, we give a constructive characterization of trees having a unique minimum global offensive alliance. Introduction Throughout this paper, G = (V, E) denotes a simple graph with vertex-set V = V (G) and edge-set E = E(G). Let G and H be two graphs with two disjoint vertex sets. Their disjoint union is denoted by G ∪ H, the disjoint union of k copies of G is denoted by kG and the disjoint union of a family of graphs G 1 , G 2 , . . . , G k is denoted by ∪ k i=1 G i . For every vertex v ∈ V (G), the open neighborhood N G (v) is the set {u ∈ V (G) \| uv ∈ E(G)} and the closed neighborhood of v is the set N G [v] = N (v) ∪ {v} . The degree of a vertex v ∈ V (G), denoted d G (v) , is the size of its open neighborhood. A vertex of degree one is called a leaf and its neighbor is called a support vertex. If v is a support vertex of a tree T , then L T (v) will denote the set of the leaves attached at v. Let L(T ) and S(T ) denote the set of leaves and support vertices, respectively, in T, and let \|L(T )\| = l (T ). As usual, the path of order n is denoted by P n , and the star of order n by K 1,n−1 . A double star S p,q is obtained by attaching p leaves at an endvertex of a path P 2 and q leaves at the second one. A subdivision of an edge uv is obtained by introducing a new vertex w and replacing the edge uv with the edges uw and wv. A subdivided star denoted by SS k is a star K 1,k where each edge is subdivided exactly once. A wounded spider is a tree obtained from K 1,r , where r ≥ 1, by subdividing at most r − 1 of its edges. For a vertex v, let C(v) and D(v) denote the set of children and descendants, respectively, of v in a rooted tree T , and let D[v] = D(v) ∪ {v}. The maximal subtree at v is the subtree of T induced by D [v], and is denoted by T v . A dominating set of a graph G is a set D of vertices such that every vertex in V − D is adjacent to some vertex in D. The domination number of G, denoted by γ (G), is the minimum cardinality of a dominating set of G. The concept of domination in graphs, with its many variations, is now well studied in graph theory. For more details, see the books of Haynes, Hedetniemi, and Slater [19,20]. Among the many variations of domination, we mention the concept of alliances in graphs that has been studied in recent years. Several types of alliances in graphs are introduced in [18], including the offensive alliance that we study here. A dominating set D with the property that for every vertex v not in D, \|N G [v] ∩ D\| ≥ \|N G [v] − D\|(1) is called global offensive alliance set of G and abbreviated GOA-set of G. The global offensive alliance number γ o (G) is the minimum cardinality among all GOA-sets of G. A GOA-set of G of cardinality γ o (G) is called γ o -set of G, or γ o (G)-set. Several works have been carried out on global offensive alliances in graphs (see, for example, [2,6], and elsewhere). Graphs with unique minimum µ-set, where µ is a some graph parameter, is another concept to which much attention was given during the last two decades. For example, graphs with unique minimum γ-set were first studied by Gunther et al. in [13]. Later this problem was studied for various classes of graphs including block graphs [7], cactus graphs [9], some cartesian product graphs [14] and some repeated cartesian products [15]. Several works on uniqueness related to other graph parameters have been widely studied, such as locating-domination number [1], paired-domination number [3], double domination number [4], roman domination number [5] and total domination number [17]. Further work on this topic can be found in [8,10,11,12,16,21,22,23] The aim of this paper is to characterize all trees having unique minimum global offensive alliance set. We denote such trees as UGOA-trees. Preliminaries results We give in this section the following observations. Some results are straightforward and so their proofs are omitted. Observation 1 Let T be a tree of order at least three and u ∈ S(T ). Then, Proof. (i) and (ii) are obvious. If (iii) is not satisfied, then all leaves attached at u would be contained in D, which is a contradiction with the minimality of D. Observation 2 Let T be a tree obtained from a nontrivial tree T ′ by joining a new vertex v at a support vertex u of T ′ . Let D and D ′ be γ o (T )-sets of T and T ′ , respectively. Then, (i) \|D ′ \| = \|D\| , ii) D ∩ V (T ′ ) is a γ o (T ′ )-set, (iii) if T is a UGOA-tree such that u is in any γ o (T ′ )-set, then T ′ is a UGOA-tree. Proof. According to Observation 1 (iii), u must be in D since l T (u) ≥ 2. i) D is clearly a GOA-set of T ′ , and then \|D ′ \| ≤ \|D\| . By Observation 1 (i), we can assume that u ∈ D ′ . Hence, D ′ can be extended to a GOA-set of T , which leads to \|D\| ≤ \|D ′ \| . Thus equality holds. ii) Since D ∩ V (T ′ ) = D is a GOA-set of T ′ with cardinality \|D\| = \|D ′ \|, we deduce that D ∩ V (T ′ ) is a γ o (T ′ )-set. iii) Item (i) together with the fact that u belongs to any γ o (T ′ ) imply that D ′ can be extended to a γ o (T )-set. Therefore, the uniqueness of D as a γ o (T )-set leads to D ′ = D, which means that D ′ is the unique γ o (T ′ ) . Observation 3 Let T be a tree obtained from a nontrivial tree T ′ different from P 2 by joining the center vertex y of the path P 3 = x-y-z at a support vertex v of T ′ . Let D and D ′ be γ o (T )-sets of T and T ′ , respectively such that each of them contains all support vertices. Then, (i) \|D ′ \| = \|D\| − 1, (ii) D ∩ V (T ′ ) is a γ o (T ′ )-set, (iii) if T is a UGOA-tree, then T ′ is a UGOA-tree. Proof. i) Since y ∈ D and v ∈ D ∩ D ′ , it follows that D − {y} is a GOA-set of T ′ and so \|D ′ \| ≤ \|D\| − 1. Moreover, since v ∈ D ′ , D ′ can be extended to a GOA-set of T by adding y. Then \|D\| ≤ \|D ′ ∪ {y}\| = \|D ′ \| + 1 and equality holds. ii) Since D ∩ V (T ′ ) = D − {y} is a GOA-set of T ′ with cardinality \|D\| − 1 = \|D ′ \| , D ∩ V (T ′ ) is a γ o (T ′ )-set. iii) Let B = {y}. In view of item (i), D ′ can be extended to a γ o (T )-set by adding the unique vertex of B. This and item (ii) together with the uniqueness of D imply that D ′ = D ∩ V (T ′ ) is the unique γ o -set of T ′ . Observation 4 Let k be a positive integer and let T be a tree obtained from a nontrivial tree T ′ by adding kP 2 joining k pairwise non-adjacent vertices of kP 2 to the same leaf v of T ′ . Let w be the support vertex adjacent to v, and let D and D ′ be γ o (T )-sets of T and T ′ , respectively. If w ∈ D ∩ D ′ , then the following three properties are satisfies. (i) \|D ′ \| = \|D\| − k, (ii) D ∩ V (T ′ ) is a γ o (T ′ )-set, (iii) if T is a UGOA-tree, then T ′ is a UGOA-tree. Proof. Let V (kP 2 ) = {x 1 , x 2 , . . . , x k , y 1 , y 2 , . . . , y k } and E(kP 2 ) = {x i y i : i = 1, 2, . . . , k}. Let v be a leaf of T ′ and w be the support vertex adjacent to v. We assume that for each i ∈ {1, . . . , k}, y i is adjacent to v in T. i) Obviously, all vertices of ∪ k j=1 {y j } are support vertices in T. Hence, in view of Observation 1 (i), we can assume that D contains all vertices of ∪ k j=1 {y j } . Therefore, since w ∈ D, D − (∪ k j=1 {y j }) is a GOA-set of T ′ , which means that \|D ′ \| ≤ D − (∪ k j=1 {y j }) = \|D\| − k. Observe that since w ∈ D ′ , D ′ can be extended to a GOA-set of T by adding all vertices of ∪ k j=1 {y j } . Hence \|D\| ≤ D ′ ∪ (∪ k j=1 {y j }) = \|D ′ \| + k and so equality holds. ii) The proof is similar to that of Observation 3(ii), by taking D ∩ V (T ′ ) = D − (∪ k j=1 {y j }). iii) The proof is similar to that of (iii) of Observation 3(iii), by taking B = ∪ p j=1 {y j }. Observation 5 Let V (T ′ ) be the vertex-set of a nontrivial tree T ′ , and let D ′ be a γ o (T ′ )-set such V (T ′ ) − D ′ has a vertex w with degree q ≥ 2 and \|N T ′ (w) ∩ (V (T ′ ) − D ′ )\| ≤ 1. Let p be a positive integer such that    p ≤ q − 1 if \|N T ′ (w) ∩ (V (T ′ ) − D ′ )\| = 0, or p ≤ q − 3 if \|N T ′ (w) ∩ (V (T ′ ) − D ′ )\| = 1.(2) Let T be a tree obtained from T ′ by adding p subdivided stars SS k 1 , . . . , SS kp (k i ≥ 2 for all i) with centers x 1 , x 2 , . . . , x p , respectively, and joining each x i (1 ≤ i ≤ p) at w. Let D be a γ o -set of T . If w and x 1 , x 2 , . . . , x p are not in D, then the following three properties are satisfied. (i) \|D ′ \| = \|D\| − p i=1 k i , (ii) D ∩ V (T ′ ) is a γ o (T ′ )-set, (iii) if T is a UGOA-tree, then T ′ is also a UGOA-tree. Proof. For i ∈ {1, . . . , p}, let S (SS k i ) be a support vertex-set of SS k i . i) Since w together with x 1 , x 2 , . . . , x p are not in D, all vertices of ∪ p i=1 S (SS k i ) must be in D. Therefore, D\ p ∪ i=1 S (SS k i ) is a GOA-set of T ′ , giving that \|D ′ \| ≤ \|D\| − p i=1 k i . On the other hand, let A = ∪ p i=1 S (SS k i ) ∪ D ′ . We have to show that A is a GOA-set of T. For this, it suffices to show that \|N T [z] ∩ A\| ≥ \|N T [z] − A\| for each z ∈ {w, x 1 , x 2 , . . . , x p }. Indeed, we have to distinguish between two cases. Case 1. z = x i , for some i ∈ {1, . . . , p}. We have then \|N T [z] ∩ A\| = \|N T [z] ∩ ∪ p i=1 S (SS k i )\| = k i ≥ 2, and \|N T [z] − A\| = \|{z, w}\| = 2. Case 2. z = w. We have then \|N T [z] ∩ A\| = q if \|N T ′ (w) ∩ (V (T ′ ) − D ′ )\| = 0, q − 1 if \|N T ′ (w) ∩ (V (T ′ ) − D ′ )\| = 1. and \|N T [z] − A\| = p + 1 if \|N T ′ (w) ∩ (V (T ′ ) − D ′ )\| = 0, p + 2 if \|N T ′ (w) ∩ (V (T ′ ) − D ′ )\| = 1. According to (2) B = ∪ p i=1 S (SS k i ) . The main result In order to characterize the trees with unique minimum global offensive alliance, we define a family F of all trees T that can be obtained from a sequence T 1 , T 2 , . . . , T r (r ≥ 1) of trees, where T 1 is the path P 3 centered at a vertex y, T = T r , and if r ≥ 2, T i+1 is obtained recursively fom T i by one of the following operations. Let A (T 1 ) = {y} . • Operation O 1 : Attach a vertex by joining it to any support vertex of T i . Let A (T i+1 ) = A (T i ) . • Operation O 2 : Attach a path P 3 = u-v-w by joining v to any support vertex of T i . Let A (T i+1 ) = A (T i ) ∪ {v} . • Operation O 3 : Let w be a support vertex of T i that satisfies one of the following two conditions. 1. l T i (w) ≥ 3, 2. \|N T i [w] ∩ A(T i )\| < \|N T i (w) ∩ (V (T i ) − A(T i )\| or * either l T i (w) = 2 and N T i (w) − A(T i ) has a vertex w t such that \|N T i (w t ) ∩ A(T i )\| ≤ \|N T i [w t ] ∩ (V (T i ) − A(T i )\| + 1, * or l T i (w) = 1 and N T i (w) − A(T i ) has two vertices w p , w q so that for l = p, q, \|N T i (w l ) ∩ A(T i )\| ≤ \|N T i [w l ] ∩ (V (T i ) − A(T i )\|+ 1. Let kP 2 be the disjoint union of k ≥ 1 copies of P 2 , and let B be a set of k pairwise non-adjacent vertices of kP 2 . Add kP 2 and attach all vertices of B to a same leaf in T i that is adjacent to w. Let A (T i+1 ) = A (T i ) ∪ B. • Operation O 4 : Let w ∈ V (T i )−A (T i ) be a vertex of degree q ≥ 2 in T i such that \|N T i (w) ∩ (V (T i ) − A (T i ))\| ≤ 1. Attach p ≥ 1 subdivided stars SS k i (k i ≥ 2 for 1 ≤ i ≤ p) with support vertex-set S (SS k i ) and of center x i by joining x i to w for all i such that p ≤ q − 1 if \|N T i (w) ∩ (V (T i ) − A (T i ))\| = 0, q − 3 if \|N T i (w) ∩ (V (T i ) − A (T i ))\| = 1. Let A (T i+1 ) = A (T i ) ∪ (∪ p i=1 S (SS k i )). Before stating our main result, we need the following lemma. Lemma 6 If T ∈ F, then A (T ) is the unique γ o (T )-set. Proof. Let T ∈ F. We proceed by induction on the number of operations, say r, required to construct T. The property is true if T is a path P 3 centered at y since A (T ) = {y} is the unique γ o (T )-set. This establishes the base case. Assume that for any tree T ′ ∈ F that can be constructed with r − 1 operations, A (T ′ ) is the unique γ o (T ′ )-set. Let T = T r with r ≥ 2 and T ′ = T r−1 . We distinguish between four cases. Case 1. T is obtained from T ′ by using Operation O 1 . Assume that T is obtained from T ′ by attaching an extra vertex at a support vertex u of T ′ . In view of Observation 1 (ii), u ∈ A(T ′ ). Hence A(T ′ ) can be extended to a GOA-set of T . By Observation 2 (i), γ o (T ) = γ o (T ′ ) , implying that A(T ′ ) is a γ o (T )-set. Applying the inductive hypothesis to T ′ , A(T ′ ) is the unique γ o (T ′ )-set. It follows that A (T ) = A (T ′ ) is the unique γ o (T )-set. Case 2. T is obtained from T ′ by using Operation O 2 . A (T ′ ) ∪ {v} is a GOA-set of T . By Observation 3 (i), γ o (T ) = γ o (T ′ ) + 1, meaning that A (T ′ ) ∪ {v} is a γ o (T )-set. The inductive hypothesis sets that A (T ′ ) is the unique γ o (T ′ )-set. Thus A (T ) = A (T ′ ) ∪ {v} is the unique γ o (T )-set. Case 3. T is obtained from T ′ by using Operation O 3 . A (T ′ )∪B is a GOA-set of T . Observation 4 (i) sets that γ o (T ) = γ o (T ′ )+k, which means that A (T ′ ) ∪ B is a γ o (T )-set. By the inductive hypothesis, A (T ′ ) is the unique γ o (T ′ )-set. Thus A (T ) = A (T ′ ) ∪ B is the unique γ o (T )-set. Case 4. T is obtained from T ′ by using Operation O 4 . A (T ′ ) ∪ (∪ p i=1 S (SS k i )) is a GOA-set of T . According to Observation 5 (i), we have γ o (T ) = γ o (T ′ ) + p i=1 k i , whence, A (T ′ ) ∪ (∪ p i=1 S (SS k i )) is a γ o (T )-set. By the inductive hypothesis, A (T ′ ) is the unique γ o (T ′ )-set. It follows that A (T ) = A (T ′ ) ∪ (∪ p i=1 S (SS k i )) is the unique γ o (T )-set. Remark that in each case, A(T i+1 ) is obtained from A(T i ) by adding all support vertices in T i+1 \T i . Hence the following corollary is immediate. Proof. It is obvious that T = K 1 is a UGOA-tree. Also, Lemma 6 states that any member of F is a UGOA-tree. Now, we prove the converse by induction on the number n of vertices of T . The converse holds trivially for n = 1 and 3 but not for n = 2 since P 2 is not a UGOA-tree. When n = 4, T is either a K 1,3 or a P 4 . Clearly P 4 is not a UGOA-tree, whilst K 1,3 is a UGOA-tree that can be obtained from a P 3 using operation O 1 , and so K 1,3 ∈ F. If n = 5, then T is either a double star S 1,2 which is not a UGOA-tree, or it is a K 1,4 or P 5 that are UGOA-tree since K 1,4 can be obtained from K 1,3 by using operation O 1 , and P 5 can be obtained from a P 3 by using operation O 3 . Therefore K 1,4 and P 5 are in F. This establishes the base case. Now, let n ≥ 6 and assume that any tree T ′ of order 3 ≤ n ′ < n with the unique γ o (T ′ )-set is in F. Let T be a tree of order n with the unique γ o (T )-set D and let s ∈ S(T ). By Observation 1 (ii), s ∈ D. If l T (s) ≥ 3, then let T ′ be the tree obtained from T by removing a leaf adjacent to s and let D ′ be a γ o (T ′ )-set. Then, clearly n ′ = \|V (T ′ )\| = n − 1 ≥ 5, and l T ′ (s) ≥ 2, so s ∈ D ′ by Observation 1 (iii). According to Observation 2 (ii), T ′ is UGOA-tree. Applying the inductive hypothesis to T ′ , we get T ′ ∈ F. Thus T is obtained from T ′ by operation O 1 , implying that T ∈ F. Assume now that for each x ∈ S(T ), l T (x) ≤ 2. Root T at a vertex r of maximum eccentricity. Let u be a support vertex of maximum distance from r and let u ′ be a leaf adjacent to u. Let v and w be the parents of u and v, respectively, in the rooted tree. We consider two cases. (3), l T (u) = 2. We claim that v ∈ S(T ). Suppose not. Then either w ∈ D and so D − {v} is a GOA-set of T with cardinality less than \|D\| , contradicting the minimality of D, or w / ∈ D and so D − {v} ∪ {w} is a γ o (T )-set, contradicting the uniqueness of D as a γ o (T )set. This completes the proof of the claim. Let T ′ = T − T u and D ′ be a γ o -set of T ′ . By Observation 1(i), we can assume that D ′ contains all support vertices in T ′ . Since \|V (T u )\| = 3, it follows that n ′ = \|V (T ′ )\| = n − 3 ≥ 3 and so T ′ = P 2 . By Observation 3(iii), T ′ is a UGOA-tree. Applying our inductive hypothesis, we get T ′ ∈ F. Thus, T can be obtained from T ′ by operation O 2 and so T ∈ F. Case 2. v / ∈ D. For all i ∈ {1, . . . , k}, let u ′ i be the unique leaf adjacent to u i (with u ′ 1 = u ′ ). We distinguish between two subcases, depending on whether w belongs to D or not. Case 2.1. w ∈ D. In view of (5), T v − {v} = kP 2 with V (kP 2 ) = {u 1 , u 2 , . . . , u k , u ′ 1 , u ′ 2 , . . . , u ′ k } and E(kP 2 ) = {u i u ′ i : i = 1, 2, . . . , k}. Let T ′ = T − (T v − {v}) . Clearly v ∈ L(T ′ ) and w ∈ S(T ′ ). If n ′ = \|V (T ′ )\| = 2, then T is a wounded spider with exactly one non-subdivided edge and in this case, it is not difficult to see that such a graph is not a UGOA-tree. Hence assume that n ′ ≥ 3. We claim the following: If l T (w) ∈ {0, 1}, then one of the two conditions holds: C 1 : \|N T [w] ∩ D\| ≤ \|N T (w) ∩ (V (T ) − D\| . C 2 : (i) either l T (w) = 1 and N T (w) − D has a vertex w t such that \|N T (w t ) ∩ D\| ≤ \|N T [w t ] ∩ (V (T ) − D)\| + 1 (ii) or, l T (w) = 0 and N T (w) − D has two vertices w p , w q such that for l ∈ {p, q}, \|N T (w l ) ∩ D\| ≤ \|N T [w l ] ∩ (V (T ) − D)\| + 1. Indeed, suppose that C 1 and C 2 are not satisfied. Assume first that l T (w) = 1, so L T (w) has exactly one vertex, say w ′ . In this case D − {w} ∪ {w ′ } is a γ o (T )-set different from D, a contradiction. Now, assume that l T (w) = 0. Since C 2 is not fulfilled, item (ii) of C 2 is satisfied for at most one vertex in N T (w) − D, say w ′′ . Then D − {w} ∪ {w ′′ } is a γ o (T )-set different from D, a contradiction. If no vertex in N T (w) − D for which item (ii) of C 2 is satisfied, then D − {w} ∪ {v} is a γ o (T )-set different from D, which leads to a contradiction again. This complete the proof of the claim. Observe that when l T ′ (w) ∈ {1, 2}, the previous claim remain true by replacing D by D ′ and T by T ′ . Thus, according to Observation 4 (iii), T ′ is a UGOA-tree. By induction on T ′ , we get T ′ ∈ F. Since T is obtained from T ′ by using operation O 3 , we directly obtain T ∈ F. Let t be the parent of w, and let X, Y and Z be the following sets Y = C(w) ∩ S(T ), X = C(w) − Y and Z = D(w) ∩ (S(T ) − Y ) . Observe that v ∈ X, u ∈ Z, N T (w) = {t} ∪ X ∪ Y and every vertex in Z plays the same role as u. Therefore by (4), we have Z ⊂ D since Z ⊂ S(T ), and by (5), every vertex in Z has exactly two neighbors such that one of them is a leaf and the other one is in X. Furthermore, as v ∈ X, u i ∈ Z for all i ∈ {1, . . . , k}, so \|Z\| ≥ k ≥ 2. Notice also that \|X\| ≥ 1 since v ∈ X. Likewise \|Y \| ≥ 1 since D is a γ o (T )-set. It is clear that Y ⊆ S(T ) and thus Y ⊆ D by Observation 1(ii). Setting Then p 1 = k. Since for all i ∈ {1, . . . , p}, x i and w are in V (T ) − D, x i must have at least two neighbors in Z. Hence d T (x i ) = p i + 1 ≥ 3. This means that for all i ∈ {1, . . . , p}, V (T x i ) induces a subdivided star SS p i of order p i + 1 centered at x i . Since w ∈ V (T ) − D, inequality (1) is valid by replacing v with w. This gives p ≤ q − 1 if t ∈ D, or p ≤ q − 3 otherwise. Let T ′ = T − ∪(∪ p i=1 T x i ) and D ′ be a γ o (T ′ )-set. Observe that T ′ contains at least one P 3 as an induced subgraph, which means that n ′ = \|V (T ′ )\| ≥ 3. For all i ∈ {1, . . . , p}, let S(SS p i ) be the support vertex-set of SS p i . Clearly ∪ p i=1 S(SS p i ) = Z and N T ′ (w) = Y ∪ {t}, so d T ′ (w) = q ≥ 2. According to Observation 1 (i), we can assume that Y ⊂ D ′ since Y ⊂ S(T ′ ). Then t is the only neighbor of w in T ′ that may not be in D ′ , that is N T ′ (w) ∩ (V (T ′ ) − D ′ ) ≤ 1. If t ∈ D ′ , then the minimality of D ′ sets that w ∈ V (T ′ ) − D ′ , because otherwise Again Observation 5(iii) sets that T ′ is a UGOA-tree. Applying the inductive hypothesis to T ′ , we deduce T ′ ∈ F. Now since T can be obtained from T ′ by operation O 4 , and finally T ∈ F. This completes the proof of Theorem 8. Open Problems The previous results motivate the following problems. 1-Characterize other UGOA-graphs. 2-Characterize trees with unique minimum defensive alliance sets (UGDA). (i) there is a γ o (T )-set that contains all support vertices of T , (ii) if D is a unique γ o (T )-set, then D contains all support vertices but no leaf, (iii) if l T (u) ≥ 2, then u belongs to any γ o -set(T ). , we have in each case \|N T [z] ∩ A\| ≥ \|N T [z] − A\| for each z ∈ {w, x 1 , x 2 , . . . , x p }. Therefore A is a GOA-set of T, giving that\|D\| ≤ \|A\| = \|D ′ \| + p i=1 k i .Hence the equality holds.ii) Using the fact that D ∩ V (T ′ ) = D\ ∪ p i=1 S (SS k i ) , this property follows in a similar manner as the proof of Observation 3(ii).(iii) This property follows in a similar manner as the proof of Observation 3(iii), by taking Corollary 7 7Let T ∈ F and S(T ) be a set of support vertices in T . Then γ o (T ) \|S(T )\| . Now we are ready to prove our main result. Theorem 8 A tree T is a UGOA-tree if and only if T = K 1 or T ∈ F. Case 1 . 1v ∈ D. If l T (u) = 1, then D ∪ {u ′ } − {u} is a γ o (T )-set, contradicting the uniqueness of D as a γ o (T )-set. Hence by According to Observation 1(ii), v / ∈ S(T ) and so l T (v) = 0. Let k = \|N T (v) − {w}\| . We have then d T (v) = k + 1 and since u ∈ N T (v) − {w}, we clearly deduce k ≥ 1. For i ∈ {1, . . . , k}, let u i ∈ N T (v) − {w} such that u 1 = u. The choice of v sets that u i ∈ S(T ), l T (u i ) ≥ 1 and so u i ∈ D for all i. ( 4 ) 4Hence by (3), we have 1 ≤ l T (u i ) ≤ 2 for all i. Assume first that l T (u j ) = 2 for some j in {1, . . . , k}. Without loss of generality, let j = 1. Then u has a further neighboru ′′ = u ′ in T. Let T ′ = T − {u ′′ } and D ′ be any γ o -set of T ′ . Clearly u ′ is the unique leaf of u in T ′ . We claim that u ∈ D ′ .Suppose not. Then u ′ and v must be in D ′ and therefore D ′′ = (D ′ \{u ′ }) ∪ {u} is a further γ o (T )-set other than D (since v belongs to D ′′ and not to D), a contradiction. This completes the proof of the claim. We have n ′ = n−1 ≥ 5. By Observation 2(iii), T ′ is a UGOA-tree. Applying our inductive hypothesis to T ′ , we get T ′ ∈ F. Hence T is obtained from T ′ by operation O 1 , implying that T ∈ F. Assume now that l T (u i ) = 1 and hence d T (u i ) = 2 for all i. By Observation 1(ii), w / ∈ S(T ) and so l T (w) = 0. Since v and w are in V (T )−D, v must have at least two neighbors in D. Hence d T (v) = k+1 ≥ 3. X = {x 1 , x 2 , . . . x p }(p ≥ 1) with x 1 = v and \|Y \| = q − 1 (q ≥ 2). Since every vertex in X plays the same role as v, x i ∈ V (T ) − D for all i ∈ {1, . . . , p}. Setting p i = \|N T (x i ) − {w}\| for i = 1, . . . , p. , we replace w by t in D ′ . By Observation 5 (ii) and (iii), we haveD ′ = D ∩ V (T ′ ). Hence t ∈ D if and only if t ∈ D ′ . Notice that if t ∈ D ′ , then N T ′ (w) ∩ (V (T ′ ) − D ′ ) is an empty-set, otherwise, t would be the unique vertex of N T ′ (w)∩(V (T ′ )−D ′ ).Thus(6)can be rewritten as follows.If N T ′ (w) ∩ (V (T ′ ) − D ′ ) = 0, then p ≤ q − 1, and if N T ′ (w) ∩ (V (T ′ ) − D ′ ) = 1, then p ≤ q − 3. Characterizations of trees with unique minimum locating-dominating sets. M Blidia, M Chellali, R Lounes, F Maffray, J. Combin. Math. Combin. Comput. 76M. Blidia, M. Chellali, R. Lounes and F. Maffray, Characterizations of trees with unique minimum locating-dominating sets, J. Combin. Math. Combin. Comput.76 (2011) 2011, 225-232. On the global offensive alliance number of a tree. M Bouzefrane, M Chellali, Opuscula Math. 29M. Bouzefrane, M. Chellali, On the global offensive alliance number of a tree, Opuscula Math. 29 (2009), 223-228. Trees with unique minimum paired domination sets. M Chellali, T Haynes, Ars Comb. 73M. Chellali and T.W Haynes, Trees with unique minimum paired dom- ination sets. Ars Comb. 73 (2004) 3-12. A characterization of trees with unique minimum double domination sets. M Chellali, T Haynes, Util. Math. 83M. Chellali and T.W Haynes, A characterization of trees with unique minimum double domination sets, Util. Math., 83 (2010) 233-242. Trees with unique Roman dominating functions of minimum weight. M Chellali, N J Rad, Discrete Math. Algorithm. Appl. 061450038M. Chellali and N.J. Rad, Trees with unique Roman dominating func- tions of minimum weight, Discrete Math. Algorithm. Appl. 06, 1450038 (2014). Independence and global offensive alliance in graphs. M Chellali, L Volkmann, Australas. J. Combin. 47M. Chellali, L. Volkmann. Independence and global offensive alliance in graphs, Australas. J. Combin., 47 (2010) 125-131. Block graphs with unique minimum dominating sets. M Fischermann, Discrete Math. 2401-3M. Fischermann, Block graphs with unique minimum dominating sets, Discrete Math. 240 (1-3) (2001), 247-251. Maximum graphs with a unique minimum dominating set. M Fischermann, D Rautenbach, L Volkmann, Discrete Math. 2601-3M. Fischermann, D. Rautenbach and L. Volkmann, Maximum graphs with a unique minimum dominating set, Discrete Math. 260 (1-3) (2003), 197-203. Cactus graphs with unique minimum dominating sets. M Fischermann, L Volkmann, Util. Math. 63M. Fischermann and L. Volkmann, Cactus graphs with unique mini- mum dominating sets, Util. Math. 63 (2003), 229-38. Unique independence, upper domination and upper irredundance in graphs. M Fischermann, L Volkmann, J. Combin. Math. Combin. Comput. 47M. Fischermann and L. Volkmann, Unique independence, upper dom- ination and upper irredundance in graphs, J. Combin. Math. Combin. Comput. 47 (2003), 237-249. Unique irredundance, domination, and independent domination in graphs. M Fischermann, L Volkmann, I Zverovich, Discrete Math. 3051-3M. Fischermann, L. Volkmann and I. Zverovich, Unique irredundance, domination, and independent domination in graphs, Discrete Math. 305 (1-3) (2005), 190-200. Maximum graphs with unique minimum dominating set of size two. M Fraboni, N Shank, Australas. J. Combin. 46M. Fraboni and N. Shank, Maximum graphs with unique minimum dominating set of size two, Australas. J. Combin. 46 (2010), 91-99. Graphs with unique minimum dominating sets. G Gunther, B Hartnell, L Markus, D Rall, Proc. 25th S.E. Int. Conf. Combin., Graph Theory, and Computing. 25th S.E. Int. Conf. Combin., Graph Theory, and Computing101G. Gunther, B. Hartnell, L. Markus and D. Rall, Graphs with unique minimum dominating sets, in: Proc. 25th S.E. Int. Conf. Combin., Graph Theory, and Computing, Congr. Numer. 101 (1994), 55-63. On unique minimum dominating sets in some cartesian product graphs. J Hedetniemi, Discuss. Math. Graph Theory. 344J. Hedetniemi, On unique minimum dominating sets in some cartesian product graphs, Discuss. Math. Graph Theory 34 (4) (2015), 615-628. On unique minimum dominating sets in some repeated cartesian products. J Hedetniemi, Australas. J. Combin. 62J. Hedetniemi, On unique minimum dominating sets in some repeated cartesian products, Australas. J. Combin. 62 (2015), 91-99. On unique realizations of domination chain parameters. J Hedetniemi, J. Combin. Math. Combin. Comput. 101J. Hedetniemi, On unique realizations of domination chain parameters, J. Combin. Math. Combin. Comput. 101 (2017), 193-211. Trees with unique minimum total dominating sets. T W Haynes, M A Henning, Discuss. Math. Graph Theory. 22T. W. Haynes and M. A. Henning, Trees with unique minimum total dominating sets. Discuss. Math. Graph Theory 22 (2002) 233-246. Alliance in graphs. S M Hedetniemi, S T Hedetniemi, P Kristiansen, J. Comb. Math. Combin. Comput. 48S.M. Hedetniemi, S. T. Hedetniemi, and P. Kristiansen, Alliance in graphs. J. Comb. Math. Combin. Comput. 48 (2004) 157-177. Fundamentals of Domination in graphs. T W Haynes, S T & Hedetniemi, P Slater, Marcel Dekker, New YorkHaynes T W, Hedetniemi S T & Slater P J, 1998, Fundamentals of Domination in graphs, Marcel Dekker, New York. T W Haynes, S T & Hedetniemi, P Slater, Domination in graphs: Advanced Topics, Marcel Dekker. New YorkHaynes T W, Hedetniemi S T & Slater P J, (1998) (Eds.), Domination in graphs: Advanced Topics, Marcel Dekker, New York, 1998. Graphs with unique maximum independent sets. G Hopkins, W Staton, Discrete Math. 57G. Hopkins and W. Staton, Graphs with unique maximum independent sets, Discrete Math. 57 (1985) 245-251. On unique independent sets in graphs. W Siemes, J Topp, L Volkmann, Discrete Math. 1311-3W. Siemes, J. Topp and L. Volkmann, On unique independent sets in graphs, Discrete Math. 131 (1-3) (1994), 279-285. Graphs with unique minimum edge dominating sets and graphs with unique maximum independent sets of vertices. J Topp, Discrete Math. 1211-3J. Topp, Graphs with unique minimum edge dominating sets and graphs with unique maximum independent sets of vertices, Discrete Math. 121 (1-3) (1993), 199-210.	[]
[ "arXiv:0707.4130v2 [hep-ex] Study of the process e + e − → ωπ 0 with the KLOE detector The KLOE Collaboration", "arXiv:0707.4130v2 [hep-ex] Study of the process e + e − → ωπ 0 with the KLOE detector The KLOE Collaboration" ]	[ "F Ambrosino \nDipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly\n", "A Antonelli \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "M Antonelli ", "F Archilli \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "C Bacci \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n\nDipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly\n", "P Beltrame \nInstitut für Experimentelle Kernphysik\nUniversität Karlsruhe\nGermany\n", "G Bencivenni \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "S Bertolucci \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "C Bini \nDipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly\n", "C Bloise \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "S Bocchetta \nDipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly\n", "V Bocci \nDipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly\n", "F Bossi \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "P Branchini \nDipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly\n", "R Caloi \nDipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly\n", "P Campana \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "G Capon \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "T Capussela \nDipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly\n", "F Ceradini \nDipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly\n", "S Chi \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "G Chiefari \nDipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly\n", "P Ciambrone \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "E De Lucia \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "A De Santis \nDipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly\n", "P De Simone \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "G De Zorzi \nDipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly\n", "A Denig \nInstitut für Experimentelle Kernphysik\nUniversität Karlsruhe\nGermany\n", "A Di Domenico \nDipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly\n", "C Di Donato \nDipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly\n", "S Di Falco \nDipartimento di Fisica dell'Università e Sezione INFN\nPisaItaly\n", "B Di Micco \nDipartimento di Fisica dell'Università \"Roma Tre\" e Sezione 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dell'INFN\nFrascatiItaly\n", "E Gorini \nDipartimento di Fisica dell'Università e Sezione INFN\nLecceItaly\n", "E Graziani \nDipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly\n", "M Incagli \nDipartimento di Fisica dell'Università e Sezione INFN\nPisaItaly\n", "W Kluge \nInstitut für Experimentelle Kernphysik\nUniversität Karlsruhe\nGermany\n", "V Kulikov \nPermanent address: Institute for Theoretical and Experimental Physics\nMoscowRussia\n", "F Lacava \nDipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly\n", "G Lanfranchi \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "J Lee-Franzini \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n\nPhysics Department\nState University of New York at Stony Brook\nUSA\n", "D Leone \nInstitut für Experimentelle Kernphysik\nUniversität Karlsruhe\nGermany\n", "M Martini \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "P Massarotti \nDipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly\n", "W Mei \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "S Meola \nDipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly\n", "S Miscetti \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "M Moulson \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "S Müller \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "F Murtas \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "M Napolitano \nDipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly\n", "F Nguyen \nDipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly\n", "M Palutan \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "E Pasqualucci \nDipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly\n", "A Passeri \nDipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly\n", "V Patera \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n\nDipartimento di Energetica dell'Università \"La Sapienza\"\nRomaItaly\n", "F Perfetto \nDipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly\n", "M Primavera \nDipartimento di Fisica dell'Università e Sezione INFN\nLecceItaly\n", "P Santangelo \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "G Saracino \nDipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly\n", "B Sciascia \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "A Sciubba \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n\nDipartimento di Energetica dell'Università \"La Sapienza\"\nRomaItaly\n", "F Scuri \nDipartimento di Fisica dell'Università e Sezione INFN\nPisaItaly\n", "I Sfiligoi \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "T Spadaro \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "M Testa \nDipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly\n", "L Tortora \nDipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly\n", "P Valente \nDipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly\n", "B Valeriani \nInstitut für Experimentelle Kernphysik\nUniversität Karlsruhe\nGermany\n", "G Venanzoni \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "R Versaci \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n", "G Xu \nLaboratori Nazionali di Frascati dell'INFN\nFrascatiItaly\n\nInstitute of High Energy Physics of Academica Sinica\nBeijingChina\n" ]	[ "Dipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly", "Institut für Experimentelle Kernphysik\nUniversität Karlsruhe\nGermany", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly", "Dipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly", "Dipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly", "Dipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly", "Institut für Experimentelle Kernphysik\nUniversität Karlsruhe\nGermany", "Dipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly", "Dipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly", "Dipartimento di Fisica dell'Università e Sezione INFN\nPisaItaly", "Dipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly", "Dipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Fisica dell'Università e Sezione INFN\nLecceItaly", "Dipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly", "Dipartimento di Fisica dell'Università e Sezione INFN\nPisaItaly", "Institut für Experimentelle Kernphysik\nUniversität Karlsruhe\nGermany", "Permanent address: Institute for Theoretical and Experimental Physics\nMoscowRussia", "Dipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Physics Department\nState University of New York at Stony Brook\nUSA", "Institut für Experimentelle Kernphysik\nUniversität Karlsruhe\nGermany", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly", "Dipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly", "Dipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Energetica dell'Università \"La Sapienza\"\nRomaItaly", "Dipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly", "Dipartimento di Fisica dell'Università e Sezione INFN\nLecceItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Scienze Fisiche dell'Università \"Federico II\" e Sezione INFN\nNapoliItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Energetica dell'Università \"La Sapienza\"\nRomaItaly", "Dipartimento di Fisica dell'Università e Sezione INFN\nPisaItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Dipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly", "Dipartimento di Fisica dell'Università \"Roma Tre\" e Sezione INFN\nRomaItaly", "Dipartimento di Fisica dell'Università \"La Sapienza\" e Sezione INFN\nRomaItaly", "Institut für Experimentelle Kernphysik\nUniversität Karlsruhe\nGermany", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Laboratori Nazionali di Frascati dell'INFN\nFrascatiItaly", "Institute of High Energy Physics of Academica Sinica\nBeijingChina" ]	[]	Using ∼ 600 pb −1 collected with the KLOE detector at DAΦNE, we have studied the production cross section of π + π − π 0 π 0 and π 0 π 0 γ final states in e + e − collisions at center of mass energies between 1000 and 1030 MeV. By fitting the observed interference pattern around M φ for both final states, we extract a measurement (preliminary) for the ratio Γ(ω → π 0 γ)/Γ(ω → π + π − π 0 ) = 0.0934 ± 0.0022. Since these two final states represent the 98% of the ω decay channels, we use unitarity to derive BR(ω → π + π − π 0 ) = (89.94±0.23)% and BR(ω → π 0 γ) = (8.40±0.19)%. Moreover, the parameters describing the e + e − → π + π − π 0 π 0 reaction around M φ are used to extract the branching fraction for the OZI and G-parity violating φ→ωπ 0 decay: BR(φ → ωπ 0 ) = (5.63 ± 0.70) × 10 −5 .	null	[ "https://arxiv.org/pdf/0707.4130v2.pdf" ]	119,671,333	0707.4130	04385c6620f48569d5941d83b714dc493bc8573e	arXiv:0707.4130v2 [hep-ex] Study of the process e + e − → ωπ 0 with the KLOE detector The KLOE Collaboration 16 May 2008 F Ambrosino Dipartimento di Scienze Fisiche dell'Università "Federico II" e Sezione INFN NapoliItaly A Antonelli Laboratori Nazionali di Frascati dell'INFN FrascatiItaly M Antonelli F Archilli Laboratori Nazionali di Frascati dell'INFN FrascatiItaly C Bacci Laboratori Nazionali di Frascati dell'INFN FrascatiItaly Dipartimento di Fisica dell'Università "Roma Tre" e Sezione INFN RomaItaly P Beltrame Institut für Experimentelle Kernphysik Universität Karlsruhe Germany G Bencivenni Laboratori Nazionali di Frascati dell'INFN FrascatiItaly S Bertolucci Laboratori Nazionali di Frascati dell'INFN FrascatiItaly C Bini Dipartimento di Fisica dell'Università "La Sapienza" e Sezione INFN RomaItaly C Bloise Laboratori Nazionali di Frascati dell'INFN FrascatiItaly S Bocchetta Dipartimento di Fisica dell'Università "Roma Tre" e Sezione INFN RomaItaly V Bocci Dipartimento di Fisica dell'Università "La Sapienza" e Sezione INFN RomaItaly F Bossi Laboratori Nazionali di Frascati dell'INFN FrascatiItaly P Branchini Dipartimento di Fisica dell'Università "Roma Tre" e Sezione INFN RomaItaly R Caloi Dipartimento di Fisica dell'Università "La Sapienza" e Sezione INFN RomaItaly P Campana Laboratori Nazionali di Frascati dell'INFN FrascatiItaly G Capon Laboratori Nazionali di Frascati dell'INFN FrascatiItaly T Capussela Dipartimento di Scienze Fisiche dell'Università "Federico II" e Sezione INFN NapoliItaly F Ceradini Dipartimento di Fisica dell'Università "Roma Tre" e Sezione INFN RomaItaly S Chi Laboratori Nazionali di Frascati dell'INFN FrascatiItaly G Chiefari Dipartimento di Scienze Fisiche dell'Università "Federico II" e Sezione INFN NapoliItaly P Ciambrone Laboratori Nazionali di Frascati dell'INFN FrascatiItaly E De Lucia Laboratori Nazionali di Frascati dell'INFN FrascatiItaly A De Santis Dipartimento di Fisica dell'Università "La Sapienza" e Sezione INFN RomaItaly P De Simone Laboratori Nazionali di Frascati dell'INFN FrascatiItaly G De Zorzi Dipartimento di Fisica dell'Università "La Sapienza" e Sezione INFN RomaItaly A Denig Institut für Experimentelle Kernphysik Universität Karlsruhe Germany A Di Domenico Dipartimento di Fisica dell'Università "La Sapienza" e Sezione INFN RomaItaly C Di Donato Dipartimento di Scienze Fisiche dell'Università "Federico II" e Sezione INFN NapoliItaly S Di Falco Dipartimento di Fisica dell'Università e Sezione INFN PisaItaly B Di Micco Dipartimento di Fisica dell'Università "Roma Tre" e Sezione INFN RomaItaly A Doria Dipartimento di Scienze Fisiche dell'Università "Federico II" e Sezione INFN NapoliItaly M Dreucci Laboratori Nazionali di Frascati dell'INFN FrascatiItaly G Felici Laboratori Nazionali di Frascati dell'INFN FrascatiItaly A Ferrari Laboratori Nazionali di Frascati dell'INFN FrascatiItaly M L Ferrer Laboratori Nazionali di Frascati dell'INFN FrascatiItaly G Finocchiaro Laboratori Nazionali di Frascati dell'INFN FrascatiItaly S Fiore Dipartimento di Fisica dell'Università "La Sapienza" e Sezione INFN RomaItaly C Forti Laboratori Nazionali di Frascati dell'INFN FrascatiItaly P Franzini Dipartimento di Fisica dell'Università "La Sapienza" e Sezione INFN RomaItaly C Gatti Laboratori Nazionali di Frascati dell'INFN FrascatiItaly P Gauzzi Dipartimento di Fisica dell'Università "La Sapienza" e Sezione INFN RomaItaly S Giovannella Laboratori Nazionali di Frascati dell'INFN FrascatiItaly E Gorini Dipartimento di Fisica dell'Università e Sezione INFN LecceItaly E Graziani Dipartimento di Fisica dell'Università "Roma Tre" e Sezione INFN RomaItaly M Incagli Dipartimento di Fisica dell'Università e Sezione INFN PisaItaly W Kluge Institut für Experimentelle Kernphysik Universität Karlsruhe Germany V Kulikov Permanent address: Institute for Theoretical and Experimental Physics MoscowRussia F Lacava Dipartimento di Fisica dell'Università "La Sapienza" e Sezione INFN RomaItaly G Lanfranchi Laboratori Nazionali di Frascati dell'INFN FrascatiItaly J Lee-Franzini Laboratori Nazionali di Frascati dell'INFN FrascatiItaly Physics Department State University of New York at Stony Brook USA D Leone Institut für Experimentelle Kernphysik Universität Karlsruhe Germany M Martini Laboratori Nazionali di Frascati dell'INFN FrascatiItaly P Massarotti Dipartimento di Scienze Fisiche dell'Università "Federico II" e Sezione INFN NapoliItaly W Mei Laboratori Nazionali di Frascati dell'INFN FrascatiItaly S Meola Dipartimento di Scienze Fisiche dell'Università "Federico II" e Sezione INFN NapoliItaly S Miscetti Laboratori Nazionali di Frascati dell'INFN FrascatiItaly M Moulson Laboratori Nazionali di Frascati dell'INFN FrascatiItaly S Müller Laboratori Nazionali di Frascati dell'INFN FrascatiItaly F Murtas Laboratori Nazionali di Frascati dell'INFN FrascatiItaly M Napolitano Dipartimento di Scienze Fisiche dell'Università "Federico II" e Sezione INFN NapoliItaly F Nguyen Dipartimento di Fisica dell'Università "Roma Tre" e Sezione INFN RomaItaly M Palutan Laboratori Nazionali di Frascati dell'INFN FrascatiItaly E Pasqualucci Dipartimento di Fisica dell'Università "La Sapienza" e Sezione INFN RomaItaly A Passeri Dipartimento di Fisica dell'Università "Roma Tre" e Sezione INFN RomaItaly V Patera Laboratori Nazionali di Frascati dell'INFN FrascatiItaly Dipartimento di Energetica dell'Università "La Sapienza" RomaItaly F Perfetto Dipartimento di Scienze Fisiche dell'Università "Federico II" e Sezione INFN NapoliItaly M Primavera Dipartimento di Fisica dell'Università e Sezione INFN LecceItaly P Santangelo Laboratori Nazionali di Frascati dell'INFN FrascatiItaly G Saracino Dipartimento di Scienze Fisiche dell'Università "Federico II" e Sezione INFN NapoliItaly B Sciascia Laboratori Nazionali di Frascati dell'INFN FrascatiItaly A Sciubba Laboratori Nazionali di Frascati dell'INFN FrascatiItaly Dipartimento di Energetica dell'Università "La Sapienza" RomaItaly F Scuri Dipartimento di Fisica dell'Università e Sezione INFN PisaItaly I Sfiligoi Laboratori Nazionali di Frascati dell'INFN FrascatiItaly T Spadaro Laboratori Nazionali di Frascati dell'INFN FrascatiItaly M Testa Dipartimento di Fisica dell'Università "La Sapienza" e Sezione INFN RomaItaly L Tortora Dipartimento di Fisica dell'Università "Roma Tre" e Sezione INFN RomaItaly P Valente Dipartimento di Fisica dell'Università "La Sapienza" e Sezione INFN RomaItaly B Valeriani Institut für Experimentelle Kernphysik Universität Karlsruhe Germany G Venanzoni Laboratori Nazionali di Frascati dell'INFN FrascatiItaly R Versaci Laboratori Nazionali di Frascati dell'INFN FrascatiItaly G Xu Laboratori Nazionali di Frascati dell'INFN FrascatiItaly Institute of High Energy Physics of Academica Sinica BeijingChina arXiv:0707.4130v2 [hep-ex] Study of the process e + e − → ωπ 0 with the KLOE detector The KLOE Collaboration 16 May 2008e + e − collisionsrare φ decaysVMDOZI violationIsospin violation Using ∼ 600 pb −1 collected with the KLOE detector at DAΦNE, we have studied the production cross section of π + π − π 0 π 0 and π 0 π 0 γ final states in e + e − collisions at center of mass energies between 1000 and 1030 MeV. By fitting the observed interference pattern around M φ for both final states, we extract a measurement (preliminary) for the ratio Γ(ω → π 0 γ)/Γ(ω → π + π − π 0 ) = 0.0934 ± 0.0022. Since these two final states represent the 98% of the ω decay channels, we use unitarity to derive BR(ω → π + π − π 0 ) = (89.94±0.23)% and BR(ω → π 0 γ) = (8.40±0.19)%. Moreover, the parameters describing the e + e − → π + π − π 0 π 0 reaction around M φ are used to extract the branching fraction for the OZI and G-parity violating φ→ωπ 0 decay: BR(φ → ωπ 0 ) = (5.63 ± 0.70) × 10 −5 . Introduction In the energy region of few tens of MeV around M φ , the e + e − → π + π − π 0 π 0 production cross section is largely dominated by the non-resonant processes e + e − → ρ/ρ ′ → ωπ 0 . However, in a region closer to M φ , a contribution from the OZI and G-parity violating decay φ→ωπ 0 is expected. This strongly suppressed decay can be observed only through the interference pattern with the previous reaction, which shows up as a dip in the production cross section as a function of the center of mass energy ( √ s). A fit to the cross section energy dependence allows us to extract the φ→ωπ 0 branching fraction (BR). There is a much more complicated interference scenario for the e + e − → π 0 π 0 γ channel. Here we expect contributions also from φ → ρπ and φ → Sγ intermediate states, where S is a scalar meson. In another paper [1] we have shown that at √ s ∼ M φ the interference between φ → Sγ and e + e − → ωπ 0 events, evaluated by fitting the M ππ -M πγ Dalitz plot, is small. Assuming this effect to be negligible to first order, a fit to the cross section interference pattern for this final state will nevertheless give information about the e + e − → ρ/ρ ′ → ωπ 0 process and the resonant decays φ → ωπ 0 and φ → ρπ 0 . Comparing with the fit to the π + π − π 0 π 0 channel, the ratio Γ(ω → π 0 γ)/Γ(ω → π + π − π 0 ) can be extracted. The KLOE detector The KLOE [2] experiment operates at DAΦNE [3], the Frascati φ-factory. DAΦNE is an e + e − collider running at a center of mass energy of ∼ 1020 MeV, the mass of the φ-meson. Equal-energy positron and electron beams collide at an angle of π-25 mrad, producing φ-mesons nearly at rest. The KLOE detector consists of a large cylindrical drift chamber, DC, surrounded by a lead-scintillating fiber electromagnetic calorimeter, EMC. A superconducting coil around the EMC provides a 0.52 T field. The drift chamber [4], 4 m in diameter and 3.3 m long, has 12,582 all-stereo tungsten sense wires and 37,746 aluminium field wires. The chamber shell is made of carbon fiber-epoxy composite and the gas used is a 90% helium, 10% isobutane mixture. These features maximize transparency to photons and reduce K L → K S regeneration and multiple scattering. The position resolutions are σ xy ∼150 µm and σ z ∼ 2 mm. The momentum resolution is σ(p ⊥ )/p ⊥ ≈ 0.4%. Vertices are reconstructed with a spatial resolution of ∼3 mm. The calorimeter [5] is divided into a barrel and two endcaps, for a total of 88 modules, and covers 98% of the solid angle. The modules are read out at both ends by photo-multipliers, both in amplitude and time. The readout granularity is ∼ and σ t = 57 ps/ E (GeV) ⊕ 50 ps, respectively. The KLOE trigger [6] uses both calorimeter and chamber information. In this analysis all the events are selected by the trigger calorimeter, requiring two energy deposits with E > 50 MeV for the barrel and E > 150 MeV for the endcaps. A cosmic veto reject events where at least two outermost EMC layers are fired. 3 The √ s dependence of e + e − → π + π − π 0 π 0 /π 0 π 0 γ cross sections As mentioned before, in the energy region below 1.4 GeV the π + π − π 0 π 0 /π 0 π 0 γ production cross sections are dominated by the non-resonant process e + e − → ρ/ρ ′ → ωπ 0 . At √ s ∼ M φ , the decay φ → ωπ 0 also contributes and interferes with the other processes. In the neutral channel there are also contributions from φ → Sγ and φ → ρπ 0 . The dependence of the cross section on the center of mass energy can be parametrized in the form [8]: σ( √ s) = σ 0 ( √ s) · 1 − Z M φ Γ φ D φ(1) where σ 0 ( √ s) is the bare cross section for the non-resonant process, Z is the complex interference parameter (i.e. the ratio between the φ decay amplitude and the non resonant processes), while M φ , Γ φ and D φ are the mass, the width and the inverse propagator of the φ meson respectively. The non-resonant cross section in this energy range increases linearly with √ s. A model independent parametrization 2will be used in this paper. σ 0 ( √ s) = σ 0 + σ ′ ( √ s − M φ ) (2)4.1 e + e − → ωπ 0 → π + π − π 0 π 0 In the π + π − π 0 π 0 analysis, data are filtered by selecting events with the expected final state signature: two tracks connected to a vertex inside a small cylindrical fiducial volume around the Interaction Point (IP) and four neutral clusters in the prompt Time Window (TW), defined as \|T γ − R γ /c\| < MIN(3.5 · σ T , 2 ns). To minimize contamination from machine background, we require also a minimal energy for the clusters (10 MeV) and a minimal angle with respect to the beam line (∼23 • ). A kinematic fit requiring total Table 1 Background channels for ωπ 0 →π + π − π 0 π 0 . Signal over background ratios after acceptance selection and χ 2 Kfit cut are reported for events collected at four-momentum conservation and time of flight (TOF) for photons improves the energy resolution. The resulting χ 2 (χ 2 Kfit ) is used to select a signal enriched (χ 2 Kfit < 50), S evts , and a background dominated (χ 2 Kfit > 50), B evts , samples. The signal analysis efficiency in the S evts sample has been evaluted by Montecarlo (MC). The resulting value ε ∼ 40% is dominated by the acceptance requirements and has a small dependence as a function of √ s. √ s ∼ M φ . Channel S/B (acc) S/B (χ 2 Kfit cut) K S K L 1 10 K + K −10 The background channels are listed in Tab. 1. The main contribution comes from φ → K S K L → π + π − π 0 π 0 and φ → K + K − with K ± → π ± π 0 , which have the same final state. The first one has also a comparable production cross section with respect to the signal at the φ peak. The other two background components (φ → ηγ with η → π + π − π 0 , and φ → π + π − π 0 ) mimic the final state signature because of additional clusters due to accidental coincidence of machine background events and/or shower fragments (splitting). In the signal enriched region, the expected contamination at √ s ∼ M φ is ∼ 12%. The signal counting on data is performed for each √ s bin by fitting the reconstructed π 0 recoil mass (M rec ) distribution for both S evts and B evts samples with MC signal and background shapes. The fit procedure is based on a likelihood function which takes into account both data and MC statistics. In Fig. 1, data-MC comparison of few relevant distributions for the most populated energy bin is shown. The results are summarized in Tab. 2 where the signal counts, the χ 2 of the fit and the visible cross section are reported for all √ s bins. A preliminary estimate of the systematic error, dominated by tracking and vertexing efficiency, has been added to the σ 4π vis error. Table 2 Signal counting, χ 2 of the fit and visible cross section for e + e − → π + π − π 0 π 0 events. The errors on σ 4π vis contains a relative systematic error contribution of 1.8%. √ s (MeV) N 4π ± δ N χ 2 /ndf σ 4π4.2 e + e − → ωπ 0 → π 0 π 0 γ The acceptance selection for π 0 π 0 γ events requires five neutral clusters with E γ ≥ 7 MeV and a polar angle \| cos θ γ \| < 0.92 in the prompt Time Window. After applying a first kinematic fit (Fit1) imposing total 4-momentum conservation, photons are paired to π 0 's, by minimizing a χ 2 built using the invariant mass of the two γγ pairs. A second kinematic fit (Fit2) imposes also constraints on the π 0 masses. The background with final state different from π 0 π 0 γ is rejected by requiring χ 2 Fit2 /Ndf ≤ 5 and ∆M γγ = \|M γγ − M π \| ≤ 5 σ γγ , where M γγ and σ γγ are evaluated using the photon momenta from Fit1. After these cuts the remaining sample is dominated by e + e − → ωπ 0 → π 0 π 0 γ and φ → S → π 0 π 0 γ events. Signal is then selected neglecting the interference between the two processes and cutting on the intermediate state mass. Since M πγ the closest mass to M ω of the two π 0 γ combinations, only events satisfying \|M πγ − M ω \| < 3 σ Mω are retained. In Tab. 3, the background channels are listed together with the S/B ratio before and after the application of the whole analysis chain. The residual 10% background contamination comes predominantly from φ → ηγ → π 0 π 0 π 0 γ events where two photons are lost or merged. In Fig. 2 data-MC comparison for events in the most populated √ s bin is shown. The ψ variable is the minimum angle between the photon and the π 0 's in the di-pion rest frame. A good agreement is observed both after acceptance selection and after applying analysis cuts. The overall analysis efficiency for the identification of the signal is evaluated by applying the whole analysis chain to signal MC events: ε ππγ ∼ 40%, almost flat in √ s. The value obtained for each bin, together with the corresponding integrated luminosity, has been applied to the signal counting to obtain the visible cross section. Results are summarized in Tab. 4; errors include statistics and background subtraction only. Table 3 Background channels for e + e − → ωπ 0 → π 0 π 0 γ. The signal over background ratio before and after the application of the analysis cuts is reported for events collected at √ s ∼ M φ. Background S/B (no cuts) S/B (all cuts) φ → Sγ → π 0 π 0 γ 1.5 35 φ → ηπ 0 γ → γγπ 0 γ 5.4 120 φ → ηγ → π 0 π 0 π 0 γ 0.04 15 φ → ηγ → γγγ 0.04 380 φ → π 0 γ 0.13 2840 Fit results and ω branching ratios extraction The measured values of visible cross section, shown in Tab. 2 and 4, are fitted with the parametrization (1), convoluted with a radiator function [7]. The free parameters are: σ i 0 , ℜ(Z i ), ℑ(Z i ) and σ ′ i , where i is the 4π or ππγ final state. In Fig. 3 data points with the superimposed fit function are shown for both channels. The preliminary values for the extracted parameters are reported in Tabs. 5. The resulting χ 2 /Ndf are 12.8/15 (P (χ 2 ) = 62%) for the fully neutral channel and 13.4/13 (P (χ 2 ) = 42%) for the other one. Cross section fit results for the e + e − → π + π − π 0 π 0 (top) and e + e − → π 0 π 0 γ (bottom) channels. Black dots are data, solid line is the resulting fit function. Table 5 Fit results for the e + e − → π + π − π 0 π 0 cross section (left) and for e + e − → π 0 π 0 γ cross section (right). Parameter (e + e − → π + π − π 0 π 0 ) σ 4π 0 (nb) 8.12 ± 0.14 ℜ(Z 4π ) 0.097 ± 0.012 ℑ(Z 4π ) −0.133 ± 0.009 σ ′ 4π (nb/MeV) 0.072 ± 0.008 Parameter (e + e − → π 0 π 0 γ) σ ππγ 0 (nb) 0.776 ± 0.012 ℜ(Z ππγ ) 0.013 ± 0.013 ℑ(Z ππγ ) −0.155 ± 0.007 σ ′ ππγ (nb/MeV) 0.0079 ± 0.0006 From the two previous measurements we obtain: σ 0 (ω → π 0 γ) σ 0 (ω → π + π − π 0 ) = 0.0956 ± 0.0022(3) Taking into account the phase space difference between the two decays [8], the ratio of the partial widths can be extracted: Γ(ω → π 0 γ) Γ(ω → π + π − π 0 ) = 0.0934 ± 0.0021(4) Since these two final states the 98% of the ω decay channels, we use the Γ(ω → π 0 γ)/Γ(ω → π + π − π 0 ) ratio and the sum of averages of the existing BR measurements on rarest decays [9] to extract the main ω branching fractions, imposing the unitarity relation: BR(ω → π + π − π 0 ) = (89.94 ± 0.23)% (5) BR(ω → π 0 γ) = (8.40 ± 0.19)% (6) with a correlation of 82%. Comparison between our evaluation and the values in PDG [9] is shown in Fig. 4. BR(φ → ωπ 0 ) evaluation The measured σ 4π 0 and Z 4π paramters of the π + π − π 0 π 0 final state are related to the BR(φ→ωπ 0 ) through the relation: BR(φ → ωπ 0 ) = σ 0 (m φ )\|Z 4π \| 2 σ φ ,(7) where σ 0 (m φ ) is the total cross section of the e + e − → ωπ 0 process and σ φ is the peak value of the production cross section for the φ resonance. Using the parameters obtained from the π + π − π 0 π 0 analysis, the Γ ee measurement from KLOE [10] for the evaluation of σ φ , and our value for BR(ω → π + π − π 0 ) we extract: BR(φ → ωπ 0 ) = (5.63 ± 0.70) × 10 −5 (8) in agreement with the previous measurement from the SND experiment [8]. ( 4 . 44×4.4) cm 2 , for a total of 2440 cells . The energy deposits are obtained from the signals amplitude while the arrival times of particles and the positions in three dimensions are obtained from the time differences. Cells close in time and space are grouped into a calorimeter cluster. The cluster energy E is the sum of the cell energies. The cluster time T and position R are energy weighed averages. Energy and time resolutions are σ E /E = 5.7%/ E (GeV) the available statistics collected at the φ peak in 2001-2002 data-taking periods, corresponding to 450 pb −1 , has been analyzed. Moreover four scan points (1010 MeV, 1018 MeV, 1023 MeV and 1030 MeV) of ∼ 10 pb −1 each and the off-peak data (1000 MeV) acquired in 2005-2006 have been included in this analysis. All runs are grouped in center of mass energy bins of 100 keV. Fig. 1 . 1Data-MC comparison for π + π − π 0 π 0 signal enriched distribution using events taken at 1019.75 MeV : (a) χ 2 Kfit (Ndf=8); (b) cosine of the angle between reconstructed π 0 's; (c) π 0 recoil mass. Black dots are data, while hatched and white histograms are MC signal and background shapes respectively, weighted by our fit results. Fig. 2 . 2Data-MC comparison for π 0 π 0 γ events taken at 1019.75 MeV. Top: normalized χ 2 of the second kinematic fit after acceptance cuts. Bottom: π 0 γ invariant mass (left), and cos ψ distribution after cutting on M πγ (right). In the upper panel all the background is grouped together, while in the lower ones the φ → Sγ contribution is shown alone. Fig. 3 . 3Fig. 3. Cross section fit results for the e + e − → π + π − π 0 π 0 (top) and e + e − → π 0 π 0 γ (bottom) channels. Black dots are data, solid line is the resulting fit function. Table 5 Fit results for the e + e − → π + π − π 0 π 0 cross section (left) and for e + e − → π 0 π 0 γ cross section (right). Fig. 4 . 4Branching fraction for the two main ω decay channels. The black square is the KLOE fit result, while the black dot is the PDG constrained fit result. The shaded regions are the 68% C.L. Table 4Signal counting and visible cross section for e + e − → π 0 π 0 γ events. . F Ambrosino, KLOE CollaborationEur. Phys. J. C. 49473KLOE Collaboration, F. Ambrosino et al., Eur. Phys. J. C 49 (2007) 473. S Guiducci, Proc. of the 2001 Particle Accelerator Conference. P. Lucas S. Webber Eds.of the 2001 Particle Accelerator ConferenceChicago, Illinois, USA353S. Guiducci et al., Proc. of the 2001 Particle Accelerator Conference (Chicago, Illinois, USA), P. Lucas S. Webber Eds., 2001, 353. . M Adinolfi, KLOE CollaborationLNF-01/016 (IRNucl. Inst. and Meth. LNF PreprintKLOE Collaboration, M. Adinolfi et al., LNF Preprint LNF-01/016 (IR) (2001), accepted by Nucl. Inst. and Meth. . M Adinolfi, KLOE CollaborationNucl. Inst. and Meth. A. 482363KLOE Collaboration, M. Adinolfi et al., Nucl. Inst. and Meth. A 482 (2002) 363. . M Adinolfi, KLOE CollaborationNucl. Inst. and Meth. A. 492134KLOE Collaboration, M. Adinolfi et al., Nucl. Inst. and Meth. A 492 (2002) 134. . M Greco, Phys. Lett. B. 318635M. Greco et al., Phys. Lett. B 318 (1993) 635. . V M Aulchenko, Jou. Exp. Th. Phys. 90927V.M. Aulchenko et al., Jou. Exp. Th. Phys. 90 (2000) 927. . W M Yao, Jou. Phys. G. 33partial update for 2008 editionW. M. Yao et al., Jou. Phys. G 33 (2006) and 2007 partial update for 2008 edition (http://pdg.web.cern.ch/pdg) . F Ambrosino, KLOE CollaborationPhys. Lett. B. 608199KLOE Collaboration, F. Ambrosino et al., Phys. Lett. B 608 (2005) 199.	[]
[ "AN A.E. LOWER BOUND FOR HAUSDORFF DIMENSION UNDER VERTICAL PROJECTIONS IN THE HEISENBERG GROUP", "AN A.E. LOWER BOUND FOR HAUSDORFF DIMENSION UNDER VERTICAL PROJECTIONS IN THE HEISENBERG GROUP" ]	[ "Terence L J Harris " ]	[]	[]	An improved a.e. lower bound is given for Hausdorff dimension under vertical projections in the first Heisenberg group. 2010 Mathematics Subject Classification. 28A78; 28A80.	10.5186/aasfm.2020.4529	[ "https://arxiv.org/pdf/1811.12559v4.pdf" ]	91,183,950	1811.12559	9616d719cb1f2cf736f757c6c6acf5779db40048	AN A.E. LOWER BOUND FOR HAUSDORFF DIMENSION UNDER VERTICAL PROJECTIONS IN THE HEISENBERG GROUP 2 Apr 2019 Terence L J Harris AN A.E. LOWER BOUND FOR HAUSDORFF DIMENSION UNDER VERTICAL PROJECTIONS IN THE HEISENBERG GROUP 2 Apr 2019 An improved a.e. lower bound is given for Hausdorff dimension under vertical projections in the first Heisenberg group. 2010 Mathematics Subject Classification. 28A78; 28A80. Introduction The aim of this work is to improve the known a.e. lower bounds for Hausdorff dimension under vertical projections in the Heisenberg group. The average behaviour of Hausdorff dimension under orthogonal projections in Euclidean space was first explored by Marstrand in 1954 [15]; many developments and generalisations have occurred since (see e.g. [16,19,10,3,17]). An effort began in [1] and [2] towards understanding the behaviour of Hausdorff dimension under projections in the Heisenberg group, which was further developed in [12] and [11] (see also [5]). One important open problem that remains is determining the a.e. behaviour of Hausdorff dimension under "vertical projections". All definitions relevant to this work will be restated here; further background is available in [1,2]. Let H be the first Heisenberg group, which as a set will be identified with R 3 = C × R (through exponential coordinates). The assumed convention for the group law on H is (z, t) * (ζ, τ ) := z + ζ, t + τ + 2 Im zζ = (z + ζ, t + τ − 2z ∧ ζ) , where ∧ : R 2 × R 2 → R is the standard wedge product on R 2 , given by (x 1 , y 1 ) ∧ (x 2 , y 2 ) = x 1 y 2 − x 2 y 1 . Define (z, t) H = \|z\| 4 + t 2 1/4 . The group H is a metric space when equipped with the left invariant metric d H , called the Korányi metric, defined by d H ((z, t), (ζ, τ )) = (ζ, τ ) −1 * (z, t) H = \|z − ζ\| 4 + \|t − τ − 2z ∧ ζ\| 2 1/4 ; (1.1) see [8] for a proof of the triangle inequality. On any compact set, this metric is bi-Lipschitz equivalent to the usual Carnot-Carathéodory metric on H [1]. For a given metric space, Hausdorff dimension is defined through the underlying distance, which for the Heisenberg group will always be the Korányi metric. Hausdorff dimension is invariant under a bi-Lipschitz change of the metric, so the main results given here will hold for Carnot-Carathéodory metric too. The horizontal and vertical projections P V θ : H → V θ and P V ⊥ θ : H → V ⊥ θ are defined for each θ ∈ [0, π) by P V θ (z, t) = (π V θ (z), 0) , P V ⊥ θ (z, t) = π V ⊥ θ (z), t − 2π V θ (z) ∧ π V ⊥ θ (z) , where π V θ : C → C denotes Euclidean projection onto the line V θ := {λe iθ : λ ∈ R}, π V ⊥ θ : C → C denotes Euclidean projection onto V ⊥ θ = {λie iθ : λ ∈ R}, the horizontal subgroup V θ is defined by V θ := {(λe iθ , 0) ∈ C × R : λ ∈ R}, and the vertical subgroup V ⊥ θ is the Euclidean orthogonal complement of V θ in R 3 : V ⊥ θ = {(λ 1 ie iθ , λ 2 ) ∈ C × R : λ 1 , λ 2 ∈ R}. The term "projection" and the formulas for P V θ , P V ⊥ θ come from the unique way of writing an element (z, t) = P V ⊥ θ (z, t) * P V θ (z, t) , as a product of an element of V ⊥ θ on the left, with an element of V θ on the right. In [1,Theorem 1.4] it was shown that for any Borel (or analytic) set A ⊆ H, (1.2) dim P V ⊥ θ A ≥      dim A if 0 ≤ dim A < 1 1 if 1 ≤ dim A < 3 2 dim A − 5 if 3 ≤ dim A ≤ 4, for a.e. θ ∈ [0, π), and it was conjectured that the lower bound dim P V ⊥ θ A ≥ dim A actually holds in the larger range 0 ≤ dim A ≤ 3. The upper limit of 3 is necessary since the vertical subgroups V ⊥ θ have Hausdorff dimension 3 (the entire Heisenberg group H has Hausdorff dimension 4). In [11], Fässler and Hovila proved (1.3) dim P V ⊥ θ A ≥ 1 + (s − 1)(s − 2) 32s 2 , for a.e. θ ∈ [0, π), s = dim A > 2, which improved (1.2) in the range 2 < dim A < 3.00348 (approximately). The main result of this work is the following lower bound. Theorem 1.1. If A ⊆ H is an analytic set with dim A > 1, then dim P V ⊥ θ A ≥ 3+dim A 4 if dim A ∈ (1, 3] 1 + dim A 6 if dim A ∈ (3, 4], for a.e. θ ∈ [0, π). This improves (1.2) and (1.3) in the range 1 < dim A < 3 + 3 11 , and gives a positive answer to Question 4.2 from [11], which asks if the a.e. lower bound of 1 can be improved for 1 < dim A ≤ 2. The proof employs some of the techniques used by Orponen and Venieri in [18] for restricted families of projections in R 3 ; the main difficulty in adapting this to the Heisenberg setting lies in finding a substitute for Marstrand's Three Circles Lemma (see [20,Lemma 3.2]). 1.1. Notation and preliminaries. Given two measure spaces X and Y , a measure ν on X and a measurable function f : X → Y , the pushforward f # ν of ν under f is a measure on Y , defined by f # ν(E) = ν(f −1 (E)) for each measurable set E ⊆ Y . For a real number t, let ⌈t⌉ denote the least integer greater than or equal to t. Let \|x\| denote the Euclidean norm of an element x ∈ R n . The Euclidean distance \|x − y\| between x and y may also be denoted by d E (x, y). For x ∈ R 3 and r > 0, let B E (x, r) and B H (x, r) be the Euclidean and Korányi balls around x of radius r. The following local Hölder condition from [4] will be useful later. c = c(R) > 0 such that \|v − w\| c ≤ d H (v, w) ≤ c\|v − w\| 1/2 , for all v, w ∈ R 3 with \|v\|, \|w\| ≤ R. In some situations, the following proposition gives a covering of a Euclidean ball by Korányi balls which is a more efficient than the single ball covering implied by the previous lemma. The following version of Frostman's Lemma provides a characterisation of Hausdorff dimension for analytic sets (see [13,16] for a proof). To state it, a subset A of a complete separable metric space X is called analytic if A is the continuous image of a Borel set B ⊆ Y , for some complete separable metric space Y . In particular, every Borel set is analytic. Lemma 1.4. Let X be a complete separable metric space, let A ⊆ X be an analytic subset of X and let s > 0. If there exists a nonzero finite Borel measure ν on A and a constant C, such that (1.4) ν(B(x, r)) ≤ r s for all r > 0 and x ∈ A, then dim A ≥ s. Conversely, if dim A > Proof of lemmas and the main theorem Most of this section is devoted to proving the lemmas from which Theorem 1.1 will follow. The first lemma of this section is an abstract version of Lemma 2.5 from [18] (see also [14,Theorem 7.2]); the proof is not too different from the Euclidean case, but is included for completeness. In the statement of the lemma, (θ, x) → π θ (x) is an arbitrary continuous function, but all statements following the proof of the lemma will specialise to the case where π θ = P V ⊥ θ is a vertical projection on H. The lemma essentially says that, given a fractal measure on a set A, if there is a quantitative restriction on how often the pushforward measure under the projection fails an s-Frostman condition, then a.e. the dimension of π θ (A) is at least s (where s may be smaller than dim A, but ideally as close to dim A as possible). Lemma 2.1. Let X, Y be metric spaces, with X compact and Y separable. Suppose that µ is a Borel probability measure on X, ν is a nonzero, finite, compactly supported Borel measure on Y , and (θ, y) → π θ (y) is a continuous function from X × Y into Y . Given s > 0, if there exist η, δ 0 > 0 such that (2.1) ν {y ∈ Y : µ {θ ∈ X : π θ# ν(B(π θ (y), δ)) ≥ δ s } ≥ δ η } ≤ δ η , for all δ ∈ (0, δ 0 ), then dim π θ (supp ν) ≥ s for µ-a.e. θ ∈ X. Remark 2.2. The proof of the lemma necessarily has a few measure-theoretic technicalities; the core part of the proof is the calculation following (2.6). Proof of Lemma 2.1. Let µ, ν, η, δ 0 , s be given. It is first shown that the sets occurring in (2.1) are measurable. For fixed x ∈ Y , and any constant c > 0, the set S := {(θ, y) ∈ X × Y : d(π θ (x), π θ (y)) < c} , is open in X × Y by continuity. Since Y is separable, the Borel sigma algebra on X × Y is equal to the one generated by the products of Borel sets [6, Lemma 6.4.2], and is therefore contained in the class of (µ × ν)-measurable sets, since µ and ν are Borel by assumption. Hence S is (µ × ν)-measurable. Therefore the function f (θ, y) = χ π −1 θ (B(π θ (x),c)) (y), is (µ × ν)-measurable, and so the function (2.2) θ → f (θ, y) dν(y) = π θ# ν(B(π θ (x), c)) is µ-measurable in θ by part (iv) of Fubini's Theorem from [9]. This proves µmeasurability of the inner part of (2.1). For the outer part, denoted by Z δ := {y ∈ Y : µ {θ ∈ X : π θ# ν(B(π θ (y), δ)) ≥ δ s } ≥ δ η } , a similar argument to that for (2.2) shows that for any δ > 0 the function (θ, y) → π θ# ν(B(π θ (y), δ)) is (µ × ν)-measurable, and hence the function y → µ {θ ∈ X : π θ# ν(B(π θ (y), δ)) ≥ δ s } is a ν-measurable function of y, by part (iii) of Fubini's Theorem from [9]. This shows that Z δ is ν-measurable. Since ν is compactly supported and π (·) is continuous, to prove the lemma it may be assumed that Y is compact. Let ǫ > 0 and let E ⊆ X be a compact set with (2.3) dim π θ (supp ν) < s − ǫ for every θ ∈ E. Any finite Borel measure on a compact metric space is inner regular, so it suffices to show µ(E) = 0. Let ǫ ′ > 0, and choose a positive δ 1 < δ0 4 small enough to ensure δ η 1 ≤ ǫ ′ . For each θ ∈ E, using (2.3) let {B(π θ (z i (θ)), δ i (θ))} ∞ i=1 be a cover of π θ supp ν by balls in Y of dyadic radii δ i (θ) < δ 1 such that ∞ i=1 δ i (θ) s < ǫ ′ . It is possible to choose the covers in such a way that the functions (2.4) π θ# ν(B(π θ (z i (θ)), c)), π θ# ν D j θ , ν π −1 θ D j θ ∩ Z δ , and (2.5) ν π −1 θ D j θ,1 \ Z δ , ν π −1 θ D j θ,2 \ Z δ , are µ-measurable in θ on E, for any c, δ > 0, for each i and for any integer j. Here D j θ := δi(θ)=2 −j B π θ (z i (θ)), 2 −j , D j θ,1 is the subset of D j θ defined as the union over those balls B π θ (z i (θ)), 2 −j in D j θ with π θ# ν B π θ (z i (θ)), 2 −j < 2 −(j−1)s , and D j θ,2 is the union of the remaining balls in D j θ , equivalently D j θ,2 = D j θ \ D j θ,1 . To verify the µ-measurability of (2.4) and (2.5), the compactness of π θ (supp ν) for each fixed θ ∈ E ensures that there is a finite subcollection (not relabelled) of balls B(π θ (z i (θ)), δ i (θ)) which cover π θ (supp ν). The union U θ of these balls is an open set, and therefore contains an open δ ′ -neighbourhood N δ ′ (π θ (supp ν)) of π θ (supp ν) for some δ ′ > 0. The compactness of Y (assumed without loss of generality) ensures that the map (θ, y) → π θ (y) is uniformly continuous on X × Y , which implies π θ ′ (supp ν) ⊆ N δ ′ (π θ (supp ν)) ⊆ U θ , for all θ ′ in a sufficiently small ball B θ around θ. Therefore the balls B (π θ (z i (θ)) , δ i (θ)) form a finite cover of π θ ′ (supp ν) for θ ′ ∈ B θ . The sets B θ cover E as θ ranges over E, so by compactness of E there is a finite subcollection {B θ1 , . . . , B θN } such that E = B θ1 ∪ B θ2 \ B θ1 ∪ · · · ∪ B θN \ ∪ N −1 i=1 B θi . The functions z i (θ) and δ i (θ) may then be taken to be constant on each part of this Borel partition of E. By the piecewise constant property and the µ-measurability of (2.2), the function π θ# ν(B(π θ (z i (θ)), c)) is µ-measurable for every i and any c > 0. This proves the µ-measurability of the first function in (2.4). Measurability of the other functions follows from a similar argument to the measurability of (2.2), using the piecewise constant property of the δ i (θ)'s and the ν-measurability of Z δ . This shows that the covers {B(π θ (z i (θ)), δ i (θ))} ∞ i=1 may be chosen to make the functions in (2.4) and (2.5) µ-measurable over E. For each θ ∈ E, (2.6) ν(Y ) ≤ j>log 2 δ1 π θ# ν D j θ , by the definition of the cover and the sets D j θ . Dividing both sides by ν(Y ) 1 and integrating over E gives µ(E) E j>\|log 2 δ1\| π θ# ν D j θ dµ(θ) ≤ j>\|log 2 δ1\| E ν π −1 θ D j θ ∩ Z 2 −(j−1) dµ(θ) (2.7) + E j>\|log 2 δ1\| ν π −1 θ D j θ,1 \ Z 2 −(j−1) dµ(θ) (2.8) + j>\|log 2 δ1\| E ν π −1 θ D j θ,2 \ Z 2 −(j−1) dµ(θ). (2.9) It remains to bound the integrals in (2.7), (2.8) and (2.9). Up to a constant the first sum, in (2.7), is bounded by δ η 1 ≤ ǫ ′ by the assumption on each Z δ in the statement of the lemma, and the choice of δ 1 . The integral in (2.8) is ǫ ′ by the condition ∞ i=1 δ i (θ) s ≤ ǫ ′ defining the cover and by the definition of D j θ,1 . It remains to bound (2.9). For each j the set (θ, y) ∈ E × Y : π θ (y) ∈ D j θ,2 is (µ × ν)-measurable by the piecewise constant property of the defining cover, so an application of Fubini's Theorem to each integral in (2.9) results in j>\|log 2 δ1\| E ν π −1 θ D j θ,2 \ Z 2 −(j−1) dµ(θ) = j>\|log 2 δ1\| Y \Z 2 −(j−1) µ θ ∈ E : π θ (y) ∈ D j θ,2 dν(y) ≤ j>\|log 2 δ1\| Y \Z 2 −(j−1) µ θ ∈ E : π θ# ν B π θ (y), 2 −(j−1) ≥ 2 −(j−1)s dν(y) j>\|log 2 δ1\| 2 −jη by definition of Z 2 −(j−1) , ǫ ′ , by the condition δ η 1 ≤ ǫ ′ imposed in the choice of δ 1 . Therefore µ(E) ǫ ′ with ǫ ′ arbitrary, and thus µ(E) = 0. This proves the lemma. The following lemma is a slightly refined version of Lemma 3.5 from [11], see also [2,Section 4]. The lemma is a kind of transversality condition, which means that in a quantitative sense the paths of two fixed, distinct points under the family of vertical projections pass each other transversally. The proof has only minor adjustments to those in [2,11] but is included for completeness. Lemma 2.3. There exists a positive constant C such that for any v, w ∈ H and any δ > 0, the set θ ∈ [0, π) : d H P V ⊥ θ (v), P V ⊥ θ (w) ≤ δ , is contained in a disjoint union of at most 41 intervals, each of length less than Cδ d H (v,w) . Proof. Fix v, w ∈ H and write v = (z, t), w = (ζ, τ ). If (2.10) \|z − ζ\| ≥ \|t − τ − 2z ∧ ζ\| 1/2 , then θ ∈ [0, π) : d H P V ⊥ θ (v), P V ⊥ θ (w) ≤ δ ⊆ θ ∈ [0, π) : \|π V ⊥ θ (z − ζ)\| ≤ δ . By writing z − ζ = \|z − ζ\|e iφ and rotating so that φ = 0, the right hand side is contained two intervals of length w) . This proves the lemma in the case of (2.10). δ \|z−ζ\| δ d H (v, If (2.10) fails, then (2.11) \|z − ζ\| < \|t − τ − 2z ∧ ζ\| 1/2 . Suppose (2.11) holds and z = ±ζ. Then d H (v, w) = \|z − ζ\| 4 + \|t − τ − 2z ∧ ζ\| 2 1/4 = \|z − ζ\| 4 + \|t − τ \| 2 1/4 ≤ 2 1/4 \|t − τ \| 1/2 = 2 1/4 t − τ − 2π V θ (z) ∧ π V ⊥ θ (z) + 2π V θ (ζ) ∧ π V ⊥ θ (ζ) 1/2 ≤ 2 1/4 d H P V ⊥ θ (v), P V ⊥ θ (w) ,(2.12) and so θ ∈ [0, π) : d H P V ⊥ θ (v), P V ⊥ θ (w) ≤ δ ⊆ θ ∈ [0, π) : d H (v, w) ≤ 2 1/4 δ . The right hand side is [0, π) if d H (v, w) ≤ 2 1/4 δ, which in this case is an interval of length π δ d H (v,w) . Otherwise the right hand side is empty and there is nothing to show. This proves the lemma in the case of (2.11) with z = ±ζ. It remains to consider the case in which (2.11) holds but z = ±ζ. Let a = t − τ − 2z ∧ ζ \|z + ζ\|\|z − ζ\| , p = z − ζ \|z − ζ\| , q = z + ζ \|z + ζ\| . Using z ∧ ζ = π V θ (z) ∧ π V ⊥ θ (ζ) − π V θ (ζ) ∧ π V ⊥ θ (z), d H P V ⊥ θ (v), P V ⊥ θ (w) ≥ t − τ − 2π V θ (z) ∧ π V ⊥ θ (z) + 2π V θ (ζ) ∧ π V ⊥ θ (ζ) 1/2 = t − τ − 2z ∧ ζ + 2π V ⊥ θ (z − ζ) ∧ π V θ (z + ζ) 1/2 = \|t − τ − 2z ∧ ζ + 2(z − ζ) ∧ π V θ (z + ζ)\| 1/2 = \|z + ζ\| 1/2 \|z − ζ\| 1/2 \|a + 2p ∧ π V θ (q)\| 1/2 . (2.13) If \|a\| ≥ 4 then (2.11) implies that (2.13) d H (v, w), and the argument is similar to the case of (2.12). Hence it may be assumed that \|a\| < 4. With this assumption, (2.11) gives d H (v, w) \|t − τ − 2z ∧ ζ\| 1/2 \|z + ζ\| 1/2 \|z − ζ\| 1/2 , and putting this into (2.13) yields d H P V ⊥ θ (v), P V ⊥ θ (w) d H (v, w) \|a + 2p ∧ π V θ q\| 1/2 . Therefore (2.14) θ ∈ [0, π) : d H P V ⊥ θ (v), P V ⊥ θ (w) ≤ δ ⊆ θ ∈ [0, π) : \|a + 2p ∧ π V θ (q)\| ≤ Kδ d H (v, w) 2 , for a sufficiently large constant K. Define F = F p,q by F (θ) = a + 2p ∧ π V θ (q), so that F ′ (θ) = 2p ∧ ∂ θ π V θ (q), F ′′ (θ) = 2p ∧ ∂ 2 θ π V θ (q) . Using π V θ (q) = q, e iθ e iθ gives ∂ θ π V θ (q) = q, ie iθ e iθ + q, e iθ ie iθ , ∂ 2 θ π V θ (q) = 2 q, ie iθ ie iθ − q, e iθ e iθ . Therefore ∂ θ π V θ (q) and 1 2 ∂ 2 θ π V θ (q) are orthonormal unit vectors in R 2 , for each θ ∈ [0, π), and so (2.15) 1 = \|p\| 2 = F ′ (θ) 2 2 + F ′′ (θ) 4 2 for all θ ∈ [0, π). It follows that for any b ∈ R, the equation F (θ) = b has at most 2 solutions in any interval of length strictly less than 1/2. To see this, let I be an interval with \|I\| < 1/2 and assume for a contradiction that F (θ) = b for has three distinct solutions in I. Then by Rolle's Theorem F ′ has two distinct zeroes in I, and by Rolle's Theorem again F ′′ has a zero θ ′′ in I. Let θ ′ be one of the zeroes of F ′ . Then by (2.15), 2 = \|F ′ (θ ′′ )\| = θ ′′ θ ′ F ′′ (θ) dθ ≤ 4\|I\| < 2, which is a contradiction. By covering the interval [0, π) with 7 intervals of length strictly less than 1/2, the equation F (θ) = b has at most 14 solutions in [0, π), for any b, and therefore the second set in (2.14) is the disjoint union of at most 15 subintervals of [0, π). Equation (2.15) implies that F ′ is 4-Lipschitz, so by using 8π < 26, these at most 15 intervals can be written as a union of at most 15 + 26 = 41 disjoint intervals I ⊆ [0, π), each of length at most 1/8, such that either \|F ′ (θ)\| > 1/2 for every θ ∈ I, or \|F ′′ (θ)\| > 3 for every θ ∈ I. Lemma 3.3 from [7] asserts that each of these intervals has length δ d H (v,w) , assuming d H (v, w) ≥ δ. If d H (v, w) ≤ δ the lemma holds trivially provided C > π, so this finishes the proof. , 5s 6 − 1 with κ < s − 1. Then there exist δ 0 , η > 0 such that ν v ∈ H : H 1 θ ∈ [0, π) : P V ⊥ θ # ν B H P V ⊥ θ (v), δ ≥ δ s−κ ≥ δ η ≤ δ η , whenever δ ∈ (0, δ 0 ). Proof. Choose (2.16) η = 1 10 4 min κ − 3(s − 1) 4 , κ − 5s 6 − 1 which is strictly positive by the assumption on κ. The choice of δ 0 will be made implicitly to eliminate implicit constants and ensure that various trivial inequalities, such as \| log δ\| ≤ δ −η , hold for δ < δ 0 . Fix δ ∈ (0, δ 0 ) and let Z = Z δ = v ∈ H : H 1 θ ∈ [0, π) : P V ⊥ θ # ν B H P V ⊥ θ (v), δ ≥ δ s−κ ≥ δ η . By dyadic pigeonholing and the Frostman condition on ν, there exists t ≥ δ 1−100η with t 1 and a ν-measurable subset Z ′ ⊆ Z satisfying ν(Z ′ ) ≥ δ η ν(Z), and (2.17) H 1 (H ′ (v)) ≥ δ 2η for all v ∈ Z ′ , where H ′ (v) is defined for any v ∈ H by H ′ (v) = θ ∈ [0, π) : ν P −1 V ⊥ θ B H P V ⊥ θ (v), 2δ ∩ A H (v, t, 2t) ≥ δ s−κ+η , and A H (v, r, R) denotes the Korányi annulus in H centred at v with inner radius r and outer radius R. By inner regularity of ν, Z ′ can also be taken to be compact. Fix v ∈ Z ′ . Using (2.17), choose three subsets H ′ j (v) ⊆ H ′ (v) separated (mod π) by a distance of at least δ 4η from each other, each contained in an interval of length δ 4η , and each with 1-dimensional measure at least δ 8η . This can be done by partitioning [0, π) into δ −4η intervals of length δ 4η , choosing the 6 intervals with the largest intersection with H ′ (v) (in terms of H 1 -measure), and then choosing 3 with gaps between them (mod π). By compactness of Z ′ , this construction can be modified to ensure that for each j, the sets H ′ j (v) are piecewise constant in v over some disjoint Borel cover of Z ′ . For each v ∈ Z ′ and v j ∈ H, define v ∼ j v j if v j ∈ P −1 V ⊥ θ B H P V ⊥ θ (v), 2δ ∩ A H (v, t, 2t) for some θ ∈ H ′ j (v). Set (2.18) α = max s − κ − 1 + 1000η s − 1 , log t log δ + 20η ∈ (0, 1). The rest of the proof will consist of verifying the following inequality: (2.19) ν(Z)t 3 δ 1000η+3(s−κ−1) ≤ ν 4 (v, v 1 , v 2 , v 3 ) ∈ Z ′ × H 1 3 : v ∼ j v j for all j, d E (ζ 2 , ℓ(ζ 1 , ζ 3 )) ≥ δ α if \|ζ − ζ 1 \|, \|ζ − ζ 3 \| ≥ t/2 max t 2s δ s 2 , min t 3s 2 δ (1−α)s 2 −1000η , t s+1 δ (1−α)(s−1)−1000η t > δ α−20η , t 3s where d E refers to the Euclidean distance, ℓ(a, b) means the infinite line through a and b, ν 4 = ν × ν × ν × ν and v = (ζ, τ ) ∈ C × R. The lemma will essentially follow by comparing the two outer parts of (2.19). The piecewise constant property of the sets H ′ j (v) ensures that the set in (2.19) is Borel measurable. To prove the lower bound of (2.19), cover the interval [0, π) with disjoint intervals of length δ/t, and fix v ∈ Z ′ , j ∈ {1, 2, 3}. Since H 1 (H ′ j (v)) ≥ δ 8η , there are at least tδ 8η−1 intervals I k = I k,j intersecting H ′ j (v), so pick some θ k = θ k,j in each intersection. Then (2.20) ν P −1 V ⊥ θ k B H P V ⊥ θ k (v), 2δ ∩ A H (v, t, 2t) ≥ δ s−κ+η , for each j and k, which follows from H ′ j (v) ⊆ H ′ (v) and the definition of H ′ (v). If j = 2, cover [0, π) with disjoint intervals J l of length δ α t . There are at least tδ −α+9η intervals J l which each intersect at least δ α−1+9η different θ k 's, since otherwise the definition of α would give the contradiction tδ 8η−1 ≤ tδ −α+9η · δ α−1 + πtδ −α δ α−1+9η < tδ 8η−1 . After removing some of the sets J l , assume that the J l 's are only those that intersect at least δ α−1+9η different θ k 's. After removing some of the sets I k , assume that all the θ k 's intersect one of these sets J l , so there are at least tδ 20η−1 sets I k . For fixed v ∈ Z ′ , v 1 , v 3 ∈ H with v ∼ 1 v 1 , v ∼ 3 v 3 , and t 2 ≤ \|ζ − ζ 1 \|, \|ζ − ζ 3 \| ≤ 2t , it will be shown that for fixed l, (2.21) ν v 2 ∈ P −1 V ⊥ θ k B H P V ⊥ θ k (v), 2δ ∩ A H (v, t, 2t) for some θ k ∈ J l : d E (ζ 2 , ℓ(ζ 1 , ζ 3 )) ≥ δ α ≥ δ α−1+11η δ s−κ . The sets in (2.20) are finitely overlapping over k by Lemma 2.3, and therefore summing (2.20) over those k with I k,2 ∩ J l = ∅ gives (2.22) ν v 2 ∈ P −1 V ⊥ θ k B H P V ⊥ θ k (v), 2δ ∩ A H (v, t, 2t) for some θ k ∈ J l δ α−1+10η δ s−κ . Hence (2.21) will follow from (2.22) combined with the observation that for t 2 ≤ \|ζ − ζ 1 \|, \|ζ − ζ 3 \| ≤ 2t, the set E := v 2 ∈ P −1 V ⊥ θ k B H P V ⊥ θ k (v), 2δ ∩ A H (v, t, 2t) for some θ k ∈ J l : d E (ζ 2 , ℓ(ζ 1 , ζ 3 )) < δ α , is contained in a Korányi ball of radius δ α−100η , therefore contributing δ (α−100η)s to the measure in (2.20), which is smaller than δ α−1+s−κ+11η by the definition of α. To prove that E is contained in the required Korányi ball, it will first be shown that the projected set F := ζ 2 ∈ R 2 : (ζ 2 , τ 2 ) ∈ A H (v, t, 2t) and π V ⊥ θ k (ζ 2 − ζ) < 2δ for some τ 2 ∈ R and θ k ∈ J l : d E (ζ 2 , ℓ(ζ 1 , ζ 3 )) < δ α , is contained in a Euclidean ball in R 2 of radius δ α−50η . To see this, fix ζ 2 ∈ F with v 2 = (ζ 2 , τ 2 ) and corresponding angle θ k given. Define ℓ(θ k ) = {ζ + λe iθ k : λ ∈ R}, so that by diam J l = δ α t and the definition of F , F ⊆ N δ α (ℓ(ζ 1 , ζ 3 )) ∩ θm∈J l N 2δ (ℓ(θ k )) ∩ B E (ζ, 2t) ⊆ N δ α (ℓ(ζ 1 , ζ 3 )) ∩ N 8δ α (ℓ(θ k )), It suffices to contain the right hand side in a ball of radius δ α−50η . This will be done by establishing a lower bound on the angle θ between the two lines. Let φ 1 ∈ H ′ 1 (v) and φ 3 ∈ H ′ 3 (v) be such that π V ⊥ φ j (ζ − ζ j ) < 2δ with j ∈ {1 , 3}, so that φ 1 and φ 3 are δ 4η separated (mod π). Write ζ 2 = λ 1 ζ 1 + (1 − λ 1 )ζ 3 + λ 2 i(ζ 1 − ζ 3 ), λ 1 , λ 2 ∈ R with \|λ 2 \|\|ζ 1 − ζ 3 \| < δ α , so that (ζ 1 − ζ 3 ) ∧ π V θ k (ζ 2 − ζ) ≥ \|(ζ 1 − ζ 3 ) ∧ (ζ 2 − ζ)\| − 8tδ ≥ \|(ζ 1 − ζ) ∧ (ζ 3 − ζ)\| − t(4δ α + 8δ) (2.23) ≥ π V φ 1 (ζ 1 − ζ) ∧ π V φ 3 (ζ 3 − ζ) − t(4δ α + 16δ) − 4δ 2 ≥ t 2 16 \|sin(φ 3 − φ 1 )\| − t(4δ α + 16δ) − 4δ 2 ≥ t 2 δ 4η 32 − t(4δ α + 16δ) − 4δ 2 (2.24) ≥ t 2 δ 5η , since t > δ α−20η by (2.18). Hence the angle θ between the two lines satisfies \|sin θ\| = (ζ 1 − ζ 3 ) \|ζ 1 − ζ 3 \| ∧ π V θ k (ζ 2 − ζ) π V θ k (ζ 2 − ζ) ≥ δ 6η . It follows that the intersection N δ α (ℓ(ζ 1 , ζ 3 )) ∩ N 2δ α (ℓ(θ k )), and therefore F , is contained in a Euclidean ball of radius δ α−50η inside R 2 . For the set E, let v 2 , v ′ 2 ∈ E. By the triangle inequality for the Korányi metric, d H P V ⊥ θ k (v 2 ), P V ⊥ θ k (v ′ 2 ) ≤ 4δ. Considering each component of the Korányi distance separately gives π V ⊥ θ k (ζ 2 − ζ ′ 2 ) ≤ 4δ, and τ 2 − τ ′ 2 − 2π V θ k (ζ 2 ) ∧ π V ⊥ θ k (ζ 2 ) + 2π V θ k (ζ ′ 2 ) ∧ π V ⊥ θ k (ζ ′ 2 ) ≤ 16δ 2 . By the identity ζ 2 ∧ ζ ′ 2 = π V θ k (ζ 2 ) ∧ π V ⊥ θ k (ζ ′ 2 ) − π V θ k (ζ ′ 2 ) ∧ π V ⊥ θ k (ζ 2 ) and the two preceding inequalities, (2.25) \|τ 2 − τ ′ 2 − 2ζ 2 ∧ ζ ′ 2 \| δ. The Euclidean projection of E down to R 2 is contained in F , and therefore in a ball of radius δ α−50η . Combining this with (2.25) gives d H (v 2 , v ′ 2 ) = \|ζ 2 − ζ ′ 2 \| 4 + \|τ 2 − τ ′ 2 − 2ζ 2 ∧ ζ ′ 2 \| 2 1/4 δ α−50η + δ 1/2 δ α−50η , for any v 2 , v ′ 2 ∈ E, by the definition of α in (2.18) and the choice of η in (2.16). This shows that the Korányi diameter of E is δ α−50η , and thus E is contained in a Korányi ball of radius δ α−100η . This implies (2.21) by (2.22), the Frostman condition on ν and the definition of α. For each j and each v ∈ Z ′ , the sets in (2.20) are finitely overlapping over k by Lemma 2.3, and therefore summing (2.20) over k gives ν {v j ∈ H : v ∼ j v j } tδ 100η+s−κ−1 . Similarly, summing (2.21) over l gives ν {v 2 ∈ H : v ∼ 2 v 2 : d E (ζ 2 , ℓ(ζ 1 , ζ 3 )) ≥ δ α if \|ζ − ζ 1 \|, \|ζ − ζ 3 \| ≥ t/2} tδ 100η+s−κ−1 . Using these two inequalities and Fubini's Theorem, gives ν 4 (v, v 1 , v 2 , v 3 ) ∈ Z ′ × H 1 3 : v ∼ j v j for all j, d E (ζ 2 , ℓ(ζ 1 , ζ 3 )) ≥ δ α if \|ζ − ζ 1 \|, \|ζ − ζ 3 \| ≥ t/2 = Z ′ {v1:v∼1v1} {v3:v∼3v3} {v2:v∼2v2 and dE(ζ2,ℓ(ζ1,ζ3))≥δ α if \|ζ−ζ1\|,\|ζ−ζ3\|≥t/2} dν 4 (v 2 , v 3 , v 1 , v) ν(Z ′ ) tδ 100η+s−κ−1 3 ≥ ν(Z)t 3 δ 301η+3(s−κ−1) , which implies the lower bound of (2.19). For the upper bound, fix v j = (ζ j , τ j ) for j ∈ {1, 2, 3}. Let A = A(v 1 , v 2 , v 3 ) := {v ∈ Z ′ : v ∼ j v j for all j, d E (ζ 2 , ℓ(ζ 1 , ζ 3 )) ≥ δ α if \|ζ − ζ 1 \|, \|ζ − ζ 3 \| ≥ t/2}. The upper bound of (2.19) will be obtained by bounding ν(A) and then integrating over v 1 , v 2 , v 3 . By the triangle inequality and Fubini, ν 4 (v, v 1 , v 2 , v 3 ) ∈ Z ′ × H 1 3 : v ∼ j v j for all j = H B H (v3,4t) B H (v3,4t) ν{v ∈ Z ′ : v ∼ j v j for all j} dν(v 1 ) dν(v 2 ) dν(v 3 ) t 3s . This proves the second case of (2.19), so it may be assumed that t > δ α−20η . Fix v ∈ A. For each j ∈ {1, 2, 3}, the inequality d H P V ⊥ θ (v), P V ⊥ θ (v j ) ≤ 2δ for some θ ∈ H ′ j (v), implies (2.26) \|τ − τ j − 2ζ ∧ ζ j \| δ, by a calculation similar to the derivation of (2.25). Hence if \|τ − τ j − 2ζ ∧ ζ j \| 1/2 ≥ \|ζ − ζ j \| for some j ∈ {1, 2, 3}, then d H (v, v j ) δ 1/2 since (2.26) corresponds to the second component of the Korányi distance, see (1.1). Therefore (2.27) ν{v ∈ A : \|τ − τ j − 2ζ ∧ ζ j \| 1/2 ≥ \|ζ − ζ j \| for some j ∈ {1, 2, 3}} δ s/2 . It remains to bound ν(A ′ ), where A ′ = {v ∈ A : \|τ − τ j − 2ζ ∧ ζ j \| 1/2 < \|ζ − ζ j \| for all j ∈ {1, 2, 3}}. Define G : H → R 3 by G(ζ, τ ) =   τ − τ 1 − 2ζ ∧ ζ 1 τ − τ 2 − 2ζ ∧ ζ 2 τ − τ 3 − 2ζ ∧ ζ 3   , so DG(ζ, τ ) = DG =   −2y 1 2x 1 1 −2y 2 2x 2 1 −2y 3 2x 3 1   , where ζ j = x j + iy j . Then A ′ ⊆ G −1 (B E (0, Cδ)) for some constant C, by (2.26). If v ∈ A ′ , then t 2 ≤ \|ζ − ζ j \| ≤ 2t for each j ∈ {1, 2, 3} by the definition of the Korányi metric. Hence if A ′ is nonempty and there exists v 0 ∈ A ′ , then by the condition d E (ζ 2 , ℓ(ζ 1 , ζ 3 )) ≥ δ α in the definition of A, ζ 2 = ζ 3 + λ 1 (ζ 1 − ζ 3 ) + λ 2 i(ζ 1 − ζ 3 ), λ 1 , λ 2 ∈ R with \|λ 2 \|\|ζ 1 − ζ 3 \| ≥ δ α . The inequality \|(ζ 1 − ζ) ∧ (ζ 3 − ζ)\| t 2 δ 4η follows similarly to the working from (2.23) to (2.24), and this gives \|(ζ 1 − ζ) ∧ (ζ 3 − ζ)\| ≥ δ 5η \|ζ 1 − ζ\|\|ζ 3 − ζ\|. Combining this with the identity \|z\| 2 \|w\| 2 = \| z, w \| 2 + \|z ∧ w\| 2 for z, w ∈ R 2 and expanding out \|( ζ 1 − ζ) − (ζ 3 − ζ)\| 2 gives \|ζ 1 − ζ 3 \| ≥ t 2 δ 5η 2 . Hence \|det DG\| = 4\|ζ 1 ∧ ζ 2 + ζ 2 ∧ ζ 3 + ζ 3 ∧ ζ 1 \| = 4 \|(ζ 1 − ζ 3 ) ∧ (ζ 2 − ζ 3 )\| = 4\|λ 2 \|\|ζ 1 − ζ 3 \| 2 ≥ 2tδ α+5η . By combining this with the formula (DG) −1 = (det DG) −1 adj DG for the inverse, where adj refers to the adjugate, the operator norm satisfies (DG) −1 t −1 δ −α−5η . Hence A ′ ⊆ G −1 (B E (0, Cδ)) ⊆ B E v 0 , C ′ t −1 δ 1−α−5η . The radius of this ball is less than 1 by the definitions of η, α in (2.16), (2.18) and the assumption t > δ α−20η . Proposition 1.3 therefore implies that A ′ can be covered by tδ −(1−α−5η) Korányi balls of radius t −1 δ 1−α−5η . Also, by Lemma 1.2 A ′ is contained in a single Korányi ball of radius t By the triangle inequality and Fubini, ν 4 (v, v 1 , v 2 , v 3 ) ∈ Z ′ × H 1 3 : v ∼ j v j for all j, d E (ζ 2 , ℓ(ζ 1 , ζ 3 )) ≥ δ α if \|ζ − ζ 1 \|, \|ζ − ζ 3 \| ≥ t/2 = H B H (v3,4t) B H (v3,4t) ν(A(v 1 , v 2 , v 3 )) dν(v 1 ) dν(v 2 ) dν(v 3 ) max t 2s δ s/2 , min t This implies the upper bound in the first part of (2.19), which finishes the proof of (2.19). If t ≤ δ α−20η , then by (2.19), ν(Z) ≤ δ 3(α(s−1)−(s−κ−1))−1001η ≤ δ η , by the definition of α in (2.18). Therefore it may be assumed that t ≥ δ α−20η . By comparing the lower and upper bounds from (2.19), (2.29) ν(Z)t 3 δ 1000η+3(s−κ−1) max t 2s δ s/2 , min t The definition of α and κ imply the inequality t 2s δ s/2 ≤ t by the definition of η in (2.16). This proves the lemma in the case 1 < s ≤ 2. In the remaining case s ≥ 2, using t 1 and taking the second term in the minimum yields ν(Z) ≤ max δ Proof of Theorem 1.1. Let A ⊆ H be an analytic set with s := dim A > 1 and let ǫ ∈ (0, s − 1). By Lemma 1.4 (Frostman), there is a nonzero, finite, compactly supported Borel measure ν on A with ν(B H (v, r)) ≤ r s−ǫ for every v ∈ H and r > 0. By Lemma 2.4 with κ = max 3(s − 1) 4 , 5s 6 − 1 > max 3(s − ǫ − 1) 4 , 5(s − ǫ) 6 − 1 there exist δ 0 , η > 0 such that ν v ∈ H : H 1 θ ∈ [0, π) : P V ⊥ θ # ν B H P V ⊥ θ (v), δ ≥ δ s−ǫ−κ ≥ δ η ≤ δ η , whenever δ ∈ (0, δ 0 ). By Lemma 2.1, for a.e. θ ∈ [0, π). dim P V ⊥ θ A ≥ dim P V ⊥ θ (supp ν) ≥ s − ǫ − κ for a.e. θ ∈ [0, π). Letting ǫ → 0 results in dim P V ⊥ θ A ≥ s − κ = For any R > 0, there exists a positive constant Proposition 1.3 ([4, Proposition 3.4]).For any R > 0, there exists a positive integer N = N (R) > 0 such that any Euclidean ball B E (v, r) with \|v\| ≤ R and r ∈ (0, 1) can be covered by at most ⌈N/r⌉ Korányi balls of radius r. Lemma 2 . 4 . 24Fix s ∈ (1, 4], let ν be a nonzero finite compactly supported Borel measure on H with sup . 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[ "Practical Approach of Knowledge Management in Medical Science", "Practical Approach of Knowledge Management in Medical Science" ]	[ "M Bohlouli \nDepartment of Computer Science and Electrical Engineering\nUniversity of Siegen\nNRW\nGermany\n", "P Uhr \nDepartment of Computer Science and Electrical Engineering\nUniversity of Siegen\nNRW\nGermany\n", "F Merges \nDepartment of Computer Science and Electrical Engineering\nUniversity of Siegen\nNRW\nGermany\n", "S Mohammad Hassani \nDepartment of Computer Science and Electrical Engineering\nUniversity of Siegen\nNRW\nGermany\n", "M Fathi \nDepartment of Computer Science and Electrical Engineering\nUniversity of Siegen\nNRW\nGermany\n" ]	[ "Department of Computer Science and Electrical Engineering\nUniversity of Siegen\nNRW\nGermany", "Department of Computer Science and Electrical Engineering\nUniversity of Siegen\nNRW\nGermany", "Department of Computer Science and Electrical Engineering\nUniversity of Siegen\nNRW\nGermany", "Department of Computer Science and Electrical Engineering\nUniversity of Siegen\nNRW\nGermany", "Department of Computer Science and Electrical Engineering\nUniversity of Siegen\nNRW\nGermany" ]	[]	Knowledge organization, infrastructure, and knowledge based activities are all subjects which help in creation of business strategies for new enterprise. In this paper, first basics of knowledge based systems are studied. Practical issues and challenges of Knowledge Management (KM) implementations are then illustrated. Finally comparison of different knowledge based projects is presented along with abstracted information on their implementation, techniques and results. Most of these projects are in the field of medical science. Based on our study and evaluation on different KM projects, we conclude that KM is being used in every science, industry and business. But its importance in medical science and assisted living projects is highlighted nowadays with the most of research institutes. Most medical centers are interested in using knowledge based services like portals and learning techniques of knowledge for their future innovations and supports.	null	[ "https://arxiv.org/pdf/2001.09795v1.pdf" ]	39,233,822	2001.09795	8234f0035cb1245782e2710cebf456b380c41cb2	Practical Approach of Knowledge Management in Medical Science M Bohlouli Department of Computer Science and Electrical Engineering University of Siegen NRW Germany P Uhr Department of Computer Science and Electrical Engineering University of Siegen NRW Germany F Merges Department of Computer Science and Electrical Engineering University of Siegen NRW Germany S Mohammad Hassani Department of Computer Science and Electrical Engineering University of Siegen NRW Germany M Fathi Department of Computer Science and Electrical Engineering University of Siegen NRW Germany Practical Approach of Knowledge Management in Medical Science Knowledge ManagementKnowledge Based SystemsPractical Knowledge ManagementKnowledge based Medical ApplicationsImplementation Issues of Knowledge Management Knowledge organization, infrastructure, and knowledge based activities are all subjects which help in creation of business strategies for new enterprise. In this paper, first basics of knowledge based systems are studied. Practical issues and challenges of Knowledge Management (KM) implementations are then illustrated. Finally comparison of different knowledge based projects is presented along with abstracted information on their implementation, techniques and results. Most of these projects are in the field of medical science. Based on our study and evaluation on different KM projects, we conclude that KM is being used in every science, industry and business. But its importance in medical science and assisted living projects is highlighted nowadays with the most of research institutes. Most medical centers are interested in using knowledge based services like portals and learning techniques of knowledge for their future innovations and supports. Introduction Knowledge Management (KM) is a concept of business administration and computer science, but it is not limited to these areas, and it brings interesting ideas which could be useful for other sciences. There are various definitions and discussions on what exactly KM is about [1, 3, 4, 5, and 6], the definition of KM and Knowledge Based Systems (KBS) are interconnected to the definition of knowledge. In practice, the terms knowledge, information and data are often used interchangeably, but giving a correct definition of knowledge is possible by distinguishing among knowledge, information and data [3]. A commonly held view with sundry minor variants is that data is raw numbers and facts, information is processed data, and knowledge is authenticated information [9 and 11]. In fact Data is unclassified and unprocessed values, a static set of transactional elements, such as 211102345, John is 6 feet tall, or structured records of transactions. Information is meaningful collection of data which has been processed and organized. Knowledge is a set of understandings and the state of knowing acquireed through experience or study to use in decision making activities. It includes facts, opinions, ideas, theories, principles, models, ignorance, awareness, familiarity, understanding, facility, and so on. Knowledge is based on data and Information. Information Systems (IS) can convert data into information, but it is not able to convert information to knowledge. KM and KBS As we mentioned before, definition of KM and KBS depend on knowledge. Alavi and Leidner [3] have defined five different perspectives for knowledge. KM involves enhancing individual's learning and understanding through provision of information. It is the process of creation, sharing, and distributing knowledge and to build and manage knowledge stocks. KM is also organized access to and retrieval of content. t is about building core competencies and understanding strategic know-how, too. [3] The role of IT in KBSs is to provide access to sources of knowledge rather than knowledge itself, gathering, storing, and transferring knowledge and to provide link among sources of knowledge to create wider breadth and depth of knowledge flows. Providing effective search and retrieval mechanisms for locating relevant information is also supported with IT in KBSs. IT can enhance intellectual capital by supporting development of individual and organizational competencies in KBSs [3]. Generally KM is any attempt to convert employees' knowledge (personal properties) to shared organizational knowledge (organizational property). It is aim to improve effectiveness and speedup, and decrease expenses of an organization. Giving correct knowledge in the suitable format and time to a proper expert or individual with reasonable costs is the aim of KM. Therefore: Knowledge is explicit and/or tacit, individual and/or organizational and authenticated information which has been acquireed through study or experience to potentially influence action, and applying expertise. KM is the practice of collecting knowledge and selectively applying it from previous experiences of decision making to current and future decision making activities with the express purpose of improving the organization's effectiveness [6]. KMS is any system which is aiming to improve the effectiveness and speedup of the organization based on the previous experiences and activities. KBS is able to create new knowledge, exchange the current knowledge, and apply all to the system. Ann Macintosh of the Artificial Intelligence Applications Institute has written a "Position Paper on Knowledge Asset Management" that identifies some of specific business factors about necessarily of KM in contemporary life, including: • Marketplaces are increasingly competitive and the rate of innovation is rising. • Reduce of staffs create a need to replace informal knowledge with formal methods. • Competitiveness reduces the size of the work force that holds valuable business knowledge. • The amount of time available to experience and acquire knowledge has diminished. • Early requirements and increasing mobility of work forces lead to loss of knowledge. • There is a need to manage increasing complexity as small operating companies are transnationals sourcing operations. • Change in strategic direction may result in the loss of knowledge in a specific area. As a result KM enables sharing of essential knowledge to complete organization tasks and provides improvements in organizations quality and human performance and competitive advantage. Practical issues and challenges of KM KM is being used in every science, industry and business. If developers and designers of KBS don't have enough information about implementation requirements of KM and KBS, their implementation results will not be as successful. In the following section we present some success factors and common failures of KM implementations. Any failure and mistake of individuals has a cost to the system. Therefore in case of any failure, the system should analyse and predicate enough information about the failure for the future scopes. By transferring acquired knowledge of failures to the current and incoming individuals, the amount of failures will be decreased in the system and therefore KM will succeed in decreasing the costs of organization. Feedback data from customers is very important in optimizing future generations of products and quality management. Therefore the system should provide a completely creative backbone to collect feedback data and keep a close relationship with customers and users. Any missing feedback data leads to unsuccessful scenarios of KM in optimizing system and quality issues. After obtaining the feedback data, the system should analyse, abstract, and preserve it. Due to staff awareness of former activities and projects some experiments are repeated in the system. As a result informing individuals about former works and projects in the organization is required. The progress of former experiments and results should be stored in a knowledge repository of the system and should be easily accessible to the individuals of the organization at any time. Due to security and trust issues, thoughts, ideas, and good functionalities are not often shared with expert. Successful KM implementation needs to create the trustworthy culture and teamwork in which professionals feel safe and secure, and can trust in the system; providing the safe and trustable culture is critical to KM success in any system. Designers and administrators' awareness of basic psychological rules lead to the defeat of KBS. Sharing of key information and progresses needs individuals' trust. Only a limited amount of people in the organizations are aware of key knowledge about their projects and works. That is because only a few people work in the key positions of projects, therefore when these people are absent, others miss key knowledge and it is more cost-intensive for an organization to reacquire the missed knowledge. When sharing of key knowledge is not possible generally, system should provide a secure process to preserve key knowledge. Security techniques such as encryption methods should be used to make it more secure. Access to this knowledge should be controlled securely and the system should allow the use of them just in specific cases, but not generally. Organizational Learning (OL) is always slow. Resulting inconveniences include delays in development of products or missing better opportunities. Therefore organizations need to employ better and more rapid technologies to accelerate their learning methods. Because of great developments in IT and their application in learning methods, it could be useful to optimize OL. Members of an organization will be disappointed when they couldn't acquire knowledge from the system. Therefore providing good access to the available knowledge resources is another requirement of KBSs. In addition to mentioned propositions, some other success factors of KM implementation are as follows: • Employee involvement • Information System (IS) infrastructure • Performance measurement • Benchmarking • Knowledge- KM in Medical Science Various fields of application, huge amounts of data and experts with specific know-how are the main reasons why the field of medical research can be seen as a precursor for other branches of applied KM. Evolutionary algorithms, fuzzy logic or neural networks have the potential of improving knowledge and can help medical practitioners in creating diagnoses for different diseases. Conducting knowledge audits are necessary to determine specific experts in a healthcare organization who have knowledge on a certain issue (e.g. disease or therapy). This process called "Knowledge Identification and Capture" and is one of the important steps in Medical KM (MKM). In the healthcare setting, knowledge creation can take place in terms of improved organizational processes and systems in hospitals, advances in medical methods and therapies, better patient relationship management practices, and improved ways of working within the healthcare organization. A healthcare organization is a collection of professional specialists who contribute to the delivery of patient care, but also often act competitively inside the organization, without being willing to transfer knowledge because of associated status and power within the organization and the society. Effective knowledge management in the Medical Application area as well as in other branches requires a "knowledge sharing" culture to be successful. Knowledge application refers to taking the shared knowledge and internalizing it within one's perspective and worldviews. Knowledge-enabling technologies which can effectively be applied to healthcare organizations are: Groupware, Intranet, Collaborative tools and Knowledge Portals. Knowledge based Medical Systems By incorporating applied KM and medical science many fields of application can be discovered. Modern medicine generates huge amounts of heterogeneous data [14] almost daily. Researchers and medical practitioners have to deal with data on one hand and with personnel know-how on the other hand. Knowledge discovery as an early step in the KM process transforms data into knowledge [14], and helps medical practitioners in that way to decrease information overload. Knowledge Management and Medical Engineering center (KMME) in corporation with some medical experts and clinical centers discovered a wide gap between raw data and concluded knowledge. Disparity of data collection and data comprehension makes computerized techniques necessary to help humans to address this problem [14]. In the following section three projects of the KMME and one another MKM project will be described briefly with their aims and benefits. StroPoS The Stroke Portal System for emergency risk patients (StroPoS) was implemented based on the concept of a corresponding portal solution [21]. The need for a portal like StroPoS results from serious influences strokes have on the association. Strokes are the fifth common cause of death in Germany in 2008 [25] and they are often affiliated with severe disablements. Early diagnosis supported by a Stroke Portal System could decrease the costs of therapy and aftercare immensely. The architecture of StroPoS System has been illustrated in Figure 1. Furthermore, a Stroke Portal System should offer a service which recognizes warnings. Concerning recognizing warnings, a Medical Call Center (MCC) with agents can help as well. The KMME developed several methods of KM within this portal which will be integrated and realized in the next steps of the project. Patients and experts can access StroPoS via Web browser. Some other components like the Risk Test (RT) and Stroke Information Unit (SIU) can be accessed by the MCC. The system is divided into two parts: Patient section and expert section. While patients have free access to information like definition and descriptions about the stroke disease, experts have to log in to get access to the expert area on the portal. After this authentication procedure they can use the CDB, which includes patient case studies, as well as the SDB for DB queries. In the database experts can find special stroke cases and can set up several parameters to localize a case. Experts can also make an anonymous case file and upload it to share with other experts. The challenge was to implement a program structure by abstracting the anonymous example health records for entering, saving and comparing different courses of disease [20]. With the RT, users can test their stroke risk. The test contains ten questions which are easy to understand and be proved by clinical partners of KBS group. Users have also the opportunity to search for special stroke clinics. To find the nearest clinic users need only to type their post code, and system will find nearest stroke clinic. 3.1.2 DiProS Disc Prolapse System (DiProS) is another knowledge-based medical application implemented by KMME at the University of Siegen. The major goal of the project is to achieve a modular and structural portal in the field of disc herniation disease. DiProS focuses on allocation of reliable and authentic information supplemented by additional knowledge for experts and individuals, regardless if they are just interested or affected by spinal disc herniation. The offered information in the portal is extended by interactive services, which allow instantaneous contact to medical practitioners and make telemedical consultation possible. Methods for estimation of individual patients are also offered in the portal. The reuse of components like the CBD or the forums saved a lot of time but it also shows that the components can be reused in other projects. Besides the well-proven components, new components have been developed, for instance, the navigation assistant which helps users to navigate through the system and find the information they need. 3.1.3 AlWiP This project is another implemented project by KMME. The major goal of the Alzheimer Wissen Portal (AlWiP) project is the early detection of Alzheimer disease indicators. Reasons, symptoms and factors of risk have to be evaluated through data and picture material (i.e. brain MRT's, course of brain alteration) in terms of correlation and integrated into the Knowledge portal as a decision support component. The whole project aims to create a knowledge management portal for the medical application of Alzheimer's disease. An image analysis of the brain, combined with information about the general state of health of patients can indicate the danger of developing Alzheimer's. (Fig. 3) With the help of the Alzheimer's and dementia specialists at the University of Cologne (the cooperation partners of the KMME), an Alzheimer's knowledge database was built for users such as family members of patients and experts, in which different forms of dementia were shown. In the discussion area, family members are able to share their experience with other concerned persons. In this way people can share their knowledge and find solution for their special problems. The portal should not replace other information sources like health personnel, expert's guides [18] or caregiver's guides [7]. In fact the portal is planned as a knowledge platform which completes these traditional resources. Beside this AlWiP is a quick and easy opportunity to get Alzheimer information on demand. For best possible user support a navigation assistant is implemented in the portal. It enables the user to ask questions like 'what is dementia'. Another new feature is that if there is reason why a user cannot read articles, the assistant can read out articles and every text file on the portal Therefore it is possible for that person to receive needed information by using the assistant. 3.1.4 OncoDoc Developed and implemented in the Service d'Oncologie Medicale Pitie-Salpetriere (Paris, France) [12], OncoDoc is a decision support system designed to provide best therapeutic recommendations for breast cancer patients. While clinical trials offer cancer patients the optimum treatment, historical accrual of such patients has not been very successful. Developed as a browsing tool of a knowledge base structured as a decision tree, OncoDoc allows physicians to control the contextual instantiation of patient characteristics to build the best formal equivalent of an actual patient [26]. Originally published as textual documents, clinical practice guidelines have poorly penetrated medical practice because their editorial properties do not allow the reader to easily solve a given medical problem at the point of care [15]. OncoDoc has been developed as a guideline-based decision support system to provide best treatment recommendations [15]. It is non-automated and allows flexibility in guideline interpretation to obtain best patient-specific recommendations at the point of care [24]. Rather than providing automated decision support, OncoDoc allows medical practitioners to control the operationalization of guideline knowledge through his hypertextual reading of a knowledge base encoded as a decision tree. In this way, physicians have the opportunity to interpret the information provided in the context of the patient, therefore, controlling the categorization to the closest matching formal patient [15]. Within the tree different parameters describe the actual state of a patient. The complete expansion of the decision tree can be seen as a patient centered repository of all theoretical clinical situations that could be met within breast cancer pathology [26]. Recommendations based on the clinical state of the patient are displayed at the lower side of the decision tree and the path through it represents the clinical profile of the patient. (Fig. 4) The System has been tested at the Institut Gustave Roussy (IGR) with a before-after study in which treatment decisions for breast cancer patients were measured before and after using the system in order to evaluate its impact upon physicians' prescribing behaviour [24]. To assess how the system could be reused in another institution which was not involved in the development process, a new experiment at IGR was conducted. Minor site-specific customizations of the knowledge base were performed. After four months, 127 cases were recorded. Results showed that there was no significant difference of physician compliance with OncoDoc (85%) when site-specific recommendations were, or not, available, although local recommendations were chosen preferably (55%), thus legitimating the adaptation [12]. Summing up, the system uses patient characteristics to discriminate between different treatments alternatives, furthermore it offers matching clinical trials for the actual case. Physicians can be supported in disseminating between next therapy steps. Conclusions The domain and benefit of KM is not limited to a specific science or industry, its techniques are being implemented in each field and industry. For example, every company wants to acquire knowledge with less time and cost, and most research and innovation is based on traditional knowledge and information. All of these are possible with only the use of KM systems and techniques. After studying different projects and their focus we conclude that KM is going to be more important and useful in medical science and assisted living, implementing knowledge based portals and knowledge learning techniques for future innovations. Of course KM is not limited to medical science, but it has had very sensitive growth and success in this area. Most medical centers and doctors want to function in real time and medicate their patients immediately, therefore research in assisted living projects is the most highlighted and this could be more efficient by employment of KM. First minutes of most medical attacks are very important for the consequences of disease. When patients are living in their private areas, direct support of the patient during the first minutes of an attack is not possible for the medical and support team. Therefore having urgent remote medicating and guidance system could be more helpful to decrease the consequences of diseases. That is the basic idea of assisted living projects which is common in most research institutes nowadays. Since some diseases need full time controlling and it is not possible for everybody to employ a full time nurse, the importance of remote and intelligent controlling systems are clear for everybody. One example of these applications could be knowledge based portals. They are able to share the knowledge of experts with individuals, exchange professional knowledge between experts, find the best fit and nearest medical centers for patients, find and introduce special cases of diseases for experts, establish panel discussion between experts and individuals, and so many other facilities. All of these services use different categories and techniques of KM. As a result KM is used in most sciences and industries because of its reasonable advantages to the systems, but nowadays it is more successful and helpful in medical sciences and most medical centers are interested in implementation and using KBSs. In this paper we have studied some failures in implementation of KM in organizations and some case studies of KM in medical science. Of course we know that challenges and success factors of KM implementations are not just limited to the what we have mentioned, but these are the most basic and mandatory conditions which every KBS should support. We think that KM will be more innovative and important given time. 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[ "a 1 properties in hadronic tau decays", "a 1 properties in hadronic tau decays" ]	[ "Ina Lorenz [email protected] ", "Emilie Passemar [email protected] ", "\nDepartment of Physics\nCenter for Exploration of Energy and Matter\nIndiana University\n47405BloomingtonINUSA\n", "\nDepartment of Physics\nIndiana University\n47408BloomingtonINUSA\n", "\nCenter for Exploration of Energy and Matter\nIndiana University\n47405BloomingtonINUSA\n", "\nTheory Center, Thomas Jefferson National Accelerator Facility\nIndiana University\n47408, 23606Bloomington, Newport News, ChicagoIN, VAUSA, USA, USA\n" ]	[ "Department of Physics\nCenter for Exploration of Energy and Matter\nIndiana University\n47405BloomingtonINUSA", "Department of Physics\nIndiana University\n47408BloomingtonINUSA", "Center for Exploration of Energy and Matter\nIndiana University\n47405BloomingtonINUSA", "Theory Center, Thomas Jefferson National Accelerator Facility\nIndiana University\n47408, 23606Bloomington, Newport News, ChicagoIN, VAUSA, USA, USA" ]	[ "38th International Conference on High Energy Physics" ]	Hadronic tau decays belong to the processes that show a resonance-like structure in the axial vector current in the 1 − 2 GeV range. This structure, often denoted as the a 1 meson, seems to show different properties in different processes. The process τ → 3πν τ allows for a clean separation of weak and strong effects and a clear production mechanism. We examine how this structure can be related to interactions between the three pions that emerge in the final state. In particular we start from the interactions between all two body combinations.	10.22323/1.282.0539	[ "https://arxiv.org/pdf/1702.05432v1.pdf" ]	119,029,114	1702.05432	e4447f31050531a4a5edfcb7b104ee08c54da72a	a 1 properties in hadronic tau decays 17 Feb 2017 3-10 August 2016 Ina Lorenz [email protected] Emilie Passemar [email protected] Department of Physics Center for Exploration of Energy and Matter Indiana University 47405BloomingtonINUSA Department of Physics Indiana University 47408BloomingtonINUSA Center for Exploration of Energy and Matter Indiana University 47405BloomingtonINUSA Theory Center, Thomas Jefferson National Accelerator Facility Indiana University 47408, 23606Bloomington, Newport News, ChicagoIN, VAUSA, USA, USA a 1 properties in hadronic tau decays 38th International Conference on High Energy Physics 17 Feb 2017 3-10 August 2016* Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ Hadronic tau decays belong to the processes that show a resonance-like structure in the axial vector current in the 1 − 2 GeV range. This structure, often denoted as the a 1 meson, seems to show different properties in different processes. The process τ → 3πν τ allows for a clean separation of weak and strong effects and a clear production mechanism. We examine how this structure can be related to interactions between the three pions that emerge in the final state. In particular we start from the interactions between all two body combinations. Hadronic tau decays belong to the processes that show a resonance-like structure in the axial vector current in the 1 − 2 GeV range. This structure, often denoted as the a 1 meson, seems to show different properties in different processes. The process τ → 3πν τ allows for a clean separation of weak and strong effects and a clear production mechanism. We examine how this structure can be related to interactions between the three pions that emerge in the final state. In particular we start from the interactions between all two body combinations. The axial vector current in the few GeV range plays a role in many processes, for example decays of B and D mesons, Higgs decays and neutrino scattering. The models employed to describe this current partly boil down to summing up Breit Wigner lineshapes and fitting to the invariant mass of the final state system, e.g. in τ → 3πν τ . The dominant resonance-like structure is denoted as a 1 meson. However, the properties of this meson, as listed in the PDG, depend on the production mechanism, and contrast significantly between πP → 3πP and the tau decay. This situation calls for a conceptual improvement of the models. On one hand, of course the Dalitz plot distributions should be considered in any resonance model. On the other hand, more information on the spin structure is available from structure functions, observables that are directy related to helicity amplitudes. In order to use this additional information we refrain from any Breit Wigner parametrization and start from the two pion interactions which are known from ππ scattering. Definitions We consider the semileptonic decay, see Fig. 2, τ(l 1 ) → ν τ (l 2 ) + π(p 1 ) + π(p 2 ) + π(p 3 ),(2.1) and its description following Refs. [1,2]. The general amplitude is M = cos θ C G F √ 2 L µ H µ . (2.2) where θ C is the Cabibbo angle. The leptonic part can be written in the standard model as L µ = u(l 2 )γ µ (1−γ 5 )u(l 1 ). For the hadronic part of the tau decay, we can write the generic matrix element state pions, the vector contribution vanishes and we consider only the axial part in the following. One possible decomposition is into form factors, as used e.g. in Ref. [2]. However, we use instead a decomposition of the matrix element into helicity amplitudes, since these have a simple expansion into partial waves. The partial waves yield a simple form of the unitarity relations that we are interested in. The helicity amplitudes can be expressed in the Mandelstam plane where s,t, u correspond to the two-by-two scattering element π(p 1 )π(p 2 )\|π(−p 3 )A µ (Q µ ) . For three body decays often denoted as Dalitz plot invariants s 1 , s 2 and s 3 , here we use H i jkl µ = π i (p 1 )π j (p 2 )π k (p 3 )\|V l µ (0) − A l µ (0)\|0 , V l µ = 1 2q τ l γ µ q, A l µ = 1 2q τ l γ µ γ 5 q,(2.s = (p 1 + p 2 ) 2 , t = (p 2 + p 3 ) 2 , u = (p 1 + p 3 ) 2 , Q 2 = s + t + u − 3M 2 π . (2.4) The center-of-mass scattering angle in each channel, θ s , θ t and θ u , respectively, are related to the Kacser function K(s) = t − u cos θ s = λ (s, M 2 π , M 2 π ) λ (s, Q 2 , M 2 π ). (2.5) The Källén function λ (a, b, c) = a 2 + b 2 + c 2 − 2(ab + bc + ca) can be written λ ab (s) = λ (s, M 2 a , M 2 b ) = [s − (M a − M b ) 2 ][s − (M a + M b ) 2 ]. (2.6) With the definitions L µν = L µ L † ν and H µν = H µ H † ν of the leptonic and hadronic tensor the differential decay rate is given by dΓ(τ → ν τ 3π) = 1 2m τ \|M \| 2 dΦ = G 2 F 4m τ cos 2 θ C L µν H µν dΦ, L µν H µν = ∑ X L X W X , (2.7) where dΦ is the phase space element. L µν and H µν , can be combined to form 16 symmetric and antisymmetric structure functions W X . One useful basis for the hadronic structure functions W X is defined via the polarization of the final state system. Consider the polarization vectors ε µ (λ ) of the three pions in their c.m. frame or the W boson in its rest frame, respectively. We can now define the helicity amplitudes A i jkl λ := π i (p 1 )π j (p 2 )π k (p 3 )\|A l µ (0)ε µ (λ )\|0 , (2.8) where the subscript denotes the helicity. The outgoing pions have the two possible physical states \|π 0 π 0 π ± and \|π + π − π ± , that can be related by their isospin structure and crossing symmetry. In the following we will consider A π 0 π 0 π ± λ (s,t, u)=A 3311 λ (s,t, u) and neglect isospin breaking. Method and parametrization We approximate the transverse amplitude similar to Refs. [3,4], A 3311 + (s,t, u) ∝ l max ∑ l=0 ∑ I (2l + 1) d l 10 (θ s ) K(s) 4s l−1 P 3311 I a +,Il (s) + d l 10 (θ t ) K(t) 4t l−1 P 3131 I a +,Il (t) + d l 10 (θ u ) K(u) 4u l−1 P 1331 I a +,Il (u) ,(3.1) where P i jmn I is the isospin projection operator. The relevant Wigner d-matrix is given by d l 10 (θ ) = − sin θ / l(l + 1)P ′ l (cos θ ), where the prime denotes a derivative of the Legendre polynomial. The above expansion results in partial waves a Il that contain no kinematical but only dynamical cuts. This allows us to relate the parts of the partial waves that contain the left-and right-hand cuts a right/le f t Il (s) in an iterative procedure suggested by Khuri and Treiman [5]. In the following we always consider a Il = a +,Il . For each channel, we can write the discontinuity as a sum of the unitarity cut in this channel and those from the crossed channel as a le f t Il Disc a Il (s) = ρ(s)t * l (s) a right Il (s) + a le f t Il (s) , (3.2) where ρ(s) = 1 − 4M 2 π /s and t l (s) is the partial wave of the two-pion system, well-known from ππ scattering. This discontinuity enters the standard dispersion relation, e.g. unsubtracted, a right Il (s) = 1 π ∞ s 0 ds ′ Disc a right Il (s ′ ) s ′ − s , s 0 = 4M 2 π . (3.3) Expanding A 3311 + (s,t, u) in the s-channel physical region, comparing to Eq. (3.1), multiplying both sides with P ′ l (z s ) and integrating over z s = cos θ s we can write a le f t Il (s) ∝ ∑ I ′ ,l ′ (2l ′ + 1) +1 −1 dz s (1 − z 2 s )P ′ l (z s ) P ′ l ′ (z t )C II ′ st a I ′ l ′ (t(s, z s )) + P ′ l ′ (z u )C II ′ su a I ′ l ′ (u(s, z s )) , (3.4) where C st/su are the standard crossing matrices, see e.g. Ref. [4]. To find a solution of this set of equations, we parametrize the transverse partial wave amplitudes similiar to Ref. [6], as a Il (s) = Ω Il (s) n−1 ∑ i c i s i + s n π ∞ s 0 ds ′ s ′n ρ(s ′ )t * l (s ′ ) Ω * Il (s ′ ) a le f t Il (s ′ ) (s ′ − s) , Ω Il (s) = exp s π ∞ s 0 ds ′ s ′ δ Il (s ′ ) s ′ − s , (3.5) where the Omnès functions Ω Il (s) contain the unitary cut in s, and we use their parametrization from Ref. [7]. The term in brackets in Eq. (3.5) contains the cuts from the crossed channels and corresponds to an n-times subtracted dispersion relation with the subtraction constants c i . In a first step the left-hand cuts can be set to zero. However, three main restrictions of this approach are relevant in our case. First, the framework relies on the assumption that two body interactions dominate. This assumption is only justified at low energy, Q 2 ≪ 1 GeV 2 . Second, the truncation of Eq. (3.1) induces an uncertainty that has to be tested in practice. Third, a precise knowledge of the individual waves decreases with increasing energy. Preliminary results Our calculation for the helicity amplitudes can directly be compared to the experimentally determined structure functions. All structure functions that are not compatible with zero according to CLEO [8] can be related to W A (s,t, u) ∝ \|A 3311 [1]. In Fig. 2 we show the structure function W A given by the CLEO collaboration in the corresponding bins and our fit result. Here we ignore the left hand cuts which corresponds to the first iteration step in a Khuri Treiman approach. For a complete analysis, see Ref. [9]. The dotted lines show the binning in Q 2 , the solid line bars correspond to bins in s and t and the red dashed line to our preliminary fit. Changing the variables by Eq. (2.4) and integrating W A (Q 2 , s,t) over s and t yields the integrated structure functions w A,int (Q 2 ) shown in Fig. 3. Here, a three body resonance-like structure occurs with zero. We thus approximate the decay rate by the transverse component [1] dN + (s,t, u)\| 2 + \|A 3311 − (s,t, u)\| 2NdQ 2 λ =1 ∝ (M 2 τ − Q 2 ) Q 2 2 1 + 2Q 2 /M 2 τ w A,int (Q 2 ). (4.1) The comparison to the decay rates from CLEO and ALEPH is given in Fig. 4. Due to the very coarse grained bins in s and t, we show the binning in Q 2 . Again, the fit does not contain a specific parametrization of the three body resonance like a Breit Wigner, but merely two body interactions. This might hint towards an interesting origin of the a 1 meson, and/or towards the necessity for (1/N)dN/ds ALEPH_2005: τ→π2π 0 CLEO, from W A (Q 2 , s, t) fit to CLEO-W A (Q 2 , s, t) Figure 4: Decay rate from a fit to CLEO structure functions, also compared to ALEPH [10]. more iterations in the partial wave procedure or to include also three body unitarity. As a feasibility study, this work shows a good first description of the structure function and the tau decay rate. Therefore for a future detailed analysis it would be desirable to obtain the Dalitz plot distributions for a direct analysis, in particular from more precise measurements by Belle and BABAR. The full Dalitz plot information will help to separate the different uncertainties, namely the knowledge of the Omnès functions, the range of applicability of the approach and the truncation error. Figure 1 : 13) where i jkl are isospin indices and τ l the Pauli matrices in isospin space. For an odd number of final Schematic rescattering of pions from the tau decay. Figure 2 :Figure 3 : 23Structure functions from CLEO[8]. and can be reproduced qualitatively by our parametrization based on two body interactions. Both figures on structures functions show a better agreement with the data for lower bin numbers or lower Q 2 values, respectively. For this kinematical region the Omnès functions are known with higher precision. 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Shi et al., Dispersive analysis of ω/φ → 3π, πγ , Phys. Rev. D91 (2015) 094029, [1409.7708]. Three-body final state interaction in η → 3π. P Guo, I V Danilkin, D Schott, C Fernández-Ramírez, V Mathieu, A P Szczepaniak, 10.1103/PhysRevD.92.0540161505.01715Phys. Rev. 9254016P. Guo, I. V. Danilkin, D. Schott, C. Fernández-Ramírez, V. Mathieu and A. P. Szczepaniak, Three-body final state interaction in η → 3π, Phys. Rev. D92 (2015) 054016, [1505.01715]. Pion-Pion Scattering and K ± → 3π Decay. N N Khuri, S B Treiman, Phys. Rev. 119N. N. Khuri and S. B. Treiman, Pion-Pion Scattering and K ± → 3π Decay, Phys. Rev. 119 (1960) 1115-1121. The decay η → 3π: study of the Dalitz plot and extraction of the quark mass ratio Q. G Colangelo, S Lanz, H Leutwyler, E Passemar, 1610.03494G. Colangelo, S. Lanz, H. Leutwyler and E. Passemar, The decay η → 3π: study of the Dalitz plot and extraction of the quark mass ratio Q, [1610.03494]. The Pion-pion scattering amplitude. IV: Improved analysis with once subtracted Roy-like equations up to 1100 MeV. R Garcia-Martin, R Kaminski, J R Pelaez, J Ruiz De Elvira, F J Yndurain, 10.1103/PhysRevD.83.074004Phys. Rev. 83740041102.2183R. Garcia-Martin, R. Kaminski, J. R. Pelaez, J. Ruiz de Elvira and F. J. Yndurain, The Pion-pion scattering amplitude. IV: Improved analysis with once subtracted Roy-like equations up to 1100 MeV, Phys. Rev. D83 (2011) 074004, [1102.2183]. Structure functions in the decay τ ± → π ± π 0 π 0 ν τ. T E Browder, CLEO collaborationhep-ex/9908030Phys. Rev. 6152004CLEO collaboration, T. E. Browder et al., Structure functions in the decay τ ± → π ± π 0 π 0 ν τ , Phys. Rev. D61 (2000) 052004, [hep-ex/9908030]. . I Lorenz, T Husek, E Passemar, M Zdráhal, A Friedland, in preparationI. Lorenz, T. Husek, E. Passemar, M. Zdráhal and A. Friedland, in preparation . Update of the ALEPH non-strange spectral functions from hadronic τ decays. M Davier, A Höcker, B Malaescu, C.-Z Yuan, Z Zhang, 10.1140/epjc/s10052-014-2803-91312.1501Eur. Phys. J. 742803M. Davier, A. Höcker, B. Malaescu, C.-Z. Yuan and Z. Zhang, Update of the ALEPH non-strange spectral functions from hadronic τ decays, Eur. Phys. J. C74 (2014) 2803, [1312.1501].	[]
[ "POSSIBLE SIGNATURE OF LOW SCALE GRAVITY IN ULTRA HIGH ENERGY COSMIC RAYS", "POSSIBLE SIGNATURE OF LOW SCALE GRAVITY IN ULTRA HIGH ENERGY COSMIC RAYS" ]	[ "R V Konoplich \nNew York University\nNew YorkUSA\n", "S G Rubin \nMoscow Engineering Physics Institute\nMoscowRussia\n\nCenter for Cosmoparticle Physics \"Cosmion\"\nMoscowRussia\n" ]	[ "New York University\nNew YorkUSA", "Moscow Engineering Physics Institute\nMoscowRussia", "Center for Cosmoparticle Physics \"Cosmion\"\nMoscowRussia" ]	[]	We show that the existence of low scale gravity at TeV scale could lead to a direct production of photons with energy above 10 22 eV due to annihilation of ultra high energy neutrinos on relic massive neutrinos of the galactic halo. Air showers initialized in the terrestrial atmosphere by these ultra energetic photons could be collected in near future by the new generation of cosmic rays experiments.	10.1134/1.1316807	[ "https://arxiv.org/pdf/astro-ph/0005225v1.pdf" ]	119,389,853	astro-ph/0005225	e4e4b27a702d61fc68c62de160db7080c09b650a	POSSIBLE SIGNATURE OF LOW SCALE GRAVITY IN ULTRA HIGH ENERGY COSMIC RAYS May 2000 December 24, 2018 R V Konoplich New York University New YorkUSA S G Rubin Moscow Engineering Physics Institute MoscowRussia Center for Cosmoparticle Physics "Cosmion" MoscowRussia POSSIBLE SIGNATURE OF LOW SCALE GRAVITY IN ULTRA HIGH ENERGY COSMIC RAYS May 2000 December 24, 2018arXiv:astro-ph/0005225v1 10 1/ √ G N ≃ 1.22 · 10 19 GeV is related to the new mass scale M s by Gauss's law: We show that the existence of low scale gravity at TeV scale could lead to a direct production of photons with energy above 10 22 eV due to annihilation of ultra high energy neutrinos on relic massive neutrinos of the galactic halo. Air showers initialized in the terrestrial atmosphere by these ultra energetic photons could be collected in near future by the new generation of cosmic rays experiments. Recently it was proposed [1] that the space is 4+n dimensional, with the Standard Model particles living on a brane. While the weakly, electromagnetically, and strongly interacting particles are confined to the brane in 4 dimensions, gravity can propagate also in extra n dimensions. This approach allows to save the gauge hierarchy problem by introducing a single fundamental mass scale (string scale) M s of the order of TeV. The usual Planck scale M P l = 1/ √ G N ≃ 1.22 · 10 19 GeV is related to the new mass scale M s by Gauss's law: M 2 P l ∼ R n M n+2 s(1) where G N is the Newton constant, R is the size of extra dimensions. It follows from (1) that R ∼ 2 · 10 −17 ( T eV M s )( M P l M s ) 2/n , cm(2) gives at n = 1 too large value, which is clearly excluded by present gravitation experiments. On the other hand n ≥ 2 gives the value R 0.25 cm, which is below the present experimental limit ∼1 cm but can be tested for the case n = 2 in gravitational experiments in near future. It can be shown that the graviton including its excitations in the extra dimensions, so-called Kaluza-Klein (KK) graviton emission, interacts with the Standard Model particles on the brane with an effective amplitude ∼ M −1 s instead of M −1 P l . Indeed, the graviton coupling to the Standard Model particle ∼ M −1 P l , the rate [2] of the graviton interaction r ∼ (M −1 P l ) 2 N , where N is a multiplicity of KK-states. Since this factor is ∼ ( √ SR) n , where √ S is the c.m. energy, then substituting R from (2) we get r ∼ M −2 s . Thus the graviton interaction becomes comparable in strength with weak interaction at TeV scale. This leads to the varieties of new signatures in particle physics, astrophysics and cosmology (see e.g. [2,3,4,5,6] ) which have already been tested in experiments or can be tested in near future. In this article we consider the possible signature of the low scale gravity in ultra high energy cosmic rays. The detection [7,8] of cosmic rays with energy above Greisen-Zatsepin-Kuzmin (GZK) cut-off of ∼ 5 · 10 19 eV presents a serious problem for interpretation. The origin of GZK cut-off [9] is due to resonant photoproduction of pions by protons on cosmic microwave background radiation which leads to a significant degradation of proton energy (about 20% for 6 Mpc) during its propagation in the Universe. Of course, proton energy does not change by many orders of magnitude if high energy protons come from the distances < 50 -100 Mpc. However, no nearby sources like active galactic nuclei have been found up to now in the arrival direction. It is difficult also to relate the observed ultra high energy events ( Fig.1 [8]) with the other particles. For example in the case of ultra high energy photons due to interaction with cosmic background radiation (γ + γ * −→ e + + e − ) the photon free mean path should be significantly less than 100 Mpc. A scenario based on direct cosmic neutrinos able to reach the Earth from cosmological distances can not reproduce the observed signatures of ultra high energy air showers occurred high in the atmosphere. Different possibilities were considered (see e.g. [10] and references therein) in order to solve this puzzle. In particular it was proposed [11,12,13] that ultra high energy neutrinos reaching the Earth from cosmological distances interact with a halo of relic light neutrinos in the Galaxy, producing due to Z, W ± boson exchange secondaries inside the galactic halo. Photons from π 0 decays and nucleons can easily propagate to the Earth and be the source of the observed ultra high energy air showers. Critical elements of models [11,12,13] are: the existence of neutrino mass in the range 0.1-10 eV and significant clustering of relic neutrinos in the halo up to 10 5 n ν , where n ν is the cosmological neutrino number density (n ν ∼ 100cm −3 ). Also the existence of ultra high energy (> 10 21 -10 23 eV) neutrino flux is necessary in order to produce multiple secondaries with energies above GZK cut-off. However if the graviton interaction comparable in strength with weak inter-action at TeV scale exists then photons can be produced directly in a reaction ν + − ν −→ g −→ γ + γ(3) due to virtual graviton exchange (Fig.2). In Standard Model process (3) occurs via loop diagram and therefore is severely suppressed. At high energies the cross section for the process (3) can be obtained immediately from that for the process e + e − −→ γγ including graviton exchange (see for example [3]) by substituting e = 0. Then dσ dz = π 16 S 3 M 8 s F 2 (1 − z 4 )(4) where √ S is c.m.s. energy, z = \|cos θ\| is the polar angle of the outgoing photon. The factor F depends on the number of extra dimensions: F = log(M 2 s /S) 2/(n − 2) , n = 2 n > 2 , at √ S << M s . In Eq.(4) it is also taken into account that primary beam of neutrinos is polarized. Integrating (4) over the polar angle and including a symmetry factor for two γ we get σ = π 20 S 3 M 8 s F 2 ≈ 7 · 10 −35 F 2 √ S T eV 6 T eV M s 8 cm 2 .(5) On can see from (5) that at TeV energies the rate of the reaction (3) is comparable with the rate of weak processes [11]. Assuming M s ∼ √ S ∼ TeV we find for example for n = 3 the following probability for the interaction of ultra high energy neutrinos inside the galactic halo: P ≈ σn G L G ∼ 10 −3 , where L G ∼ 100 Kpc is the size of the galactic neutrino halo, n G ∼ 10 5 n ν is the neutrino number density in the galactic halo. This probability is significantly greater than the probability of ultra high energy neutrino interaction in terrestrial atmosphere [14]. Let us note that nearby galaxies also can be sources of additional ultra high energy photons due to neutrino interaction with relic neutrinos of galactic halos [12,13]. TeV range in c.m.s. corresponds to the energy of extragalactic neutrino flux E ≈ 10 22 − 10 23 eV since E ≈ S 2m ≈ 5 · 10 22 √ S T eV 2 10eV m eV (6) where m is neutrino mass. Photon distribution in reaction (3) in laboratory system is given by dσ d(ω/E) = 8πF 2 m 3 E 3 M 8 s ω E (1 − ω E )[(1 − ω E ) 2 + ( ω E ) 2 ](7) where ω >> m is photon energy. This distribution is shown in Fig.3. It follows from (7) that photons are produced in the reaction (3) mainly within the energy range 0.2E ω 0.8E with an average energy ≈ E/2. Therefore existence of low scale gravity at TeV scale or above could lead to the direct production of photons with energy ω > 10 22 eV (at these energies the mean interaction length for pair production for photons in the radio background is ≈ 1 − 10M pc [15]). Such photons can be hardly produced in standard weak interaction processes because in last ones photons appear as a result of cascade processes significantly reducing photon energy in comparison with the initial neutrino energy. For example, as it was shown in [11] final energy of photons produced due to cascade processes can be by 10-100 times less than the energy of the initial neutrino flux. Of course photons with the energy ∼ 10 23 eV could be produced in cascade processes induced by neutrinos of the energy > 10 24 − 10 25 eV but from the observations of cosmic rays we know that cosmic ray fluxes decrease with the energy as E −3 , and therefore the probability of such events is significantly suppressed. Fluxes of ultra high energy cosmic rays at the Earth are very small Φ ∼ 0.03km −2 sr −1 yr −1 . Until now only about 60 events were collected with energies above GZK cut-off. However in near future improved Fly's Eye (7000 km 2 sr) [8] will allow to detect about 20 events/yr. It seems possible that such detector could collect rare ultra energetic photons (ω > 10 22 eV ). The detection of such events could be an indication that these ultra high energy photons were produced in ν − ν annihilation in the galactic halo due to effects of low scale gravity at TeV scale. Authors thank D.Fargion for interesting discussions on ultra high energy cosmic rays. 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[]	[ "Instantons ", "A D Large N C Limit " ]	[]	[]	In this contribution we discuss our current understanding of instanton effects in the large Nc limit of QCD. We argue that the instanton liquid can have a smooth large Nc limit which is in agreement with scaling relations derived from Feynman diagrams. We also discuss certain limits of QCD, like the case of high baryon density, in which the Witten-Veneziano relation can be derived from QCD and is saturated by instantons.	10.1142/9789812701725_0003	[ "https://arxiv.org/pdf/hep-ph/0412215v2.pdf" ]	119,347,313	hep-ph/0412215	75ee341c53d784aecb64d54622ea685f31e706cc	2004 Instantons A D Large N C Limit 2004 In this contribution we discuss our current understanding of instanton effects in the large Nc limit of QCD. We argue that the instanton liquid can have a smooth large Nc limit which is in agreement with scaling relations derived from Feynman diagrams. We also discuss certain limits of QCD, like the case of high baryon density, in which the Witten-Veneziano relation can be derived from QCD and is saturated by instantons. Introduction In the limit in which the masses of the light up, down and strange quarks are taken to zero, and the masses of the heavy quarks are taken to infinity QCD is a parameter free theory. This is one of the aspects of QCD that make it such a beautiful theory, as it implies that all dimensionful numbers, like the masses and radii of hadrons, can be expressed in terms of a single dimensionful quantity, Λ QCD . It also implies, however, that there is no expansion parameter that can be used to perform systematic calculations. Many years ago 't Hooft suggested to consider the limit in which the number of colors, N c , is large and to use 1/N c as an expansion parameter 1 . In order to keep the QCD scale parameter fixed we have to take the N c → ∞ limit with the 't Hooft parameter λ = g 2 N c constant. In the large N c limit the perturbative expansion in Feynman diagrams is replaced by an expansion in the genus of the two-dimensional Riemann surface spanned by the diagrams. This result fits very well with the idea that the large N c limit of Yang-Mills theory is equivalent to a string theory. For N = 4 SUSY Yang-Mills theory an explicit realization of this idea is provided by the AdS/CFT correspondence, but in the case of QCD the precise form of the string theory is not known. An interesting problem arises if we consider the fate of the axial anomaly in the large N c limit. For this purpose we add a θ term L = ig 2 θ 32π 2 G a µνG a µν ,(1) to the QCD lagrangian. The θ term is a total derivative and in perturbation theory physics is independent of θ. Witten suggested that non-perturbative effects generate θ-dependence in the pure gauge theory and that the topological susceptibility, χ top = d 2 E dθ 2 θ=0 ,(2) is O(1) in the large N c limit 2,3 . This suggestion was originally based on the fact that perturbative contributions to the topological charge correlator scale as N 0 c in the large N c limit, see Fig. 1a. Recently, Witten provided additional evidence for this conjecture using the AdS/CFT correspondence 3 . The scaling behavior χ top ∼ N 0 c was also observed in lattice simulations of pure gauge QCD for 4 N c = 2, . . . , 6. Naive N c counting implies that the contribution of fermions to χ top is subleading in 1/N c . We know, however, that there is no θ-dependence in QCD with massless fermions. This implies that the topological susceptibility receives a contribution related to fermions that cancels the pure gauge result, see Fig. 1b. Witten argued that this apparent contradiction can be resolved if the mass of the η ′ meson scales as N −1/2 c in the large N c limit. Witten and Veneziano derived a relation between the mass of the η ′ and the topological susceptibility in pure gauge theory 5,6,7,8 f 2 π m 2 η ′ = 2N F χ top \| no quarks .(3) Using χ top = O(1) and f 2 π = O(N c ) we observe that indeed m 2 η ′ = O(1/N c ). The θ-dependence of vacuum energy is related to topological properties of QCD. In the semi-classical approximation these features can be described in terms of instantons. Instantons are localized field configurations that carry topological charge 9 Q top = g 2 32π 2 d 4 x G a µνG a µν = ±1.(4) If the coupling is small then the density of instantons scales as exp(−8π 2 /g 2 ). In this limit instantons form a dilute, weakly interacting gas. The topological susceptibility is χ top = lim V →∞ Q 2 top V ≃ N V ,(5) where N = N + + N − is the number of instantons and anti-instantons and V is the volume. In 1978 Witten pointed out that this result implies that the contribution of instantons to the topological susceptibility scales as exp(−1/g 2 ) ∼ exp(−N c ) which seems to contradict the assumption 2 χ top = O(1). This argument is a little oversimplified, since instantons in QCD come in all sizes, and only small instantons are exponentially suppressed. We will come back to this problem in Sect. 4. Before we do so, we would like to comment on phenomenological consequences of equ. (5). Using the experimental values of f π and m η ′ the Witten-Veneziano relation implies χ top ≃ (200 MeV) 4 for N c = 3. If the topological susceptibility is saturated by a dilute gas of instantons, this value corresponds to a density (N/V ) ≃ 1 fm −4 . An estimate of the typical instanton size can be obtained by using the perturbative instanton size distribution and integrating it up to the phenomenological value of (N/V ). This leads to a value of ρ ≃ 1/3 fm. These two numbers form the basis of a successful picture of the QCD vacuum, usually called the instanton liquid model 10,11,12 . The instanton model not only accounts for topological properties of the QCD vacuum, but also describes chiral symmetry breaking and the correlation functions of light hadrons 13,14,15 . Topological properties of the QCD vacuum have also been studied in lattice QCD. It was found that the topological susceptibility in pure gauge QCD is 16 χ top ≃ (200 MeV) 4 , as predicted by the Witten-Veneziano relation. It was also observed that the topological susceptibility is stable under cooling, and appears to be dominated by semi-classical configurations. Lattice simulations also seem to confirm the values of the key parameters of the instanton liquid 17 , (N/V ) ≃ 1 fm −4 and ρ ≃ 1/3 fm. Instantons and the Witten-Veneziano relation Before we discuss the N c scaling behavior we would like to study the mechanism for topological charge screening and the mass of the η ′ in the instanton model. We will assume that instantons are small, ρΛ QCD ≪ 1, and that the instanton liquid is dilute, ρ 4 N/V ≪ 1. As we shall see below, these assumptions can be rigorously justified in the case of QCD at large baryon density. At zero density, however, this is a model assumption. The partition function of the instanton ensemble can be written as 18,19,20 Z = N+,N− µ N++N− 0 N + !N − ! N++N− i d 4 z i exp (−S ef f ) .(6) where µ 0 is the partition function of a single instanton and the effective action is given by S ef f = i d 4 x 2N f f π η 0 Q + d 4 x L(η 0 , η 8 ).(7) Here, the topological charge density is Q(x) = Q i δ(x − z i ) and L(η 0 , η 8 ) is the flavor singlet sector of the pseudoscalar meson lagrangian L = 1 2 (∂ µ η) 2 + (∂ µ η 8 ) 2 + 1 2 4 3 m 2 K − 1 3 m 2 π η 2 8 + 1 2 2 3 m 2 K + 1 3 m 2 π η 2 0 + 2 √ 2 3 m 2 π − m 2 K η 0 η 8 .(8) The meson lagrangian arises from bosonizing the fermionic interaction between instantons. The pion decay constant f π is determined by the solution of a saddle point equation, see Sect. IV.G. in the review article 14 for more details. The meson masses m π and m K satisfy Gell-Mann-Oakes-Renner relations. The partition function equ. (6) describes a system of charges interacting through the exchange of almost massless eta mesons. The physics of this system is very similar to that of a Coulomb gas. We expect, in particular, that topological charge gets screened and that the eta meson acquires a mass. We can show this explicitly by performing the sum in equ. (6) and expanding the resulting cosine function to second order in the fields. The topological charge correlator is given by where D(m, x) = mK 1 (mx)/(4π 2 x) is the euclidean space propagator of a scalar particle. The delta-function is the contribution from a single instanton, while the other terms are the contribution of the screening cloud. The η and η ′ mass satisfy the Witten-Veneziano relation Q(x)Q(0) = N V δ 4 (x) − 2N f f 2 π N V cos 2 (φ)D(m η ′ , x) + sin 2 (φ)D(m η , x) ,(9)u L d L d R L/R u R −1 q q ∆ L L R R ∆ G If 2 π m 2 η ′ + m 2 η − 2m 2 K = 2N F N V ,(10) and φ is the η − η ′ mixing angle. The coefficient of the delta-function is the topological susceptibility in the pure gauge theory. The topological susceptibility in the full theory can be calculated using equ. (9). The result is χ top = f 2 π 2N f m 2 top 1 − ( 4 3 m 2 K − 1 3 m 2 π )m 2 top ( 4 3 m 2 K − 1 3 m 2 π )m 2 top + 2m 2 K m 2 π − m 4 π ,(11) where m 2 top = m 2 η ′ + m 2 η − 2m 2 K . The result shows that the topological susceptibility vanishes if any of the quark masses m u = m d or m s is zero. This can be seen by using m π = 0 for m u = m d = 0 and m 2 π = 2m 2 K for m s = 0. Instantons and the Witten-Veneziano relation: Large baryon density The partition function described in the last section provides a simple and intuitive description of the η ′ prime mass and topological charge screening. The problem is that the theory is based on the assumption that the instanton liquid is dilute and weakly interacting. In this case the screening length l ∼ m −1 η ′ is much larger than the typical distance between charges. In N c = 3 QCD, however, the η ′ is heavy and the screening length is very short. Instead of ρ ≪ (N/V ) 1/4 ≪ m −1 η ′ we have ρ ∼ m −1 η ′ < (N/V ) 1/4 . There is an interesting limit of QCD in which the dilute instanton liquid description can be rigorously justified. This is the case of QCD at large baryon density. It has long been known that large instantons are suppressed if the baryon density (or the temperature) is large. In the last few years it has also become clear that chiral symmetry remains broken at large baryon density 21 . This implies that there is a flavor singlet Goldstone boson, and that in the limit m q → 0 the mass of this mode is due to the anomaly. In the following we shall discuss the case of two colors and flavors 22,23,24 , but the results can be generalized to other values of N c and N f . At large baryon density the axial U (1) symmetry is broken by a diquark condensate ǫ αβ ψ α L ψ β L = − ǫ αβ ψ α R ψ β R . Here, α, β are spinor indices. The condensate is a color and flavor singlet. The U (1) A Goldstone mode corresponds to fluctuations of the relative phase of the left and right handed condensates. The effective lagrangian for the singlet Goldstone boson is L = f 2 P (∂ 0 φ) 2 − v 2 (∂ i φ) 2 − V (φ).(12) The decay constant and Goldstone boson velocity are not related to instantons and can be determined in perturbation theory. At leading order the result is 22 f 2 P = µ 2 8π 2 , v 2 = 1 3 ,(13) where µ is the baryon chemical potential. The potential V (φ) vanishes in perturbation theory but receives contributions from instantons, see Fig. 2. We find V (φ) = −A P cos(φ + θ),(14) where θ is the QCD theta angle. If the chemical potential is big, µ ≫ Λ QCD , large instantons are suppressed and the coefficient A P can be determined in perturbation theory. The result is 22,23 A P = C 2,2 6π 4 4π g ∆ µ 2 2π 2 2 8π 2 g 2 4 Λ µ 8 Λ −2 (15) with C Nc,N f = 0.466 exp(−1.679N c )1.34 N f (N c − 1)!(N c − 2)! .(16) At large µ the superfluid gap ∆ can also be determined in perturbation theory. The result is 25,26,27,28 ∆ = 512π 4 µ g 5 exp − 2π 2 g(µ) − π 2 + 4 16 .(17) Using equ. (13)(14)(15)(16)(17) we can determine the mass of the pseudoscalar Goldstone boson. We have m 2 P = A P 2f 2 P .(18) Note that we have not used the large N c limit in order to derive this Witten-Veneziano relation. The result is exact in the limit µ ≫ Λ QCD even if N c = 2, 3. Also note that A P is the second derivative of the effective potential with respect to θ at θ + φ = 0 and is equal to the density of instantons. The vacuum energy, however, is determined by minimizing V with respect to φ and is independent of θ. This implies that χ top is zero, as expected for QCD with massless fermions. Equ. (18) predicts the density dependence of the flavor singlet Goldstone boson mass, see Fig. 3. This prediction can be tested using lattice simulations. We should note that formally, the prediction is only valid for µ ≫ Λ QCD but it is interesting to note that the result extrapolates to m P ∼ 1 GeV at zero baryon density. The Instanton liquid at large N c In this section we describe a study of the instanton ensemble in QCD for different numbers of colors 29 . We consider the partition function of a system of instantons in pure gauge theory Z = 1 N I !N A ! NI +NA I [dΩ I n(ρ I )] exp(−S int ).(19) Here, Ω I = (z I , ρ I , U I ) are the collective coordinates of the instanton I and n(ρ) is the semi-classical instanton distribution function 30 n(ρ) = C Nc 8π 2 g 2 2Nc ρ −5 exp − 8π 2 g(ρ) 2 ,(20)C Nc = 0.466 exp(−1.679N c ) (N c − 1)!(N c − 2)! ,(21)8π 2 g 2 (ρ) = −b log(ρΛ), b = 11 3 N c .(22) We have denoted the classical instanton interaction by S int . If the instanton ensemble is sufficiently dilute we can approximate the instanton interaction as a sum of two-body terms, S int = IJ S IJ . For a well separated instanton-anti-instanton pair the interaction has the dipole structure 10 S int = − 8π 2 g 2 4ρ 2 I ρ 2 A R 4 IA \|u\| 2 1 − 4 cos 2 θ .(23) Here ρ I,A are instanton radii and R IA is the instanton-anti-instanton separation. The relative color orientation is characterized by a complex fourvector u µ = 1 2i tr(U IA τ + µ ), where U IA = U I U † A depends on the rigid gauge transformations that describe the color orientation of the individual instanton and anti-instanton and τ + µ = ( τ , −i). We have also defined the relative color angle cos 2 θ = \|u ·R\| 2 /\|u\| 2 . The dipole interaction is valid if R 2 IA ≫ ρ I ρ A . A strongly overlapping instanton-anti-instanton pair is not a semiclassical field configuration and we do not know how to treat it correctly. In practice we have chosen to deal with this problem by including a short range repulsive core S core = 8π 2 g 2 \|u\| 2 f ρ 2 I ρ 2 J R 4 IJ .(24) The precise form of the function f (x) is not very important. What is important for the N c scaling is that the core is proportional to the classical action N c =3 N c =4 N c =5 N c =6 ¼ ½ ¾ ¿ ¼ ½ ¾ ¿ ½¼ ½½ ½¾ ´ µ Ö ÖÖ ÖÖ Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö Ö · · ·· ·· · · · · · · · ·· · ·· ···· ·· · · · · · · · · · · 8π 2 /g 2 and that it includes the factor \|u\| 2 which ensures that instantons in commuting SU (2) subgroups of SU (N c ) do not interact. The partition function equ. (19) is quite complicated and in general has to be analyzed using numerical methods. Before we describe numerical results we present a variational bound. Diakonov and Petrov 12 proposed to approximate the partition sum in terms of a variational single instanton distribution µ(ρ). For this ansatz the partition function reduces to Z 1 = 1 N I !N A ! NI +NA i dΩ I µ(ρ I ) = 1 N I !N A ! (V µ 0 ) NI +NA(25) where µ 0 = dρ µ(ρ). The exact partition function is Z = Z 1 exp(−(S − S 1 )) ,(26) where S is the full action, S 1 = log(µ(ρ)) is the variational estimate and the average . is computed using the variational distribution function. The partition function satisfies the bound Z ≥ Z 1 exp(− S − S 1 ),(27) which follows from convexity. The optimal distribution function µ(ρ) is determined from a variational principle, (δ log Z)/(δµ(ρ)) = 0, where Z is computed from equ. (27). One can show that the variational result for the free energy F = − log(Z)/V provides an upper bound on the true free energy. In order to compute the variational bound we have to determine the average interaction S 1 . In the original work 12 the authors used the instanton interaction in the sum ansatz. The result is S int = 8π 2 g 2 γ 2 ρ 2 I ρ 2 J , γ 2 = 27 4 N c N 2 c − 1 π 2 .(28) For the "dipole plus core" interaction the result has the same dependence on N c but the numerical coefficient is different. Note that the interaction contains a factor N c /(N 2 c − 1) ∼ 1/N c which reflects the probability that two random instantons overlap in color space. Since the classical action scales as S 0 ∼ 1/g 2 we find that the average interaction between any two instantons is O(1). Applying the variational principle, one finds µ(ρ) = n(ρ) exp −βγ 2 N ρ 2 V ρ 2 ,(29) where β = β(ρ) is the average instanton action and ρ 2 is the average size. We observe that the single instanton distribution is cut off at large sizes by the average instanton repulsion. The instanton density and average size are given by N V = Λ 4 C Nc β 2Nc Γ(ν)(βνγ 2 ) −ν/2 2 2+ν ,(30)ρ 2 = νV βγ 2 N 1/2 , ν = b − 4 2 .(31) These results imply that N V ∼ N c , ρ ∼ N 0 c .(32) Note that the instanton action is O(N c ) in the large N c limit, i.e. instantons remain semi-classical. The total density is not exponentially suppressed because an exponentially large factor associated with different instanton embeddings cancels the exponential suppression from the action. Also note that the density scales like the number of commuting subgroups of SU (N c ) and the instanton packing fraction ρ 4 N/(V N c ) is independent of N c . Numerical results are shown in Figs. 4-6. Fig. 4 shows the instanton size distribution. We observe that small instantons are suppressed as N c → ∞, but there is a critical size for which the number is independent of N c and the total number scales as N c . The results are consistent with the idea that the size distribution slowly approaches a delta function 31,32,33,34 . We also show lattice results reported by Teper at this meeting 4 . The lattice results are clearly consistent with our model calculation. Fig. 5 shows the instanton density, the chiral condensate and the topological susceptibility. The instanton density and the chiral condensate scale as N c . The topological susceptibility, on the other hand, goes to a constant as N c → ∞. This means that χ top does not satisfy the relations χ top ∼ (N/V ) expected for a dilute gas of instantons. In Fig. 6 we show the masses of the pion, the rho meson, and the eta prime meson. The results are consistent with the expectation m 2 ρ ∼ N 0 c and m 2 η ′ ∼ 1/N c . Π(x)/Π 0 (x) N c =3 N c =4 N c =5 N c =6 Application: Scalar mesons and the large N c limit We saw in the previous section that the instanton liquid is consistent with the standard large N c scaling relations. What is maybe even more important is that instantons can be used to understand corrections to the leading order large N c results. It is well known, for example, that the OZI rule does not work equally well in all channels. The OZI rule is very well satisfied in the vector channel; the mixing is close to ideal, and the rho and omega meson are almost degenerate. In the scalar and pseudoscalar channels, on the other hand, the OZI rule is badly violated. The eigenstates in the pseudoscalar sector are close to flavor, not mass, eigenstates and the mass difference between the pion and the eta prime meson is large. In the scalar sector we find a heavy iso-vector state, the a 0 , but a light iso-scalar, the σ-meson, which is strongly coupled to ππ states. Jaffe suggested that the unusual properties of the light scalar mesons could be explained by assuming a large (qq)(qq) admixture 35,36 . He observed that the spectrum of the flavor nonet obtained by coupling two antitriplet scalar diquarks is inverted as compared to a standard qq nonet, and contains a light isospin singlet, a strange doublet, and a heavy triplet plus singlet with hidden strangeness. This compares very favorably to the observed light sigma, the strange kappa, and the heavier a 0 (980) and f 0 (980). It also explains why the a 0 and f 0 are strongly coupled to KK and πη. Instantons are important because they can account for the observed pattern of OZI violating effects 37,38 . There are no direct instanton effects in the vector channel, and as a result OZI violation is small. In the scalar and pseudoscalar channels, on the other hand, instantons lead to strong flavor mixing. We have recently examined instanton contributions to scalar meson correlation functions in more detail 39 . For some earlier work on the subject we refer the reader to 40 . Fig. 7 shows the correlation functions in the sigma (qq) and a 0 (qτ a q) channel. For N c = 3 we find a light ∼ 600 MeV sigma state and a heavy ∼ 1 GeV a 0 meson. When the number of colors is increased the light sigma state disappears and for N c = 6 the sigma mass is also in the ∼ 1 GeV range. We have also determined the off-diagonal sigma-pi-pi correlation function (qq)(0)(qγ 5 τ a q) 2 (x) . We find that for N c = 3 the sigma is strongly coupled to two-pion states, but for N c > 3 the coupling becomes much smaller. These results are in agreement with a study based on chiral lagrangians 41 . Supersymmetric gauge theories In Sect. 4 we argued that the large N c scaling behavior of the instanton contribution to QCD correlation functions agrees with the scaling of perturbative Feynman diagram. This result was based on fairly general arguments, but it did involve one important assumption regarding the effective instanton interaction that cannot be rigorously justified at this time. In order to study this problem in more detail it useful to consider supersymmetric generalizations of QCD. In this contribution we would like to review some results that are relevant to SUSY gluodynamics (N = 1 SUSY gauge theory). This theory is interesting because it exhibits confinement and a fermion bilinear condensate. It was also recently argued that there is a new large N c limit, called the orientifold large N c expansion, which relates the quark condensate in N f = 1 QCD to the gluino condensate in SUSY gluodynamics 48,49 . SUSY gluodynamics is defined by the lagrangian L = − 1 4g 2 G a µν G a µν + i g 2 λ aα D αβ λ aβ ,(33) where λ is a Weyl fermion in the adjoint representation and α,β are spinor indices. The theory has a U (1) A symmetry λ → e iφ λ which is broken by the anomaly. A discrete Z 2Nc subgroup is non-anomalous. The Z 2Nc symmetry is dynamically broken by the gluino condensate 1 16π 2 Tr[λλ] = Λ 3 exp 2πik N c .(34) Here, k = 0, 1, . . . , N c − 1 labels the N c different vacua of theory and Λ is the scale parameter defined by Λ 3 = µ 3 1 g 2 (µ) exp − 8π 2 g 2 (µ)N c .(35) The instanton solution in SUSY gluodynamics has 4N c bosonic zero modes (4 translations, 1 scale transformation and 4N c − 5 rigid gauge rotations) and 2N c fermion zero modes. Reviews of the SUSY instanton calculus can be found in 50,51,52 . The collective coordinate measure is 2 3Nc+2 π 2Nc−2 Λ 3Nc (N c − 1)!(N c − 2)! d 4 z dρ 2 (ρ 2 ) 2Nc−4 d 2 η d 2 ζ d Nc−2 ν d Nc−2ν ,(36) where z denotes the instanton position and ρ is the instanton size. The Grassmann spinors η α and ζ α parameterize the so-called supersymmetric and superconformal zero modes, and the Grassmann numbers ν,ν parameterize the superpartners of the rigid gauge rotations. There is no direct instanton contribution to the gluino condensate but the value of λλ can be extracted from an indirect instanton calculation. The standard method consists of adding N f = N c − 1 quark flavors to the theory, and to consider the limit in which the squark fields have a large expectation value. In this case instantons are small and we can reliably compute their contribution to the superpotential. The result is 53,54 W N f =Nc−1,Nc ef f = (Λ Nc−1,Nc ) b0 det N f (Q fQf ′ )(37) where Q f , f = 1, . . . , N f are quark superfields, b 0 = 3N c − N f is the first coefficient of the beta function, and Λ N f ,Nc is the scale parameter. Supersymmetry guarantees that equ. (37) is correct even if the squark vev is not large. The extra quark fields can be decoupled by sending the vev to infinity. The result is W 0,Nc ef f = N c Λ 3 0,Nc(38) from which one can determine the gluino condensate equ. (34). In SUSY gluodynamics there is a direct instanton contribution to the (λλ) Nc correlation function. The result is independent of the relative coordinates and given by 54 1 (16π 2 ) Nc Tr[λλ] . . . Tr[λλ] = 2 Nc Λ 3Nc (N c − 1)!(3N c − 1) .(39) It is tempting to extract the gluino condensate from the N c 'th root of equ. (39). This is sometimes called the strong-coupling instanton (SCI) calculation, in contrast to the weak-coupling instanton (WCI) result equ. (34). The SCI result disagrees with the WCI by a factor that scales as N c in the large N c limit. It is not entirely clear why the SCI calculation fails, but the problem is likely related to the fact that the groundstate breaks a Z Nc symmetry. This implies that the theory has to be defined carefully in order to pick out a unique ground state. In the WCI calculation the ground state is implicitly selected through the decoupling procedure. As an alternative to the decoupling procedure one can define SUSY gluodynamics by compactifying the theory on R 3 × S 1 . Here, both fermions and bosons obey periodic boundary conditions and one can show that the gluino condensate is independent of the size of the compactified dimension. In the compactified theory the gauge symmetry is spontaneously broken by a non-zero expectation value of the gauge field A 4 . The corresponding gauge invariant order parameter is the Polyakov line (the holonomy) along the compact dimension. Instantons with a non-trivial holonomy were constructed by Kraan and van Baal 55 . It turns out that these objects have constituent monopoles (dyons) with fractional topological charge S dyon = 8π 2 g 2 N c , Q dyon = 1 N c .(40) Each of these dyons has a pair of gluino zero modes that contribute directly to the gluino condensate 56,57 1 16π 2 Tr[λλ] dyon = 1 N c Λ 3 .(41) Summing over the N c constituent dyons one recovers the WCI result for the gluino condensate. This result supports the old idea that in the large N c limit the relevant field configurations are not instantons but instantons constituents with fractional topological charge 58 . These configurations are unsuppressed because they have action S ∼ O(1). Conclusions We have argued that it is possible for the instanton liquid model to have a smooth large N c limit which is in agreement with scaling relations derived from Feynman diagrams. In this limit the density of instantons grows as N c whereas the typical instanton size remains finite. Interactions between instanton are important and suppress fluctuations of the topological charge. As a result the U (1) A anomaly is effectively restored even though the number of instantons grows with N c . Using variational arguments and numerical simulations we have shown that this scenario does not require fine tuning. It arises naturally if the instanton ensemble is stabilized by a classical repulsive core. In this case we obtain a picture in which the instanton density is large but the instanton liquid remains dilute because instantons are not strongly overlapping in color space. Further investigations will have to show whether this scenario is indeed correct, but the lattice measurements of the instanton size distribution reported by Teper 4 at this meeting are certainly encouraging. We also emphasized that instantons provide a simple explanation of the observed pattern of OZI rule violating effects. Violations of the OZI rule are large in channels like the scalar and pseudoscalar mesons that receive direct instanton contributions. Instantons may also play a role in explaining regularities in hadron spectra that go beyond the naive quark model, such as diquark clustering in mesons and exotic baryons. Finally we stressed that there are important lessons to be learned from generalizations of QCD. QCD at high baryon density provides a beautiful and rigorous realization of the instanton mechanism for generating the eta prime mass. Supersymmetric gluodynamics is an example for a theory in which instantons provide the essential non-perturbative input for the calculation of the fermion condensate. This calculation can now be linked, thanks to the orientifold large N c limit, to the quark condensate in N f = 1 QCD. Figure 1 . 1The diagrams on the left and right show the large Nc scaling of typical diagrams that contribute to the topological charge correlator in pure gauge QCD and in QCD with light fermions. Figure 2 . 2The figure on the left shows the instanton induced quark interaction in N F = 2 QCD at non-zero baryon density. Scattering on particle-hole qq −1 pairs leads to gauge field screening and suppresses large instantons. The figure on the right shows the instanton contribution to the vacuum energy at large baryon density. The squares denote insertions of the diquark condensates ψψ and ψψ . conf) [arb.sc.] 2∆ Λ conf [arb.sc.] Figure 3 . 3Flavor singlet pseudoscalar Goldstone boson mass in Nc = 2 QCD. The solid line marked by squares is the result of the instanton calculation. For comparison, the solid line marked by circles shows an estimate based on the assumption m 2 P f 2 P ∼ σ 2 , where σ is the string tension. The overall scale of this curve is not known. We also show the energy gap 2∆ and the confinement scale. Figure 4 . 4The figure on the left shows the instanton size distribution obtained from numerical simulations of the instanton ensemble in pure gauge QCD for different numbers of colors. The figure on the right shows lattice results reported by Teper at this meeting. The ⋆ + •• symbols correspond to Nc = 2, 3, 4, 5. Figure 5 . 5Nc dependence of the instanton density, the topological susceptibility and the quark condensate from numerical simulations of the instanton liquid in pure gauge QCD. Figure 6 . 6Nc dependence of the pion, the rho, and the eta prime mass from numerical simulations of the instanton liquid in pure gauge QCD. The dotted lines are fits to m 2 ρ ∼ const, m 2 η ′ ∼ 1/Nc and m 2 π ∼ c 0 + c 1 /Nc. Figure 7 . 7Correlation functions in the σ (qq) and a 0 (qτ a q) channel for different numbers of colors. The correlators are normalized to the free correlation functions and were calculated in a pure gauge instanton ensemble. The fact that the a 0 correlator becomes negative is an artefact of the quenched approximation which disappear as Nc → ∞. Figure 8 . 8Instanton contribution to the superpotential in SUSY gluodynamics with N f = Nc − 1 quark flavors. The dots denote (2Nc − 4) pairs of quark ψ and gluino λ zero modes. There is a significant literature on instanton effects in supersymmetric gauge theories at large N c . Instantons in the AdS/CFT correspondence (N = 4 SUSY gauge theory) were studied in a series of papers by Bianchi et al and Dorey et al 42,43,44 . The instanton contribution to the Seiberg-Witten superpotential in N = 2 SUSY gauge theory was originally studied by Finnell and Pouliot 45 and later generalized to arbitrary N c by Klemm et al 46 and Douglas and Shenker 47 . 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[ "Emergence of Bulk CsCl Structure in (CsCl) n Cs + Cluster Ions", "Emergence of Bulk CsCl Structure in (CsCl) n Cs + Cluster Ions" ]	[ "Andrés Aguado \nDepartamento de Física Teórica\nUniversidad de Valladolid\n47011ValladolidSpain\n" ]	[ "Departamento de Física Teórica\nUniversidad de Valladolid\n47011ValladolidSpain" ]	[]	The emergence of CsCl bulk structure in (CsCl)nCs + cluster ions is investigated using a mixed quantum-mechanical/semiempirical theoretical approach. We find that rhombic dodecahedral fragments (with bulk CsCl symmetry) are more stable than rock-salt fragments after the completion of the fifth rhombic dodecahedral atomic shell. From this size (n=184) on, a new set of magic numbers should appear in the experimental mass spectra. We also propose another experimental test for this transition, which explicitely involves the electronic structure of the cluster. Finally, we perform more detailed calculations in the size range n=31-33, where recent experimental investigations have found indications of the presence of rhombic dodecahedral (CsCl)32Cs + isomers in the cluster beams.	10.1103/physrevb.62.13687	[ "https://arxiv.org/pdf/physics/0008235v1.pdf" ]	119,348,053	physics/0008235	387bef030cde40dd4c6e62e9cd0757dbf291dc84	Emergence of Bulk CsCl Structure in (CsCl) n Cs + Cluster Ions 29 Aug 2000 Andrés Aguado Departamento de Física Teórica Universidad de Valladolid 47011ValladolidSpain Emergence of Bulk CsCl Structure in (CsCl) n Cs + Cluster Ions 29 Aug 2000arXiv:physics/0008235v1 [physics.atm-clus] The emergence of CsCl bulk structure in (CsCl)nCs + cluster ions is investigated using a mixed quantum-mechanical/semiempirical theoretical approach. We find that rhombic dodecahedral fragments (with bulk CsCl symmetry) are more stable than rock-salt fragments after the completion of the fifth rhombic dodecahedral atomic shell. From this size (n=184) on, a new set of magic numbers should appear in the experimental mass spectra. We also propose another experimental test for this transition, which explicitely involves the electronic structure of the cluster. Finally, we perform more detailed calculations in the size range n=31-33, where recent experimental investigations have found indications of the presence of rhombic dodecahedral (CsCl)32Cs + isomers in the cluster beams. I. INTRODUCTION A general goal of cluster physics is to study the emergence of bulk behavior right from the molecular limit, by building clusters of increasing size and following the size evolution of selected properties. From the theoretical point of view, this ambitious plan has been largely impeded because of the slow and nonmonotonic size evolution observed in many properties. The predicted cluster structures are not simply related to the corresponding bulk structures in many cases, which precludes the possibility of a meaningful extrapolation to the bulk limit. Moreover, cluster structure is difficult to determine theoretically due to the huge increase in the number of isomers with size, and experimentally due to the small number of scatterers compared with the bulk case. Nevertheless, recent advances involving ion mobility measurements, 1-4 electron diffraction from trapped cluster ions, 5,6 or the use of photoelectron spectra as a fingerprint of structure 7 have been successful in elucidating the structures of several ionic and covalent clusters. Abundance patterns obtained from the mass spectra of binary ionic clusters like the alkali halides and alkaline-earth oxides point towards a prompt establishment of bulk rock-salt symmetry. [8][9][10][11] Theoretical calculations have shown, however, that small sodium iodide and lithium halide clusters adopt ground state structures based on the stacking of hexagonal rings. 12,13 In the case of alkaline-earth oxide clusters, the large and coordination-dependent values of the oxide polarizabilities favor the formation of structures with a large proportion of ions in surface sites, inducing a delay in the emergence of bulk structural properties. [14][15][16] Turning to the alkali halides, bulk CsCl, CsBr, and CsI crystallize in the CsCl-type structure, while both experimental mass spectra 8,10 and theoretical calculations 17 indicate that small clusters of those materials adopt ground state structures which are fragments of a rock-salt lattice. This implies that there has to be a structural phase transition as the cluster size is increased. Ion mobility measurements performed by Löffler 4 suggest that (CsCl) n Cs + cluster ions with n=32 are specially compact, which might be explained by the presence of isomers with the shape of a perfect three-shell rhombic dodeca-hedron (that is with bulk CsCl symmetry) in the cluster beam. The electron diffraction experiments performed recently in the group of Parks 6 show that there is a substantial proportion of isomers with bulk CsCl symmetry for the same size. In this theoretical work we analyce the above mentioned size-induced phase transition in (CsCl) n Cs + cluster ions. We consider only those sizes that correspond to geometrical shell closings for the CsCl-type (perfect rhombic dodecahedra with n=32,87,184,335,552) and rock-salt (perfect cubes with n=13,62,171,364,665) structural series. In doing so, we try to avoid any nonmonotonic size evolution in the calculated properties. In the upper part of Fig. 1 we display the relative number of atoms with a given coordination as a function of N −1/3 , where N=2n+1 is the total number of atoms in the cluster. In the lower part we show the number of atoms with nonbulk coordination relative to the total number of surface atoms. For the largest sizes considered the proportion of bulklike atoms is dominant, and within the surface the proportion of face-like atoms is already much larger than those of edge and vertex-like atoms. From those sizes to the bulk, the only meaningful size evolution of these proportions will be a slow approach to zero of the face-like atoms. We thus expect to capture all the physical information relevant to the phase transition by studying this set of clusters and the corresponding bulk phases, which have been studied both with the same theoretical model. In this way inaccuracies related to the use of different methodologies are avoided and a meaningful extrapolation to the bulk limit can be done. 18 In a second part of the work, we explicitely analyze the structures adopted by (CsCl) n Cs + cluster ions in the size range n=31-33, in order to explain the experimental findings of Refs. 4 and 6. The rest of the paper is organized as follows: Section II includes just a brief description of the theoretical model employed in the calculations, as a full account of it has been given in previous publications. 12,15,16 In Section III we present and discuss the results of the calculations, and Section IV summarizes the main conclusions. II. THEORETICAL MODEL Cluster energies have been obtained by performing Perturbed Ion (PI) plus polarization calculations. This is a well tested method that describes accurately both bulk 19 and cluster 15,16 limits. Its theoretical foundation lies in the theory of electronic separability. [20][21][22] Very briefly, the cluster wave function is broken into local group functions (ionic in nature in our case) that are optimised in a stepwise procedure. In each iteration, the total energy is minimized with respect to variations of the electron density localized in a given ion, with the electron densities of the other ions kept frozen. In the subsequent iterations each frozen ion assumes the role of nonfrozen ion. When the self-consistent process finishes, 12 the outputs are the total cluster energy and a set of localized wave functions, one for each geometrically nonequivalent ion of the cluster. This procedure leads to a linear scaling of the computational effort with cluster size, which allows the investigation of large clusters with an explicit inclusion of the electronic structure. The cluster binding energy can be decomposed into ionic additive contributions E bind clus = R∈clus (E R add − E R 0 ),(1) being E R add the contribution of the ion R to the total cluster energy and E R 0 the energy of the ion R in vacuo. In this way the contribution of ions with different coordinations to the binding energy can be separately analyzed, which is particularly convenient for our study. Each additive energy can be decomposed in turn as a sum of deformation and interaction terms E bind clus = R∈clus (E R def + 1 2 E R int ),(2) where E R def is the self-energy of the ion R, measured relative to the vacuum state, and E R int contains electrostatic, exchange and repulsive overlap energy terms. 12,15 The polarization contribution to the cluster binding energy is not computed in the actual version of the PI code, as it assumes (for computational simplicity) that the electronic charge distribution of each ion in the cluster is spherically symmetric. Thus, a polarization correction to the PI energy is computed semiempirically as described in Refs. 15,16. Bulk polarizabilities are used for both Cs + and Cl − ions. 23 This is a good approximation for the Cs + cations. The main effect on the anion polarizabibities when passing from the bulk to a cluster environment is an increase of the polarizabilities of those ions located on the cluster surface, due to the lower average coordination compared to the bulk. However, we have checked that our main conclusions are not affected by an increase in the surface chloride polarizabilities as large as 10-20 %, which are typical values for halides. 14 The short-range induction damping parameters have been obtained through the scaling procedure validated in Ref. 24. The reliability for cluster calculations of the mixed quantum-mechanical/semiempirical energy model thus obtained has been checked and shown to be high in previous publications. 15,16 III. RESULTS A. The rock-salt to CsCl-type structural transition Fig. 2 shows the size evolution of the binding energy per ion. First of all, we note that the PI model properly reproduces the stability trend in the bulk, predicting the CsCl structure as the most stable one. This is a tough problem for semiempirical methods, as Pyper 25 has shown that a full account of the coordination number dependence of the self-energy and overlap contributions is necessary to obtain the correct ground state structure. The values of the binding energy, plotted as a function of N −1/3 , lie neatly on a straight line. The regression coefficients obtained from a fit are 0.9998 in all cases if we exclude from the fitting the NaCl-type cluster with n=13, which is the smallest one. We have calculated after the fitting procedure the energy of the 5×5×7 cuboid (also included in Fig. 2), and checked that it lies on the fitted NaCl-type energy curve. This shows that a consideration of perfect cubes (or cuboids) on one hand, and rhombic dodecahedra on the other hand removes the nonmonotonic behavior from the size evolution of the binding energies. Our results predict that the rhombic dodecahedra become definitely more stable after the completion of the fifth shell of atoms, that is for n=184. The fourshell rhombic dodecahedron and the 5×5×7 cuboid are essentially degenerate, so both of them will contribute to the enhanced abundance observed experimentally for n=87. 10 We have not found any experimental mass spectrum for values of n as high as 184, but we predict that a new set of magic numbers, corresponding to the closing of rhombic dodecahedral atomic shells, should emerge from this size on. The magic numbers corresponding to the closing of perfect cubic shells will probably not disappear still at that specific size from the mass spectra, because they do not coincide with the CsCl shell closings, and complete cubes can remain more stable than incomplete rhombic dodecahedra until larger values of n are reached. Polarization has little influence on these general results, and only affects the energetic ordering of the two essentially degenerate isomers mentioned above. Now we turn to an analysis of the physical factors responsible for this transition. In Fig. 3 we show the binding energy per ion, averaged over subsets of ions with a fixed coordination. The contribution of bulklike ions to the binding energy favors always the formation of CsCl-type structures. However, the contribution of facelike ions favors the formation of rock-salt fragments. As soon as the proportion of bulk ions is larger than that of surface atoms, which occurs after the completion of the fifth rhombic dodecahedral atomic shell, fragments of the CsCl-type lattice become more stable. The energy contribution of those ions in edge positions is approximately the same for both structural families except for the smallest clusters; finally, corner atoms favor the CsCltype structures, but their small relative number results in a very small contribution to the total energy for those sizes where the transition occurs. Fig. 3 has reduced the structural phase transition in (CsCl) n Cs + cluster ions to an essentially bulk effect. By this we mean that CsCl-type structures become more stable as soon as the proportion of bulklike atoms is dominant. To complete our discussion we have then to address the stability question in the bulk. This is more easily understood by analyzing the reasons why other alkali halides like NaCl or CsF do not crystallize in the CsCl-type lattice. The largest contribution to the binding energy of an ionic crystal is the Madelung energy term E M = AM Re , with A M the Madelung constant and R e the equilibrium interionic distance. The Madelung constant of the CsCl-type lattice (1.762675) is larger than that of the rock-salt lattice (1.747565), so were the value of R e the same for both structures, the CsCl-type would always be more stable. We have solved for the electronic structure of NaCl and CsF crystals in the CsCl-type structure at a nonequilibrium value of the interionic distance, chosen in such a way that the Madelung energy term is exactly the same as in the corresponding rock-salt lattice at equilibrium. In the case of NaCl, E add (Na + ) favors the CsCl-type structure, but E add (Cl − ) largely favors the rock-salt phase. The main reason is the large anion-anion overlap at that artificial distance, that is the Na + cation is so small compared to the Cl − anion that eight anions can not be packed efficiently around a cation. In CsF the situation is reversed, and it is the cation-cation overlap that is too large. This demonstrates that the stability situation in the bulk is a purely packing effect: in CsCl, CsBr and CsI, the large value of the cation-anion size ratio allows for an equilibrium interionic approach in the CsCl-type structure close enough as to obtain a Madelung energy term more negative than in the rocksalt phase, without a large overlap interaction between like ions. The same is true for the bulklike ions in the clusters studied, and so when those ions begin to dominate the energetics, the bulklike fragments become more stable. We have made a prediction above that can be tested experimentally, namely the emergence of a new set of magic numbers from n=184 on. Here we propose another, perhaps more indirect, experimental test. In Fig. 4, the eigenvalues of the 3p orbitals of Cl − (with opposite sign) are plotted as a function of N −1/3 . We have a band of eigenvalues for each size because the anions occupy nonequivalent positions in the clusters. As the clusters under study are formed by closed shell ions whose wave functions are strongly localized, it can be assumed that an electron is extracted from a specific localized orbital when the cluster is ionized. This is the lowest bound 3p orbital, which corresponds always to a chloride anion with a low coordination. Thus the dashed lines represent the size evolution of the vertical ionization potential IP (in the Koopmans' approximation) for both structural families. For the rock-salt series, that size evolution is approximately linear in N −1/3 , but for the CsCl-type series it shows a more or less oscillating behavior, which should be detected in experimental measurements of the vertical IP if rhombic dodecahedra actually are the ground state structures from a given size on. We can explain these different electronic behaviors in a very simple way: in the rock-salt clusters the eight corner sites are always occupied by Cs + cations. The weakest bound electron corre-sponds always to a Cl − anion with fourfold coordination, namely anyone of those closer to the corner cation sites. On the other hand, rhombic dodecahedra have fourteen corner sites. When the number of atomic shells is even, all these sites are occupied by Cs + cations, but when that number is odd, eight of them are cationic sites and the other six anionic sites. Thus the nonmonotonic behavior of the vertical IP is due to the different local coordination of the Cl − anion to which the weakest bound electron is attached as the number of atomic shells increases. B. Structures of (CsCl)nCs + (n=31-33) and comparison to experiment We finish our study with an explicit consideration of (CsCl) n Cs + clusters in the size range n=31-33, the range covered in the experiments of Löffler 4 and Parks. 6 Specifically, we have considered the most compact 7×3×3, 4×4×4 and 5×4×3 rock-salt structures, and the threeshell rhombic dodecahedron, with some atoms added or removed from different positions. The binding energies are shown in Table I. The ground state (GS) structure of (CsCl) 31 Cs + is a complete 7×3×3 cuboid. The 4×4×4 fragment with an anion removed from a corner position is slightly less stable, and the lowest energy rhombic dodecahedral isomer we have obtained has a still lower stability. For n=32, the complete three-shell rhombic dodecahedron becomes the GS isomer. All the different incomplete rock-salt fragments have a smaller binding energy. For n=33, the different rock-salt isomers are essentially degenerate, but the CsCl-type structure is found again at a higher energy. This sequence of GS structures for (CsCl) n Cs + clusters is consistent with the experimental findings. 4,6 The relative mobility is a local maximum for n=32, as the perfect three-shell rhombic dodecahedron is evidently more compact than the complete 7×3×3 (CsCl) 31 Cs + cuboid or any of the incomplete rock-salt structures obtained for n=33. Also, the energetical ordering of the isomers is consistent with the large proportion of CsCl-type isomers found for n=32 in the electron diffraction experiments. IV. SUMMARY We have reported a computational study of the sizeinduced rock-salt to CsCl-type structural phase transition in (CsCl) n Cs + cluster ions. For this purpose, the Perturbed Ion (PI) method, supplemented with a semiempirical account of polarization effects, has been employed. Only cluster ions with an atomic closed-shell configuration have been considered in order to avoid nonmonotonic behavior in the calculated properties. Moreover, we have employed the same theoretical model to study both cluster and bulk limits, which allows for a meaningful extrapolation strategy. The main result is that rhombic dodecahedral isomers become definetely more stable than rock-salt structures after the completion of the fifth rhombic-dodecahedral atomic shell, that is for a size n=184. Thus, it is predicted that a new set of magic numbers, reflecting the establishment of the new structural symmetry, should emerge from that size on. The size evolution of the vertical ionization potential of the cluster ions should also be a good experimental fingerprint of the transition. In order to explain the nature of the transition, an analysis of the binding energy into ionic components has been performed. The result is quite simple: bulklike ions always prefer to have a CsCl-type environment, even for the smallest cluster sizes (this has been shown to be a purely packing effect), while surfacelike atoms prefer to adopt rock-salt structures. The transition occurs as soon as the proportion of bulklike atoms is large enough to dominate the energetics of the whole cluster. One of the possibilities advanced by Parks consistent with his experimental results 6 is the existence of isomers with mixed symmetry. Our results indicate that the formation of isomers with a CsCl-type core and a rock-salt-type surface could be energetically favored, if the strain accumulated in the bonds at the interface region separating both phases can be conveniently relaxed. This point deserves further investigation. The structures adopted by (CsCl) n Cs + cluster ions have been more carefully studied in the size range n=31-33, which has been covered in the experimental investigations. Our results are consistent with the experimental findings, and show that the three-shell rhombic dodecahedron is the lowest energy isomer for n=32. ACKNOWLEDGMENTS This work has been supported by DGES (Grant PB98-0368) and Junta de Castilla y León (VA70/99). The author is indebted to J. M. López for a careful reading of the manuscript. Tables. Table I Binding energy, in eV/ion, of the different rock-salt and CsCl-type structures for the size range n=31-33. Captions of Captions of Figures. Figure 1 Size evolution of the number of atoms with a given coordination, relative to the total number of atoms (upper half) or to the total number of surface atoms (lower half). The left half refers to CsCl-type symmetry and the right half to rock-salt symmetry. Figure 2 Size evolution of the binding energy per ion for both CsCl-type and rock-salt structural families, with (lower half) and without (upper half) the inclusion of polarization corrections. The value of N −1/3 at the transition point has been indicated with an arrow. Figure 3 Size evolution of the binding energy contributions from ions with different coordinations. Full circles represent ions in the CsCl-type structures and squares represent ions in the rock-salt structures. Figure 4 Size evolution of the 3p orbital eigenvalues of chloride anions. The dashed line represents the variation of the vertical ionization potential in the Koopmans' approximation with size. Binding Energy (eV) M. F. Jarrold, J. Phys. Chem. 99,11 (1995). . M Maier-Borst, P Löffler, J Petry, D Kreisle, Z. Phys. D. 40476M. Maier-Borst, P. Löffler, J. Petry, and D. Kreisle, Z. Phys. D 40, 476 (1997). . P Dugourd, R R Hudgins, M F Jarrold, Chem. Phys. Lett. 267186P. Dugourd, R. R. Hudgins, and M. F. 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[ "A PARAMETRIZED NON-EQUILIBRIUM WALL-MODEL FOR LARGE-EDDY SIMULATIONS", "A PARAMETRIZED NON-EQUILIBRIUM WALL-MODEL FOR LARGE-EDDY SIMULATIONS" ]	[ "S Hickel \nFaculty of Aerospace Engineering\nTU Delft\nNL\n", "E Touber \nDepartment of Mechanical Engineering\nImperial College\nUK\n", "J Bodart \nISAE\nUniversité de Toulouse\nFrance\n", "J Larsson \nDepartment of Mechanical Engineering\nUniversity of Maryland\nUSA\n" ]	[ "Faculty of Aerospace Engineering\nTU Delft\nNL", "Department of Mechanical Engineering\nImperial College\nUK", "ISAE\nUniversité de Toulouse\nFrance", "Department of Mechanical Engineering\nUniversity of Maryland\nUSA" ]	[]	Wall-models are essential for enabling large-eddy simulations (LESs) of realistic problems at high Reynolds numbers. The present study is focused on approaches that directly model the wall shear stress, specifically on filling the gap between models based on wall-normal ordinary differential equations (ODEs) that assume equilibrium and models based on full partial differential equations (PDEs) that do not. We develop ideas for how to incorporate non-equilibrium effects (most importantly, strong pressure-gradient effects) in the wall-model while still solving only wall-normal ODEs. We test these ideas using two reference databases: an adverse pressure-gradient turbulent boundary-layer and a shock/boundary-layer interaction problem, both of which lead to separation and re-attachment of the turbulent boundary layer.	10.13140/2.1.1281.6324	[ "https://arxiv.org/pdf/1511.09356v1.pdf" ]	640,511	1511.09356	f08de8692bb1add8ed828e64ab11794896f1075d	A PARAMETRIZED NON-EQUILIBRIUM WALL-MODEL FOR LARGE-EDDY SIMULATIONS 30 Nov 2015 S Hickel Faculty of Aerospace Engineering TU Delft NL E Touber Department of Mechanical Engineering Imperial College UK J Bodart ISAE Université de Toulouse France J Larsson Department of Mechanical Engineering University of Maryland USA A PARAMETRIZED NON-EQUILIBRIUM WALL-MODEL FOR LARGE-EDDY SIMULATIONS 30 Nov 2015arXiv:1511.09356v1 [physics.flu-dyn] Wall-models are essential for enabling large-eddy simulations (LESs) of realistic problems at high Reynolds numbers. The present study is focused on approaches that directly model the wall shear stress, specifically on filling the gap between models based on wall-normal ordinary differential equations (ODEs) that assume equilibrium and models based on full partial differential equations (PDEs) that do not. We develop ideas for how to incorporate non-equilibrium effects (most importantly, strong pressure-gradient effects) in the wall-model while still solving only wall-normal ODEs. We test these ideas using two reference databases: an adverse pressure-gradient turbulent boundary-layer and a shock/boundary-layer interaction problem, both of which lead to separation and re-attachment of the turbulent boundary layer. INTRODUCTION Large-eddy simulations (LES) have become part of the basic toolkit for fundamental fluids research. However, the "near-wall problem" of requiring essentially DNS-type grid resolution in the innermost layer of turbulent boundary layers has effectively prevented LES from being applied to many realistic turbulent flows (cf. Piomelli & Balaras, 2002). The solution is to model (rather than resolve) the inner part of turbulent boundary layers, say the innermost 10-20% of the boundary layer thickness δ . By doing this, the grid resolution in an LES is set solely by the need to resolve the remaining outer layer. There are essentially two different classes of methods that follow this approach. In hybrid LES/RANS and detached eddy simulation (DES), the unsteady evolution equations with an eddy viscosity term are solved everywhere in the domain. The eddy viscosity is then taken from some RANS-type model in the inner layer and some LES-type subgrid model in the outer layer everywhere else in the flow. A second class of methods instead models the wall-stress directly. The LES domain is then defined formally as extending all the way to the wall, while an auxiliary set of equations is solved in an overlapping layer covering the innermost 10-20% of δ . These auxiliary equations are forced by the LES at their upper boundary, and feed the computed wall shear stress and heat transfer back to the LES. The focus in this study is exclusively on the second class of methods. The most obvious models for the wall stress assume equilibrium; that is, they neglect both the convective and pressure-gradient terms in addition to the wall-parallel diffusive terms. For constant-density flows above the viscous layer, these models yield the famous log-law. While the wall-model only models the innermost 10-20% of δ (and the LES directly resolves the outermost 80-90%), there has been a long interest in removing the assumption of equilibrium from the wall-model, in hopes of making wallmodeled LES capable of more accurate predictions in the presence of flow separation, etc. One approach to including non-equilibrium effects was pioneered by Balaras et al. (1996), who solved the thin boundary layer equations (including convection and pressure-gradient, neglecting only wall-parallel diffusion) as a wall-model. From a practical point-of-view, the main drawback of this approach is that a partial differential equation (PDE) must be solved as the wall-model. Thus a full grid with neighbor connectivity is needed in the near-wall layer, in addition to the already existing LES grid. This is a serious obstacle if one seeks to implement the wall-model in an unstructured code for complex geometries. In fact, one could argue that any new wall-model (of the wall-stress kind) will be broadly adopted only if it involves at most connectivity (i.e., derivatives) in the wall-normal direction and time, both of which are easily implemented in a general unstructured code framework. The challenge, therefore, is to include non-equilibrium effects without the need for wall-parallel derivatives. Hoffmann & Benocci (1995) and Chen et al. (2013) included the pressure-gradient and the temporal term but excluded the convective term 1 . Since the pressure-gradient is constant throughout the wall-modeled layer, and imposed from the LES, this approach does not require wall-parallel derivatives within the wall-model. Wang & Moin (2002) and subsequently Catalano et al. (2003) went one step further and retained only the pressure-gradient term in calculations of the flow over a trailing edge and a circular cylinder, respectively. Neither of these approaches is satisfactory, for reasons to be shown below. The objective of the present study is to develop a wallmodel that includes non-equilibrium effects while still requiring only numerical connectivity in, at most, the wallnormal and temporal directions. a) The 8-degree IUSTI shock/turbulent-boundary-layer interaction (labeled STBLI throughout) experiment of Dupont et al. (2006), with wall-resolved reference LES by Touber & Sandham (2009). Visualization of the instantaneous temperature. b) The adverse pressure-gradient turbulent boundary-layer (labeled APGTBL throughout) studied in Hickel & Adams (2008). Visualization of the instantaneous coherent structures (Q-criterion) and the separation bubble(u 1 = 0 iso-surfaces). Towards this end, the present paper will: 1. Argue and show that past attempts at including or neglecting the temporal, convective and pressuregradient terms independently are inconsistent, in the sense that the temporal and convective terms jointly describe the evolution of a fluid particle and that the pressure-gradient and convective terms largely balance outside of the viscous sublayer. 2. Argue and show that the convective term can be parametrized in terms of outer layer LES quantities, thereby eliminating the need for wall-parallel derivatives in the wall-model. TIME-FILTERED EQUATIONS When implemented in an LES code, the wall-model is continuously forced by the LES at the upper boundary of the wall-modeled domain (say, at height h wm ). To give accurate results, h wm should be within the inner part of the boundary layer, so about 10-20% of the boundary layer thickness δ or less. For accuracy, the grid spacing in the LES needs to be sufficiently small compared to h wm (Kawai & Larsson, 2012), which implies that h wm should not be chosen too small. The continuous forcing by the LES at the top boundary means that the wall-model operates in an unsteady mode. The relatively large (RANS-type) eddy viscosity in the wall-model acts as a low-pass filter; therefore, the solution in the wall-modeled layer will be unsteady with primarily low frequencies. This is approximately accounted for in the analysis below by applying a low-pass filter to the wall-resolved LES databases, specifically a top-hat fil-ter with characteristic width τ defined as u(t; τ) = 1 τ τ/2 −τ/2 u(t − t ′ )dt ′ .(1) With density-weighting, the associated Favre filter is u = ρu/ρ. Application of this filter to the streamwise momentum equation yields to leading order ∂ t u+ u j ∂ j u− ∂ 2 µ(T ) ∂ 2 u ρ ≈ − ∂ 1 p ρ − ∂ j (ρ u j u − u j u ) ρ modeled , (2) where wall-parallel diffusion has been neglected, as well as terms due to nonlinearity in the temperature-dependence of the viscosity. The streamwise, wall-normal and spanwise directions (perhaps defined locally) are denoted by subscripts 1, 2 and 3, respectively. For brevity, the streamwise velocity is interchangeably labeled u or u 1 . TEST CASES As a first step in this study, data from two reference large-eddy simulations is used to assess the new ideas in an a priori manner. The two reference LESs used sufficiently fine grids to fully resolve the viscous near-wall layer in a quasi-DNS sense. The first case considered is wall-resolving LES (Touber & Sandham, 2009) of a shock/turbulent-boundarylayer interaction (labeled STBLI throughout) consistent with the flow conditions of the IUSTI experiment of Dupont et al. (2006). An oblique shock wave generated by an 8-degree wedge impinges on a Ma = 2.3 (2) for the STBLI case at stations 3 (upper row) and 4 (lower row), filtered using time-scales τU ∞ /δ ≈ 0.0 (left column) and τU ∞ /δ ≈ 8.5 (right column). Pressure-gradient ( ) , viscous term ( ), resolved convection in all three directions ( ), and unresolved wall-normal Reynolds stress ( ). flat-plate turbulent-boundary layer with a displacementthickness Reynolds number of Re δ 1 = 21000. The test case provides regions where equilibrium assumptions are supposed to hold and regions with strong non-equilibrium effects. A visualization of this flow is shown in Figure 1a) based on the instantaneous temperature for the LES of Touber & Sandham (2009). The second test case is the incompressible nonequilibrium turbulent flat-plate boundary-layer flow of Hickel & Adams (2008) with a displacement-thickness Reynolds number going from Re δ 1 = 1000 to 30000. Due to the strong non-equilibrium conditions, which result from a constant adverse pressure-gradient imposed at the upper domain boundary, the mean velocity profiles of this boundary layer flow do not follow the classic logarithmic law of the wall. The adverse pressure-gradient decelerates the flow and eventually leads to a highly unsteady and massive flow separation, which is not fixed in space and covers more than a third of the computational domain. The separated flow region and instantaneous coherent structures are visualized in Figure 1b) through an instantaneous iso-surface of u 1 = 0 and iso-surfaces of the Q-criterion, respectively. This case is labeled APGTBL throughout this paper. CONSISTENCY WITH THE BERNOULLI EQUATION A typical wall-model essentially solves Eq. (2) with the unresolved convective term parametrized using an eddy viscosity model. The by far most common approach is to assume equilibrium, i.e., to neglect the temporal, resolved convective and pressure-gradient terms. This is exact only for Couette flow, but is a good approximation in many cases. Hoffmann & Benocci (1995) and, later on, two studies coming out of the Center for Turbulence Research (Wang & Moin, 2002;Catalano et al., 2003) retained the pressure-gradient term in Eq. (2) but neglected the convective term. One objective of this paper is to point out that this is inconsistent. Consider a flow with a non-zero pressure gradient. In the limit of weak turbulence, for flow in a straight line sufficiently far from the wall such that viscous effects are negligible, Eq. (2) should reduce to the socalled "Euler-s" equation, or, more familiarly (after integration along a streamline), to the Bernoulli equation. In other words, a non-zero pressure-gradient is accompanied by accelerating/decelerating flow, which causes a non-zero streamwise convective term. Therefore, if the pressuregradient term is explicitly included in the wall-model, then the streamwise convective term must also be included to satisfy this minimal consistency requirement. Evidence of this is shown in Figures 2 and 3, which show selected terms in the streamwise momentum equation (2) for the two test cases. Results are shown both unfiltered and filtered in time for the first test case, roughly mimicking the effect of the RANS-type eddy viscosity. The filtering hardly affects the viscous and pressure-gradient terms at all, which is to be expected given their essentially linear nature. Note also that not all terms are shown in the figures; thus the sum of all lines is not exactly zero. The pressure-gradient is essentially balanced by the time-filtered convection term (i.e., the convection that can be resolved by a wall-model) in most of the outer part of the boundary layer, and only within the viscous region does this approximate balance between convection and pressure- CONVECTIVE PARAMETRIZATION The previous section argued and showed that inclusion of the pressure-gradient term implies that the convective term must also be included for consistency reasons. The convective term, however, includes derivatives in the wall-parallel directions, which implies that a regular grid with full connectivity in all directions is needed to solve the wall-model. While this can be done relatively easily for academic test cases (cf. Balaras et al., 1996;Wang & Moin, 2002;Kawai & Larsson, 2010), it is hard to imagine an implementation in a general-purpose code with an unstructured grid topology. Therefore, it is crucial to remove the need for wall-parallel derivatives in the wall-model. This can be done by parameterizing the convective term u j ∂ j u in terms of outer layer quantities, which are available from the LES. The most straight-forward parametrization stems directly from the results shown and discussed in Section above. Since u j ∂ j u is essentially balanced by the pressure-gradient −∂ 1 p/ρ above the viscous layer, it follows directly that one can approximate u j ∂ j u ≈ − ∂ 1 p ρ h wm · y/y pg , y < y pg 1 , y ≥ y pg(3) where y pg is the point where viscous effects start damping the streamwise convective term (akin to the thickness of a Stokes layer). For an attached turbulent boundary layer, the value of y pg should be specified in viscous (plus) units for validity across different Reynolds numbers. Since the purpose of this study is to enable wall-models to capture separating flows, we instead set this parameter in viscous pressure-gradient scaling y ⋆ pg = y pg (ρ w \|∂ 1 p\|/µ 2 w ) 1/3 ; a fixed value of y ⋆ pg = 4 is used throughout here, with little attempts made at finding an optimal value. Equation (3) implies that the net effect of convection and pressure-gradient is zero above y pg . Thus this parametrization predicts that the logarithmic slope of the mean velocity is independent of the pressure-gradient, but that the additive intercept constant is not (if a regular mixing-length eddy viscosity model is used, such as Eq. (5)). A second potential parametrization of the convective term is to assume that the streamwise component u∂ 1 u is dominant in the unsteady type of boundary layer flow that occurs in a wall-model, and then to assume that the vertical shape of the derivative ∂ 1 u can be modeled by the shape of the velocity profile u itself. In other words, to approximate the convective term as u j ∂ j u ≈ u u\| h wm α u j ∂ j u h wm ,(4) where the convective term at height h wm is taken from the LES. We found that this approach leads to good predictions that depend only weakly on the precise values of the free parameter α; throughout this study α = 3/2 is used. These parameterizations of the convective term are tested a priori on the APGTBL case in Figure 4. First, note that the infinite time-filtering for this test case implies that w∂ 3 u = 0. Secondly, while the v∂ 2 u term is not insignificant, it is small compared to the streamwise component u∂ 1 u. The parametrization in terms of velocity, Eq. (4), gives a very reasonable agreement with the wall-resolved LES, while the parametrization in terms of pressure gradient, Eq. (3), only captures the gross features. However, as will be seen below, this is not the complete story. A PRIORI VALIDATION To assess the two proposed parameterizations of the convective term, a different type of a priori test is performed. Data from a height h wm above the wall is taken from the wall-resolved reference LES databases and used as the top boundary condition for the wall-model equations; these equations are then solved, and the wall stress is extracted and compared to the actual wall stress in the reference LES databases. The wall-model is defined by Eq. (2), with the unresolved convective term (the last term) modeled using an eddy-viscosity hypothesis and Eq. (5), and where the sum of the temporal and convective terms (the first two terms) is modeled using either (3) or (4). In the present study, the simple mixing-length model µ t,wm = κρy τ w ρ 1 − exp −y + A + 2 ,(5) with κ = 0.41 and A + = 17 is used. A finite volume approach is used to discretize the equations, and convergence is achieved using a Newton-type iterative procedure. Identical formulations are used for both databases with identical parameters, and compared with an equilibrium wall-model as well as the non-equilibrium wallmodel of Duprat et al. (2011). The results are shown in Figure 5. In the APGTBL case, all of the different wall-models under-predict the skin friction; this could be due to the low Reynolds number in this case (Re τ ≈ 400 at station 1), for which the wall-model parameters are not optimal. If the pressure-gradient is included without any additional modeling (as done by Hoffmann & Benocci, 1995;Wang & Moin, 2002;Catalano et al., 2003;Chen et al., 2013), the skin friction is further under-predicted, and separation occurs much too early. Including both the pressure-gradient and a parametrized convective term gives a model which satisfies the necessary "Bernoulli consistency." Despite this, and despite producing impressive a priori agreement of the vertical profiles in Fig. 4, the parametrization based on the velocity profile, Eq. (4), gives disappointing results, hardly better than without the convective term at all. In contrast, the parametrization based on the pressure-gradient, Eq. (3), gives excellent results for the STBLI case and reasonable results for the APGTBL case, albeit a bit worse than the results of the basic equilibrium model. We also note that the parametrization based on the velocity profile is more sensitive to numerical convergence issues, while the parametrization based on the pressure-gradient was found to be numerically robust. The model of Duprat et al. (2011) gives disappoint- ing results, essentially no different from the basic equilibrium model for the APGTBL case and with severely overpredicted wall shear for the STBLI case. SUMMARY Wall-models are essential for enabling the use of largeeddy simulations on realistic problems at high Reynolds numbers. The present study is focused on approaches that directly model the wall shear stress, specifically on filling the gap between models based on wall-normal ODEs that assume equilibrium and models based on full PDEs that do not. Ideas for how to incorporate non-equilibrium effects (most importantly, strong pressure-gradient effects) in the wall-model while still solving only wall-normal ODEs are developed and tested using two reference databases computed using wall-resolved LES: an adverse pressuregradient turbulent boundary-layer and a shock/boundarylayer interaction problem, both of which lead to boundarylayer separation and re-attachment. First, it is pointed out that the convective term and the pressure-gradient term must be treated consistently with each other, since a non-zero pressure-gradient is almost necessarily associated with a non-zero convective acceleration; these terms will have offsetting contributions in most cases. The bottom line is that these terms should either be retained or neglected jointly, not independently as done in several prior studies (e.g., Hoffmann & Benocci, 1995;Wang & Moin, 2002;Catalano et al., 2003). Similarly, since the temporal and convective terms jointly describe the acceleration of a fluid particle in its Lagrangian frame, for consistency these two terms must be treated in the same way as well. Next, it is argued that a non-equilibrium wallmodel in ODE-form requires that the convective terms be parametrized using LES data from the top of the wallmodeled layer. Two forms of this parametrization are proposed: one based on the pressure-gradient, one based on the velocity profile and the LES velocity gradient. When assessed a priori using the reference databases, no clear conclusion is reached: the pressure-based parametrization can capture only the gross features of the convective term, whereas the second parametrization based on the velocity profile gives a very good agreement with the wall-resolved LES data. However, when used to compute the skin friction in the two test cases, the model based on the pressuregradient appears superior: the predicted skin friction is very close to the reference one for the shock/boundary-layer in-teraction case, but slightly under-predicted for the adverse pressure-gradient case. Figure 1 : 1Test cases and interrogation stations used in this study. Figure 2 : 2Averaged terms in Eq. Figure 3 : 3Averaged terms in Eq. (2) for the APGTBL case at stations 1 to 4 using the time-scale τ → ∞. Pressure-gradient ( ) , viscous term ( ), resolved convection in all three directions ( ), and unresolved wall-normal Reynolds stress ( ). All quantities are given in outer scaling, i.e., scaled by the local boundary-layer edge velocity. gradient break down. Figure 4 : 4Parametrization of the convective term in terms of the pressure-gradient and velocity profile for the APGTBL case at stations 1, 3, 5 and 8 using the time-scale τ → ∞. Approximate u j ∂ j u from Eq. (3) ( ), Eq. (4) ( ) , exact u∂ 1 u (• ), and exact u j ∂ j u ( ). Figure 5 : 5A-priori study: Friction coefficient obtained using various non-equilibrium formulations and compared to existing models. Exact from reference LES (• ), equilibrium model ( ), model of Duprat et al. (2011) ( ), adding ∂ 1 p by itself ( ), adding ∂ 1 p + Eq. (3) with (y * pg = 4) ( ), adding ∂ 1 p + Eq. (4) ( ). Throughout this paper, temporal term refers to ∂ t u i while convective term refers to u j ∂ j u i . Twolayer approximate boundary conditions for large-eddy simulations. E Balaras, C Benocci, U Piomelli, AIAA J. 346BALARAS, E., BENOCCI, C. & PIOMELLI, U. 1996 Two- layer approximate boundary conditions for large-eddy simulations. AIAA J. 34 (6), 1111-1119. Numerical simulation of the flow around a circular cylinder at high reynolds numbers. P Catalano, M Wang, G Iaccarino, P Moin, Int. J. Heat Fluid Flow. 24CATALANO, P., WANG, M., IACCARINO, G. & MOIN, P. 2003 Numerical simulation of the flow around a circu- lar cylinder at high reynolds numbers. Int. J. Heat Fluid Flow 24, 463-469. Wall modeling for implicit largeeddy simulation and immersed-interface methods. Z L Chen, S Hickel, A Devesa, J Berland, N A Adams, 10.1007/s00162-012-0286-6Theor. Comput. Fluid Dyn. CHEN, Z. L., HICKEL, S., DEVESA, A., BERLAND, J. & ADAMS, N. A. 2013 Wall modeling for implicit large- eddy simulation and immersed-interface methods. Theor. Comput. Fluid Dyn., DOI: 10.1007/s00162-012-0286-6. Space and time organization in a shock-induced separated boundary layer. P Dupont, C Haddad, J F Debieve, J. Fluid Mech. 559DUPONT, P., HADDAD, C. & DEBIEVE, J. F. 2006 Space and time organization in a shock-induced sepa- rated boundary layer. J. Fluid Mech. 559, 255-277. A wall-layer model for large-eddy simulations of turbulent flows with/out pressure gradient. C Duprat, G Balarac, O Metais, P M Congedo, O Brugiere, Phys. Fluids. 2315101DUPRAT, C., BALARAC, G., METAIS, O., CONGEDO, P. M. & BRUGIERE, O. 2011 A wall-layer model for large-eddy simulations of turbulent flows with/out pres- sure gradient. Phys. Fluids 23, 015101. Implicit LES applied to zero-pressure-gradient and adverse-pressure-gradient boundary-layer turbulence. S Hickel, N A Adams, Int. J. Heat Fluid Flow. 29HICKEL, S. & ADAMS, N. A. 2008 Implicit LES applied to zero-pressure-gradient and adverse-pressure-gradient boundary-layer turbulence. Int. J. Heat Fluid Flow 29, 626-639. Approximate wall boundary conditions for large eddy simulations. 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Fluid Mech. 34PIOMELLI, U. & BALARAS, E. 2002 Wall-layer models for large-eddy simulations. Annu. Rev. Fluid Mech. 34, 349-374. Large-eddy simulation of low-frequency unsteadiness in a turbulent shockinduced separation bubble. E Touber, N Sandham, Theor. Comp. Fluid Dyn. 23TOUBER, E. & SANDHAM, N. 2009 Large-eddy simula- tion of low-frequency unsteadiness in a turbulent shock- induced separation bubble. Theor. Comp. Fluid Dyn. 23, 79-107. Dynamic wall modeling for large-eddy simulation of complex turbulent flows. M Wang, P Moin, Phys. Fluids. 147WANG, M. & MOIN, P. 2002 Dynamic wall modeling for large-eddy simulation of complex turbulent flows. Phys. Fluids 14 (7), 2043-2051.	[]
[ "Nonequilibrium critical dynamics of the ferromagnetic Ising model with Kawasaki dynamics", "Nonequilibrium critical dynamics of the ferromagnetic Ising model with Kawasaki dynamics" ]	[ "Florent Krzaka \nClaude Godrèche Service de Physique de l'État Condensé\nDipartimento di Fisica, INFM (UdR Roma I and SMC center)\nCEA Saclay\nUniversità di Roma \"La Sapienza\", P. A. Moro 291191, 00185Gif sur Yvette cedex, RomaFrance, Italy\n", "La \nClaude Godrèche Service de Physique de l'État Condensé\nDipartimento di Fisica, INFM (UdR Roma I and SMC center)\nCEA Saclay\nUniversità di Roma \"La Sapienza\", P. A. Moro 291191, 00185Gif sur Yvette cedex, RomaFrance, Italy\n", "Federico Ricci-Tersenghi \nClaude Godrèche Service de Physique de l'État Condensé\nDipartimento di Fisica, INFM (UdR Roma I and SMC center)\nCEA Saclay\nUniversità di Roma \"La Sapienza\", P. A. Moro 291191, 00185Gif sur Yvette cedex, RomaFrance, Italy\n" ]	[ "Claude Godrèche Service de Physique de l'État Condensé\nDipartimento di Fisica, INFM (UdR Roma I and SMC center)\nCEA Saclay\nUniversità di Roma \"La Sapienza\", P. A. Moro 291191, 00185Gif sur Yvette cedex, RomaFrance, Italy", "Claude Godrèche Service de Physique de l'État Condensé\nDipartimento di Fisica, INFM (UdR Roma I and SMC center)\nCEA Saclay\nUniversità di Roma \"La Sapienza\", P. A. Moro 291191, 00185Gif sur Yvette cedex, RomaFrance, Italy", "Claude Godrèche Service de Physique de l'État Condensé\nDipartimento di Fisica, INFM (UdR Roma I and SMC center)\nCEA Saclay\nUniversità di Roma \"La Sapienza\", P. A. Moro 291191, 00185Gif sur Yvette cedex, RomaFrance, Italy" ]	[]	We investigate the temporal evolution of a ferromagnetic system of Ising spins evolving under Kawasaki dynamics from a random initial condition, in spatial dimensions one and two. We examine in detail the asymptotic behaviour of the two-time correlation and response functions. The linear response is measured without applying a field, using a recently proposed algorithm. For the chain at vanishingly small temperature, we introduce an accelerated dynamics which has the virtue of projecting the system into the asymptotic scaling regime. This allows us to revisit critically previous works on the behaviour at large time of the two-time autocorrelation and response functions. We also analyse the case of the two-dimensional system at criticality. A comparison with Glauber dynamics is performed in both dimensionalities, in order to underline the similarities and differences in the phenomenology of the two dynamics.	10.1088/1742-5468/2004/04/p04007	[ "https://export.arxiv.org/pdf/cond-mat/0401334v2.pdf" ]	21,214,733	cond-mat/0401334	ba6e160cd86b337010952acddc6019c9c5504b03	Nonequilibrium critical dynamics of the ferromagnetic Ising model with Kawasaki dynamics 5 Apr 2004 Florent Krzaka Claude Godrèche Service de Physique de l'État Condensé Dipartimento di Fisica, INFM (UdR Roma I and SMC center) CEA Saclay Università di Roma "La Sapienza", P. A. Moro 291191, 00185Gif sur Yvette cedex, RomaFrance, Italy La Claude Godrèche Service de Physique de l'État Condensé Dipartimento di Fisica, INFM (UdR Roma I and SMC center) CEA Saclay Università di Roma "La Sapienza", P. A. Moro 291191, 00185Gif sur Yvette cedex, RomaFrance, Italy Federico Ricci-Tersenghi Claude Godrèche Service de Physique de l'État Condensé Dipartimento di Fisica, INFM (UdR Roma I and SMC center) CEA Saclay Università di Roma "La Sapienza", P. A. Moro 291191, 00185Gif sur Yvette cedex, RomaFrance, Italy Nonequilibrium critical dynamics of the ferromagnetic Ising model with Kawasaki dynamics 5 Apr 2004(Dated: January 19, 2004)numbers: 0570Ln6460My7540Gb We investigate the temporal evolution of a ferromagnetic system of Ising spins evolving under Kawasaki dynamics from a random initial condition, in spatial dimensions one and two. We examine in detail the asymptotic behaviour of the two-time correlation and response functions. The linear response is measured without applying a field, using a recently proposed algorithm. For the chain at vanishingly small temperature, we introduce an accelerated dynamics which has the virtue of projecting the system into the asymptotic scaling regime. This allows us to revisit critically previous works on the behaviour at large time of the two-time autocorrelation and response functions. We also analyse the case of the two-dimensional system at criticality. A comparison with Glauber dynamics is performed in both dimensionalities, in order to underline the similarities and differences in the phenomenology of the two dynamics. I. INTRODUCTION The kinetics of ferromagnetic spin systems evolving after an initial quench from a high temperature disordered initial condition to a final temperature, equal or below the critical temperature, is a well investigated field (see e.g. ref. [1] for a review). However, only recently has the emphasis been put on the dynamics of two-time quantities, such as the correlation and response functions, or the fluctuation-dissipation violation ratio, with the aim of quantifying the distance of the system to equilibrium during its temporal evolution [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] (see e.g. [19] for a brief review). Most of these studies focus on the non-conserved order parameter case, where at each time step a single spin is updated according to the rules of Glauber dynamics. An overall coherent picture of this field has by now emerged, though some controversies remain [20,21,22,23]. Much fewer studies have been devoted so far to the same questions, namely the temporal evolution of two-time quantities, for the case of conserved order parameter dynamics. Restricting to a system of discrete spins, the rules of Kawasaki dynamics [24] now consist in choosing two adjacent opposite spins, and exchange them with a rate depending on the energy difference between the initial and final configurations. In the recent past the question of the long-time behaviour of the autocorrelation function for conserved dynamics has already been addressed [25,26,28]. In particular predictions have been given for the values of the autocorrelation exponents λ and λ c governing the decay of the autocorrelation function at large temporal separations, respectively in the low temperature phase, and at criticality. Finally, a very recent work addresses the question of the response for the Ising-Kawasaki chain in the low temperature scaling regime [29]. The prediction made in this reference states that the fluctuation-dissipation plot (that is the relationship between integrated response and correlation) of this case is identical to that of the Glauber non-conserved case, which is itself known analytically [8,9]. This result is rather surprising because it would imply some kind of "super universal" behaviour in the relationship between correlation and response in the asymptotic regime. The aim of the present work is to revisit and extend these former studies. We investigate the behaviour of the two-time correlation and response functions for an Ising spin system, both in one and two dimensions, with Hamiltonian H = − i,j σ i σ j(1) where i, j are nearest neighbours, evolving under Kawasaki dynamics after a quench of the system from a disordered high temperature initial condition to the critical temperature. We first give the method used in this paper in order to compute the linear response. This method, due to Chatelain [30], and later on clarified in [31], will be used both for the one-dimensional and two-dimensional cases. We then describe the rules of an accelerated dynamics for the Ising-Kawasaki chain corresponding to the formal limit T → 0, which is nevertheless faithful, i.e. reproduces exactly the results that would be obtained with the usual rules of Kawasaki dynamics for vanishingly small temperatures. We finally present the results of extensive numerical computations of the autocorrelation and response functions, first for the case of a chain, then for the two-dimensional system at criticality. II. MEASURING THE LINEAR RESPONSE WITHOUT APPLYING A FIELD IN KAWASAKI DYNAMICS In this section we give an analytical expression of the response function to an infinitesimal field, for a ferromagnetic system evolving under Kawasaki dynamics, and show how this quantity can be measured. We follow the lines of reasoning of refs. [30,31], which are devoted to the same question for single-spin flip dynamics. A. Kawasaki dynamics with heat bath rule Hereafter, time t is discrete and counts the number of spin exchange attempts, and not the number of Monte Carlo sweeps. In order to define a response function, an external perturbing field h i is applied on any site i, and, in presence of the field, the Hamiltonian is changed to H − h i σ i . Kawasaki rules consist in updating a pair of two opposite adjacent spins σ i = −σ j (we will always omit the δ σ i ,−σ j factor in the following), with heat bath probabilities P(σ i = σ, σ j = −σ) = exp[βσ(h W ij + h i − h j )] 2 cosh[β(h W ij + h i − h j )] ,(2) where β is the inverse temperature, and the Weiss field h W ij takes into account the effect of neighbours on the couple of spins which are updated. For a generic 2-spins interaction Hamiltonian we have h W ij = k∈∂i\j J ik σ k − l∈∂j\i J jl σ l ,(3) where ∂i\j represents the set of neighbours of i, with j excluded. For example, in one dimension, with Hamiltonian (1), we have h W i,i+1 = σ i−1 − σ i+2 . B. Response function Following strictly the notation of ref. [31], we consider systems made of N Ising spins, where the autocorrelation and the response functions are defined as C(t, s) = 1 N N i=1 σ i (t)σ i (s) , R(t, s) = 1 N N i=1 ∂ σ i (t) ∂h i (s) ,(4) with · representing the average over thermal histories. We concentrate on the integrated response function, or susceptibility χ(t, s) = T t s du R(t, u) ,(5) where the temperature T has been added in the usual definition in order to simplify the notation and to have a well defined expression in the T → 0 limit. Denoting by I(t) and J(t) the indices of the two spins to be updated at time t, the expectation value of the k-th spin at time t is given by σ k (t) = Tr σ(t ′ ) σ k (t) t t ′ =1 W I(t ′ )J(t ′ ) σ(t ′ )\| σ(t ′ −1) ,(6) where σ is the vector of the N spins configuration, the trace is over all the histories σ(t ′ ) with 1 ≤ t ′ ≤ t, and the transition probability are given by W ij ( σ\| τ ) = exp[βσ i (h W ij + h i − h j )] 2 cosh[β(h W ij + h i − h j )] k =i,j δ σ k ,τ k .(7) Note that h W ij ( σ) = h W ij ( τ ), because the Weiss field does not depend on the value of spins at sites i and j. Since the transition probability W ij only depends on the perturbing fields on sites i and j, one has ∂W ij ( σ\| τ ) ∂h k h=0 = βW ij ( σ\| τ ) δ i,k (σ i − σ W ij ) + δ j,k (σ j + σ W ij ) ,(8) where we have defined σ W ij ≡ tanh(βh W ij ). Now, if on site k an infinitesimal probing field h k is switched on at time s (i.e. h k (t) = h θ(t − s)), all transition probabilities with index k (and only these ones) will depend on the perturbing field for times larger than s. Then differentiation of eq. (6) with respect to this field yields the integrated response χ lk (t, s) = T ∂ σ l (t) ∂h k h=0 = Tr σ(t ′ ) σ l (t) t t ′ =1 W I(t ′ ),J(t ′ ) σ(t ′ )\| σ(t ′ −1) t u=s+1 δ I(u),k σ k (u) − σ W I(u)J(u) (u) + δ J(u),k σ k (u) + σ W I(u)J(u) (u)(9) which can be simply written as a correlation function χ lk (t, s) = σ l (t) ∆σ k (t, s) ,(10) where ∆σ k (t, s) = t u=s+1 δ I(u),k σ k (u) − σ W I(u)J(u) (u) + δ J(u),k σ k (u) + σ W I(u)J(u) (u) .(11) C. Computing the linear response in a simulation Let us note that calculating the linear response in a numerical simulation with no perturbing field using eq. (10) is as easy as measuring a correlation function. One has to keep track of the vector ∆σ k (t, s) that should be updated for all times between s and t (i.e. when the "ghost" field is switched on). At each of these times, one has to: • compute h W ij and σ W ij ≡ tanh(βh W ij ); • update spins σ i and σ j according to the heat bath probability (with no external field) P(σ i = σ, σ j = −σ) = exp(βσh W ij ) 2 cosh(βσh W ij ) ; • increment ∆σ i , ∆σ j as ∆σ i → ∆σ i + σ i − σ W ij ,(12)∆σ j → ∆σ j + σ j + σ W ij .(13) It is important to note at this point that contributions to the increment ∆σ i come either when the updated spin flips, or when it does not, keeping its previous value. This is the reason why one can speak of a "ghost" field in this method (see above). An illustration of this fact is encountered in section III B below. The use of eq. (10) allows one to compute, in a single run, the integrated response (5) (zero-field cooled magnetization), as well as the thermoremanent magnetization, ρ(t, s) = T s 0 du R(t, u),(14) both for many different values of s. III. ACCELERATED DYNAMICS OF THE ISING-KAWASAKI CHAIN IN THE ZERO TEMPERATURE LIMIT It is well known that Kawasaki dynamics at zero temperature rapidly brings the onedimensional system to a blocked state, where the distance between any couple of domain walls is at least 2 [32,33]. In such a situation any spin-exchange move would cost an energy ∆E = 4, and is therefore forbidden. For example ↑↑↑↑↓↓↓↓ → ↑↑↑↓↑↓↓↓ would create two domain walls. Such a process is called an evaporation [32]. However, at an infinitesimal temperature T = ε, the evolution may eventually undergo such an evaporation with probability 1/(1 + e 4/ε ), hence on a time scale O(e 4/ε ), which diverges for ε → 0. After each evaporation process the dynamics proceeds by diffusion, that is by moves with energy cost ∆E = 0 (e.g. ↓↓↓↑↓↓↓↓ → ↓↓↓↓↑↓↓↓), until a condensation process occurs, corresponding to a move with energy cost ∆E = −4, which brings the system back to a new blocked state (e.g. ↓↓↓↑↓↑↑↑ → ↓↓↓↓↑↑↑↑). The diffusion and condensation processes take place on scales of time O(1), i.e. much smaller than the typical time between two consecutive evaporations, and therefore can be considered as instantaneous if time is counted in units of τ = e 4/ε , when ε → 0. We exploit the strong separation of time scales between evaporation and diffusion/condensation processes, and we simulate the dynamics of the Ising-Kawasaki chain in the limit Moreover, for T → 0, eqs. (12,13) can be simplified to ∆σ i → ∆σ i + {2σ i , σ i , 0}, and ∆σ j → ∆σ j +{2σ j , σ j , 0}, for evaporation, diffusion and condensation processes, respectively. T = ε → 0 Between two evaporation processes, there is however an additional contribution to take into account (see eq. (17) below), which is not apparent in eqs. (12,13). This is a direct consequence of the fact, mentioned above, that ∆σ k (t, s) gets contributions of updated spins, even when they do not flip. Consider a blocked configuration, where all the spins are aligned with their local fields, σ i = sign(h W ij ) and σ j = − sign(h W ij ). Since the time spent in each blocked state becomes infinite for ε → 0, a non trivial limit for the integrated quantity ∆σ is generated. Indeed, working at temperature T = ε, and, noting that in a blocked state h W ij = 2σ i , we have, σ W ij = tanh(h W ij /ε) ≈ sign(h W ij ) 1 − 2e −2\|h W ij \|/ε = σ i 1 − 2 τ ,(15) hence σ i − σ W ij ≈ 2σ i τ .(16) Then the summation over τ /n MCS of the last quantity gives a finite limit when ε → 0 and τ → ∞. In practice this means that, just before leaving a blocked state by an evaporation process, all the ∆σ i for spins close to a domain wall must be updated with the following rule ∆σ i → ∆σ i + 2σ i n .(17) Some final comments on the accelerated dynamics described in this section are in order. • This dynamics is faithful, i.e. it reproduces exactly the results that would be obtained by a standard Monte Carlo simulation at finite temperature T = ε, in the limit ε → 0. This equivalence will be illustrated below on the example of the mean length of domains. It is also very efficient, since it requires much less computational efforts than the standard Monte Carlo dynamics. • Its definition can be extended to any spatial dimension. • Finally, one may wonder how this dynamics compares to the effective dynamics of refs. [32,34], which is only defined in one dimension. The spirit of the later is to trace upon all events occurring between the instant of time a spin detaches from a domain and that when it reaches the neighbouring domain. The accelerated dynamics introduced in this section does not trace upon these events. However doing so -in one dimension only-would lead to the dynamics of [32,34], and allow a faster computation of the average domain length L(t), and with little more work, of the autocorrelation. In contrast, tracing upon these events is much more subtle for the computation of the response, and would deserve further study. Otherwise stated, it is not clear to us for the time being whether the method of [32,34] can be used for the computation of the response. The difficulty comes from the fact that ∆σ gets contributions even when spins are updated without changing their value. IV. RESULTS FOR THE DYNAMICS OF THE ISING-KAWASAKI CHAIN IN THE ZERO-TEMPERATURE LIMIT In this section we report the results of extensive numerical simulations, using the methods described in the previous sections. We are interested in the behaviour of observables in the low-temperature scaling regime defined by 1 ≪ t ≪ t eq , where the equilibration time t eq is related, at inverse temperature β = 1/ε, to the equilibrium correlation length ξ ≈ e 2β /2 by t eq ∼ ξ 5 [35]. This regime is naturally attained by the accelerated dynamics. The mean domain length is related to the energy E(t) of the chain at time t by L(t) = 2 1 + E(t) .(18) It is well known that, in the low-temperature scaling regime, L(t) scales as where τ = e 4β ∼ ξ 2 . This scaling is illustrated in ref. [35], where for increasing values of τ (i.e. decreasing temperatures) a linear master curve is found when L(t) is plotted against L(t) ∼ t τ 1/3 ,(19)(t/τ ) 1/3 . We measured L(t) using the accelerated dynamics of section III. Since time is measured in units of τ in such a scheme, the curve thus found is the limiting master curve that would have been found by the conventional means of [35] in the limit T → 0. We indeed checked that the amplitude of the law (19), L(t) = A (t/τ ) 1/3 , found in [35] was in agreement, though less precise, than that found in the present work, A ≃ 2.29. Figure 1 depicts the master curve thus obtained in the T → 0 limit. The system size is N = 2 12 = 4096, and the number of samples are reported in Table I. (We also checked for the absence of any finite-size effects by simulating few N = 2 16 samples.) The inset of figure 1 shows the growth of L(t) in a log-log scale. The convergence to the slope 1/3 is slow, mainly because of the presence of an offset at initial times. This offset corresponds to the first blocked state reached by the system. Remind that the system is B. Autocorrelation A well-known fact of the kinetics of coarsening with non-conserved dynamics is that, in the low-temperature scaling regime, and for large temporal separations (1 ≪ s ≪ t ≪ t eq ), the two-time autocorrelation C(t, s) decays as [1] C(t, s) ∼ L(t) L(s) −λ ,(20) and, for the particular case where s = 0, as C(t, 0) ∼ L(t) −λ ,(21) defining the autocorrelation exponent λ [36]. The case of conserved dynamics is more complex. For quenches to temperatures below Given the prediction of ref. [25] that, in the low-temperature phase, C(t, 0) ∼ L(t) −λ , with λ = d, Yeung et al. conclude that, for d = 1 (and more generally for low dimensions), the behaviour (21) holds for s small, while for s in the scaling regime the behaviour (20) should be replaced by C(t, s) ∼ L(t) L(s) −λ ′(22) with a different exponent λ ′ > λ. The prediction above therefore implies that for the Ising-Kawasaki chain at vanishingly small temperature the curve of the autocorrelation for two values of s, one being taken in the short-time regime, the other one in the scaling regime, should cross at some later time t * . This simple observation was not noticed by the authors of [28]. The question therefore arises of what is the behaviour of the autocorrelation C(t, s) for later times t ≫ t * . Hereafter we suggest the following scenario, based on the reasonable hypothesis that the two curves mentioned above do not cross. Define more precisely t * (s) by L(t * ) −λ ∼ L(t * ) L(s) −λ ′ ,(23) i.e. t * (s) ∼ s λ ′ /(λ ′ −λ) . Then, there are two scaling regimes: • the intermediate scaling regime of Yeung et al., for t ≪ t * , where (22) holds, • the ultimate scaling regime, for t ≫ t * , where C(t, s) ∼ L(t) −λ .(24) Therefore the following scaling law should hold Outcomes from our simulations are compatible with these predictions. Figure 2 shows the autocorrelation function as a function of L(t), for the values of s given in Table I. The presence of two different scaling regimes is evident, although the intermediate regime is not very clean on small s data, and the ultimate regime is not reached for large s data. Figure 3 is a plot of the scaling function g(x) defined above. We find λ ′ ≃ 2.5, with λ = 1. C(t, s) = L(t) −λ g L(t) λ ′ −λ L(s) λ ′ ,(25) Let us note that before attaining the intermediate scaling regime there is yet another temporal regime, clearly visible on figures 2 and 3, which takes place at very short times. One can show that in the scaling variable of figure 3 this very short time regime becomes smaller when s increases, and therefore irrelevant in the s → ∞ limit. Indeed one finds numerically -see figure 4-that in the very short time regime the autocorrelation function can be written as follows time regime is s 2/3 . This timescale is much smaller than the timescale of the intermediate regime, which is s. In preparation of section V, let us mention that, at criticality, dynamical scaling predicts that C(t, 0) ∼ L(t) −λc ,(27) defining the critical autocorrelation exponent λ c [2,3]. We have also, for s in the scaling regime, C(t, s) ∼ L(s) −2β/ν L(t) L(s) −λc(28) where β and ν are the usual static critical exponents (β = 1/4, ν = 1, in 2D). This form holds for both non-conserved and conserved dynamics, with the possibility that, for the latter, the exponent appearing in (28) be not the same as in (27), as discussed in section V. For conserved dynamics at criticality, refs. [25,26,27] predict λ c = d, while statements made in [28] on the long-time behaviour of the autocorrelation are less precise. A last comment is in order. At criticality, there are no well-defined growing domains. The interpretation of the growing length in eqs. (27) and (28) the system looks critical. The present situation of a system evolving after a quench from high temperature to T = T c is nevertheless usually referred to as "critical coarsening". By convenience we shall still call the length L(t) the mean domain size. Note that for the Ising chain the magnetization exponent β = 0, hence there is no distinction to be made between the two behaviours (20) and (28). C. Response and fluctuation-dissipation plot Let us recall that for the zero-temperature non-conserved (Glauber) case, the two-time correlation [1] and response functions are known analytically [8,9]. As a consequence, the fluctuation-dissipation ratio X(t, s), defined by [5] R(t, s) = X(t, s) T ∂C(t, s) ∂s , can be obtained in closed form. In the scaling regime, it reads X(t, s) ≈ 1 2 1 + s t , yielding, for large temporal separations, the limiting ratio X ∞ = lim s→∞ lim t→∞ X(t, s) = 1 2 . In the scaling regime, the integrated response functions ρ(t, s) and χ(t, s) only depend on t/s, or equivalently on C, ρ(C) = 1 √ 2 − χ(C) = √ 2 π arctan 1 √ 2 tan πC 2 .(29) Alternatively the limiting ratio can be extracted from ρ(C), as X ∞ = lim C→0 ρ(C) C .(30) Note that, in contrast to the generic two-dimensional case considered in section V, for the Ising chain, the integrated response function does not bear any dependence in s because the exponent β = 0. We now turn to the Ising-Kawasaki chain. Figure 5 depicts a plot of the integrated response χ(t, s) against the correlation C(t, s), in the limit of vanishingly small temperature, for the values of the waiting time s given in Table I However this limiting curve lies above the theoretical Glauber curve (29), in contradiction with the prediction made in ref. [29], stating that the fluctuation-dissipation plot for the Ising-Kawasaki chain is identical to that of the Ising-Glauber chain, for vanishingly small temperature. We are thus led to critically review ref. [29]. Glauber curve (29). The authors of ref. [29] therefore induce that the data obtained for vanishingly small temperatures and increasing values of s should eventually converge to eq. (29). Noting that the values of (T, s) mentioned above correspond, in units of e 4β , to s = 7.2, s = 6.6, and s = 33 respectively, we see on figure 5 that, indeed, for this range of values of s, the data points fall not too far away from the Glauber curve. However, since this holds neither for smaller nor for larger values of s (in units of e 4β ), we conclude that the apparent identity between Kawasaki and Glauber curves observed in [29] is just a coincidence due to the range of values considered in this reference. The existence, in the intermediate regime, of a non trivial scaling limit for the response implies, as a corollary, a non trivial value of X ∞ , when the ratio x = t/s → ∞. A precise numerical value of this quantity is however difficult to obtain from ZFC data. At present we can not exclude the possibility that X ∞ is the same for both Glauber and Kawasaki dynamics. Consider now the ultimate regime (t ≫ t * (s)). This regime corresponds to C < C * (s) where C * (s) ≡ C(t * (s), s) ∼ s −λλ ′ /(λ ′ −λ) , with λλ ′ /(λ ′ − λ) ≃ 1. 66. This is reminiscent of the situation encountered at criticality for the two-dimensional Ising-Glauber system [10]. In this case the fluctuation-dissipation theorem holds except in a region C < C * (s) ∼ s −ac , where a c = 2β/νz c ≃ 0.115, which vanishes for increasing values of s. However this mechanism does not prevent the occurrence of a non trivial limiting ratio X ∞ [10]. (See section V.) By analogy, we expect a non trivial value of X ∞ in the ultimate regime, a priori different from that obtained in the intermediate regime. Note however that since the exponent λλ ′ /(λ ′ −λ) is larger than 1, C * (s) is decreasing very fast, and, as a consequence, the regime where X ∞ could be measured is hardly reachable in a numerical simulation. V. RESULTS FOR THE CRITICAL DYNAMICS OF THE 2D ISING-KAWASAKI MODEL The aim of this section is to investigate the critical coarsening of a two-dimensional system of spins evolving under Kawasaki dynamics from a random initial condition. We will, as in the previous section, and inspired by the results found there, examine in turn the behaviour of the two-time correlation function, then of the two-time response function. A. Mean domain length The numerical study of the long-time behaviour of C(t, s = O(1)) for the critical dynamics of the 2D Ising-Kawasaki model is well established [26]. Scaling is observed, i.e. eq. (27) holds, with a decay exponent λ c = 2, confirming the prediction of [25]. The mean domain size itself is observed to grow as L(t) ∼ t 1/zc , with z c = 4−η = 15/4 [26]. A confirmation of these results is provided by figure 6, where the mean size of domains is extracted by the excess energy with respect to the equilibrium energy at T c , E eq = 1/ √ 2, i.e. L(t) = (E(t) − E eq ) −1 . This method has been already used for low temperature coarsening, either for conserved [37], or non conserved dynamics [36]. It is however the first time that it is used at criticality, where in general the growing length scale L(t) is obtained from the position of the first zero of the equal-time correlation function. Our reason to do so lies in the fact that for critical coarsening dynamical scaling holds, as it does for low temperature coarsening, and that therefore there exists only one single growing length scale in the system. In order to compare the qualitative behaviour of both conserved and non-conserved dynamics at T c , we take a series of snapshots at instants of times where similar domain sizes were reached in both dynamics (see figure 7). Since for non-conserved dynamics z c ≈ 2.17, in 2D, conserved dynamics is much slower. It is interesting to note the overall similarity of the snapshots at corresponding instants of time. B. Autocorrelation The study of the behaviour of C(t, s) when the waiting time s is deep in the scaling regime is largely unexplored. We first measured the autocorrelation as a function of L(t) for different values of s (see fig. 8). As in the one-dimensional case, we observe a seemingly In order to assess this point, we define, by analogy with the one-dimensional case (see eq. (25)), the scaling function g c as C(t, s) = L(t) −λc g c L(t) λ ′ c −λc L(s) λ ′ c −2β/ν ,(31) with g c (x → 0) ∼ x −1 in the intermediate regime, while in the ultimate regime g c (x → ∞) should converge to a constant. Figure 9 depicts the scaling function obtained using λ c = 2 and λ ′ c = 3.5. In view of this figure it is reasonable to conclude again in favour of the existence of these two scaling regimes, defined by the relative magnitude of t with respect to the crossover timescale t * (s) ∼ s (λ ′ c −2β/ν)/(λ ′ c −λc) . Otherwise stated, t ≪ t * (s) in the intermediate regime, while t ≫ t * (s) in the ultimate regime. We have, with the values of the exponents given above, and 2β/ν = 1/2, t * (s) ∼ s 2 . C. Response and fluctuation-dissipation plot We now turn to the response. Following [10], we choose to compute the thermoremanent magnetization (14) (TRM). Due to the fast increase of the crossover timescale, t * (s) ∼ s 2 , and to the extreme difficulty to have precise data for the response at very long times, the results presented below only concern the intermediate regime. These results are contained in figures 10, 11, and 12. In the intermediate regime we assume for the TRM a scaling form similar to that of the correlation function, equation (28), that is ρ(t, s) ∼ L(s) −2β/ν L(t) L(s) −λ ′ c .(32) This scaling form is well verified by our numerical data, as illustrated by figure 10, which depicts plots of the rescaled correlation and response functions, both for Glauber and Kawasaki dynamics. For the latter, it is interesting to note that, when s increases, the master curve is attained from above for the correlation, and from below for the response, indicating that asymptotically the two scaling functions should have the same algebraic decay with exponent λ ′ c . Plots of the scaling functions of autocorrelation and response for the two-dimensional Ising-Glauber model at criticality first appeared in [10]. The value of the exponent measured in the present work is slightly smaller than that found in [10]. Another representation of the same data is given in figure 11, which depicts the parametric plot of the rescaled response against the rescaled correlation, both for Glauber and Kawasaki dynamics. This figure shows that the two dynamics lead to the same phenomenology. Indeed, define a crossover scale for the autocorrelation by C * (s) = C(2s, s) [10], corresponding, for s large enough, to a value of the abscissa on figure 11 approximately equal to 0.47 for Glauber dynamics, and to 0.27 for Kawasaki dynamics. Then, on the right part of the plots with respect to these values, the slope of the two curves is equal to one, in agreement with the fluctuation-dissipation theorem. The left part of the plots corresponds to the scaling regime, with a crossover towards a non trivial slope at the origin, equal to the limiting violation ratio X ∞ . In order to extract the numerical estimate of these slopes from these data, we plot them in two different fashions, as shown on figure 12. Since ρ(t, s) ≈ X ∞ C(t, s) (see (30) Error bars are not shown in the latter, in order to improve readability. For Glauber dynamics X ∞ seems to be attained linearly in s/t , which leads, by taking the intercept with the yaxis, to the prediction X Glauber ∞ ≈ 0.33, in agreement with the estimates given in recent works [15,30]. Note that for the spherical model the correction to X ∞ is exactly in s/t [10]. Finally, though it is difficult to be conclusive on the sole basis of figure 12, the latter gives VI. CONCLUSION Let us summarize the most salient points of this study. We extend the method of [30,31] to the case of Kawasaki dynamics. We introduce a new method for the investigation of the low-temperature scaling regime of the Ising-Kawasaki chain. We define rules for an accelerated dynamics, which are both faithful and efficient. This method can be extended to higher dimensions. We find new results concerning the behaviour at large time of the autocorrelation function for the critical Kawasaki dynamics in both dimensionalities, and demonstrate the existence of an ultimate regime, which was overlooked in previous studies. We believe this regime to be also present in the behaviour of the response, though hardly accessible with present computer capabilities. As a corollary, we expect the existence of a different value of the limiting ratio X ∞ in the ultimate regime, which would require time scales which are presently unreachable. In the course of this study we were led to question the validity of the results of ref. [29] concerning the fluctuation-dissipation plot for the Ising-Kawasaki chain. The evidence, claimed in [29], for the identity of the fluctuation-dissipation plots for Glauber and Kawasaki dynamics is seemingly coincidental, and lies in the range of values used in this reference. For the two-dimensional system at criticality, the fluctuation-dissipation plots obtained from the two dynamics are phenomenologically similar, but not quantitatively identical. In particular, the limiting violation ratios X ∞ are found to be different. It is harder though to be conclusive in the two-dimensional case than in the one-dimensional one. In the former, the scaling region defined by C(t, s) < C * (s) ∼ s −0.115 is small, while in the latter, since β = 0, it identifies to the whole range 0 ≤ C(t, s) ≤ 1. In the intermediate scaling regime numerical data are compatible with the simple form λ ′ c = d + 3/2 for the autocorrelation exponent. It would be interesting to assess the validity of this hypothesis. A theoretical explanation of the existence of the ultimate regime, both for the autocorrelation and for the response, is needed. Finally, a natural extension of the present work is the study of the behaviour of the autocorrelation and response in the low-temperature phase (T < T c ) of the two-dimensional kinetic Ising model with Kawasaki dynamics. with the following accelerated dynamics. Time in this accelerated dynamics is counted in units of τ = e 4/ε . A. Rules for the accelerated dynamics As long as the system is not in a blocked state, we use the usual T = 0 Kawasaki dynamics where only spin exchanges with ∆E ≤ 0 are accepted. During these lengths of time, since the only processes occurring, diffusion and condensation, are actually instantaneous on the scale τ = e 4/ε , time is not increased at all. When the system is in a blocked state (with a number n of domain walls), we choose randomly a domain wall, we exchange the two spins on the sides of the domain wall (evaporation), and we increase the time by ∆t = 1/n. The choice for ∆t can be understood as follows. In a Monte Carlo simulation at T = ε, the probability of accepting an evaporation process being approximately equal to e −4/ε , on average, a number O(e 4/ε ) of tries will be necessary before a success. In each Monte Carlo sweep (MCS) n tries are made, since the only couples of spins which satisfy the requirement σ i = −σ j are those around a domain wall. Thus the typical number of MCS done before an evaporation process takes place is τ /n, i.e. ∆t = 1/n in units of τ . B. Rules for updating ∆σ during the accelerated dynamics Being a quantity integrated over time, ∆σ k (t, s) gets contributions from both fast and slow processes. L/ 3 FIG. 1 : 31(t) vs. t 1Mean domain length L(t) against t 1/3 for the Ising-Kawasaki chain obtained by the accelerated dynamics for vanishingly small temperature. Inset: same data on a log-log scale. each choice of the number of evaporation processes done before the measurement of the autocorrelation and response, the corresponding value of the waiting time s and the number of different thermal histories are reported. prepared at time t = 0 in a random configuration. It is then evolved under T = 0 Kawasaki dynamics, until it reaches the first blocked state at time t = 0 + in units of τ . This first blocked state has a mean domain length L(0 + ) = 4.15886(15). T c , Yeung et al.[28] find bounds on the autocorrelation exponent which depend on the value of the smallest time s. For s = 0, λ ≥ d/2, where d is the dimensionality of space, while for values of s in the scaling regime, λ ≥ d/2 + 2 for d ≥ 2, and λ ≥ 3/2 for d = 1. with g(x → 0 )FIG. 2 : 02∼ x −1 in the intermediate regime, while in the ultimate regime g(Two-time autocorrelation function of the Ising-Kawasaki chain at vanishingly small temperature. Full line has slope −λ = −1, while dotted ones have slope −λ ′ = −2.5. FIG. 3 : 3Scaling function for the two-time autocorrelation of the Ising-Kawasaki chain at vanishingly small temperature (see text). FIG. 4 : 4is the typical size over which Scaling of the autocorrelation function at very short times. FIG. 5 : 5Integrated response against autocorrelation for the Ising-Kawasaki chain at vanishingly small temperature. . Most of the plot corresponds to the intermediate regime (t ≪ t * (s)), since the ultimate regime is reached only for the two smallest s values and very small correlations C(t, s) 0.03 (seefigure 2). Restricting to the first regime, we observe a slow convergence of the data to a limiting curve when s increases. A similar plot of the integrated response against correlation is presented in this reference, for three sets of values of T and s: (T = 0.48, s = 3 · 104), (T = 0.70, s = 2 · 103), and (T = 0.70, s = 104). For these values of (T, s) the data follow rather closely, at least in a range of values of the correlation, the theoretical FIG. 6 : 62D Ising-Kawasaki model at criticality: C(t, 0) and L(t). FIG. 7 :FIG. 8 : 78Snapshots of configurations after a quench from T = ∞ to T = T c in a 500 2 Ising spin system. The four snapshots on the left correspond to Glauber dynamics, while the four on the right to Kawasaki dynamics. Times have been chosen in order to have similar values for L(t).different decay exponent as soon as s is large enough, i.e. in this regime eq. 2D Ising-Kawasaki model at criticality: C(t, s) against L(t) for different values of s. CFIG. 9 : 9(t,s)L(t) λ c L(t) λ c '-λ c L(s) 2β/ν /L(s) λ c ' λ c =2 λ c '2D Ising-Kawasaki model at criticality: scaling function for the two-time autocorrelation. FIG. 10 : 102D Ising model at criticality: scaling functions of correlation and response. (left) Glauber dynamics. The full lines have slope λ c = −1.54. (right) Kawasaki dynamics. The dotted lines have slope λ ′ c = −3.5. ), we plot the ratio ρ/C, first versus L(s)/L(t) (figure a), then versus the ratio of times s/t (figure b). FIG. 11 : 11more evidence in favour of different values for the limiting ratios corresponding to the two dynamics, with X Kawasaki ∞ 2D Ising model at criticality: parametric plots of rescaled response versus rescaled correlation for Glauber and Kawasaki dynamics, using the data of figure 10. The full line at the origin has slope 0.33. 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[ "Model of fluorescence intermittency of single colloidal semiconductor quantum dots using multiple recombination centers", "Model of fluorescence intermittency of single colloidal semiconductor quantum dots using multiple recombination centers" ]	[ "Pavel A Frantsuzov \nDepartment of Physics\nUniversity of Notre Dame\nNotre Dame\n46556INUSA\n", "Sándor Volkán-Kacsó \nDepartment of Physics\nUniversity of Notre Dame\nNotre Dame\n46556INUSA\n", "Bolizsár Jankó \nDepartment of Physics\nUniversity of Notre Dame\nNotre Dame\n46556INUSA\n" ]	[ "Department of Physics\nUniversity of Notre Dame\nNotre Dame\n46556INUSA", "Department of Physics\nUniversity of Notre Dame\nNotre Dame\n46556INUSA", "Department of Physics\nUniversity of Notre Dame\nNotre Dame\n46556INUSA" ]	[]	We present a new physical model resolving a long-standing mystery of the power-law distributions of the blinking times in single colloidal quantum dot fluorescence. The model considers the nonradiative relaxation of the exciton through multiple recombination centers. Each center is allowed to switch between two quasi-stationary states. We point out that the conventional threshold analysis method used to extract the exponents of the distributions for the on-times and off-times has a serious flaw: The qualitative properties of the distributions strongly depend on the threshold value chosen for separating the on and off states. Our new model explains naturally this threshold dependence, as well as other key experimental features of the single quantum dot fluorescence trajectories, such as the power-law power spectrum (1/f noise).	10.1103/physrevlett.103.207402	[ "https://arxiv.org/pdf/0906.3260v3.pdf" ]	32,258,201	0906.3260	05d8e99e2eaf2f3087a3c43078bdd89cff1aab0d	Model of fluorescence intermittency of single colloidal semiconductor quantum dots using multiple recombination centers 16 Nov 2009 Pavel A Frantsuzov Department of Physics University of Notre Dame Notre Dame 46556INUSA Sándor Volkán-Kacsó Department of Physics University of Notre Dame Notre Dame 46556INUSA Bolizsár Jankó Department of Physics University of Notre Dame Notre Dame 46556INUSA Model of fluorescence intermittency of single colloidal semiconductor quantum dots using multiple recombination centers 16 Nov 2009(Dated: November 16, 2009) We present a new physical model resolving a long-standing mystery of the power-law distributions of the blinking times in single colloidal quantum dot fluorescence. The model considers the nonradiative relaxation of the exciton through multiple recombination centers. Each center is allowed to switch between two quasi-stationary states. We point out that the conventional threshold analysis method used to extract the exponents of the distributions for the on-times and off-times has a serious flaw: The qualitative properties of the distributions strongly depend on the threshold value chosen for separating the on and off states. Our new model explains naturally this threshold dependence, as well as other key experimental features of the single quantum dot fluorescence trajectories, such as the power-law power spectrum (1/f noise). Substantial progress has been made recently in the study of long range correlations in the fluctuations of the emission intensity (blinking) in single colloidal semiconductor nanocrystals (QD) [1,2,3,4,5], nanorods [6], nanowires [7] and some organic molecules [8]. By introducing an intensity threshold level to separate bright (on) and dark (off) states, Kuno et al. [2] found that the on-and off-time distributions in QDs exhibit a spectacular power-law dependence over 5-6 orders of magnitude in time. p on/off (t) ∼ t −m(1) As discovered later by Shimizu et al. [3], the powerlaw on-time distribution is cut off at times ranging from a few seconds to 100 s, depending on the dot and its environment. During the past eight years or so the truncated power-law form of the blinking on-time distributions was confirmed by many experimental groups (see [4,5,9,10] and references therein), but its microscopic origin remains a mystery. Remarkably, there are no "set" values for the on-time and off-time exponents. They are scattered in the region from 1.2 to 2.0. Similar on-and off-time distributions were found recently for the other blinking systems mentioned above: semiconductor nanorods(NRs) [6,11], nanowires (NWs) [7] and organic dyes [8]. The generality of the phenomenon is rather intriguing. We argued that there must be a common underlying mechanism responsible for the long time correlated fluorescence intermittency detected in all these systems [12]. Most theoretical explanations of the QD blinking [2,3,13,14,15] are based on the Efros/Rosen charging mechanism [16]. The mechanism attributes onand off-states to a neutral and a charged QD, respectively. The light-induced electronic excitation in the charged QD is quenched by a fast Auger recombination process. A number of experimental results indicate, however, that there are no unique bright (on) and dark (off) states of the QD, but a continuous set of emission intensities [17,18,19]. One can therefore suggest an alternative mechanism of the blinking, assuming slow fluctuations in the non-radiative recombination rate of the excited state [20,21,22]. In order to gain further insight into the possible blinking mechanism, we performed an extensive analysis of the on-and off-distributions of actual single QD fluorescence trajectories. Our procedure is different from the conventional ones, as we applied the Maximum Likelihood Estimator (MLE) method to find the best Gammadistribution p(t) ∼ t −m exp(−t/T ) fit for the set of on and off durations. The MLE approach [23] gives an unbiased estimation for the parameters of the power-law distribution with minimal statistical error. These properties are crucial and allow for the investigation of a single trajectory. Our approach, in contrast with procedure used by Hoogenboom et al. [23], allows us to find optimal values for not only for m, but for the truncation time T as well. The fluorescence trajectories we investigated were obtained by Protasenko and Kuno and have already been analyzed by others [24,25]. Our fitting procedure is performed repeatedly for a number of threshold values for each trajectory. In all the cases the off-time distribution truncation time is found too long to be detected. Also, the threshold dependence of the distributions was all but ignored until now. The only exception is the recent observation made on nanorods (not QDs) by Drndic and her coworkers [11]. In any case, the fundamental nature of this dependence was not revealed until now. An example of the threshold dependency of the power law exponent (slope on log-scale) and on-truncation time for a singe QD trajectory is presented in Fig 1. While we investigated a large number of trajectories, we have deliberately chosen for this paper one with clearly visible telegraph noise-like features and well-defined on and off maxima in the intensity histogram [see inset in Fig. 1(b)]. As it is evident from Fig 1, even for this apparently ideal case, the distribution parameters are strongly threshold dependent. While the majority of the analyzed trajectories are not like telegraph noise, we mention that all show similar threshold dependence. The on-time truncation time decreases monoton-ically with increasing of the threshold. This trend is the same for most single QD fluorescence trajectories we analyzed. The scaling of the slope as a function of the threshold is more complicated. The exponent of the offtime distribution shows several extrema, whereas the ontime exponent has a minimum as the threshold value is varied. We wish to emphasize that dependence of on-and off-time exponents on threshold can qualitatively change from one trajectory to another. We interpret this strong dependence on threshold as a clear indication that the standard trajectory analysisbased on the separation between on and off events with a somewhat arbitrary threshold -is not quite adequate, and the trajectories should be analyzed over the full range of threshold parameter. It also explains wide distribution of the exponents found by different groups. As shown below, one of the key results of this paper is to exploit the threshold dependence of the trajectory parameters to retrieve important information about the physical mechanism of the fluctuations. The power spectrum of the fluorescence trajectory of a single QD has a power law form [26,27] S I (ω) ∼ ω −l where l is close to 1. Therefore, we can consider the QD blinking process as an example of single particle 1/f (flicker) noise. The generally accepted phenomenological model for the electrical 1/f noise generation in solids is that of electrical transport in the presence of an environment consisting of multiple stochastic two-level systems (TLS) [28,29]. In the case of QD blinking we suggest a similar physical model based on a TLS environment [30]. In our model the non-radiative relaxation of the QD excitation occurs via trapping of holes to one of the N quenching centers, followed by a non-radiative recombination with the remaining electron. Each of these quenching centers could be dynamically switched between inactive and active conformations. The two conformational states differ in their ability to trap holes: the hole trapping rate is much larger in the active conformation than it is in the inactive state. Recent studies of trapping rates in the single QDs [32] showed that the number of hole traps on the surface and on the core/shell interface is in order of 10. Interestingly, we find that we only need a similar number of recombination centers in order to reproduce the basic features of the fluorescence trajectories. A possible microscopic origin of the conformation change in the recombination center could be due to the light-induced jumps of the surface or interface atom between two quasi-stable positions. The surface atoms in such a small object as colloidal QD can be found in a variety of local crystal configurations. Consequently, we can expect a wide distribution of switching rates. The non-radiative trapping rate in our model can therefore be expressed as k t (t) = N i=1 k i σ i (t) + k 0 .(2) For each TLS the stochastic variable σ i (t) randomly jumps between two values 0 and 1, corresponding to inactive and active conformations, respectively. Furthermore, k i is the trapping rate in the active configuration, and k 0 is the background non-radiative relaxation rate. The time distribution functions for the σ i = 0 → 1 transitions and σ i = 1 → 0 transitions for the i-th TLS are exponential and can be characterized by the transition rates γ + i and γ − i , respectively. While in the simplest model the transition rates for the individual TLS are constants (non-interacting TLSs), we will show that a more general case of the interacting TLS systems must also be considered. The power spectral density of the process (2) within the non-interacting TLS model is a sum of Lorentzians S k (ω) = 1 π N i=1 γ + i γ − i γ + i + γ − i k 2 i ω 2 + (γ + i + γ − i ) 2 .(3) The number of parameters in the above expression can be drastically reduced if the experimental constraint of 1/f noise spectrum is imposed. Indeed, after choosing k i = k and γ + i = γ − i = γ i ≡ γ 0 a i , where a ≪ 1, one can effectively fit the spectrum in Eq. (3) with 1/ω in the frequency region γ N ≪ ω ≪ γ 1 [28]. Assuming low excitation intensities and steady-state conditions for the fermionic degrees of freedom, the quantum yield Y (t) is given by [20]: Y (t) = k r k r + k t (t) ,(4) where k r is the radiative relaxation rate. Let us now show that our suggested model of fluorescence fluctuations exhibits strong threshold dependence of the on-and off-time distribution parameters. The problem of finding these distributions for the stochastic process Y (t) with known properties and threshold value y is equivalent to the well-known crossing problem [33]. There are only few cases when this problem can be solved exactly [34]. Fortunately, our present model can be reduced to such an exactly solvable case. The system at any moment t could be completely described by the configuration Σ = {σ 1 , . . . , σ N }. Clearly, there are 2 N different configurations. A random walk in the given configuration space is a Markovian stochastic process. The vector P containing probabilities of all configurations P Σ satisfies the Master equation d dt P (t) =Ŵ P (t)(5) where the transition matrixŴ contains the following nonzero elements W Σ + i Σ − i = γ + i , W Σ − i Σ + i = γ − i , W ΣΣ = − Σ ′ =Σ W Σ ′ Σ , where Σ − i = {σ 1 , . . . , σ i = 0, . . . , σ N } and Σ + i = {σ 1 , . . . , σ i = 1, . . . , σ N } for each given Σ. The nonradiative relaxation rate for given configuration Σ can be expressed by Eq. (2), which allows us to find the corresponding emission intensity level Y Σ from Eq. (4). Let us introduce a threshold value for the quantum yield y. By definition, the QD is in the bright (on) state if Y (t) ≥ y and dark (off) state otherwise. For each threshold level y all configurations can be separated to a bright group, satisfying a condition Y Σ ≥ y and a dark group. The vector of probabilities can be presented in the form P = P b P d , where vectors P b and P d contain probabilities of bright and dark configurations, respectively. The transition matrix can be recast in block formŴ = Ŵ bbŴbd W dbŴdd . The expressions for the normalized on-time and off-time distribution functions in this notations are well-known [35] p on (t) = 1,Ŵ db exp(Ŵ bb t)Ŵ bd P e 1,Ŵ bd P e where a, b denotes the scalar product, 1 is the unity vector and P e is the equilibrium probabilities vector, satisfying a stationary conditionŴ P e = 0. We found that the on-time and off-time distributions generated by Eqs. (6) can be fitted by a power law function (1) (see the insert in Fig. 2a). Beyond a certain off/on time value the power law behavior sharply changes to exponential asymptotic behavior exp(−t/T ). In our analysis, this value of T is defined as a truncation time. We performed simulations of on-time and off-time distributions for the model system of non-interacting TLS. This relatively simple model reproduces the general trend seen experimentally in the truncation times: The on-time truncation decreases and the off-time truncation increases when the threshold value goes up. While this simple, non-interacting TLS model is useful in illustrating our procedure, it cannot reproduce the threshold dependence of the exponents. The slope of the on-time distribution monotonically increases with the threshold value, when the off-time exponent has an opposite trend. In order to make our model more realistic, we introduce interaction between TLS in the simplest mean-field form (similar to Ref. [31]). The interaction is characterized by the parameter α, whereas the bias for an individual TLS is parameterized by β: γ ± i = γ i exp ± α N i=1 (σ i − 1/2) ± β(7) Fig. 2 provides the numerical calculation results for the interacting model with the following parameters: N = 10, γ 1 = 1, a = 10 −1/2 , k r /k = 1, k 0 = 0, α = 0.27 and β = −0.13. As seen from this figure, the threshold dependence of the truncation times keep the same trend as for noninteracting case. In contrast, the slopes now show a non-monotonic threshold dependence reproducing qualitatively the experimental behavior shown in Fig.1a. The insert in Fig. 2b shows that the interacting TLS model is capable of generating the two-maximum intensity distribution seen in Fig 1b. The relative ease with which our simple phenomenological model captured the experimental trend gives us hope that the model can be used to extract interaction parameters for the TLS environment. These parameters could provide useful experimental constraints on future microscopic models for the TLS environment of a variety of systems showing fluorescence intermittency. The model proposed here also explains recent observations of the non-blinking dots. Furthermore, a similar model can be constructed for the fluorescence intermittency seen in quantum wires. The details for these results will be published elsewhere. In conclusion, the phenomenological model we proposed in this paper succeeds in qualitatively explaining the key experimental facts characterizing long-correlated fluorescence intensity fluctuations of the single colloidal quantum dots: (1) the truncated power-law distributions for on-and off-times obtained by the commonly used threshold procedure; (2) the strong threshold dependence of the distribution parameters m and T and wide range of the the extracted exponents; (3) the 1/f noise form of the power spectrum of the intensity fluctuations; (4) the continuous distribution of emission intensities and excitation lifetimes; (5) the weak temperature dependence of the fluorescence intermittency due to the light-driven character of the TLS switching process. We would like to thank Dr. Vladimir Protashenko and especially Professor Masaru Kuno for many useful conversations and for providing us with high quality experimental data. We would also like to acknowledge the support of the Institute for Theoretical Sciences, the Department of Energy, Basic Energy Sciences, and the National Science Foundation via the NSF-NIRT grant No. ECS-0609249. . 1: (color online) The threshold dependence of the ontime (red squares) and off-time (blue circles) distribution exponents (a) and on-time distribution truncation time (b) obtained from the experimentally measured single QD fluorescence trajectory. Error bars show the standard deviation values. Insert b: a part of the trajectory and an intensity histogram. − 1 p 1off (t) = 1,Ŵ bd exp(Ŵ dd t)Ŵ db P e 1,Ŵ db P e −1 FIG. 2 : 2(color online) The theoretical threshold dependence of the on-time (red squares) and off-time (blue circles) distribution exponents (a) and truncation times (b) for the interacting TLS model(7). Insert a: the on-and off-time distribution functions at the threshold value y=0.25. Insert b: probability distribution function (PDF) of the quantum yield. . 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[ "Space-charge distortion of transverse profiles measured by electron-based Ionization Profile Monitors and correction methods", "Space-charge distortion of transverse profiles measured by electron-based Ionization Profile Monitors and correction methods" ]	[ "D Vilsmeier *[email protected] \nGSI Helmholtz Centre for Heavy Ion Research\nJohann Wolfgang Goethe-University\nPlanckstr. 160323, 64291Frankfurt, Frankfurt am Main, DarmstadtGermany, Germany\n", "M Sapinski \nGSI Helmholtz Centre for Heavy Ion Research\nJohann Wolfgang Goethe-University\nPlanckstr. 160323, 64291Frankfurt, Frankfurt am Main, DarmstadtGermany, Germany\n", "R Singh \nGSI Helmholtz Centre for Heavy Ion Research\nJohann Wolfgang Goethe-University\nPlanckstr. 160323, 64291Frankfurt, Frankfurt am Main, DarmstadtGermany, Germany\n" ]	[ "GSI Helmholtz Centre for Heavy Ion Research\nJohann Wolfgang Goethe-University\nPlanckstr. 160323, 64291Frankfurt, Frankfurt am Main, DarmstadtGermany, Germany", "GSI Helmholtz Centre for Heavy Ion Research\nJohann Wolfgang Goethe-University\nPlanckstr. 160323, 64291Frankfurt, Frankfurt am Main, DarmstadtGermany, Germany", "GSI Helmholtz Centre for Heavy Ion Research\nJohann Wolfgang Goethe-University\nPlanckstr. 160323, 64291Frankfurt, Frankfurt am Main, DarmstadtGermany, Germany" ]	[]	Measurements of transverse profiles using Ionization Profile Monitors (IPMs) for high brightness beams are affected by the electromagnetic field of the beam. This interaction may cause a distortion of the measured profile shape despite strong external magnetic field applied to impose limits on the transverse movement of electrons. The mechanisms leading to this distortion are discussed in detail. The distortion itself is described by means of analytic calculations for simplified beam distributions and a full simulation model for realistic distributions. Simple relation for minimum magnetic field scaling with beam parameters for avoiding profile distortions is presented. Further, application of machine learning algorithms to the problem of reconstructing the actual beam profile from distorted measured profile is presented. The obtained results show good agreement for tests on simulation data. The performance of these algorithms indicate that they could be very useful for operations of IPMs on high brightness beams or IPMs with weak magnetic field.	10.1103/physrevaccelbeams.22.052801	[ "https://arxiv.org/pdf/1811.00371v2.pdf" ]	54,083,670	1811.00371	666cbf2f59ca92e028dbadcbfa1af56facec2b16	Space-charge distortion of transverse profiles measured by electron-based Ionization Profile Monitors and correction methods 19 Nov 2018 D Vilsmeier *[email protected] GSI Helmholtz Centre for Heavy Ion Research Johann Wolfgang Goethe-University Planckstr. 160323, 64291Frankfurt, Frankfurt am Main, DarmstadtGermany, Germany M Sapinski GSI Helmholtz Centre for Heavy Ion Research Johann Wolfgang Goethe-University Planckstr. 160323, 64291Frankfurt, Frankfurt am Main, DarmstadtGermany, Germany R Singh GSI Helmholtz Centre for Heavy Ion Research Johann Wolfgang Goethe-University Planckstr. 160323, 64291Frankfurt, Frankfurt am Main, DarmstadtGermany, Germany Space-charge distortion of transverse profiles measured by electron-based Ionization Profile Monitors and correction methods 19 Nov 2018(Dated: November 21, 2018) 1 Measurements of transverse profiles using Ionization Profile Monitors (IPMs) for high brightness beams are affected by the electromagnetic field of the beam. This interaction may cause a distortion of the measured profile shape despite strong external magnetic field applied to impose limits on the transverse movement of electrons. The mechanisms leading to this distortion are discussed in detail. The distortion itself is described by means of analytic calculations for simplified beam distributions and a full simulation model for realistic distributions. Simple relation for minimum magnetic field scaling with beam parameters for avoiding profile distortions is presented. Further, application of machine learning algorithms to the problem of reconstructing the actual beam profile from distorted measured profile is presented. The obtained results show good agreement for tests on simulation data. The performance of these algorithms indicate that they could be very useful for operations of IPMs on high brightness beams or IPMs with weak magnetic field. I. INTRODUCTION The principle of beam profile measurement using Ionization Profile Monitors (IPMs) is seemingly very simple. The beam particles ionize the residual gas. The products of the ionization -electrons or ions -are extracted towards a position-sensitive detector using the guiding electric field (also referred to as 'clearing' or 'external' field) provided by electrodes. The distribution of the particles on the detector ideally corresponds to a projection of the transverse distribution of the beam density. This simple and straightforward principle, illustrated in Fig. 1, is non-destructive for the beam and thus an appealing application for synchrotrons and storage rings. In this context IPMs are installed in many hadron accelerators [1] and even investigated for usage on electron machines [2]. Figure 1: Sketch of an IPM together with read-out system. The electric and magnetic guiding fields are aligned, perpendicular to the beam direction as well as to the read-out system [3]. However already for medium-intensity beams the movement of electrons or ions is easily affected by the transient beam electric field. Because of their lower mass, electrons are removed much faster from the influence of bunch fields, however the effect of this fields on their trajectories is proportionally larger. Usually ion-based devices are preferred due to simpler assembly (no need of ion-trapping [4]), smaller impact of fringe fields of neighbouring magnets and lack of background signal from electrons generated from various other sources [5]. However the influence of beam fields plays a major role for high-brightness beams and thus electron-based IPMs are preferred over ion-based devices, mainly because the electron movement can be easily confined to small gyroradius by applying external magnetic field. Another argument in favor of electron-based devices is the fact that they are more suitable for time-resolved measurements because the electrons have much smaller time-of-flights to reach the detector. Distortions of the measured beam profile using ion-based IPMs are either mitigated by increasing the electric guiding field or corrected by empirical mathematical models [6][7][8][9]. For operation of electron-based IPMs with magnetic field it was first proposed to tune the magnetic field strength such that the electrons perform exactly one gyration between the beam and the detector [10]. However this method does not work for high-brightness beams, in which electrons get a significant momentum transfer not only transverse to the detector plane but also in the direction towards the detector. Therefore increasing the magnetic field in order to reduce the magnitude of gyroradius is the typical counteraction against the distortion of the measured beam profiles in that case. This is also because retrospective profile correction methods, similar to the ones developed for ion-based IPMs, do not exist for electron-based devices. One reason for that is the more complex particle movement in the presence of a combination of electric and magnetic fields. This movement and therefore the profile distortion, strongly depend on the beam fields and hence on bunch charge, transverse and longitudinal bunch shape as well as bunch spacing. The first beam space-charge induced distortions in an electron-based IPM operated with magnetic guiding field were observed in the Large Hadron Collider (LHC) at beam energies of 4 TeV and magnetic guiding field of 0.2 T [11]. Following studies have analyzed this phenomenon and estimated a magnetic field strength of 1 T required to suppress profile distortions [12][13][14]. However such large field strengths pose a technological as well as financial challenge, especially because of the relatively large gap between the poles needed to fit the detector vacuum chamber. Besides beam space-charge induced profile distortion there are various other phenomena which can have an effect on measured profiles. These are briefly discussed below however not addressed any further throughout this study: 1. Ionization by bunch field. At extreme field strengths the gas ionization cross-section is modified due to the Stark shift of energy levels and at even higher fields the gas gets ionized by the collective bunch electric field [15], rather than by interaction with single beam particles. 2. Burnout of the rest gas. Under some conditions, the ionization rate of the rest gas can be higher than the replenishment of gas in the volume of the beam leading to non-proportionality between beam density and amount of ionization events. This can be counterbalanced by, for instance, using gas jet [16]. 3. Guiding field non-uniformity. If the electric and magnetic guiding fields are nonuniform, it will additionally impact the resulting particle trajectories [17]. Also if the guiding fields are not perfectly aligned with respect to each other this introduces an additional disturbance. 4. Gas ionization from synchrotron radiation and the influence of wake fields on particle trajectories may constitute additional sources of profile distortion however these effects have not been observed or studied up to now. Some of the mentioned effects emerge from the influence of the particle beam and depend mainly on the amplitude of the beam electric field and the duration of the corresponding pulse. An overview of various effects in dependence on these two parameters is shown in Fig. 2 alongside the beam parameters of several hadron accelerators, the most powerful laser system up-to-date (LFEX), 3rd generation light source (ALBA) and advanced free electrons laser (SwissXFEL). The peak fields in waist of LFEX laser beam reaches 1 × 10 7 MV m −1 , well beyond the limit of gas ionization by bunch electric field. In comparison, the highest field in an accelerator -a focused SwissXFEL beam -will be reaching about 2 × 10 4 MV m −1 . All hadron machines are currently below 10 MV m −1 but the bunch lengths are about 3 orders of magnitude larger than those of LFEX and XFEL. Fig. 2 also indicates two effects related to beam space-charge interaction: electrons being trapped by the beam fields exceeding the electric guiding field and electron velocities reaching into the relativistic regime. Depending on the acquisition system in use, additional distortion of the measured distributions may occur. The electron or ion detection is usually based on amplification of the signal using Multi-Channel Plate (MCP) and either conversion of resulting electrons to light using a phosphor screen followed by a camera or direct conversion to electric signals using anode-strips. Such systems may suffer from non-uniformity of MCP gain, Phosphor deterioration or point-spread function of the optical system positioned after the Phosphor. Recent developments focus on use of hybrid silicon pixel detector instead of MCP/Phosphor or MCP/anode-strip combination and hence are able to sidestep these issues [18]. II. PROFILE DISTORTION In the following sections we will focus on profile distortion effects related to the electron movement which is governed by their interaction with the present electromagnetic fields. For the scope of this contribution, we assume perfectly uniform and aligned guiding fields and do not consider gas target and acquisition system related effects which were mentioned in the previous section. The coordinate system used throughout this study is x, along the measured profile (horizontal), y, towards the acquisition system (vertical) and z, along the beam (longitudinal). For illustration purposes we refer in this section to the data obtained for an example case which has been studied by means of simulations. The corresponding parameters are given in Table I and correspond to a possible LHC flat top configuration (nominal value of bunch charge is about 1.3 · 10 11 , but charges above 2 · 10 11 are expected for HL-LHC). The LHC beam case is well studied [12][13][14][19][20][21]. As another example we will shortly discuss the SIS-100 IPM system for proton beam at flat top. The electron movement is initialized by the ionization process and further influenced by the interaction with the guiding fields and the electromagnetic field of the particle beam (space-charge interaction). The relevant effects can be summarized by the following three aspects : • Ionization momenta -The initial momenta of electrons, obtained during the ionization process, clearly influence their further trajectories. Due to the interaction with the magnetic guiding field an initial momentum component along the beam axis results in a transverse displacement of the subsequent gyromotion while a transverse momentum component results in a longitudinal displacement. This effect becomes more pronounced for small guiding field strengths as well as for small beam sizes where the magnitude of the gyromotion increases relative to the beam dimensions. For relativistic beams the electrons are mainly ejected transverse to the beam direction [22]. • Space-charge interaction -The interaction of electrons with the electric field of the particle beam may significantly alter their trajectories, resulting in noticeable distortions of measured profiles. This interaction cannot be treated analytically without introducing significant approximations and hence must be studied by the means of numerical simulations. As this effect correlates with the magnitude of the beam electric field, it becomes especially significant for high intensity and high energy beams. However also small magnetic guiding fields may not be sufficient to counteract the beam field interaction and hence provoke an increase in gyroradii. • Gyromotion -The electrons perform a spiral movement under the influence of the external magnetic guiding field. Depending on the magnitude of the gyroradius an electron may experience a significant displacement with respect to its center of gyration. This effect is especially significant for small beam sizes since the magnitude of the displacement increases relative to the beam size. Often the magnetic guiding field is sufficient for suppressing profile distortion however in extreme cases the space-charge interaction can become so strong, with electric fields of up to a few MV m −1 (typical extraction fields are around 50 kV m −1 ), that electrons are (a) trapped inside the space-charge region since the beam electric field outweighs the electric guiding field around the center of the bunch, and (b) forced on significantly different trajectories, resulting in a vast increase of gyroradii up to a few millimeter. Since the above effects depend on the various parameters of the particle beam it is often not obvious to determine an appropriate strength for the magnetic guiding field which allows to sufficiently suppress the gyroradius increase. Having in mind the above-mentioned effects we may subdivide the IPM volume into two regions: • Space-charge region -This is the region close to the beam center where the interaction with the beam's electromagnetic field has a major influence. The gyro-velocity of electrons are subject to a perpetual oscillation induced by the ExB-drift along the beam and the magnitude of this oscillation is altered when the beam center approaches and recedes from the positions of the electrons. Because of the complex shape of the electromagnetic fields the electron motion exhibits other electromagnetic drifts as well as for example drifts due to the time dependence of the beam field (polarization drift) [23]. For estimating the dimensions of this region one can consider the ExB-drift velocity as an indicator for space-charge action. The region in which this ExB-drift velocity is more than 1% of the undisturbed gyro-velocity, is of the order of a few millimeters. • Detector region -This is the region close to the acquisition system where the beam Note that as the bunch recedes from the electrons' positions the space-charge region shrinks in time. While this region can be quite large during the bunch center passing, the transverse electric field quickly diminishes according to the bunch's line density. Therefore also electrons that are ionized near the bunch's center will eventually end up performing a pure gyromotion as in the detector region. Fig. 3 shows the bunch electric field as it decreases towards the detector (y) and along the beam axis (z) and hence illustrates the separation in space-charge and detector region. While the two dependencies are shown separately, their collective effect results in a much stronger decrease as the bunch recedes from the electrons' positions and hence the effective space-charge region is smaller than indicated by the single dependencies. This is because the beam velocity is significantly larger than the electrons' velocities along the beam axis and hence the spatial dependence (z) can be thought of as a time-dependence (t). Considering an electron's initial position during ionization and final position during detection, x 0 and x f respectively, the overall displacement ∆x = x f − x 0 can be ascribed to three different effects: 1. ∆x 1 -Displacement of the electron's gyro-center x c 0 with respect to the initial position, Figure 3: Evolution of the transverse bunch electric field E x at x = σ x in direction towards the detector (y) and during beam passage (z, t), for the parameters in Table I. The x-position is fixed to 1 σ x and y-and z-position are fixed to zero if not varied. due to its initial velocity along the beam: ∆x 1 = x c 0 − x 0 . Electrons that are ejected transverse to the longitudinal beam axis, in horizontal direction, merely suffer from a displacement of their gyrocenter along that axis, i.e. not affecting their horizontal position which determines the transverse profile. If they are ejected along the beam however there will be a corresponding displacement transverse to it. This type of displacement is typically around a few tens of microns and can be assessed with differential ionization cross sections which usually depend on the projectiles charge state and energy. In relativistic cases transverse ejection dominates and so the resulting displacement is mainly along the beam [22]. 2. ∆x 2 -Displacement of the space-charge altered gyrocenter x c s with respect to the initial gyrocenter: ∆x 2 = x c s − x c 0 . This shift is found to be induced by the magnetic field of the beam. In addition the longitudinal electric field of the beam induces an ExB-drift in horizontal direction which adds up as an additional displacement of the gyrocenter. However for relativistic beams the exposure to the longitudinal field is short and hence this drift tends to be negligible. The net effect of this type of displacement is around a few microns even for high beam currents. 3. ∆x 3 -Displacement of the final position with respect to the stable gyrocenter due to the nature of the gyromotion: ∆x 3 = x f − x c s . The electrons perform a gyromotion in the plane above the detector and in case this motion spans over multiple elements of the detecting system, the electrons might be detected on any of them. If the moment of detection is not correlated to the phase of the gyromotion then the resulting displacement is random. Since this type of displacement is proportional to the resulting gyroradii it depends on the magnitude of the preceding space-charge interaction and can be around some hundreds of microns up to a few millimeter. These effects only consider displacements in horizontal direction, along the measured profile. In fact the electrons experience an additional significant displacement in longitudinal direction, along the beam, due to the ExB-drift induced by the transverse electric field of the beam E x . However such drifts can be neglected in case the physical situation is similar along the beam axis (i.e. uniform guiding fields, similar gas pressure). Table I). The term "undisturbed trajectory" refers to the case if there were no space-charge interaction, i.e. only pure gyromotion. Because the initial and final gyroradius is about 100 µm the corresponding part of the trajectory appears as a flat ellipse due to large scale of the z-drift. The "shifted gyrocenter" is induced by the transverse electric field of the beam and acts as a starting point for the polarization drift; this shift is described by equation Eq. (6). "Bunch center passing" indicates the moment when the electron and the bunch center are aligned with respect to z-position. The "initial" and "final gyrocenter" are separated by a distance of 0.5 µm. In order to get an analytic estimate of the interaction we assume a simplified case in which the longitudinal electric field is zero and the transverse electric field is approximated by a linearly increasing field E x = E · x where E < 0 (this corresponds to the electric field inside a uniformly charged cylinder bunch of positive ions). For a Gaussian bunch shape this approximation is also well applicable at the center of the bunch as can be seen from Fig. 5 (neglecting the y-and z-dependence of the field). We also consider no bunch magnetic field since it is small compared to the magnetic guiding field B. The electric guiding field is acting in y-direction and hence has no influence on the motion in the xz-plane. In this simplified scenario the particle motion in the xz-plane is obtained as: x(t) = v x0 Ω sin(Ωt) +q Ω 2 (Bv z0 − Ex 0 ) cos(Ωt) + 1 +q E Ω 2 x 0 −q B Ω 2 v z0 (1) z(t) = −q B Ω 2 v x0 cos(Ωt) +q 2 B Ω 3 (Bv z0 − Ex 0 ) sin(Ωt) + E B +q E 2 Ω 2 B x 0 −q E Ω 2 v z0 t (2) + z 0 +q B Ω 2 v x0ẋ (t) = v x0 cos(Ωt) −q Ω (Bv z0 − Ex 0 ) sin(Ωt) (3) z(t) =q B Ω v x0 sin(Ωt) +q 2 B Ω 2 (Bv z0 − Ex 0 ) cos(Ωt) + E B +q E 2 Ω 2 B x 0 −q E Ω 2 v z0(4) where we have used the following abbreviations (q is the elementary charge and m is the electron mass):q ≡ q m (5a) Ω 2 ≡q 2 B 2 −qE > 0 , Ω = √ Ω 2 (5b) Eq. (1) shows that the particle performs an oscillating movement in x-direction while Eq. (2) shows a similar oscillation as well as an ExB-drift in z-direction. The absolute value of the gyro-velocity is given by \|v xz (t)\| = ẋ(t) 2 +ż(t) 2 . From Eq. (3) and Eq. (4) we can see that this velocity expression contains terms sin 2 (Ωt), cos 2 (Ωt) and sin(Ωt) · cos(Ωt) which are π-periodic as well as terms sin(Ωt) and cos(Ωt) which are 2π-periodic. Fig. 7 shows the gyro-velocity for an example particle subject to the realistic electric field of a Gaussian bunch. The expected twofold periodicity is clearly visible while an additional damping effect occurs due to the longitudinal field dependence when the beam recedes from the particle's position. The minimum and maximum of the velocity become apparent when considering the two-dimensional trajectory (x(t), z(t)). At one turn-around point the velocities from ExB-drift and the oscillating movement are aligned and add up, resulting in a large velocity while at the other turn-around point they are opposite resulting in a smaller velocity (or even a backward drift). Such behavior can as well be observed from simulations as described below. Fig. 8 shows the velocities of various particles for different starting points x 0 subject to the realistic field of a Gaussian charge distribution. One can observe a variation of the amplitude modulation as well as different periods of the oscillation depending on the starting point. This is due to the non-linear shape of the Gaussian-bunch electric field for positions farther away from the bunch center (compare the electric field plot in Fig. 5). Table I). Analyzing Eq. (1) to Eq. (4) we can infer various parameters of the resulting electron motion. Shift of gyration center From Eq. (1) one can read the center of oscillation x c = 1 +q E Ω 2 x 0 −q B Ω 2 v z0(6) which contains an additional term corresponding to the transverse bunch electric field. This shift however is compensated due to the time-dependence of the electric field and the corresponding polarization drift as shown in Appendix A. The same effect can be observed from simulations as well. This results in the initial and final gyrocenter to be aligned with respect to each other as shown in Fig. 4. Note that a displacement of the gyrocenter with respect Table I). to the initial position x 0 however occurs due to a non-zero initial velocity component along the beam v z0 . This shift is not compensated for by any other effects. Displacement due to gyromotion In the detector region, without the influence of the beam electric field, the electrons perform a pure gyromotion with gyroradius proportional to the planar velocity. Because of the velocity oscillations in the space-charge region, described by Eq. (3) and Eq. (4) as well as The resulting gyromotion implies possible displacements to positions different from the center of the gyromotion. This effect is discussed in detail in [13,14]. Time-of-flight In order to obtain an estimate for the final gyroradius of electrons we need to derive an estimate for their time-of-flight until they reach away from the influence of disturbing Figure 9: Distribution of gyro-velocity ratios, final over initial velocity, due to space-charge interaction (parameters corresponding to Table I). beam fields. Since the above analytic considerations are valid only inside the bunch, we set the space-charge region as the region inside the bunch and the detector region as the region outside the bunch. This assumption neglects modifications of the particles' velocities outside the bunch while in reality they may remain affected by the bunch electric field at these positions as the space-charge region may extend farther out. A more complex description would be to treat the decreasing part of the field outside the bunch by another linear field model, similarly to the one inside, and to reuse the above derived equations for that case by applying appropriate coordinate transformations. However since this approach is significantly more complex we stick with the former approach as it turns out to be sufficient as indicated by the following simulation results. For estimating the time-of-flights one cannot simply rely on the time the bunch needs to recede from the electron but one needs to consider the electron leaving the bunch volume (i.e. the space-charge region) in the transverse direction as well. In order to obtain a more accurate estimate for this time-of-flight in the vertical direction we need to take the bunch electric field and the electric guiding field into account. We do so by simplifying the longitudinal time dependence by an exponential dependency instead of the Gaussian dependency, in order to relax the explicit time dependence. The spatial dependence is linear as before. The corresponding equation of motion is: y(t) =qE g −qEy exp − \|t\| σ z(7) where E g is the electric guiding field strength. For a detailed derivation of the resulting motion (y(t),ẏ(t)) consider Appendix B. In order to estimate the time-of-flights we need to find the roots of the functions f 1 (t) = x(t) 2 + y(t) 2 − R where R is the radius of the cylinder bunch and x(t) is given by Eq. (1) and Table I. The simulations were performed using the Virtual-IPM simulation tool [24]. The estimated distributions show agreement in the shape as well as the mean and variance and hence suggest the applicability of the derived equations. These equations, even though obtained for simplified bunch shape, can be used to further study the phenomenon of profile distortion as shown in the following section. f 2 (t) = z(t) − (t − 4σ z )βc where β = v/c, C. Magnetic field tuning and scaling for realistic beams Since IPMs find application in different circumstances including a variety of beam parameters it is useful to establish a relation between the beam parameters and the magnetic field required to suppress profile distortion. Such a relation allows for uncomplicated checks when designing new devices. Different criteria for a sufficient field strength are possible such as absolute thresholds ensuring that most of the electrons' gyroradii are below an acceptable threshold (e.g. requiring a certain fraction to be below some percentage of the bunch width or to be smaller than the detector resolution) or relative thresholds that ensure that the increase of measured profile standard deviation remains below some percent threshold. For the further investigations we will use the latter criterion. We establish a fit of the three most significant beam parameters relevant for profile distortion, the transverse and longitudinal bunch size and the bunch charge: B min, 1% = N a σ b z σ c t · d + f σ e t(8) where a, b, c, d, e, f are fit parameters, σ 2 t is the variance of the (symmetric) transverse beam distribution to be measured in units units of millimeter, σ 2 z is the variance of the bunch Simulation parameters corresponding to Table I because this would be useful in investigation of emittance blow in modern colliders [25]. In a subsequent fit this threshold is converted to a parameter. For verifying the relationship and to infer the parameters we use the analytic considerations from the previous section as well as simulations with three different bunch shapes: Gaussian, Uniform and Parabolic Ellipsoid. For simplicity the bunch shapes are considered to be rotational symmetric around the beam axis (σ t ≡ σ x = σ y ). The charge density distributions of these shapes are given by: Gaussian ∝ exp − x 2 + y 2 2σ 2 t − z 2 2σ 2 z Uniform      ∝ exp − z 2 2σ 2 z , if x 2 + y 2 ≤ σ 2 t , with radius σ t ≡ 0 , otherwise Parabolic Ellipsoid      ∝ 1 − x 2 +y 2 σ 2 t − z 2 σ 2 z , if x 2 +y 2 σ 2 t + z 2 σ 2 z ≤ 1, with semi-axes σ z , σ t ≡ 0 , otherwise(9) We only consider relativistic beams since for low energy beams the ionization cross sections take on substantially different shapes [26] and the generated electrons are longer exposed to the beam electric field due to the reduced velocity and hence effects from the longitudinal electric field start to play a role. Hence for the fitting relativistic β ≥ 0.99 is assumed. The relationship Eq. (8) is fitted with simulation data obtained with the Virtual-IPM simulation tool, for the parameters given in Table II. These parameters correspond to the Gaussian bunch shape. For the other bunch shapes the parameters are scaled where appropriate in order to minimize the squared difference between initial beam distributions; for more details consider Appendix D. The resulting scaling factors are given in Table II as well. Table III shows the fit parameters obtained for the various bunch shapes as well as the quality of fit. They show agreement among each other and also with the results obtained for the analytic derivations. Applying the formula to the example case in Table I we obtain a minimum required field of 0.975 T which is in conformity with the estimated value obtained from previously run simulations with a different simulation tool [13]. Using the same formula for set of beam parameters corresponding to the planned IPM on SIS-100 proton beam (29.9 GeV energy, 2 × 10 13 charges, σ t = 2.17 mm, σ z = 15 ns), the minimum required field is estimated to 76 mT while the current design foresees a field of 84 mT [27]. A previous derivation of minimum required magnetic field strength that was done for related circumstances, however considering the gyroradius increase as an intermediate quantity instead of the final profile standard deviation, shows a similar dependence on the beam parameters [12]. In order to assess the effect of a chosen threshold for standard deviation increase τ = σ m σ −1 t − 1, with measured (simulated) standard devation σ m , on the minimum required magnetic field we include this parameter as well in a subsequent fit of the form: B min = N a σ b z σ c t τ g · d + f σ e t τ h(10) The results are similar to the ones obtained before and are shown in Table IV. Another way to include the threshold in the formula is to model the remaining fit parameters as dependent on the threshold: B min = N a(τ ) σ b(τ ) z σ c(τ ) t · d(τ ) + f (τ ) σ e(τ ) t(11) This formulation captures potential variations in the beam parameter dependence for varying thresholds τ . The results however, as shown in Table V, confirm that the effect of varying threshold is mainly a scaling factor. These results are in agreement with the τ f scaling shown from Eq. (10) and Table IV. III. PROFILE CORRECTION The first remedy against the space-charge profile distortion is to increase the applied magnetic guiding field, such that the gyroradii are bounded by user defined limits as shown by Eq. (8), Eq. (10) and Eq. (11) with corresponding parameters given by, respectively, Table III, Table IV and Table V. These relations are fit for the broad parameter range in Table II. Most magnetic IPM designs fall into this category and further analysis is not required. However for some extreme scenarios very large magnetic fields are required [13], which are both expensive to obtain and occupy significant space in already cramped synchrotrons. As technology advances more applications for high energy and high brightness beams are to be expected, putting current magnetic IPM designs to test. For these scenarios, correction mechanisms to obtain a measure of actual profile from the distorted profile, have been studied. Foremost of them was the study of a hardware electron "sieve", which aims to filter electrons based on their gyroradius before they reach the acquisition system, and a numerical reconstruction on the resulting "sieved" profiles can be performed [14]. Though the study with simulations gave promising results, the sieve was found to be rather complex and thick structure difficult to integrate in machine vacuum. Another approach was parametric curve fitting of the distorted profile with analytic functions and correlation of the beam width with those parameters [13]. The fit result however suffered from too few available data points and hence complicating establishing a general dependence between the parameters. Lately, approaches to record the inverse mapping between distorted profile to the original profile or the second moment of the original profile as a function of space charge parameters has been introduced either in form of look-up tables (LUT) [28] and supervised learning [20]. All the aforementioned correction methods rely on well understood and benchmarked IPM simulations. In this section, the problem of profile correction is generally introduced. Following that, several sub-approaches in the supervised learning scheme are mentioned and an extension to a full profile reconstruction from distorted profiles with arbitrary initial profile shapes is shown. A. Problem description In case of negligible space-charge interaction the distortion can be described via convolution of beam profile P beam with a point-spread function (PSF) to obtain the measured profile P measured [13]. This PSF depends on the initial velocities of electrons and can assessed by means of differential ionization cross sections. The PSF itself describes the electron transport in the IPM. It corresponds to the probability that an electron is detected on a certain position displaced from its ionization position and can be obtained from considering the time that an electron spends above the various bins of the detector [14]. This PSF is independent of the position along the initial profile. If the PSF depends on the position along the profile -as in case of space-charge influence on electron movement -a convolution cannot be used to describe the profile deformation anymore. As a more general transformation a matrix multiplication can be used instead: P measured,i = M ij · P beam,j(12) where M ij is the probability that an electron which was generated at position j is collected at position i. Comparing with convolution for the space-charge free case, the matrix M contains the PSF as columns, shifted across the rows with the PSF center at the diagonal (M ij = PSF[i − j]) . A fast iterative procedure to correct measured profiles using the simulated M −1 matrices is discussed in [28]. A newer approach of supervised machine learning is to deduce a set of rules for mapping measured profiles to their original counterparts. The idea is to infer a corresponding set of rules by providing distorted profiles alongside other relevant beam parameters and to map these to the original distribution of the beam. Fig. 11 shows a schema of profile correction with the aid of simulation data. B. Data generation by IPM simulations A recent joint effort between laboratories led to the development of a generic simulation tool called "Virtual-IPM" [24]. The Virtual-IPM tool has been used for simulating the movement of electrons inside the IPM, in presence of transient beam fields. Table VI shows the parameter ranges that are used for the simulations in order to span the relevant parameter space. The single bunches are modeled by three-dimensional Gaussian charge distributions. The electric field of bunches is computed via an analytic formula for a twodimensional Gaussian charge distribution in the transverse plane [29] while the longitudinal dependency is taken into account by rescaling the field with the beam's line density. The longitudinal field component is neglected. This approximation is justified because the beam is highly relativistic and the longitudinal dimension of bunches is significantly larger than their transverse dimensions and therefore the electric field is mainly acting in the transverse plane. The advantage of the analytic formula, as compared to a numerical Poisson solver, is that it is much faster in computing the electric field and it does not suffer from discretization effects. Both the electric and the magnetic field of the beam are taken into account. The external electric and magnetic guiding fields are modeled to be uniform within the field cage. The initial velocities of electrons are generated according to a double differential cross section for a Hydrogen target [22]. The passage of only a single bunch is simulated because the extraction times for electrons are only a few nanoseconds for the given electric guiding field while the bunch spacing is 25 ns. The output of the simulations is summarized by histograms with 9.9 mm range and 55 µm bin size, representing the electrons' positions at the moment of ionization, in the following referred to as initial, and at the moment of detection, in the following referred to as final. 55 µm resolution corresponds to a hybrid-pixel detector type which has been successfully operated in the PS IPMs [18]. Eq. (10) has the interesting property that for a given measured standard deviation of σ m and a given magnetic field B the formula encodes the corresponding beam profile standard deviation σ t which gives rise to the distortion via the threshold parameter τ . Specifically the beam profile standard deviation is the root of the function f (σ t ) = d · N a σ b z σ c t σ −1 t σ m − 1 g + f σ e t σ −1 t σ m − 1 h − B = 0(13) on the interval (0, σ m ), which can be computed by means of numerical methods. We test this method on the simulation data prepared according to Table VI by using bisection method [30] with 0.1 µm tolerance. The resulting residuals have an overall mean and standard deviation of, respectively, 0.357 µm and 3.21 µm. The residuals plot Fig. 12 shows that the quality varies with beam profile standard deviation. Nevertheless most results are within 2% accuracy. Previous attempts of reconstructing the initial beam profile standard deviation value from Table VI. measured profiles using Machine Learning methods have shown a better performance while being fitted on a much smaller parameter range [20]. Eq. (10) on the other hand has been fitted on a range which spans multiple orders of magnitude. D. Reconstruction of complete profiles with machine learning Relevant for the presented problem are supervised regression models for predicting continuous variables. In general a supervised machine learning (ML) model represents an algorithm f , a mapping from input x to output y p , called decision function, that is specified by a set of parameters θ ≡ {θ i }. The inputs and outputs are exactly opposite to the data preparation stage as shown in Figure 8. While the structure of such an algorithm (e.g. the number of parameters \|θ\|) depends on its hyper-parameters and is fixed, the goal is to tune θ i such that the corresponding function f (x \| θ) describes the output best. The quality of this description is typically assessed by a so called loss function L(x, y \| θ) → R + which measures the deviation of the predictions y p from the target output values y. Minimizing this loss function is used as a criterion for optimization of the model parameters θ. In order to find a suitable ML algorithm for a given problem their performances are compared. For doing so the data set is split in three subsets corresponding to training, validation and testing. The training set is used to fit the particular ML algorithms in order to determine the optimal parameters θ. The validation set is used for assessing the performance of an algorithm once it is fitted. Validating on a distinct data set prevents effects of over-fitting the algorithm to that particular training set and hence ensures generalization of the algorithm. The test similarly aims to prevent a tuning bias towards the validation set, resulting from the multiple iterations corresponding to hyperparameter tuning. The test set is used to assess the final performance that is to be expected once a ML algorithm with a specific configuration has been chosen. Full profile reconstruction Previous works have studied the usage of machine learning models for establishing a relationship between measured profile and original beam profile standard deviation [19,20]. A very recent approach investigated reconstruction of the complete profile shape from measured profiles [21]. This approach used an additional global data transformation that is derived from the training data in order to converge during the fitting procedure. Here we present a novel approach that works only with per-profile normalization, providing a more consistent way of data preparation. a. Model architecture In order to establish a mapping between measured profiles together with bunch length and bunch intensity data to the original beam profiles we recall Eq. (12). This equations states that the process of profile distortion can be described by a matrix multiplication M of the beam profile P beam . In general this matrix is dependent on the beam parameters and hence the beam profile itself: M → M (σ z , N p , P beam ). We suppose the matrix is invertible and that the inverse matrix M −1 similarly depends on the measured profile: M −1 → M −1 (σ z , N p , P measured ). This means that the process of profile distortion does not lose any information and the original profile shape is still encoded in the measured data. Then the original profile can be obtained by means of a matrix multiplication M −1 with the measured profile P measured . Here we identify the task of the neural network as the generation of the inverse matrix in dependence of the measured profile as well as the additional beam parameters bunch length and bunch intensity. Table VII. For identifying optimal configurations we monitored the final validation loss as well the ratio of initial to final validation loss in order to account for different starting losses. In a first step consensus about the batch size was found. With this optimal value fixed during a second and third iteration the optimal number of nodes and learning rate was identified respectively. The number of nodes however was found to have no significant influence on performance, if large enough, as shown in Fig. 15. The final configuration was batch size 236 (evenly divides the training set of 13 452 samples), learning rate 8.882 × 10 −3 and number of nodes in the first dense layer equal to 42. We used He-uniform initializer [31] and mean squared error (MSE) as the loss function. b. Data preparation Data preparation is an important aspect of any machine learning analysis and helps the fitting procedure to converge. Also it needs to be ensured that the way of data preparation is compatible with real application requirements. The previous study on full profile reconstruction [21] used per-profile normalization followed by a per-feature normalization which was computed over the full training set. In the current study we focus on a more intuitive approach that only uses per-profile normalization. For data preparation we use the following steps: viation of resulting profile standard deviation was ∆σ x /σ x = 2.01 % ± 1.00 % which is about one order of magnitude larger than what has been obtained from previous attempts with machine learning models directly targeting the profile standard deviation [20]. Table VIII shows the performance on the test set compared to the previously developed algorithm, the latter of which clearly performs better. The neural network has been trained on Gaussian profiles only however in real world applications the actual beam shapes might deviate from this ideal scenario. Hence an impor- Generalized Gaussian, used for instance in linac beam profile description [28]: β 2αΓ(1/β) exp − \|x − µ\| α β(14) with shape parameters α, β, µ; β = 2, α = √ 2σ x corresponds to a normal distribution. Γ denotes the gamma function [32]. Q-Gaussian, used in studies of tails of hadron beams: √ β C q 1 − (1 − q)βx 2 1 1−q(15) with shape parameters β, q; q = 1, β = σ −2 x /2 corresponds to a normal distribution. C q is a normalization factor to provide unit integral. The parameters of the non-Gaussian shapes correspond to the ranges shown in Table VI but are rescaled in order to match the Gaussian distribution in their limits. For the Generalized Gaussian α x,y,z = σ x,y,z and for the Q-Gaussian β x,y,z = σ −2 x,y,z /2. Fig. 16 shows an example profile for each profile shape together with the predicted reconstruction. Note that the Q-Gaussian profiles for q = 2 are not affected by distortion however they are significantly wider than the training profiles. The overall performance, as shown in Table IX and Fig. 17, decreases about one order of magnitude as compared to the Gaussian profiles. The new approach however shows improved performance as compared with the previous approach. Especially on the very wide Q-Gaussian (q = 2) profile shapes, the new approach doesn't suffer from "blowing up" the results as much, but instead manages to preserve the undistorted profiles to a better degree. The obtained results imply that the neural network identifies the underlying distortion mechanism rather than memorizing a specific profile shape which it has been trained on. The results were obtained using various scientific Python packages, including Keras [33], Matplotlib [34], NumPy [35], Pandas [36], SciPy [37], Tensorflow [38] and Virtual-IPM [24]. Uniform, Gaussian and Parabolic shapes. Following that we addressed the problem of reconstruction of the real beam profile from distorted measurement. First an approach based on the previously developed formulas for minimum magnetic field is presented which allows to infer the beam profile standard deviation from measured profile standard deviation on a broad range of beam parameters. The second approach addressed the problem of complete profile shape reconstruction and was realized by using a dedicated machine learning model involving neural network. The performance of this method was tested on various realistic profile shapes, including Gaussian, Q-Gaussian and Generalized Gaussian shapes. By fitting the model only with Gaussian profiles we showed that the approach generalizes as well to beam shapes significantly different than normal distributions. A. COMPENSATION OF GYRATION CENTER SHIFT Eq. (6) indicates the shift of the gyration center with respect to the undisturbed case (for which E = 0). Using Eq. (5) this becomes: x c = ω 2 Ω 2 x 0 − v z0 ω(16) with ω =qB the gyrofrequency for the undisturbed case E = 0. For the derivation of Eq. (1) -Eq. (4) the bunch electric field was assumed constant, i.e. E = const. In order to incorporate the time-dependence due to bunch movement we can consider a series of partial solutions (x n (t), z n (t)) each of which is valid on an interval [t n , t n+1 ] for which the time dependence remains approximately constant. These partial solutions are of the form: x n (t) = a n Ω n sin(Ω n t) − b n Ω n cos(Ω n t) + c ṅ x n (t) = a n cos(Ω n t) + b n sin(Ω n t) z n (t) = −a n ω Ω 2 n cos(Ω n t) − b n ω Ω 2 n sin(Ω n t) + E n B c n t + d ṅ z n (t) = a n ω Ω n sin(Ω n t) − b n ω Ω n cos(Ω n t) + E n B c n(17) where E n ≡ E(t n ) and a 0 = v x0 b 0 = − ω Ω 0 v z0 − E 0 B x 0 c 0 = ω 2 Ω 2 0 x 0 − v z0 ω d 0 = z 0 + ω Ω 2 0 v x0(18) Note that c 0 represents the (shifted) gyration center. In order to link the solutions and their coefficients together we require continuity at the boundary of time intervals: (x n (t n ),ẋ n (t n ), z n (t n ),ż n (t n )) ! = (x n−1 (t n ),ẋ n−1 (t n ), z n−1 (t n ),ż n−1 (t n )) Using Eq. (17) cos(Ω n t n ) sin(Ω n t n ) 0 0 − ω Ω 2 n cos(Ω n t n ) − ω Ω 2 n sin(Ω n t n ) En B t n 1 ω Ωn sin(Ω n t n ) − ω Ωn cos(Ω n t n ) En B 0               a n b n c n d n        =        x n−1 (t n ) x n−1 (t n ) z n−1 (t n ) z n−1 (t n )       (20) The matrix on the left hand side has determinant −Ωω −1 = 0 and hence a unique solution for the coefficients (a n , b n , c n , d n ) exists. Inverting the matrix we obtain:        Ω 2 n −ω 2 Ωn sin(Ω n t n ) cos(Ω n t n ) 0 ω Ωn sin(Ω n t n ) ω 2 −Ω 2 n Ωn cos(Ω n t n ) sin(Ω n t n ) 0 − ω Ωn cos(Ω n t n ) ω 2 Ω 2 n 0 0 − ω Ωn ωΩ 2 n −ω 3 Ω 2 n t n ω Ω 2 n 1 ω 2 −Ω 2 n Ω 2 n t n       (21) The parameter c n encodes the gyration center shift and is given by the third line of the inverse matrix: c n = ω 2 Ω 2 n x n−1 (t n ) − ω Ω 2 nż n−1 (t n )(22) Inserting Eq. (17) Since for n → ∞ we have Ω n → ω it follows that c n → x 0 − v z0 ω −1 which is the undisturbed gyration center. Hence the initial shift, encoded in c 0 , is compensated for t → ∞. Actually this limit is already reached for E → 0 which is the case once the bunch has receded from the electron's position. B. DERIVATION OF TIME-OF-FLIGHT FOR UNIFORM CHARGE DISTRIBU-TION We define a ≡ qEg m , b ≡ q\|E\| m and thus obtain: y(t) = a − by exp \|t\| σ z(24) We solve the differential equation for t ≥ 0 (denoted as y > ) since for t ≤ 0 (denoted as y < ) we can then reuse this solution together with the transformation t → −t. For t ≥ 0 the solution is: y(t) = ησ z J 0 (u(t)) G 3,0 2,4 u(t) 2 − ησ z Y 0 (u(t)) G 1,3 2,0 u(t) 2 4 + k 1 J 0 (u(t)) + k 2 Y 0 (u(t)) (25) where η ≡ aσ z π, u(t) ≡ 2σ z √ b exp Meijer G-functions [32]. By using the relations d dx G 3,0 2,4 {x} = − J 0 (2 √ x) x d dx G 1,3 2,0 {x} = − 2Y 0 (2x) x(26) we confirm that the terms which contain the derivatives of the Meijer G-function cancel each other. We then obtain forẏ(t): y(t) = −u(t)[ησ z J 1 (u(t)) G 3, For the t ≤ 0 case we apply t → −t and reuse the above solution. Note that the derivativeu(t) changes its sign due to the transformation. For t 0 < 0 we obtain k > 1 , k > 2 from the continuity condition at t = 0: y < (0) ! = y > (0) andẏ < (0) ! =ẏ > (0)). C. METHODS FOR MINIMUM MAGNETIC FIELD COMPUTATION For the computation of minimum magnetic field strength from analytic formulae we used the following methods. For each configuration (beam parameters + IPM parameters including magnetic field) 1000 particles were sampled from the initial beam distribution (sampling x-and y-position as well as the ionization time at z = 0 since the guiding fields are assumed to be uniform and the resulting profile is integrated along z). The initial momenta are sampled from double differential cross sections [22]. For each particle its time-of-flight (TOF) until leaving the bunch region, described by x(t) 2 +y(t) 2 ≤ R 2 and z(t) ≤ (t−4σ z )βc, was computed as the minimum between the TOF in transverse direction, considering Eq. (1) and Eq. (25), as well as in z-direction, using the bunch velocity and the electron velocity in z-direction via Eq. (2) and Eq. (4). We then use Eq. (3) and Eq. (4) in order to compute the final transverse velocity \|v xz (t of )\| = v x (t of ) 2 + v z (t of ) 2 from which we calculate the gyroradius given the magnetic guiding field. We then consider gyration around the central point indicated by Eq. (1) and a random position of detection within the gyromotion range with probabilities proportional to the corresponding point-spread function (PSF) [14]. In order to increase the statistics, for each particle 100 final positions were sampled according to the corresponding PSF, all corresponding to the same initial position. This results in a total of 100 000 data points for computing the standard deviation of the beam profile and the measured profile respectively. Each configuration was scored according to the relative deviation of measured standard deviation with respect to standard deviation of the beam profile λ = (σ m −σ b )σ −1 b , where λ denotes the score and σ m , σ b denote the measured and beam profile standard deviation respectively. A particular magnetic field strength is considered sufficient if λ ≤ 0.01. The magnetic field strength B is computed by finding the root of the function λ(B) − 0.01 using the bisection method [30] with a tolerance of 1 mT. D. SCALING FACTORS FOR DIFFERENT BUNCH SHAPES Different bunch shapes are compared to a Gaussian distribution and their free parameters are adjusted such that the resulting distribution has a minimal least square difference to the corresponding Gaussian distribution: arg min η +∞ −∞ (ρ(x\|η) − N (0, σ)) p dx (29) where ρ(x\|η) represents a particular bunch shape parametrized by η. Least squares difference (p = 2) is used for the Uniform bunch shape and p = 4 is used for the Parabolic Ellipsoid shape for numerical stability. The L-BFGS-B solver from the scipy.optimize package [37] is Figure 2 : 2Amplitude and duration of transient beam electric fields generated in various scientific apparatus. Approximate boundaries indicating appearance of phenomena like electron trapping or relativistic regime are shown. The estimates were done for a magnetic guiding field of 0.2 T. The dashed vertical line indicates the gyro-period for electrons moving in the 0.2 T magnetic field. The present study focuses on application of electron-based IPMs in high-brightness hadron machines such as the Large Hadron Collider (LHC) or Super Proton Synchrotron (SPS) at CERN or the SIS-100 under construction at FAIR. For these machines the beamspace induced profile distortion becomes a prominent factor that potentially hinders the successful operation of electron-based IPMs. This type of profile distortion as well as possible mitigation strategies are the focus of the present study. fields diminished to a negligible magnitude. This means the electron movement are solely subject to the electric and magnetic guiding fields and hence the electrons perform a pure gyromotion planar to the detector while being accelerated towards it. The characteristics of this gyromotion (gyroradius and gyrocenter position) depend on the previous beam space-charge interaction. Fig. 4 4shows an example trajectory which exhibits the different electromagnetic drifts, the separation in space-charge and detector region, as well as the different kind of displacements ∆x {1,2,3} mentioned above. The increase in gyroradius is also clearly visible. Fig. 5 5shows the velocity increase of electrons which is induced by the space-charge interaction and consequentially leads to a deformation of measured profiles as shown byFig. 6. The mean of initial velocities is 9.94 × 10 5 m s −1 which corresponds to a gyroradius of 28.3 µm. The total displacement of an electron is compound of the three above-mentioned stages: ∆x = ∆x 1 + ∆x 2 + ∆x 3 . Regarding the ExB-drift due to non-zero longitudinal field component E z one can consider the fact that the transverse field is scaled with relativistic factor of γ ≈ 7000. Since the ExB-drift-velocity is proportional to the electric field strength and the longitudinal drifts typically are of the order of a few millimeter the expected drift distance transverse to the beam axis is expected to be at least γ −1 times smaller, resulting in drift distances in the sub micrometer regime.B. Description for uniform beam distributionsElectron trajectories are mainly influenced by the transverse electric field of the beam and the magnetic guiding field. This interaction alters the trajectories in a complex way which cannot be predicted analytically for any realistic beam distribution without further assumptions. Figure 4 : 4Example trajectory obtained for r 0 = (1.5σ x , 0, 0), v 0 = 10 6 · (−1, 0, 1)m s −1 and generated 2σ z before the bunch center (beam parameters corresponding Figure 5 : 5Velocity increase, given as the ratio of velocity at detection and velocity at ionization, plotted against the initial horizontal position of electrons. The transverse electric field in x-direction, measured at y = z = 0 at the center of the bunch, is overlaid.Beam parameters corresponding toTable I. Figure 6 : 6Simulated example profiles for beam parameters corresponding toTable Iusing magnetic guiding field strengths of 100 mT and 200 mT respectively. Figure 7 : 7Position and velocity trajectory for an example particle with r 0 = (2σ x , 0, 0), v 0 = 10 6 √ 2 −1 · (1, 1, −1)m s −1 and generated 0.5σ z before the bunch center (beam parameters corresponding to Figure 8 : 8Gyro-velocity evolution for different starting positions r 0 = (x 0 , 0, 0). Initial velocity is v 0 = 10 6 √ 2 −1 · (1, 1, −1)m s −1 and electrons are generated σ z /2 before the bunch center (beam parameters corresponding Fig. 7 and 7Fig. 8, the electrons are likely to end up with an increased velocity in the detector region and consequentially with an increased gyroradius.Fig. 9shows the distribution of gyro-velocity change. Around 90% of electrons end up with an increased gyro-velocity due to space-charge interaction and hence end up with increased gyroradii. with c the speed of light; whatever value of t is smaller determines the moment when the electron leaves the bunch. The final parameters are then obtained by plugging this value back into the analytic velocity equations Eq. (3)and Eq.(4). For more details on the computation consider Appendix C. Fig. 10 10shows the gyroradius distributions computed the analytic estimation for different magnetic field strengths, compared with the results obtained from complete simulations including all effects by using realistic Gaussian bunch fields. The beam and device parameters correspond to Figure 10 : 10Comparison of gyroradius distributions calculated from analytic equations with distributions obtained from complete simulations with realistic Gaussian bunch fields. and the magnetic guiding field is varied.line density in units of nanosecond and N is the number of charges per bunch in units of 1 × 10 12 . B min, 1% is the minimum magnetic field required to obtain σ measured t with at least 1% accuracy with respect to σ beam t . The first term in Eq. (8) corresponds to the space-charge interaction while the second term is attributed to the effect of the initial velocity distribution which, as a constant effect, has a relatively increasing effect on smaller beam profiles; hence the dependence on σ t . The accuracy threshold was chosen to be a rather challenging 1% Figure 11 : 11The physical process is modelled by a simulation tool and the inverse process is approximated by the chosen machine learning algorithm or stored parameter dependent inverse mapping matrix.The transformation matrix depends on the beam parameters M ij = M ij (N p , σ x , σ z ) and for a given set of parameters it can be obtained by simulations. Performing simulations over a grid of space-charge distortion parameters one can establish a look-up table of matrices M and their inverses M −1 . Figure 12 : 12Residuals for the beam profile standard deviation reconstruction based on Eq. (13) and tested with simulations corresponding to Figure 13 : 13Fig. 14 sketches the procedure and shows the involvement of the machine learning model. Since the final prediction is obtained via a matrix multiplication (linear transformation) the neural network can be regarded as a generator model which produces linear models.The architecture of the neural network is depicted inFig. 14. The output layer, rep-Schematic of the architecture of a feed-forward multi-layer perceptron. Figure 14 : 14Left: Sketch of the full profile reconstruction procedure. Right: Schema of matrix generation by neural network. resenting the flattened matrix, uses ReLU activation function since the resulting matrix entries need to be greater than or equal to zero. The output is reshaped into a matrix and the columns are normalized to one since signal preservation is another constraint. We performed hyper-parameter search over the batch size, learning rate and the number of nodes in the first dense layer. The corresponding values are shown in Figure 15 : 15Comparison of neural network performance for different widths of the first dense (hidden) layer. Other hyper-parameters were fixed to initializer = He-uniform, batch size = 236, learning rate = 8.882 × 10 −3 . 1 . 1Measured profiles are cropped to the range [−3.90 mm, 3.90 mm]. 2. Measured bunch length and number of charges per bunch are divided by, respectively, 1.2 ns and 2.1 × 10 11 .3. Measured profiles are normalized to unit integral.4. Measured profile bins are set to zero for values smaller than 5% of the peak value.The previous approach on the other hand computed a global valid range by considering a 5% peak-threshold region of the largest profile in the training set which was [−0.93 mm, 0.93 mm]. All profile bins outside that region were dropped and not used for the analysis. Hence, for comparison with the previous approach, we also apply this transformation in addition, by dropping data and predictions outside that valid range. However during training we use the full range available.c. ResultsWe used mean squared error (MSE) as a loss function and monitored the mean absolute error (MAE) as an additional metric. For Gaussian profiles we also compare the deviation of standard deviation of the target beam profiles and predicted profiles. Fitting converged after 25 epochs, yielding mean and standard deviation of MSE = 5.12 × 10 −7 ± 2.49 × 10 −7 , MAE = 6.44 × 10 −4 ± 1.70 × 10 −4 on the test data set. The de- Figure 16 : 16Example profiles for non-Gaussian beam shapes together with reconstruction. These examples are the ones with largest MSE error after reconstruction for each set. The Q-Gaussian profiles with q = 2 (bottom right corner) are significantly wider than the training profiles. 2 Figure 17 : 217) Q-Gaussian q = 0.6 (d) Q-Gaussian q = Performance on non-Gaussian beam shapes. The Q-Gaussian profiles with q = 2 (bottom right corner) are significantly wider than the training profiles.IV. SUMMARY Distortions of measured beam profiles by electron-based magnetic Ionization Profile Monitors is a relatively new issue affecting the successful operation of these devices at some present and future accelerators operated for high-brightness beams. We discussed the underlying effects and their magnitude for this type of distortion in detail at the example of a possible LHC flat top configuration, supported by both analytic calculations as well as complete numerical simulations. Analytic descriptions for the simplified case of a uniform bunch shape are derived and benchmarking of these results with complete simulations was performed by using a well established simulation tool. Simple formulae for computing the minimum magnetic field strength required to avoid space-charge induced profile distortion from relevant beam parameters are presented. They are validated against the derived analytic results for uniform beam as well as for complete simulations of various realistic beam shapes, including , J n , Y n denote the Bessel functions of first and second kind respectively and G (u 0 ) Y 0 (u 0 ) J 1 (u 0 ) Y 1 (u− ηJ 0 (u 0 ) G 3 + ηY 0 (u 0 ) G 1 − ηJ 1 (u 0 ) G 3 + ηY 1 (u 0 ) G 1 0001010030011031010 we can derive the values for k 1 , k 2 (setting u 0 ≡ u(t 0 )) Table I : IBeam and device parameters for the example case which corresponds to a possible LHC flat top configuration.Parameter Value Beam particle type Protons Energy/u 6.5 TeV Bunch population 2.1 × 10 11 ppb Bunch length (4σ z ) 0.9 ns Bunch width (1σ x ) 270 µm Bunch height (1σ y ) 360 µm Electrode distance 85 mm Applied voltage 4 kV Magnetic field 0.2 T Number of sim. electrons 100 000 Time step size 0.3125 ps A. Effects related to electron movement Table II : IIBeam parameter ranges for simulation of the different bunch shapes. Scaling factors for minimizing the squared difference between the initial one-dimensional profiles of the different shapes are indicated as well. Each parameter follows a logarithmic distribution over 20 points resulting in a total of 8000 data points.Bunch -width σ t -length σ z -charge N Gaussian 0.2 mm to 20 mm 0.3 ns to 300 ns 1 × 10 9 to 1 × 10 13 Uniform × 1.76 × 1 × 1 Parabolic Ellipsoid × 2.44 × 2.44 × 1 Table III : IIIFit parameters for Eq. (8) for different bunch shapes as well as for the analytic considerations of section II B. The quality of fit in form of mean squared error (MSE) is given as well. Table IV : IVFit parameters for equation Eq. (10) for the Gaussian bunch shape. The quality of fit in form of mean squared error (MSE) is given as well.a b c d e f g h MSE [T 2 ] 0.592 0.559 1.052 0.020 0.938 0.005 0.384 0.479 3.79 × 10 −4 Table V : VFit parameters for equation Eq. (11) for the Gaussian bunch shape. The quality of fit in form of mean squared error (MSE) is given as well. For τ = 1 % the results slightly deviate from the ones inTable IIIbecause the underlying simulation data was regenerated.Threshold [%] a b c d e f MSE [T 2 ] Table VI : VIParameters for the simulation. The bunch population, length, width and height are varied, resulting in a total of 21 021 data samples. The parameter ranges roughly correspond to the LHC flat top.Parameter Values Beam particle type Protons Energy/u 6.5 TeV Bunch population 1.1 × 10 11 ppb to 2.1 × 10 11 ppb Bunch length (4σ z ) 0.9 ns to 1.2 ns Bunch width (1σ x ) 270 µm to 370 µm Bunch height (1σ y ) 360 µm to 600 µm Electrode distance 85 mm Applied voltage 4 kV Magnetic field 0.2 T Number of sim. electrons 1 000 000 Time step size 0.3125 ps C. Reconstruction of beam profile standard deviation Table VII : VIIHyper-parameters that have been searched over and their values. Eachhyper-parameter set is randomly sampled from the given distribution. Batch size is log-sampled with base 2 and learning rate and number of nodes are log-sampled with base 10. A total of 1000 samples were scanned over. Hyper-parameter Range Batch size 1 to 2048 Learning rate 1 × 10 −2 to 1 × 10 −6 Number of nodes 10 to 2288 Table VIII : VIIIPerformance of old and new approach on the test data set, containing 4205samples.H H H H H H H H H Method Loss MSE [10 −7 ] MAE [10 −4 ] ∆σ/σ [%] meas. rec. rec. rec. Old 370 ± 253 0.32 ± 0.11 1.28 ± 0.21 0.15 ± 0.11 New 5.12 ± 2.49 6.44 ± 1.70 2.01 ± 1.00 tant requirement is that the reconstruction algorithm still works even if applied to different beam shapes. In order to verify this generalization we tested the performance on various non-Gaussian shapes, sampled from Generalized Gaussian distributions and Q-Gaussian distributions. The charge distributions are given by the following equations. Table IX : IXTest performance on various non-Gaussian beam shapes shown in Eq.(14) andEq. 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[ "Approximating with Gaussians", "Approximating with Gaussians" ]	[ "Craig Calcaterra [email protected] \nMetropolitan State University\n\n", "Axel Boldt \nMetropolitan State University\n\n" ]	[ "Metropolitan State University\n", "Metropolitan State University\n" ]	[]	Linear combinations of translations of a single Gaussian, e −x 2 , are shown to be dense in L 2 (R). Two algorithms for determining the coefficients for the approximations are given, using orthogonal Hermite functions and least squares. Taking the Fourier transform of this result shows low-frequency trigonometric series are dense in L 2 with Gaussian weight function.	null	[ "https://arxiv.org/pdf/0805.3795v1.pdf" ]	17,163,048	0805.3795	efb472e9d91d3db6ac56b780840a03a3f7fb061e	Approximating with Gaussians May 26, 2008 Craig Calcaterra [email protected] Metropolitan State University Axel Boldt Metropolitan State University Approximating with Gaussians May 26, 2008Hermite seriesGaussian functionlow-frequency trigono- metric series AMS Subject Classifications: 41A3042A3242C10 Linear combinations of translations of a single Gaussian, e −x 2 , are shown to be dense in L 2 (R). Two algorithms for determining the coefficients for the approximations are given, using orthogonal Hermite functions and least squares. Taking the Fourier transform of this result shows low-frequency trigonometric series are dense in L 2 with Gaussian weight function. 1 Linear combinations of Gaussians with a single variance are dense in L 2 L 2 (R) denotes the space of square integrable functions f : R → R with norm f 2 := R \|f (x)\| 2 dx. We use f ≈ g to mean f − g 2 < . The following result was announced in [4]. Theorem 1 For any f ∈ L 2 (R) and any > 0 there exists t > 0 and N ∈ N and a n ∈ R such that f ≈ N n=0 a n e −(x−nt) 2 . Proof. Since the span of the Hermite functions is dense in L 2 (R) we have for some N f ≈ /2 N n=0 b n d n dx n e −x 2 .(1) Now use finite backward differences to approximate the derivatives. We have for some small t > 0 N n=0 b n d n dx n e −x 2 ≈ /2 b 0 e −x 2 + b 1 1 t e −x 2 − e −(x−t) 2 + b 2 1 t 2 e −x 2 − 2e −(x−t) 2 + e −(x−2t) 2 + b 3 1 t 3 e −x 2 − 3e −(x−t) 2 + 3e −(x−2t) 2 − e −(x−3t) 2 + · · · = N n=0 b n 1 t n n k=0 (−1) k n k e −(x−kt) 2 .(2) This result may be surprising; it promises we can approximate to any degree of accuracy a function such as the following characteristic function of an interval with support far from the means of the Gaussians e −(x−nt) 2 which are located in [0, ∞) at the points x = nt. The graphs of these functions e −(x−nt) 2 are extremely simple geometrically, being Gaussians with the same variance. We only use the right translates, and they all shrink precipitously (exponentially) away from their means. a n e −(x−nt) 2 ≈ characteristic function? Surely there is a gap in this sketchy little proof ? No. We will, however, flesh out the details in section 2. The coefficients a n are explicitly calculated and the L 2 convergence carefully justified. But these details are elementary. We include them in the interest of appealing to a broader audience. Then is this merely another pathological curiosity from analysis? We probably need impractically large values of N to approximate any interesting functions. No, N need only be as large as the Hermite expansion demands. Certainly this particular approach depends on the convergence of the Hermite expansion, and for many applications Hermite series converge slower than other Fourier approximations-after all, Hermite series converge on all of R while, e.g., trigonometric series focus on a bounded interval. Hermite expansions do have powerful convergence properties, though. For example, Hermite series converge uniformly on finite compact subsets whenever f is twice continuously differentiable (i.e., C 2 ) and O e −cx 2 for some c > 1 as x → ∞. Alternately if f has finitely many discontinuities but is still C 2 elsewhere and O e −cx 2 the expansion again converges uniformly on any closed interval which avoids the discontinuities [15], [16]:. If f is smooth and properly bounded, the Hermite series converges faster than algebraically [7]. Then is the method unstable? Yes, there are two serious drawbacks to using Theorem 1. 1. Numerical differentiation is inherently unstable. Fortunately we are estimating the derivatives of Gaussians, which are as smooth and bounded as we could hope, and so we have good control with an explicit error formula. It is true, though, that dividing by t n for small t and large n will eventually lead to huge coefficients a n and round-off error. There are quite a few general techniques available in the literature for combatting round-off error in numerical differentiation. We review the well-known n-point difference formulas for derivatives in section 6. 2. The surprising approximation is only possible because it is weaker than the typical convergence of a series in the mean. Unfortunately f (x) = ∞ n=0 a n e −(x−nt) 2 Theorem 1 requires recalculating all the a n each time N is increased. Further, the a n are not unique. The least squares best choice of a n are calculated in section 3, but this approach gives an ill-conditioned matrix. A different formula for the a n is given in Theorem 3 which is more computationally efficient. Despite these drawbacks the result is worthy of note because of the new and unexpected opportunities which arise using an approximation method with such simple functions. In this vein, section 4 details an interesting corollary of Theorem 1: apply the Fourier transform to see that low-frequency trigonometric series are dense in L 2 (R) with Gaussian weight function. Calculating the coefficients with orthogonal functions In this section Theorem 3 gives an explicit formula for the coefficients a n of Theorem 1. Let's review the details of the Hermite-inspired expansion f (x) = ∞ n=0 b n d n dx n e −x 2 claimed in the proof. The formula for these coefficients is b n := 1 n!2 n √ π R f (x) e x 2 d n dx n e −x 2 dx. Be warned this is not precisely the standard Hermite expansion, but a simple adaptation to our particular requirements. Let's check this formula for the b n using the techniques of orthogonal functions. n!2 n √ π H n (x) e −x 2 /2 : n ∈ N is a well-known basis of L 2 (R) and is orthonormal since R H m (x) H n (x) e −x 2 dx = n!2 n √ πδ m,n .(3) This means given any g ∈ L 2 (R) it is possible to write g (x) = ∞ n=0 c n 1 √ n!2 n √ π H n (x) e −x 2 /2(4) (equality in the L 2 sense) where c n := 1 √ n!2 n √ π R g (x) H n (x) e −x 2 /2 dx ∈ R. The necessity of this formula for c n can easily be checked by multiplying both sides of (4) by H n (x) e −x 2 /2 , integrating and applying (3). However, we want f (x) = ∞ n=0 b n d n dx n e −x 2 so apply this process to g (x) = f (x) e x 2 /2 . But f (x) e x 2 /2 may not be L 2 integrable. If it is not, we must truncate it: f (x) e f (x) e x 2 /2 χ [−M,M ] (x) = ∞ n=0 c n 1 √ n!2 n √ π H n (x) e −x 2 /2 so f (x) χ [−M,M ] (x) = ∞ n=0 c n (−1) n √ n!2 n √ π (−1) n H n (x) e −x 2 = ∞ n=0 b n d n dx n e −x 2 where c n = 1 √ n!2 n √ π R f (x) e x 2 /2 χ [−M,M ] (x) H n (x) e −x 2 /2 (x) dx = 1 √ n!2 n √ π R f (x) χ [−M,M ] (x) H n (x) dx so we must have b n = c n (−1) n √ n!2 n √ π = 1 n!2 n √ π R f (x) χ [−M,M ] (x) e x 2 d n dx n e −x 2 dx.(5) Now the second step of the proof of Theorem 1 claims that the Gaussian's derivatives may be approximated by divided backward differences d n dx n e −x 2 ≈ 1 t n n k=0 (−1) k n k e −(x−kt) 2 in the L 2 (R) norm. We'll use the "big oh" notation: for a real function Ψ the statement " Ψ (t) = O (t) as t → 0 " means there exist K > 0 and δ > 0 such that \|Ψ (t)\| < K \|t\| for 0 < \|t\| < δ. Proposition 2 For each n ∈ N and p ∈ (0, ∞)   R d n dx n e −x 2 − 1 t n n k=0 (−1) k n k e −(x−kt) 2 p dx   1/p = O (t) . Proof. In Appendix 6 the pointwise formula is derived: d n dx n g (x) = 1 t n n k=0 (−1) k n k g (x − kt)− t (n + 1)! n k=0 (−1) k n k k n+1 g (n+1) (ξ k ) where all of the ξ k are between x and x + nt. Therefore the proposition holds with g (x) = e −x 2 since g (n+1) (ξ k ) is integrable for each k. This is not perfectly obvious because we don't have explicit formulae for the ξ k . But the tails of g (n+1) vanish exponentially, the continuity of g (n+1) guarantees a finite maximum on the bounded interval between the tails, and \|ξ k − x\| < k \|t\|. Continuing the derivation of the coefficients a n we now have for sufficiently small t = 0 f ≈ N n=0 b n 1 t n n k=0 (−1) k n k e −(x−kt) 2 = N k=0 N n=k b n (−1) k t n n k e −(x−kt) 2(6) In the last equality we just switched the order of summation (see [9], section 2.4 for an overview of such tricks). Combining (5) and (6) we have Theorem 3 For any f ∈ L 2 (R) and any > 0 there exist N ∈ N and t 0 > 0 such that for any t = 0 with \|t\| < t 0 f ≈ N n=0 a n e −(x−nt) 2 for some choice of a n ∈ R dependent on N and t. If f (x) e x 2 /2 is integrable, then one choice of coefficients is a n = (−1) n n! √ π N k=n 1 (k−n)!(2t) k R f (x) e x 2 d k dx k e −x 2 dx. If f (x) e x 2 /2 is not integrable, replace f in the above formula with f · χ [−M,M ] where M is chosen large enough that f − f · χ [−M,M ] 2 < . Remark 4 The approximation in Theorem 3 also holds on C [a, b] with the uniform norm since the Hermite expansion is uniformly convergent on C 2 [a, b] (see [15], [16]) and the finite difference formula's error term from Appendix 6 converges to 0 uniformly as t → 0 + . The Stone-Weierstrass Theorem does not apply in this situation because linear combinations of Gaussians with a single variance do not form an algebra. Remark 5 As a consequence of Theorem 3 for any > 0 the closed linear span of e −(x−s) 2 : s ∈ [0, ) is L 2 (R). It is even sufficient to replace [0, ) with i 2 j : i, j ∈ N ∩ [0, ). Let's explore some concrete examples in applying Theorem 3. Choose an interesting function with discontinuities and some support negative: f (x) := (x − 1) 2 χ [−1,2] (x) := (x − 1) 2 0 for x ∈ [−1, 2] otherwise and observe graphically: f (x) := (x − 1) 2 χ [−1,2] (x) Hermite series N = 20 Hermite N = 40 Theorem 3 N = 20, t = .05 Theorem 3 N = 20, t = .01 Theorem 3 N = 40, t = .01 The Hermite approximation is slowed by discontinuities, but does converge. The next choice of f is continuous but not smooth. f (x) := (sin x) χ [−π,π] (x) Hermite expansion N = 10 Hermite expansion N = 20 Theorem 3 N = 10, t = .01 Theorem 3 N = 20, t = . 05 Theorem 3 N = 20, t = .01 In section 6 we review a standard technique accelerating this convergence in t. In our experiments, though, we've found the Hermite expansion is generally the bottleneck, not the round-off error of the derivative approximations for e −x 2 . Hermite expansion N = 60 Hermite expansion N = 100 Hermite expansion N = 120 We need about 120 terms before visual accuracy is achieved for this simple function. There is a host of methods in the literature for improving convergence of the Hermite expansion, but generally we have better success with functions that are smooth and bounded [7]. Theorem 3 gives a formula for the coefficients a n but this formula is not unique, and in fact is not "best" according to the classical continuous least squares technique. Least squares approximation N = 5, t = .01 Theorem 3 approximation N = 5, t = .01 In least squares we minimize the error function E 2 (a 0 , ..., a N ) := R f (x) − N n=0 a n e −(x−nt) 2 2 dx by setting ∂E2 ∂aj = 0 for j = 0, ..., N and solving for the a n . These N + 1 linear equations are called the normal equations. The matrix form of this system is M − → v = − → b where M is the matrix M = π 2 e − " k 2 +j 2 − (k+j) 2 2 « t 2 N j,k=0 and − → v = [a j ] N j=0 and − → b =   R f (x) e −(x−jt) 2 dx   N j=0 M is symmetric and invertible, so we can always solve for the a n . But these least squares matrices are notorious for being ill-conditioned when using nonorthogonal approximating functions. The Hilbert matrix is the archetypical example. The current application is no exception since the matrix entries are very similar for most choices of N and t, so round-off error is extreme. Choosing N = 7 instead of 5 in the graphed example above requires almost 300 significant digits. 4 Low-frequency trig series are dense in L 2 with Gaussian weight For f ∈ L 2 (R, C) define the norm f 2,G := R \|f (x)\| 2 e −x 2 dx 1/2 . Write f ≈ ,G g to mean f − g 2,G < . Theorem 6 For every f ∈ L 2 (R, C) and > 0 there exists N ∈ N and t 0 > 0 such that for any t = 0 with \|t\| < t 0 f (x) ≈ ,G N n=0 a n e −intx for some choice of a n ∈ C dependent on N and t. Proof. We use the Fourier transform with convention F [f ] (s) = 1 √ 2π R f (x) e −isx dx. F is a linear isometry of L 2 (R, C) with F e −αx 2 = 1 √ 2α e − s 2 4α , F [f (x + r)] = e −irs F [f (x)] and F [g * h] = √ 2πF [g] F [h] . where * is convolution. Let f ∈ L 2 and we now show f 2 (x) := 1 √ 2π e −x 2 * F −1 [f ] (x) ∈ L 2 . Notice g := F −1 [f ] ∈ L 2 and f 2 2 2 = R R 1 √ 2π g (x − y) e −y 2 dy 2 ds ≤ 1 2π R R \|g (x − y)\| 2 e −2y 2 dyds = c W t0 \|g\| 2 1 = c g 2 1 = c g 2 2 = c f 2 2 < ∞ for some c > 0. Here W t [h] is the solution to the diffusion equation for time t and initial condition h. (The notation W refers to the Weierstrass transform.) The reason for the third equality in the previous calculation is that W t maintains the L 1 integral of any positive initial condition h for all time t > 0 [17]. Now approximate the real and imaginary parts of f 2 with Theorem 3. Then we get 1 √ 2π e −x 2 * F −1 [f ] (x) ≈ N n=0 a n e −(x−nt) 2 a n ∈ C and applying F gives Proof. On [a, b] the Gaussian is bounded and so the norms with or without weight function are equivalent. Apply Theorem 6 to f ∈ L 2 ([a, b] , R) and choose t such that N t < ω to get f ≈ N n=0 Re (a n ) cos (ntx) + Im (a n ) sin (ntx) where a n = (−1) n n!2π N k=n 1 (k−n)!(2t) k R e −x 2 * F −1 [f ] (x) e x 2 d k dx k e −x 2 dx. Applying Remark 5 to this result shows even discrete sets of positive frequencies that approach 0 make the span of the corresponding sine and cosine functions equal toL 2 ([a, b] , R). Finally, low-frequency cosines span the even functions: We'd like to conclude right now that the b n = 0 or b n ≈ 0, but that is not true. However, every function g on [−b, b] may be written uniquely as a sum of even and odd functions g = g e + g o g e (x) = g (x) + g (−x) 2 g e (x) = g (x) − g (−x) 2 and so g ≈ h ⇒ g e ≈ h e . Therefore f = f e ≈ N n=0 a n cos (ntx) + b n sin (ntx) e = N n=0 a n cos (ntx) . Beware this last result; it's not as strong as Fourier approximation. The coefficients for the sine functions calculated above may be large; the proposition merely promises the linear combination of the sine terms is small. Using least squares, however, will have vanishing sine coefficients. Origins and generalizations The mathematical inspiration for Theorem 1 comes from geometrical investigations in infinite dimensional control theory. We noticed that function translation and vector translation in L 2 (R) do not commute. Specifically, "function translation" is a flow on the infinite dimensional vector space L 2 (R) given by the map F : L 2 (R) × R → L 2 (R) where F t (f ) (x) := f (x + t). "Vector transla- tion" in the direction of g ∈ L 2 (R) is the flow G : L 2 (R) × R → L 2 (R) where G t (f ) := f + tg. Taking for example g (x) := e −x 2 and composing F and G we see F t • G t = G t • F t since for f ≡ 0 F t • G t (f ) (x) = te −(x+t) 2 while G t • F t (f ) (x) = te −x 2 . Notice however the key fact F t • G t − G t • F t t 2 (f ) → d dx e −x 2 as t → 0 In finite dimensions the commutator quotient above gives the Lie bracket [X, Y ] of the vector fields X and Y which generate the flows F and G, respectively. A fundamental result in finite-dimensional control theory states that the reachable set via X and Y is given by the integral surface to the distribution made up of iterated Lie brackets starting from X and Y (Chow's Theorem, which is an interpretation of Frobenius' Foliation Theorem, see [13], e.g.). The idea we are exploiting is that iterated Lie brackets for our flows F and G will give successive derivatives of the Gaussian, whose span is dense in L 2 (R). Consequently, the reachable set via F and G from f ≡ 0 should be all of L 2 (R). That is to say, sums of translates and multiples of one Gaussian (with fixed variance) can approximate any integrable function. Unfortunately this program doesn't automatically work on the infinite dimensional vector space L 2 (R) since the function translation flow is not generated by a simple vector field on L 2 (R). So instead of studying vector fields, we consider flows as primary. The fundamental results can be rewritten and still hold in the general context of a metric space [3]. Then other functions besides g (x) = e −x 2 can be checked to be derivative generating and other flows may be used in place of translation. E.g., Fourier approximation is achieved using dilation F : L 2 (R, C) × R → L 2 (R, C) where F t (f ) (x) := f (e t x) and G t (f ) (x) := f (x) + te ix . This gives us a general tool for determining the density of various families of functions. Another opportunity for generalizing the results of this paper presents itself with the observation that Hermite expansions are valid for functions defined on C or R n and in spaces of tempered distributions; and divided differences works in all of these spaces as well. Note also that while the results of section 2 work for uniform approximations of continuous functions on finite intervals (Remark 4), this is an open question for low-frequency trigonometric approximations. The results of this paper can be ported to the language of control theory where we can then conclude the system u t = c 1 (t) u x + c 2 (t)e −x 2(7) is bang-bang controllable with controls of the form c 1 , c 2 : R + → {−1, 0, 1}. Theorem 3 drives the initial condition f ≡ 0 to any state in L 2 under the system (7), but may be nowhere near optimal for approximating a function such as e −(x+10) 2 , since it uses only Gaussians e −(x+s) 2 with choices of s << 10. Finally, interpreting Theorem 1 in terms of signal analysis, we see a Gaussian filter is a universal synthesizer with arbitrarily short load time. Let G (x) := 1 √ π e −x 2 . A Gaussian filter is a linear time-invariant system represented by the operator W (f ) (x) := (f * G) (x) = 1 √ π R f (y) e −(s−x) 2 dy. Notice if you feed W a Dirac delta distribution δ t (an ideal impulse at time x = t) you get W (δ t ) = G (x − t) . Then Theorem 1 gives Corollary 9 For any f ∈ L 2 (R) and any > 0 and any τ > 0 there exists t > 0 and N ∈ N with tN < τ such that f ≈ W N n=0 a n δ nt for some choice of a n ∈ R. Feed a Gaussian filter a linear combination of impulses and we can synthesize any signal and arbitrarily small load time τ . The design of physical approximations to an analog Gaussian filter are detailed in [6], [11]. Appendix: Approximating higher derivatives The results in this paper may be much improved with voluminous techniques available from numerical analysis. E.g., [8] gives an algorithm which speeds the calculation of sums of Gaussians, and [10] explores Hermite expansion acceleration useful in step 1 of the proof of Theorem 1. This section is devoted to reviewing methods which improve the error in step 2, approximating derivatives of the Gaussian with finite differences. We also derive the error formula used in Proposition 2. Above we approximated derivatives with the formula d n dx n f (x) = 1 t n n k=0 (−1) n−k n k f (x + kt) gives round-off error as t → 0 + + O (t) truncation error . (8) The Nörlund-Rice integral may be of interest for extremely large n as it avoids the calculation of the binomial coefficient by evaluating a complex integral. In this section, though, we devote our attention to deriving n-point formulas; these formulas decrease round-off error by increasing the number of evaluations f (x + kt)-this shrinks the truncation error without sending t → 0. In approximating the kth derivative with an n + 1 point formula f (k) (x) ≈ 1 t k n i=0 c i f (x + k i t) we wish to calculate the coefficients c i . In the forward difference method, the k i = i, but keeping these values general allows us to find the coefficients for the central or backward difference formulas just as easily. The following method for finding the c i was shown to us by our student Jeffrey Thornton who rediscovered the formula. Taylor's Theorem has f (x + k i t) = n j=0 (k i t) j j! f (j) (x) + (k i t) n+1 (n + 1)! f (n+1) (ξ i ) for some ξ i between x and x + k i t. From this it follows n i=0 c i f (x + k i t) =        f (x) tf (x) . . . t n f (n) (x) t n+1        T           1 1 · · · 1 k 0 k 1 · · · k n k 2 0 2! k 2 1 2! · · · k 2 n 2! . . . . . . . . . . . . k n 0 n! k n 1 n! · · · k n n n! k n+1 0 f (n+1) (ξ0) (n+1)! k n+1 1 f (n+1) (ξ1) (n+1)! · · · k n+1 n f (n+1) (ξn) (n+1)!                c 0 c 1 . . . which is possible since the k i are different, so the matrix is invertible, as is seen using the Vandermonde determinant det = Π 0≤i<j≤n (k j − k i ) Π 2≤i≤n i! . Then we must have n i=0 c i f (x + k i t) =        f (x) tf (x) . . . t n f (n) (x) t n+1        T             0 . . . 1 (k-th position) . . . 0 1 (n+1)! n i=0 c i k n+1 i f (n+1) (ξ i )             = t k f (k) (x) + t n+1 (n + 1)! n i=1 c i k n+1 i f (n+1) (ξ i ) . Therefore f (k) (x) = 1 t k n i=0 c i f (x + k i t) + Error for c i which satisfy (9) where Error = − t n+1−k (n + 1)! n i=0 c i k n+1 i f (n+1) (ξ i ) . This Error formula shows how truncation error may be decreased by increasing n without shrinking t, thus combatting round-off error at the expense of increased computation of sums. The coefficients in (8) are obtained by solving M for the c i with k i chosen as k i = i. Thornton also points out that the k i may be chosen as complex values when f is analytic (as is the case with our Gaussians). This gives us another opportunity to mitigate round-off error, since a greater quantity of regularly-spaced nodes k i can be packed into an epsilon ball around zero in the complex plane than on the real line. As final note we mention there have been numerous advances to the present day in inverting the Vandermonde matrix. We mention only the earliest application to numerical differentiation [14] which gives a formula in terms of the Stirling numbers. Remember the following properties of the Hermite polynomials H n ([16], e.g.). Define H n (x) := (−1) n e x 2 d n dx n e −x 2 . The set of Hermite functions h n (x) := 1 x 2 /2 χ [−M,M ] (x) is L 2 for any M < ∞ and f · χ [−M,M ] ≈ /3 f for a sufficiently large choice of M . Now we get new c n as follows 3 Our last examples in this section illustrate how convergence is faster for functions which are smooth and "clamped off", Calculating the coefficients with least squares Theorem 1 promises any L 2 function can be approximated f (x) ≈ N n=0 a n e −(x−nt) 2 . the fact that e −s 2 /4 > e −s 2 . This result is surprising, even in the context of this paper, because for instance, series of the form N n=−N a n e −i(x+nt) for all t and a n are not dense in L 2 and in fact only inhabit a 4-dimensional subspace of the infinite dimensional Hilbert space[3].Corollary 7 On any finite interval [a, b] for any ω > 0 the finite linear combinations of sine and cosine functions with frequency lower than ω are dense in L 2 ([a, b] , R). Proposition 8 8On any finite interval [0, b] for any ω > 0 the finite linear combinations of cosine functions with frequency lower than ω are dense in L 2 ([0, b] , R). Proof. Let f ∈ L 2 ([0, b] , R) and extend it as an even function on [−b, b]. Now use the previous corollary to write f ≈ N n=0 a n cos (ntx) + b n sin (ntx) . Representation and Control of Infinite Dimensional Systems. Alain Bensoussan, Springer2nd ed.Alain Bensoussan, et al., "Representation and Control of Infinite Dimen- sional Systems," 2nd ed., Springer, 2006. The Weierstrass Transform and Hermite Polynomials. G G Bilodeau, Duke Mathematical Journal. 292G. G. Bilodeau, The Weierstrass Transform and Hermite Polynomials, Duke Mathematical Journal, Vol. 29, No. 2, 1962. Craig Calcaterra, arXiv:math/0608416Foliating Metric Spaces, preprint. Craig Calcaterra, Foliating Metric Spaces, preprint, arXiv:math/0608416, 2006. Linear Combinations of Gaussians with a Single Variance are dense in L 2. Craig Calcaterra, Proceedings of the World Congress on Engineering. the World Congress on EngineeringCraig Calcaterra, Linear Combinations of Gaussians with a Single Variance are dense in L 2 , Proceedings of the World Congress on Engineering, 2008. Synthesis and Reactance of 4-poles. S Darlington, J. Math. & Phys. 18S. Darlington, Synthesis and Reactance of 4-poles, J. Math. & Phys., 18, pp. 257-353, 1939. Gaussian-Response Filter Design. Milton Dishal, Electrical Communication. 361Milton Dishal, Gaussian-Response Filter Design, Electrical Communication, Volume 36, No. 1, pp. 3-26, 1959. . David Gottlieb, Steven Orszag, Numerical Analysis of Spectral Methods. SIAMDavid Gottlieb and Steven Orszag, "Numerical Analysis of Spectral Meth- ods," SIAM, 1977. A New Version of the Fast Gauss Transform, Documenta Mathematica, Extra Volume ICM. Leslie Greengard, Xiaobai Sun, IIILeslie Greengard and Xiaobai Sun, A New Version of the Fast Gauss Trans- form, Documenta Mathematica, Extra Volume ICM 1998, III, pp. 575-584. Donald Knuth, Concrete Mathematics. Addison-Wesley2nd ed.Donald Knuth, "Concrete Mathematics," 2nd ed., Addison-Wesley, 1994. Greg Leibon, Daniel Rockmore & Gregory Chirikjian, A Fast Hermite Transform with Applications to Protein Structure Determination, Proceedings of the 2007 international Workshop on Symbolic-Numeric Computation. New York, NYACMGreg Leibon, Daniel Rockmore & Gregory Chirikjian, A Fast Hermite Transform with Applications to Protein Structure Determination, Proceed- ings of the 2007 international Workshop on Symbolic-Numeric Computation, ACM, New York, NY, pp. 117-124, 2007. J Madrenas, M Verleysen, P Thissen, J L Voz, A CMOS Analog Circuit for Gaussian Functions, IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing. 43J. Madrenas, M. Verleysen, P. Thissen, and J. L. Voz, A CMOS Analog Circuit for Gaussian Functions, IEEE Transactions on Circuits and Systems- II: Analog and Digital Signal Processing, Vol. 43, No. 1, 1996. A First Course in Numerical Analysis. Anthony Ralston, Philip Rabinowitz, McGraw-HillAnthony Ralston and Philip Rabinowitz, "A First Course in Numerical Analysis," McGraw-Hill, 1978. Mathematical Control Theory. Eduardo D Sontag, Springer-Verlag2nd EdEduardo D. Sontag, "Mathematical Control Theory," 2nd Ed., Springer- Verlag, 1998. A Spitzbart, N Macon, Numerical Differentiation Formulas. 64A. Spitzbart and N. Macon, Numerical Differentiation Formulas, The American Mathematical Monthly, Vol. 64, No. 10, pp. 721-723, 1957. M H Stone, Developments in Hermite Polynomials. 29M. H. Stone, Developments in Hermite Polynomials, The Annals of Math- ematics, 2nd Ser., Vol. 29, No. 1/4, pp. 1-13, 1927-1928. . Gabor Szegö, Orthogonal Polynomials. American Mathematical Society3rd ed.Gabor Szegö, "Orthogonal Polynomials," American Mathematical Society, 3rd ed., 1967. The Heat Equation. David Widder, Pure and Applied Mathematics. 67Academic PressDavid Widder, "The Heat Equation," Pure and Applied Mathematics, Vol. 67. Academic Press, 1975.	[]
[ "Rational curves with one place at infinity ", "Rational curves with one place at infinity " ]	[ "Abdallah Assi " ]	[]	[]	Let K be an algebraically closed field of characteristic zero. Given a polynomial f (x, y) ∈ K[x, y] with one place at infinity, we prove that either f is equivalent to a coordinate, or the family (f λ ) λ∈K has at most two rational elements. When (f λ ) λ∈K has two rational elements, we give a description of the singularities of these two elements.	10.1007/978-3-319-05681-4_14	[ "https://arxiv.org/pdf/1206.6189v2.pdf" ]	2,370,132	1206.6189	cfa792828b08ddfda9aa9ffe35645dd638a86fc1	Rational curves with one place at infinity * 21 Oct 2013 Abdallah Assi Rational curves with one place at infinity * 21 Oct 2013arXiv:1206.6189v2 [math.AG] Let K be an algebraically closed field of characteristic zero. Given a polynomial f (x, y) ∈ K[x, y] with one place at infinity, we prove that either f is equivalent to a coordinate, or the family (f λ ) λ∈K has at most two rational elements. When (f λ ) λ∈K has two rational elements, we give a description of the singularities of these two elements. Introduction and notations Let K be an algebraically closed field of characteristic zero, and let f = y n + a 1 (x)y n−1 + . . . + a n (x) be a monic reduced polynomial of K[x] [y]. For all λ ∈ K, we set f λ = f − λ. Hence we get a family of polynomials (f λ ) λ∈K . We shall suppose that f λ is a reduced polynomial for all λ ∈ K. Let g be a nonzero polynomial of K[x] [y]. We define the intersection multiplicity of f with g, denoted int(f, g), to be the rank of the K-vector space K[x] [y] (f, g) . Note that int(f, g) is also the x-degree of the y-resultant of f and g. Let p = (a, b) ∈ V (f ) ∩ V (g), where V denotes the set of zeros in K 2 . By settingx = x − a,ȳ = y − b, we may assume that p = (0, 0). We define the intersection multiplicity of f with g at p, denoted int p (f, g), to be the rank of the K- vector space K[[x, y]] (f, g) . Note that int(f, g) = p∈V (f )∩V (g) int p (f, g). We define the local Milnor number of f at p, denoted µ p (f ), to be the intersection multiplicity int p (f x , f y ), where f x (resp. f y ) denotes the x-derivative (resp. the y-derivative) of f . We set µ(f ) = p∈V (f ) µ p (f ) and µ = int(f x , f y ) and we recall that µ = λ∈K µ(f λ ) = λ∈K p∈V (f λ ) µ p (f λ ). Let q be a point in V (f ) and assume, after possibly a change of variables that q = (0, 0). The number of places of f at q, denoted r q , is defined to be the number of irreducible components of f in K[[x, y]]. Assume, after possibly a change of variables, that deg x a i (x) < n − i for all i = 1, . . . , n (where deg x denotes the x-degree). In particular f has one point at infinity defined by y = 0. 2 Curves with one place at infinity Let h f (x, y, u) = u n f ( x u , y u Let the notations be as in Section 1., in particular f = y n + a 1 (x)y n−1 + . . . + a n (x) is a monic reduced polynomial of K[x, y]. Let R(x, λ) = P 0 (λ)x i + . . . + P i (λ) be the y-resultant of f λ , f y . We say that (f λ ) λ∈K is d-regular (discriminant-regular) if P 0 (λ) ∈ K * . Note that (f λ ) λ∈K is d-regular if and only if int(f λ , f y ) = i for all λ ∈ K. Suppose that (f λ ) λ∈K is not d-regular, and let λ 1 , . . . , λ s be the set of roots of P 0 (λ). We set I(f ) = {λ 1 , . . . , λ s }, and we call [5]). I(f ) the set of d-irregular values of (f λ ) λ∈K . Let A f = s k=1 (i − int(f − λ k , f y )). For all λ ∈ K − I(f ), we have int(f λ , f y ) = µ + n − 1 + A f , where µ = int(f x , f y ) (seeNote that A f = λ∈K (i − int(f λ , f y )), in particular (f λ ) λ∈K is d-regular if and only if A f = 0. On the other hand, given a ∈ K, if int(f a , f y ) = µ + n − 1, then either (f λ ) λ∈K is d-regular or I(f ) = {a}. Assume that deg x a k (x) < k for all k = 1, . . . , n, in such a way that y = 0 is the only point at infinity of f . Proposition 2.1 (see [2] and [3]) Let the notations be as above and assume that f has one place at infinity, i.e. the projective curve defined by the homogeneous equation h f (x, y, u) = f ( x u , y u )u n is analytically irreducible at the point at infinity (1 : 0 : 0). We have the following • For all λ ∈ K, f − λ has one place at infinity. • The family (f λ ) λ∈K is d-regular. In particular, int(f λ , f y ) = µ + n − 1 for all λ ∈ K. • If µ = 0, then deg x a n (x) divides n and there exists an automorphism σ of K 2 such that σ(f ) is a coordinate of K 2 . Let the notations be as above. If δ p (resp. δ ∞ ) denotes the order of the conductor of f at p ∈ V (f ) (resp. at the point at infinity), then 2δ p = µ p + r p − 1 (resp. 2δ ∞ = µ ∞ + r ∞ − 1) (see [7]). Assume that f is an irreducible polynomial, and let g(f ) be the genus of the normalized curve of V (f ). By the genus formula we have: 2g(f ) + ( p∈V (f ) 2δ p ) + 2δ ∞ = (n − 1)(n − 2). Now int(f, f y ) = µ + n − 1 + A(f ), where A(f ) is a nonnegative integer and A(f ) = 0 if and only if (f λ ) λ∈K has at most one d-irregular value at infinity. On the other hand, the local intersection multiplicity of f with f y at the point at infinity is µ ∞ + n − 1. In particular µ + µ ∞ = (n − 1)(n − 2), consequently, if µ(f ) = p∈V (f ) µ p , and µ(f ) = µ − µ(f ), then 2g(f ) + ( p∈V (f ) 2δ p ) + 2δ ∞ = µ(f ) + µ(f ) + µ ∞ + A(f ). We finally get: ( * * ) 2g(f ) + p∈V (f ) (r p − 1) + r ∞ − 1 = µ(f ) + A(f ) in particular g(f ) = p∈V (f ) (r p − 1) + r ∞ − 1 = 0 if and only if A(f ) = µ(f ) = 0. Roughly speaking, f is a rational unibranch curve (at infinity as well as at finite distance) if and only if the pencil (f λ ) λ∈K has at most one d-irregular value at infinity and for all λ = 0, f λ is a smooth curve. Under these hypotheses, Lin-Zaidenberg Theorem implies that f is equivalent to a quasihomogeneous curve Y a − X b with gcd(a, b) = 1 (see [6]). Note that these hypotheses are satisfied when r ∞ − 1 = 0 = µ. Hence we get the third assertion of Proposition 2.1. since in this case, min(a, b) = 1 and f is equivalent to a coordinate Rational one place curves Let f = y n + a 1 (x)y n−1 + . . . + a n (x) be a polynomial of K[x, y] and let the notations be as in Sections 1 and 2. Assume that f has one place at infinity, i.e. r ∞ = 1. If f is rational, then it follows from the equality (*) of Section 2 that p∈V (f ) (r p − 1) = µ(f ). We shall prove the following: Theorem 3.1 Assume that f has one place at infinity and let (f λ ) λ∈K be the pencil of curves defined by f . If f is rational, then exactly one of the following holds: i) For all λ ∈ K, f λ is rational, and σ(f ) is a coordinate of K 2 for some automorphism σ of K 2 . ii) The polynomial f − λ is rational for at most one λ 1 = 0, i.e. the pencil (f λ ) λ∈K has at most two rational elements. We shall prove first the following Lemma: K-vector space K[[x]][y] (H x , H y ) ). We have the following: i) µ (0,0) ≥ r − 1. ii) If r ≥ 3, then µ (0,0) > r − 1. iii) If r = 2 and µ (0,0) = r − 1 = 1, then (H 1 , H 2 ) is a local system of coordinates at (0, 0). Proof. We have int (0,0) (H, H y ) = µ (0,0) + N − 1, but int (0,0) (H, H y ) = r i=1 int(H i , H iy ) + 2 i =j int (0,0) (H i , H j ) = r i=1 int[(H ix , H iy ) + deg y H i − 1] + 2 i =j int (0,0) (H i , H j ) hence µ (0,0) + N − 1 = ( r i=1 int(H ix , H iy )) + N − r + 2 i =j int (0,0) (H i , H j ).that s i=1 (µ p i + r i − 1) = µ, in particular µ ≤ s i=1 2µ p i = 2µ(f ), hence µ(f ) ≥ µ 2 . If f λ 1 is rational for some λ 1 = 0, then the same argument as above implies that µ(f λ 1 ) ≥ µ 2 . This is possible only for at most one λ 1 = 0, hence ii) follows immediately. The following proposition characterizes the case where the pencil (f λ ) λ∈K has exactly two rational elements. Proposition 3.3 Let the notations be as in Theorem 3.1. and assume that the pencil (f λ ) λ∈K has exactly two rational elements f and f λ 1 . We have µ(f ) = µ(f λ 1 ) = µ 2 , furthermore, given a singular point p of V (f ) (resp. V (f λ 1 )), f (resp. f λ 1 ) has two places at p and µ p (f ) = 1 (resp. µ p (f λ 1 ) = 1). In particular, f (resp. f λ 1 ) has exactly µ 2 singular points. Proof. It follows from the proof of Theorem 3.1. that µ(f ) ≥ µ 2 and that µ(f λ 1 ) ≥ µ 2 . Clearly this holds only if µ(f ) = µ(f λ 1 ) = µ 2 . Let p be a singular point of V (f ). We have µ p = r p − 1, hence, by Lemma 3.2. ii), r p ≤ 2. But µ p > 0, hence r p = 2 and µ p = 1. This implies that f has µ 2 singular points. Clearly the same holds for f λ 1 . The results above imply the following: Proposition 3.4 Assume that f has one place at infinity and let (f λ ) λ∈K be the pencil of polynomials defined by f . Assume that f is a rational polynomial and that µ(f ) > 0. Let p 1 , . . . , p s be the set of singular points of f . We have the following i) If r p i = 1 (resp. r p i ≥ 3) for some 1 ≤ i ≤ s, then f is the only rational point of the pencil (f λ ) λ . ii) If r p i = 2 for all 1 ≤ i ≤ s but s = µ 2 , then f is the only rational element of the pencil (f λ ) λ . Proof. This is an immediate application of Theorem 3.1. and Proposition 3.3. i) f = g + λ 1 for some λ 1 ∈ K , and f is equivalent to a coordinate, i.e. σ(f ) is a coordinate of K 2 for some automorphism σ of K 2 . ii) f = g + λ 1 for some λ 1 ∈ K * , µ(f ) = µ(g) = int(f x , f y ) 2 > 0, and f (resp. g) has int(f x , f y ) 2 singular points with two places at each of them. iii) int(f, g) > 0, i.e. f, g meet in a least one point of K 2 . Proof. The polynomial f (resp. g) has one place at infinity. If int(f, g) = 0, then f = ag + λ 1 , a, λ 1 ∈ K * . Since f and g are monic, then a = 1. Hence g and g + λ 1 are two rational elements of the pencil (f λ ) λ∈K . Now apply Theorem 3.1. and Proposition 3.3. Remark 3.6 Let (x(t), y(t)) = (t 3 − 3t, t 2 − 2) and (X(s), Y (s)) = (s 3 + 3s, s 2 + 2), and let f (x, y) = res t (x − x(t), y − y(t)) (resp. g(x, y) = res s (x − X(s), y − Y (s))). We have (x(t) − X(s), y(t) − Y (s)) = K[t, s], hence int(f, g) = 0. In fact, f (x, y) = y 3 − x 2 − 3y + 2 = −x 2 + (y + 2)(y − 1) 2 and g(x, y) = y 3 − x 2 − 3y − 2 = −x 2 + (y − 2)(y + 1) 2 , hence f = g + 4. The genus of a generic element of the family (f λ ) λ is 1, and f, f − 4 are the two rational elements of this family. Note that µ = 2 and µ(f ) = µ(f − 4) = 1. This example shows that the bound of Theorem 3.1. is sharp. Remark 3.7 Let (f λ ) λ∈K be a pencil of polynomials of K[x, y] and assume that f − λ is irreducible for all λ ∈ K. If the generic element of the pencil is rational, then for all λ ∈ K, f − λ is rational and irreducible. In this case, by [8], f has one place at infinity and σ(f ) is a coordinate of K 2 for some automorphism σ of K. Assume that the genus of the generic element of the pencil (f λ ) λ∈K is greater than or equal to one. Similarly to the case of curves with one place at infinity, it is natural to address the following question: Question: Is there an integer c ∈ N such that, given a pencil of irreducible polynomials (f λ ) λ∈K , if µ + A f > 0, then the number of rational elements in the pencil is bounded by c? Lemma 3. 2 2Let H = y N + a 1 (x)y N −1 + . . . + a N (x) be a non zero reduced polynomial of K[[x]][y], and let H = H 1 . . . H r be the decomposition of H into irreducible components of K[[x]][y]. Let µ (0,0) denotes the Milnor number of H at (0, 0) (i.e. µ (0,0) is the rank of the Finally we have µ (0,0) = ( r i=1 int(H ix , H iy )) − r + 1 + 2 i =j int (0,0) (H i , H j ) . Now for all1 ≤ i ≤ r, int (0,0) (H ix , H iy ) ≥ 0 and i =j int (0,0) (H i , H j ) ≥ C r 2 = r(r − 1) 2 , hence µ (0,0) ≥ r(r − 1) − (r − 1) = (r − 1) 2 and i), ii) follow immediately. Assume that r = 2. If µ (0,0) = r − 1, then int (0,0) (H 1x , H 1y ) = int (0,0) (H 2x ,H2y ) = 0 and int (0,0) (H 1 , H 2 ) = 1. This implies iii) Proof of Theorem 3.1.. If µ(f ) = 0, then µ = 0 and by Proposition 2.1., σ(f ) is a coordinate of K 2 for some automorphism σ of K 2 . Assume that µ(f ) > 0 and let p 1 , . . . , p s be the set of singular points of V (f ). Let r i denotes the number of places of f at p i for all 1 ≤ i ≤ s. By Lemma 3.2., for all 1 ≤ i ≤ s, µ p i ≥ r i − 1, on the other hand, equality (*) of Section 2 implies Proposition 3. 5 5Let f = g be two monic polynomials of K[x][y] and assume that f, g are parametrized by polynomials of K[t]. Under these hypotheses, exactly one of the following conditions holds: ). The local equation of f at infinity is nothing but F (y, u) = 2000 Mathematical Subject Classification: 14H20 † Université d'Angers, Mathématiques, 49045 Angers ceded 01, France, e-mail:[email protected] Visiting address: American University of Beirut, Department of Mathematics, Beirut 1107 2020, Lebanon h f (1, y, u) ∈ K[[u]][y]. We define the Milnor number of f at infinity, denoted µ ∞ , to be therank of the K-vector space K[[u]][y] (F u , F y ) . We define the number of places at infinity of f , denoted r ∞ , to be the number of irreducible components of F (y, u) in K[[u]][y]. Acknowledgments: The author would like to thank the referee for the valuable and helpful comments. On the semigroup of a meromorphic curve. S S Abhyankar, Proceedings, International Symposium on Algebraic Geometry. International Symposium on Algebraic GeometryKyoto1S.S. Abhyankar.-On the semigroup of a meromorphic curve, Part 1, in Proceedings, Inter- national Symposium on Algebraic Geometry, Kyoto (1977), 240-414. -Puiseux expansion and generalized Tschirnhausen transformation. S S Abhyankar, T T Moh, Crelle Journal. 260S.S. Abhyankar and T.T. Moh.-Newton-Puiseux expansion and generalized Tschirnhausen transformation, Crelle Journal, 260 (1973) 47-83. -Puiseux expansion, and generalized Tschirnhausen transformation II. S S Abhyankar, T T Moh, Crelle Journal. S.S. Abhyankar and T.T. Moh.-Newton-Puiseux expansion, and generalized Tschirnhausen transformation II, Crelle Journal, 261 (1973), 29-54. Sur l'intersection des courbes méromorphes. A Assi, C. R. Acad. Sci. Paris Sér. I Math. 3297A. Assi.-Sur l'intersection des courbes méromorphes, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), n 0 7, 625-628. Meromorphic plane curves. A Assi, Math. Z. 2301A. Assi.-Meromorphic plane curves, Math. Z. 230 (1999), n 0 1, 16-183. An irreducible simply connected algebraic curve in C 2 is equivalent to a quasihomogeneous curve. V Lin, M Zaidenberg, Dokl. Akad. Nauk SSSR. 5V. Lin and M. Zaidenberg.-An irreducible simply connected algebraic curve in C 2 is equiv- alent to a quasihomogeneous curve, Dokl. Akad. Nauk SSSR, 271 (1983), n 0 5, 1048-1052. Singular points of complex hypersurfaces. J Milnor, Ann. of Math. Studies. 61Univ. PressJ. Milnor.-Singular points of complex hypersurfaces, Ann. of Math. Studies, 61, Princeton, Univ. Press., Princeton, NJ, 1968. W Neumann, P Norbury, Nontrivial rational polynomials in two variables have reducible fibres. 58W. Neumann and P. Norbury.-Nontrivial rational polynomials in two variables have re- ducible fibres, Bull. Austral. Math. Soc., Vol 58 (1998), 501-503. O Zariski, Le probleme des modules pour les branches planes, Lectures at Centre de Mathématiques, Ecole Polytechnique, Notes by F. Kmety and M. Merle. O. Zariski.-Le probleme des modules pour les branches planes, Lectures at Centre de Mathématiques, Ecole Polytechnique, Notes by F. Kmety and M. Merle, 1973.	[]
[ "A MODULAR ARCHITECTURE FOR CREATING MULTIMODAL AGENTS A PREPRINT", "A MODULAR ARCHITECTURE FOR CREATING MULTIMODAL AGENTS A PREPRINT" ]	[ "Thomas Baier [email protected] \nFaculty of Humanities Vrije\nFaculty of Humanities Vrije\nUniversiteit Amsterdam Amsterdam\nThe Netherlands\n", "Selene Baez Santamaria [email protected] \nFaculty of Humanities Vrije\nUniversiteit Amsterdam Amsterdam\nThe Netherlands\n", "Piek Vossen \nUniversiteit Amsterdam Amsterdam\nThe Netherlands\n" ]	[ "Faculty of Humanities Vrije\nFaculty of Humanities Vrije\nUniversiteit Amsterdam Amsterdam\nThe Netherlands", "Faculty of Humanities Vrije\nUniversiteit Amsterdam Amsterdam\nThe Netherlands", "Universiteit Amsterdam Amsterdam\nThe Netherlands" ]	[]	The paper describes a flexible and modular platform to create multimodal interactive agents. The platform operates through an event-bus on which signals and interpretations are posted in a sequence in time. Different sensors and interpretation components can be integrated by defining their input and output as topics, which results in a logical workflow for further interpretations. We explain a broad range of components that have been developed so far and integrated into a range of interactive agents. We also explain how the actual interaction is recorded as multimodal data as well as in a so-called episodic Knowledge Graph. By analysing the recorded interaction, we can analyse and compare different agents and agent components.	10.48550/arxiv.2206.00636	[ "https://arxiv.org/pdf/2206.00636v1.pdf" ]	249,240,497	2206.00636	5077024ade3165851afc561aeadd2b6a4650a1a5	A MODULAR ARCHITECTURE FOR CREATING MULTIMODAL AGENTS A PREPRINT June 2, 2022 Thomas Baier [email protected] Faculty of Humanities Vrije Faculty of Humanities Vrije Universiteit Amsterdam Amsterdam The Netherlands Selene Baez Santamaria [email protected] Faculty of Humanities Vrije Universiteit Amsterdam Amsterdam The Netherlands Piek Vossen Universiteit Amsterdam Amsterdam The Netherlands A MODULAR ARCHITECTURE FOR CREATING MULTIMODAL AGENTS A PREPRINT June 2, 2022Multimodal agents · Multimodal interaction · Data sharing The paper describes a flexible and modular platform to create multimodal interactive agents. The platform operates through an event-bus on which signals and interpretations are posted in a sequence in time. Different sensors and interpretation components can be integrated by defining their input and output as topics, which results in a logical workflow for further interpretations. We explain a broad range of components that have been developed so far and integrated into a range of interactive agents. We also explain how the actual interaction is recorded as multimodal data as well as in a so-called episodic Knowledge Graph. By analysing the recorded interaction, we can analyse and compare different agents and agent components. This paper explains the overall architecture and implementation of the event-bus and demonstrates how it renders interaction data to the EMISSOR format and eKGs. We also explain how different interactive agents can be created within the same platform using various components. Currently, we implemented a simple Eliza agent 4 as well as a more complex Leolani agent [Vossen et al., 2019] 5 . Different agents still render compatible multimodal data from their interactions within our framework. Therefore, our platform can be used to compare agents by analysing the rendered data, both in EMISSOR and as an eKG. This paper is further structured as follows. In Section 2, we explain the overall architecture of the platform and its connection to EMISSOR and eKGs. Section 3 describes the components that are currently available. Section 4 explains how you can create your own agent either by combining existing components or by defining your own component. All our code is available on Github under the Apache2.0 open source license from: https://github.com/leolani. 2 Event-Bus architecture Figure 1 shows a schematic representation of the event-bus architecture. The central component is the event-bus itself, to which signals and interpretations are pushed either by backend components (at the left side) or interpretation components (at the right side). The event-bus is represented as a vertical (gray) bus to emphasize that it has a temporal dimension where events are processed in a sequence. The labels on the arrows between the event-bus and the components denote the type of event transmitted. Vertical lines in the event-bus symbolize different topics used by the individual components to define the event flow through the application. The topic labels components are connected to are denoted in italic next to the arrows. Examples of backend components are microphones, audio controllers, and cameras. The sensors connected to backend components can be directly linked to the system or obtained from remote servers. This schema shows how a microphone produces an audio signal with a start and stop point. Some examples of interpretation components are shown on the right side of the event-bus. The VAD component at the top applies voice-activity detection to an audio signal published by the microphone. It reports back to the event-bus whether the audio is human speech, annotating the corresponding segment in the audio. The speech-to-text (STT) component applies speech recognition to any audio signal annotated as human speech and publishes the transcript as a text signal to the event-bus. The text signal is picked up by an implementation of Figure 1: Schematic representation of the event-bus architecture showing backend components to read from sensors and fill the queue in the event-bus and interpretation components for annotating published signals. In this schema, only audio signals are processed with an Eliza module for generating a response. Interaction data is recorded from events by the EMISSOR component. the Eliza chatbot [Weizenbaum, 1966], which checks it for triggers and returns a response as another text signal. Finally, the text-to-speech (TTS) component picks up the text response and converts it to an audio signal. The backend speaker listens to the event-bus for any audio signals to output. An audio controller between the microphone and speaker mutes the mic when speaking and vice-versa. Any data produced by the sensors or the components as signals is stored in a backend container. Enabling reasoning The main application can function with just the event-bus and some simple components, taking in signals and generating signals as a response. This behavior, however, is primarily reactive and not reflective of the information conveyed during an interaction. If we want an agent capable of reasoning over the interpretations, we first need to capture the interaction, including metadata and signals, together with their annotations and the knowledge conveyed such that we can reason over it. For this, we included specific components to connect the event-bus data to EMISSOR and the eKG. Figure 2 shows the interconnections across the three components: event-bus, EMISSOR and the eKG. The three components are aligned such that each interaction in the event-bus generates a corresponding multimodal signal representation in EMISSOR and, if the interpretation yields knowledge, also an RDF triple in the eKG. EMISSOR stores interactions as scenarios with metadata in JSON files for signals in each modality: images, audio, text, and RDF triples. Further details are described in Báez Santamaría et al. [2021]. The metadata grounds the signals to a temporal ruler and defines any annotation of segments within a signal as interpretations. The signals themselves are stored as separate files on disk. We consider the RDF triples extracted from text as a modality stored on disk in EMISSOR, but they are also pushed to a separate triple store functioning as the eKG. In the eKG model, interactions are initialized as instances of situational eps:contexts grounded in time and space. Contexts correspond to scenarios in EMISSOR. Within a context, a series of sem:events are represented, which are grounded in time like the signals in EMISSOR. We distinguish conversation and perception events. Conversation: Conversation events in the eKG are structured as grasp:Chat instances containing grasp:Utterances in which participants of the interaction make grasp:Statements or formulate questions. The information conveyed in grasp:Statements is stored in gaf:Claim named graphs attributed to speakers (grasp:wasAttributedTo) that hold certain perspectives (grasp:Attribution). These perspectives are properties such as grasp:CertaintyValue, grasp:PolarityValue, grasp:SentimentValue, grasp:EmotionValue. 6 As such the model can express that people believe or deny certain facts with some certainty at moments in time. Claims correspond to text signals in EMISSOR, which are derived from audio signals. Perceptions: In contrast, perception events in the eKG are structured as grasp:Visual instances containing grasp:Detectionss which could be objects, or people. These detections are considered to be instances of grasp:Experiences. Similar to conversation events, the information obtained from these grasp:Experiences is stored in gaf:Claim named graphs with triples for the output of object/face-recognition components. Therefore, the information is grasp:wasAttributedTo sensors, and the grasp:Attributions are restricted to grasp:CertaintyValues only. Perception events therefore reflect the awareness of an agent of objects and people in the context. Perceptions correspond with image signals in EMISSOR. As is shown in Figure 2, the event-bus supports specific components that render the data for EMISSOR and for the eKG. These components can also push data, such as annotations or triples, to the event-bus itself so that other components can take these as input for further processing. The event-bus is thus not only populated by the sensors but also by interpretation components. In the next section, we give an overview of all the components that are currently available in our platform and we explain in more detail how they interact. After that, we explain how new components can be added and how you can create your own agent. Components available The task of interaction is broken down into a large number of smaller tasks operating on specific input events taken from the event-bus and pushing their output back onto the event-bus. So far, we developed the following components, grouped by their underlying input signal: 1. Audio signal voice activity detection 7 classifying an audio signal as speech, output is an annotated audio signal pushed to the event-bus and saved to EMISSOR as an audio annotation. 6 Currently, we use NLP techniques to extract such values from the text generated by speech-recognition. In future work, modules will be added that extract these from the audio signal or face expressions as well 7 https://github.com/leolani/cltl-vad automatic speech recognition 8 transforming speech to text, three different models can be chosen which generate a text output signal to the event-bus and EMISSOR. speaker identification (under development) identifying the speaker based on a sample of their voice, output is a speaker's identity (as an IRI and their name), and an audio signal annotated with the source pushed to the event-bus and EMISSOR but also registered in the eKG as an encounter. 2. Text signal mention detection 9 detecting names and potential objects in a text signal, output is an annotated signal with named entities and object mentions, pushed to the event-bus and EMISSOR. mention identification 10 identifying instances in the knowledge graph for mention annotations in a text signal, output is an identifier (IRI), pushed to the event-bus and EMISSOR. Corresponding RDF triples (IRI gaf:denotedBy gaf:utterance#offset) are pushed to the eKG. triple extraction 11 detecting RDF triples in a text signal; three different modules can be selected; output are RDF triples pushed to the event-bus, EMISSOR and the eKG. query extraction 12 detecting a question in a text signal and converting this into a SPARQL query; output is a SPARQL query send to the eKG and possibly other knowledge graphs. emotion detection 13 detecting basic emotions in a text signal; output is an emotion label pushed to the event-bus and EMISSOR as an annotation and possibly as a perspective to the eKG. gesture generation 14 returns a gesture that is an appropriate response to the emotion annotations in a text signal; output is an emotion label pushed to the event-bus and EMISSOR as an annotation and possibly as a perspective to the eKG. RDF signal knowledge representation for claims 15 posting RDF claims to the eKG; output is JSON-LD as responses to the changes in the eKG by digesting new claims pushed to the event-bus. knowledge representation for queries 16 posting SPARQL queries to the eKG or other graphs; output are RDF triples as the result of the query pushed to the event-bus. response generation 17 verbalising qualitative reflections on changes in the eKG (responses); output is a natural language response as a text signal, pushed to the event-bus and stored in EMISSOR. language generation 18 verbalising SPARQL query results; output is natural language as a text signal, pushed to the event-bus and stored in EMISSOR. Image signal object recognition 19 detecting multiple objects in images; output is bounding boxes and object labels as an annotated image signal, pushed to the event-bus and stored in EMISSOR. face detection 20 detecting human faces in an image, the output are bounding boxes with the estimated age and gender as an annotated image, pushed to the event-bus and stored in EMISSOR. face identification 21 identifying people from their face, the output is an identifier (IRI and a label as name), pushed to the event-bus and stored in EMISSOR. Encounters are also registered in the eKG as perception events. Processing audio signals The components that process the audio signal were discussed in detail in Section 2 when explaining Figure 1. In a nutshell, the voice-activity-detection component detect portions of human speech in audio, which the automaticspeech-recognition transcribes. Eventually, these components push a text signal to the event-bus, which is rendered from an audio signal for text processing components. Processing text and rdf signals The components that operate on text signals are more complex. They generate different interpretations and types of output signals and combine with components that operate on RDF triple signals. Various Natural Language Processing (NLP) modules are called within different components, among which spaCy [Vasiliev, 2020], NLTK, StanfordNLP [Angeli et al., 2015] and various transformer models. In addition to their basic processing, triple-extraction components output various annotations, for example, mentions of people and objects for which referring expressions need to be resolved to people and objects known. Referring expressions can be names (Carl), common noun phrases (the waiter), or ambiguous pronouns. By reasoning over the context and previous encounters, mention-identification components aim to establish the referent of these expressions, possibly involving the human interlocutor in case of doubt (e.g., by posting a question as a text signal "Which waiter?"). Resolved expressions are converted to IRIs incorporated in triples. These can be registrations of mentions, e.g., Carl talking about Carla, or as part of triples expressing a property from an individual: leolaniWorld:Carla leolaniWorld:live-in leolaniWorld:Amsterdam. Whenever a gaf:Claim is posted to the eKG, the knowledge-representation-for-claims component generates a reflective response to the claim. This involves running various pre-coded SPARQL queries to detect knowledge gaps, conflicts, uncertainties, novelties, analogies, and possible generalizations. The response-generation component selects a result to formulate a response attempting to improve the eKG, e.g., fill gaps or resolve uncertainty. The JSON triples are verbalized as natural language text in a text signal. The previous route through the event-bus applies to signals classified as statements. If a text signal in the event-bus is classified as a question, the query-extraction component extracts a SPARQL query from the text signal that the knowledge-representation-for-queries component posts this to the eKG (or any other triple store). The languagegeneration component takes the query's result (one, many, or none triples) to verbalize it as a text signal. Different answers are generated according to the type and number of query results. Furthermore, answers consider the status of the knowledge, e.g. how certain, who is the source, and when was it mentioned. In addition, some components only annotate mentions or emotions expressed in the text (without necessarily being involved in a triple). The mention-detection component detects things that (can) exist in the physical world, which are entities that the object recognition can classify. Mentions are saved as annotations of the text signal in EMISSOR and pushed to the event-bus for further processing. Next, the mention-identification component picks up mentions and tries to resolve its identity given the entities registered in the eKG, e.g. by checking names, the contexts, or resolving pronouns' co-reference. Identities are pushed to the event-bus and registered as gaf:Mentions in the eKG. Possibly, a new identity can be established and a new IRI is stored in the eKG with the properties that can be inferred. The emotion-detection component interprets a text signal and pushes an emotion to the event-bus and as annotation in EMISSOR. Emotions expressed by the interlocutor represent potential perspectives of the source on the situation for the eKG. The same component can be applied to text signals of the agent. Especially when creating a response to new information, the corresponding emotion can be expressed by the gesture-generation component or audio properties such as volume and pitch. Processing image signals Finally, the object recognition and face-detection components take image signals as input to detect objects and faces with bounding boxes. These are saved as annotations of image signals in EMISSOR and pushed to the event-bus. From the event-bus, the face-identification component establishes the identity of people by comparing it with the faces of known people. Unknown faces are saved as new identities with inferred properties, such as gender and age. A new identity pushed to the event-bus is picked up by a component that asks for a person's name. The image-signal-processing components either produce annotations pushed to the event-bus and/or encoded in EMISSOR or produce an IRI with properties: name, age, gender, and registered perception event in the eKG. Sample input-output signal sequences We show some schematic representations of the sequence of input and output events with their payload type and input/output topics denoted above the arrows 22 . The above components become active when the input-topic requirements are met. As such they do not represent higher-level goals or targets but are data-driven, i.e. always respond to the presence of input topics in the event-bus. The event-bus architecture however also allows to define higher-level intentions that represent behaviour of the agent to fulfill a given task or reach a goal. These intentions can be prioritised and remain active until resolved. For each intention, we can specify which components are actively listening to the event-bus and which follow-up intentions are published once a task is completed or a certain goal is reached. The above components thus are only activated within these intentions. We implemented the following high-level intentions in the platform: Greeting After detecting a new human face in an image signal, the agent tries to identify the person to great him/her by name or to get to know a new person by asking for a name. Giving consent Ask people's permission to keep the data and share it for research. If no consent is given, the agent will remove the episodic data (EMISSOR and eKG) after the interaction. Eliza Takes text signals as input and generate a text response using the Eliza trigger patterns and hard-coded responses. About agent Takes text signals as input that contain a question about the agent, and generates a text signal that informs the participant about the agent. What do you see Takes text signals as input that contain a question about the visual context and generates a text signal that describes what has been seen in the recent context (where recency is defined in the configuration). Leolani Interaction with an eKG as described in the beginning of this section, relating interpretations to identities (IRIs) and triples [Vossen et al., 2019]. Blenderbot Process text signals with Blenderbot [Roller et al., 2020]. Blenderbot is a generative model trained with different conversational data to generate a human-like response. Goodbye When a face is no longer detected or the text signal contains a goodbye clue it ends the scenario and says goodbye to the participant. Both components and higher-level intentions can be created and combined freely. Partly, this provides control over certain goals or tasks to be completed but it also allows for flexible and spontaneous behavior. In the next section, we explain how you can design your own agent by adding components and high-level intentions. All the code for the platform with the currently developed components are available on Github: https://github.com/leolani under Apache2.0 license. A good starting point for building an agent is the repository: https://github.com/leolani/ cltl-combot. Developing your own agent As components are self-contained and loosely coupled through the event-bus, developing an agent is reduced to composing the necessary components and defining a flow of events, which is done by configuring the input and output topics of the components appropriately. Adding, replacing and mixing components To enable compatibility between components, we use EMISSOR also as the preferred data format of the payload carried by the events. Signal components publish EMISSOR Signals, eventually split up into a start and stop event. Interpretation components publish EMISSOR Mentions, defining segments and their annotations. Like this, any component that can process e.g. EMISSOR text signals or interpretations of a certain type, can be integrated into an agent by simply configuring it to listen to a topic that provides events with a text signal or the desired type of interpretation as payload. Furthermore, data from components that provide EMISSOR-based event payloads can be directly recorded in EMISSOR by a central storage component without additional adaption. As long as component implementations process the same type of events, one implementation can simply be swapped for another. Even multiple implementations of the same component can be included in the same application, either for comparison or to increase performance. In the current version of the platform, we have for example various components for extracting triples and for speech recognition that can be applied simultaneously to increase recall or precision through voting. The source of signals and annotations is included in all EMISSOR data structures, which allows downstream components to determine the origin of an event. Our modular architecture aims for easy integration of components that were not designed for our platform. The overhead to turn a software application into a component compatible with our platform consists of converting input and output of the application from and to EMISSOR format and connecting it to the event-bus. This can usually be achieved by adding these steps on top of existing code, without the need for modification. Application types By choosing different implementations of the event-bus (local vs. remote), agent implementations can reach from e.g. a monolithic Python application to a containerized setup, running each component as a separate process on a single machine, for instance using docker-compose 23 , or even running in a cluster consisting of many physical machines using e.g. Kubernetes 24 . These different setups do not require any code adaption of the components itself, as those only communicate through the event-bus interface, which remains identical. In a containerized setup, our architecture also allows mixing components from different platforms, e.g. using Python next to Java components, or components running different Python environments in the same agent to resolve dependency conflicts, which is a common problem in software development. Out of the box For the components listed in Section 3 we provide Python implementations, each in its own repository published in the Leolani Github organization 25 . Each component can be packaged as Python package and needs to provide a service module containing the EMISSOR and event-bus integration, which can be run from a Python application. Packages also provide the possibility to reuse component code across components. In the future we plan to add Docker configurations to enable running each component in a Docker container. Our current agent implementations are realized as monolithic Python applications, which initialize and run the service modules from the each of the included component packages, and use a local implementation of the event-bus. We provide a Python implementation of the backend server which can be run on Linux/OS X/Windows platforms 26 , as well as a backend server that can be run on Pepper robots 27 . Our agents can be used as a skeleton to build further agents, and in the future we will provide docker-compose and Kubernetes setups based on component Docker 28 images to run the agents as fully containerized applications. In the containerized setting we will be able to run components either locally or remotely in the cloud. To collect all code needed for an agent we created parent repositories for each agent which contain all of the agent's components as git submodules. We provide build tooling to package each of our Python components, share the packages between components and setup Python environments with all dependencies needed to run the agent application and individual components. This process can be executed centrally from the parent repository using our build setup. Following this pattern, new agents can be created also outside of our organisation, mixing components under different governance. To build components, the https://github.com/leolani/cltl-combot provides infrastructure libraries for eventbus integration, configuration management and resource management (locking). The cltl.combot.infra.event module provides our event-bus interface with a local implementation as well as an implementation based on the kombu 29 library supporting the AMQP 30 protocol. This allows to connect our framework to various available messaging servers like for instance RabbitMQ 31 . Furthermore, cltl.combot.infra.topic_worker.TopicWorker provides a utility class to listen to a configurable set of topics and process incoming events sequentially in a single thread, such that only the actual processing function, accepting a single event as input, needs to be implemented by the user of the library. Our own components are structured to separate functionality, using the cltl namespace, from the event-bus integration, using the cltl_service namespace. A simple example is presented in Appendix B, C and D. For a full example including configuration and resource management we provide a template component at https://github.com/leolani/cltl-template with more detailed code templates, makefiles for our build setup and in the future a Dockerfile template. Conclusions We described a platform for creating interactive agents which also captures the interaction at the signal level, the interpretation level and the integrated knowledge level. Central to our platform is an event-bus on which signals and interpretations can be published in temporal sequence as so-called topics. By defining components as services that take certain topics as input and produce other topics as output, we have the flexibility to engineer any combination of components to create agent behaviour. Our platform also supports the specification of higher-level intentions that can be carried out to fulfill a task or goal. Such intentions define which low-level components should be active to achieve this. Multimodal interaction is extremely complex and rich. The agents that are currently built through our platform are far from what is needed to deal with all facets of human interaction. A lot of research and development is needed not only on individual components but also on their interaction and integration. Our platform will specifically help developing such complex integrated agents. In the near future, we will use the platform to create different agents that can be used in interaction experiments. The rendered multimodal data and eKGs can be analysed and compared to assess and evaluate the agents and their components. As such, we hope it will function as a laboratory for agent development and testing. A EMISSOR multimodal data representation Figure 2 : 2Schematic overview of relations across data structures in EMISSOR, the episodic Knowledge Graph and the event-bus. 22 Topic names are exemplary and can be configured in the application Figure 3 : 3Visualization of an interaction in the EMISSOR data format. Figure 3 , 3taken from[Báez Santamaría et al., 2021], shows a visualization of four modalities (text, audio, visual, and knowledge). Signals are grounded in a temporal container on the horizontal axis, with bars marking alignments through the temporal ruler. Red boxes mark segments annotated as mentions of objects (pills and the table). Text segments highlighted in red are annotated as mentions of triples. The upper graphs represent corresponding triples from eKG generated from the annotated source modalities along the temporal sequence. The visual modality shows two different camera viewpoints (left is what Carl sees and right is what Leolani sees) concatenated side by side. https://www.dbpedia.org 3 Note that the agent's memory does not forget anything and is limitless 4 https://github.com/leolani/eliza-parent 5 https://github.com/leolani/leolani-mmai-parent https://docs.docker.com/compose 24 https://kubernetes.io 25 https://github.com/leolani 26 https://github.com/leolani/cltl-backend 27 https://github.com/leolani/cltl-backend-naoqi 28 https://www.docker.com https://github.com/celery/kombu 30 https://www.amqp.org 31 https://www.rabbitmq.com Emissor: A platform for capturing multimodal interactions as episodic memories and interpretations with situated scenario-based ontological references. Selene Báez Santamaría, Thomas Baier, Taewoon Kim, Lea Krause, Jaap Kruijt, Piek Vossen, Proceedings of the 1st Workshop on Multimodal Semantic Representations (MMSR). the 1st Workshop on Multimodal Semantic Representations (MMSR)Selene Báez Santamaría, Thomas Baier, Taewoon Kim, Lea Krause, Jaap Kruijt, and Piek Vossen. Emissor: A platform for capturing multimodal interactions as episodic memories and interpretations with situated scenario-based ontological references. Proceedings of the 1st Workshop on Multimodal Semantic Representations (MMSR), pages 56-77, 2021. Modelling context awareness for a situated semantic agent. Piek Vossen, Lenka Bajčetić, Selene Báez Santamaria, Suzana Basić, Bram Kraaijeveld, Proceedings of 11th International and Interdisciplinary Conference on Modeling and Using Context, CONTEXT 2019. 11th International and Interdisciplinary Conference on Modeling and Using Context, CONTEXT 2019Piek Vossen, Lenka Bajčetić, Selene Báez Santamaria, Suzana Basić, and Bram Kraaijeveld. Modelling context awareness for a situated semantic agent. In Proceedings of 11th International and Interdisciplinary Conference on Modeling and Using Context, CONTEXT 2019, 2019. Eliza-a computer program for the study of natural language communication between man and machine. Joseph Weizenbaum, Communications of the ACM. 91Joseph Weizenbaum. Eliza-a computer program for the study of natural language communication between man and machine. Communications of the ACM, 9(1):36-45, 1966. Leveraging linguistic structure for open domain information extraction. Yuli Vasiliev, ; Gabor Angeli, Melvin Jose Johnson Premkumar, Christopher D Manning, Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing. the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language ProcessingLong Papers1Natural Language Processing with Python and SpaCy: A Practical IntroductionYuli Vasiliev. Natural Language Processing with Python and SpaCy: A Practical Introduction. No Starch Press, 2020. Gabor Angeli, Melvin Jose Johnson Premkumar, and Christopher D Manning. Leveraging linguistic structure for open domain information extraction. In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 1: Long Papers), pages 344-354, 2015. Stephen Roller, Emily Dinan, Naman Goyal, Da Ju, Mary Williamson, Yinhan Liu, Jing Xu, Myle Ott, Kurt Shuster, Eric M Smith, arXiv:2004.13637Recipes for building an open-domain chatbot. arXiv preprintStephen Roller, Emily Dinan, Naman Goyal, Da Ju, Mary Williamson, Yinhan Liu, Jing Xu, Myle Ott, Kurt Shuster, Eric M Smith, et al. Recipes for building an open-domain chatbot. arXiv preprint arXiv:2004.13637, 2020.	[ "https://github.com/leolani.", "https://github.com/leolani/cltl-vad", "https://github.com/leolani", "https://github.com/leolani/", "https://github.com/leolani/cltl-combot", "https://github.com/leolani/cltl-template", "https://github.com/leolani/eliza-parent", "https://github.com/leolani/leolani-mmai-parent", "https://github.com/leolani", "https://github.com/leolani/cltl-backend", "https://github.com/leolani/cltl-backend-naoqi", "https://github.com/celery/kombu" ]
[ "Constraining axion inflation with gravitational waves from preheating", "Constraining axion inflation with gravitational waves from preheating" ]	[ "Peter Adshead \nDepartment of Physics\nUniversity of Illinois at Urbana-Champaign\n61801UrbanaIllinoisUSA\n", "John T Giblin Jr\nDepartment of Physics\nKenyon College\n43022GambierOhioUSA\n\nCERCA/ISO\nDepartment of Physics\nCase Western Reserve University\n44106ClevelandOhioUSA\n", "Mauro Pieroni \nInstituto de Física Teórica UAM/CSIC\nCalle Nicolás Cabrera 13-15E-28049Cantoblanco, MadridSpain\n\nDepartamento de Física Teórica\nUniversidad Autónoma de Madrid (UAM)\nCampus de Cantoblanco28049MadridSpain\n\nTheoretical Physics\nBlackett Laboratory\nImperial College\nSW7 2AZLondonUK\n", "Zachary J Weiner \nDepartment of Physics\nUniversity of Illinois at Urbana-Champaign\n61801UrbanaIllinoisUSA\n" ]	[ "Department of Physics\nUniversity of Illinois at Urbana-Champaign\n61801UrbanaIllinoisUSA", "Department of Physics\nKenyon College\n43022GambierOhioUSA", "CERCA/ISO\nDepartment of Physics\nCase Western Reserve University\n44106ClevelandOhioUSA", "Instituto de Física Teórica UAM/CSIC\nCalle Nicolás Cabrera 13-15E-28049Cantoblanco, MadridSpain", "Departamento de Física Teórica\nUniversidad Autónoma de Madrid (UAM)\nCampus de Cantoblanco28049MadridSpain", "Theoretical Physics\nBlackett Laboratory\nImperial College\nSW7 2AZLondonUK", "Department of Physics\nUniversity of Illinois at Urbana-Champaign\n61801UrbanaIllinoisUSA" ]	[]	We study gravitational wave production from gauge preheating in a variety of inflationary models, detailing its dependence on both the energy scale and the shape of the potential. We show that gauge preheating generically leads to a large gravitational wave background that contributes significantly to the effective number of relativistic degrees of freedom in the early Universe, N eff . We demonstrate that the efficiency of gravitational wave production is correlated with the tensor-to-scalar ratio, r. In particular, we show that efficient gauge preheating in models whose tensor-to-scalar ratio would be detected by next-generation cosmic microwave background experiments (r 10 −3 ) will either be detected through its contribution to N eff or ruled out. Furthermore, we show that bounds on N eff provide the most sensitive probe of the possible axial coupling of the inflaton to gauge fields regardless of the potential.	10.1103/physrevd.101.083534	[ "https://arxiv.org/pdf/1909.12842v1.pdf" ]	203,593,680	1909.12842	4ba3a81f869c2d1c2daa016108384f6f8ade2875	Constraining axion inflation with gravitational waves from preheating Peter Adshead Department of Physics University of Illinois at Urbana-Champaign 61801UrbanaIllinoisUSA John T Giblin Jr Department of Physics Kenyon College 43022GambierOhioUSA CERCA/ISO Department of Physics Case Western Reserve University 44106ClevelandOhioUSA Mauro Pieroni Instituto de Física Teórica UAM/CSIC Calle Nicolás Cabrera 13-15E-28049Cantoblanco, MadridSpain Departamento de Física Teórica Universidad Autónoma de Madrid (UAM) Campus de Cantoblanco28049MadridSpain Theoretical Physics Blackett Laboratory Imperial College SW7 2AZLondonUK Zachary J Weiner Department of Physics University of Illinois at Urbana-Champaign 61801UrbanaIllinoisUSA Constraining axion inflation with gravitational waves from preheating We study gravitational wave production from gauge preheating in a variety of inflationary models, detailing its dependence on both the energy scale and the shape of the potential. We show that gauge preheating generically leads to a large gravitational wave background that contributes significantly to the effective number of relativistic degrees of freedom in the early Universe, N eff . We demonstrate that the efficiency of gravitational wave production is correlated with the tensor-to-scalar ratio, r. In particular, we show that efficient gauge preheating in models whose tensor-to-scalar ratio would be detected by next-generation cosmic microwave background experiments (r 10 −3 ) will either be detected through its contribution to N eff or ruled out. Furthermore, we show that bounds on N eff provide the most sensitive probe of the possible axial coupling of the inflaton to gauge fields regardless of the potential. We study gravitational wave production from gauge preheating in a variety of inflationary models, detailing its dependence on both the energy scale and the shape of the potential. We show that gauge preheating generically leads to a large gravitational wave background that contributes significantly to the effective number of relativistic degrees of freedom in the early Universe, N eff . We demonstrate that the efficiency of gravitational wave production is correlated with the tensor-to-scalar ratio, r. In particular, we show that efficient gauge preheating in models whose tensor-to-scalar ratio would be detected by next-generation cosmic microwave background experiments (r 10 −3 ) will either be detected through its contribution to N eff or ruled out. Furthermore, we show that bounds on N eff provide the most sensitive probe of the possible axial coupling of the inflaton to gauge fields regardless of the potential. I. INTRODUCTION While the qualitative predictions of inflation are wellmotivated and well-understood [1][2][3][4][5], there is as yet no unique, complete model which connects inflation to the standard model (SM) of particle physics. A crucial component of such a model is the subsequent reheating process [6][7][8][9], which must realize a phase transition from the cold post-inflationary state to the hot Big Bang. In this phase, the inflaton decays into other species to repopulate the Universe and begin the radiation-dominated era. During an initial stage of preheating, the coherent oscillation of the inflaton field induces explosive production of bosons via parametric or tachyonic resonance (see Refs. [10,11] for reviews). The rapid production of inhomogeneities during this phase generically sources a significant gravitational wave background [12][13][14][15][16][17][18][19][20][21][22][23]. Furthermore, preheating can have important consequences for observable predictions of inflationary models in the cosmic microwave background (CMB): the evolution of the equation of state after inflation affects the mapping of fluctuations' present-day length scales to the times those scales first left the horizon during inflation. In particular, if preheating is sufficiently efficient then the onset of radiation domination occurs nearly instantaneously after inflation ends. Most of the early work on preheating focused on models which couple the inflaton to another canonical scalar field [12,[24][25][26][27][28][29][30][31][32][33], with recent studies exploring nonminimally coupled scalar fields [34][35][36][37][38]. Alternatively, the inflaton could couple directly to gauge fields [39][40][41][42][43][44][45][46]. One * [email protected] † [email protected] ‡ [email protected] § [email protected] such coupling of particular theoretical and phenomenological interest is that of a (pseudoscalar) inflaton φ coupled to the Chern-Simons density FF of a gauge field [47][48][49]. From the theoretical point of view, the (approximate) shift symmetry of an pseudoscalar inflaton (axion) inherently protects its potential from large corrections, ensuring the flatness required for a successful inflationary phase. The phenomenology of these models is extremely rich, with possible observable signatures including the production of sizable non-Gaussianities [50][51][52][53], observable gravitational waves [51,52,[54][55][56][57], primordial black holes [53,[58][59][60][61][62], µdistortions [55,63], primordial magnetic fields [49,[64][65][66][67][68], and the generation of the baryon asymmetry [57,[69][70][71][72]. Preheating into gauge fields via a Chern-Simons coupling was first studied within the context of chaotic inflation in Ref. [40], and the model's viability for magnetogenesis was explored in Ref. [41]. Subsequently, we extended this work in Ref. [46] to compute the gravitational wave spectrum produced by the dynamics of gauge preheating. The surprising result was that the (over-) production of gravitational radiation provides the strongest probe of (or constraints on) the coupling scale between the axion and gauge fields. As detailed in Ref. [41] (and Section II C below), next-generation CMB experiments will probe the radiation content of the Universe to a precision sufficient to rule out most of the interesting region of parameter space in these models. Here and in Ref. [73] we use this result to put important constraints on the axial coupling between the inflaton and gauge fields in a variety of wellmotivated inflationary models. While the chaotic inflation scenario considered in Ref. [46] models the inflaton's coherent oscillations about the minimum of its potential to leading order, this model is disfavored at the 95% confidence level, primarily by constraints on the tensor-to-scalar ratio [5]. To more completely understand the role gauge preheating could play in constraining axion inflation, we investigate the dependence of preheating dynamics and gravitational wave production on the details of the inflationary potential. While the efficiency of preheating and the production of gravitational waves is qualitatively generic, both the energy scale of inflation and the shape of the potential significantly alter the quantitative results as we demonstrate below. This paper is organized as follows. In Section II we review models in which an axion or pseudoscalar inflaton is coupled to an Abelian gauge field and introduce the constraints on stochastic backgrounds of gravitational waves from the CMB. After describing the numerical prescription employed for simulations in Section III, we outline some analytic estimates in Section IV. We investigate the effect of the energy scale of inflation and the shape of the inflationary potential on preheating and gravitational wave production in Sections V A and V B, respectively. In Section VI we draw our conclusions. In Appendix A we detail the equations of motion of the system and summarize a linear analysis thereof. Appendix B details our procedure for setting accurate initial conditions. Finally, in Appendix C we verify that the Universe remains radiation dominated after preheating to establish the robustness of the gravitational-wave transfer function (for the models we consider). II. GAUGE FIELDS DURING AND AFTER AXION INFLATION In this section we introduce the models of interest and briefly review the dynamics of axially coupled gauge fields during and after axion inflation, relegating many details to Appendix A. A. Models of inflation We consider the action for a pseudoscalar inflaton φ minimally coupled to gravity and to an Abelian gauge field A µ , S = d 4 x √ −g M 2 pl 2 R − 1 2 ∂ µ φ∂ µ φ − V (φ) − 1 4 F µν F µν − α 4f φF µνF µν ,(1) where f is a mass scale (the axion decay constant), α is a dimensionless coupling constant, and F µν ≡ ∂ µ A ν − ∂ ν A µ is the field strength tensor of a U(1) gauge theory. 1 We do not identify the gauge field as, e.g., that of standard 1 The extension to different Abelian gauge groups (such as N copies of of a U(1) gauge group) is straightforward. Conversely, the extension to non-Abelian gauge groups is non-trivial. See for example [74][75][76][77][78][79]. model hypercharge, nor do we include any charged fields in our model. The dual tensor is defined bỹ F µν = 1 2 µναβ F αβ .(2) where µνρσ is the Levi-Civita symbol with convention 0123 = 1/ √ −g. We set c = = k B = 1 and denote by M pl = 1/ √ 8πG N the reduced Planck mass. The background spacetime is the mostly-plus, conformal Friedmann-Lemaître-Robertson-Walker (FLRW) metric, ds 2 = a(τ ) 2 −dτ 2 + dx 2 ,(3) where the conformal time coordinate τ is related to the cosmic time coordinate via dτ = dt/a. Throughout, primes denote derivatives with respect to conformal time, e.g., a ≡ ∂a/∂τ , and dots denote derivatives with respect to cosmic time, e.g.,ȧ ≡ ∂a/∂t. We use repeated lower indices to indicate contractions with the Kronecker delta. In this work, we are interested in the effects of varying the shape of the potential V (φ) on the efficiency of preheating and the subsequent gravitational wave production. For this purpose we consider various classes of inflationary potentials (the most representative among the ones typically considered by Planck [5]), reporting our parameter choice(s) and the corresponding scalar tilt n s and tensor-to-scalar ratio r (evaluated at a pivot scale which exited the horizon 60 e-folds before the end of inflation). It is worth stressing that in the context of standard inflationary model building most of these models assume the inflaton to be a scalar field. Conversely, for the models described by Eq. (1) the inflaton is pseudoscalar. 2 Specifically, we explore the following classes of models: 1. Chaotic-like monomial models, which have the potential V (φ) = m 4−n \|φ\| n .(4) Notice that for n = 2 this simply reduces to chaotic inflation [4], the case we consider here. For n = 2 the Planck normalization of the scalar power spectrum [5] sets m 2 = 1.9 × 10 −11 M 2 pl , with n s = 0.966 and r = 0.13. Another set of models, natural inflation, have potentials of the form V (φ) = V 0 (1 + cos (φ/v)) [84]. In this case, with v = √ 8πM pl the Planck normalization sets V 0 = 5.9 × 10 −10 M 4 pl , for which n s = 0.952 and r = .033. However, we find that the results in this case are virtually identical to those for chaotic inflation, so we omit them below. 2. Starobinsky-like models, which have potentials given by [85] 3 V (φ) = V 0 1 − exp \|φ\| v 2 .(5) For v = 10 M pl /3 the Planck normalization sets V 0 = 6.2 × 10 −10 M 4 pl , in which case n s = 0.969 and r = 0.016. The class of α-attractor models, motivated by supergravity [89,90] (typically divided into E-models and T-models) have exponentially flat potentials whose steepness is controlled by the parameter α, related to the curvature of the Kähler manifold. Since by varying this parameter is possible to interpolate between chaotic-like models and Starobinsky-like models, we omit the results for this class. 3. Monodromy inflation, corresponding to the potential V (φ) = µ 3 φ 2 + φ 2 c − φ c .(6) This is part of a broad class of string-theorymotivated models in which large (super-Planckian) field displacement is obtained by wrapping the inflaton trajectory into a series of (sub-Planckian) fundamental circuits [91][92][93]. Note that deep in the inflationary phase (φ φ c ) the potential is well approximated by a linear potential (n = 1) of the class in Eq. (4). For φ c = M pl /10 the Planck normalization sets µ = 6.0 × 10 −4 M pl , for which n s = .975 and r = .067. 4. Hilltop-like models, with potentials given by [94] V (φ) = V 0 1 − \|φ\| v p 2 ,(7) where p ≥ 2. We consider a variety of p and v and list the corresponding model fits and predictions in Table I. 5. D-brane models, with potentials of the form [95][96][97][98] V (φ) = V 0 1 − v \|φ\| p 2 .(8) Fixing p = 2 and v = M pl /2, the Planck normalization sets V 0 = 7.5 × 10 −11 M 4 pl , in which case n s = 0.975 and r = 2.2 × 10 −3 . B. Gauge fields on the axion background The equations of motion that result from the variation of Eq. (1) are difficult to solve analytically due to the nonlinear interactions of the axion and gauge field. Furthermore, the homogeneous axion background sources a tachyonic instability in the gauge field which quickly renders a linear analysis invalid. In this subsection, we briefly outline the well-known linear treatment of the gauge-field fluctuations in the inflationary phase and during early stages of resonance. Full details may be found in Appendix A. In a homogeneous axion background, the helical polarizations of the gauge field obey [40] A ± (k) + k (k ∓ 2Hξ) A ± (k) = 0,(9) where we define the instability parameter ξ ≡ α 2f φ H(10) and the conformal Hubble parameter, H = a /a. The interaction between the gauge field and the homogeneous, rolling axion leads to the exponential growth of one of the polarizations of the gauge field, understood as an imaginary ("tachyonic") effective frequency ω ∼ k(k − 2Hξ) (assuming ξ > 0) for modes k < 2Hξ. The polarizations are amplified (relative to the conformally-invariant radiation solution A rad,± ) by a factor A ± A rad,± ∼ e π 2 ξ± π 2 ξ .(11) The amplification is controlled by the parameter ξ, which depends on the inflaton velocity, φ /H. The largest effects therefore occur near the end of inflation and during reheating when the inflaton velocity is largest. Modes with wavenumbers in the band 1 k/H 2ξ are significantly enhanced by the axion. Shortly after their production during reheating these modes can rescatter, generating the subdominant helicity as well as inflaton quanta (see Fig. 2). We use numerical simulations to capture these nonlinear dynamics in later sections. C. Gravitational waves and ∆N eff The tachyonic production of gauge field modes leads to large anisotropic stresses which results in the copious production of gravitational waves [46]. These gravitational waves form a stochastic background with frequencies near the Hubble scale during reheating, corresponding to f ∼ 10 9 Hz today 4 for inflation occurring near the scale of grand unification, Λ ∼ 10 16 GeV. Gravitational waves with wavelengths that are much shorter than the Hubble scale contribute energy density that gravitates like radiation. During preheating, significant energy density can be deposited into subhorizon gravitational waves which, as in Ref. [46], may be constrained by CMB and BBN measurements of the total radiation density in species beyond the standard model, ∆N eff = N eff − 3.046. The net energy density in gravitational waves is given by Ω gw,0 h 2 = d ln k 1 ρ 0 dρ gw,0 d ln k .(12) By conservatively assuming that all extra radiation density present during the formation of the CMB (beyond the standard model) is comprised of gravitational waves, a bound on N eff directly constrains Ω gw,0 h 2 via [99] Ω gw,0 h 2 Ω γ,0 h 2 = 7 8 4 11 4/3 ∆N eff ,(13) where the present energy density in photons is Ω γ,0 h 2 = 2.47 × 10 −5 . From the Planck limit, \|∆N eff \| 0.33 [100,101], we obtain a bound Ω gw,0 h 2 1.85 × 10 −6 . Nextgeneration CMB experiments (e.g., CMB-S4) will probe ∆N eff ≤ .03 at 1σ and ∆N eff ≤ .06 at 2σ [102], improving the upper limit by an order of magnitude to Ω gw,0 h 2 1.68 − 3.36 × 10 −7 .(14) A separate analysis [103] using Planck data provides a stronger constraint, Ω gw,0 h 2 < 1.2 × 10 −6 at 95% confidence. The same study projected that next-generation satellite missions COrE [104] and Euclid [105] will place 2σ bounds of ∆N eff < .013 or Ω gw,0 h 2 < 7.6 × 10 −8 . In the results below, we explore the relationship between gravitational wave production at these levels and the preheating process itself. The first direct detection of gravitational waves by LIGO [106] prompts consideration of direct gravitational wave detectors as alternative probes of stochastic gravitational wave backgrounds. In particular, the presentlyoperating LIGO [107] and VIRGO [108] detectors (with KAGRA [109] expected to join this network of detectors in the near future), as well as future missions like LISA [110] and the Einstein Telescope [111], could detect stochastic gravitational wave backgrounds generated in the very early Universe. In order detect a stochastic gravitational wave background with a direct gravitational wave detector, the corresponding signal must have a sufficiently large signal-to-noise ratio (SNR), given by [99,112] SNR 2 = T fmax fmin df Ω gw Ω s 2 (15) ≡ T fmax fmin df Ω gw 4π 2 S n f 3 /(3H 2 0 ) 2 ,(16) where Ω s is used to denote the detector sensitivity (expressed as a cosmological parameter), S n is the strain sensitivity of the instrument, T denotes the observation time, and f min and f max are respectively the minimal and maximal frequencies to which the instrument is sensitive. The fact that Ω s is proportional to f 3 (times the strain) makes direct detection of high frequency gravitational waves difficult; however, if inflation occurred at a lower scale, therefore producing a lower-frequency signal, direct detection is possible [14,15]. At the same time, current methods (see, for example, [113][114][115]) for the direct detection of high-frequency gravitational waves are unfortunately not expected to be sensitive enough to place interesting bounds on stochastic backgrounds of cosmological gravitational waves. III. NUMERICAL METHODS Our numerical approach is very similar to that employed in Ref. [46]. We numerically integrate the classical equations of motion of the inflaton, Eq. (A22), and the gauge fields, Eq. (A19), in an FLRW background governed by Eqs. (A1) and (A2). Specifically, we discretize the evolution equations onto a 3-D, regularly-spaced grid with periodic boundary conditions and step the coupled system of equations through time using the fourth-order Runge-Kutta method. Spacetime expansion is implemented self-consistently by computing the current energy density and pressure, Eqs. (A3) and (A4), averaged over the simulation volume. Numerical treatments of gauge fields, compared to those for scalar fields, require particular care. Specifically, numerical evolutions of gauge fields must be stable with respect to their constraints. These constraints are Gauss's law, Eq. (A20), and the prescribed gauge condition (in our case Lorenz gauge, ∂ µ A µ = 0). One numerical method for evolving gauge theories is lattice gauge theory, which recasts the gauge fields as a system of "link variables" representing the discrete connection between adjacent lattice sites. The advantage of this method is that discrete gauge invariance is an exact symmetry of the discrete system. The system of dynamical equations of motion that results from the direct variation of the discrete lattice action yields an evolution scheme which preserves the gauge condition and (degree of satisfaction of) Gauss's law. However, as noted by [45,116], an appropriately gauge-invariant representation of the axial coupling termF µν F µν renders the equations of motion implicit, requiring a computationally-expensive iterative solution technique. We take an alternative approach and evolve the Euler-Lagrange equations of the continuum theory for all of the components of the gauge field A µ in Lorenz gauge. That is, rather than evolve the evolution equations of a discretized theory, we numerically integrate the equations of motion of the continuum theory by discretizing the dynamical equations themselves (as often employed for scalar fields). While gauge invariance is not exact in this case, the critical property required for robust results is stability. One can recast the equations of motion such that constraint violations, while dynamical, remain bounded in time [117,118]. Doing so requires evolving additional, redundant degrees of freedom, which in our case means evolving all four components of the gauge potential, A ν = ∇ 2 A ν + η βν α f ∂ α φ 1 2 ε αβρσ F ρσ ,(17) obtained by applying the Lorenz gauge choice, ∂ µ A µ = 0, to Eq. (A18). The results of [39,40] demonstrate the stability of the satisfaction of the gauge constraint under both the axial coupling considered here and a dilatonic coupling to the gauge-field kinetic term. Recent work used lattice-gauge-theory-based simulations employing an iterative scheme [45], the results of which reproduce those of Ref. [40] (and those presented in Section V A below). In addition, in Ref. [44] we compared simulations of preheating into three U(1) gauge fields using the approach detailed above as well as preheating into a set of SU(2) gauge fields (with the gauge self-coupling tuned so that internal interactions are negligible) using lattice gauge theory, finding near-perfect agreement between the two methods. There are two differences in our numerical implementation relative to that in Ref. [46]. First, we have improved our procedure for obtaining the power spectra of the gauge fields at the end of inflation. To capture the tachyonic enhancement of (one polarization of) the gauge fields during inflation, we numerically evolve the background spacetime, the homogeneous mode of the inflaton, and linearized equations of motion of the gaugefield polarizations. In contrast to Refs. [40,46], here we more carefully account for the backreaction of the gauge fields onto the background quantities. At the range of couplings we explore, the approximations used to derive the analytic results Eqs. (A35) and (A36) (that \|ξ\| 4 and is constant) are inaccurate during the final few e-folds of inflation. While these approximations are valid earlier during inflation and at larger couplings (the regimes for which Eqs. (A35) and (A36) were developed [57,119]), they overestimate the backreaction of the gauge fields onto the background at the end of inflation. At the largest couplings we consider here (e.g., α/f ∼ 15 M −1 pl for chaotic inflation), these errors artificially offset the end of inflation by ∼ .5 e-folds. As a result, the use of Eqs. (A35) and (A36) to compute the duration of inflation contributes errors to the relationship between present-day length scales and the number of e-folds before the end of inflation when those scales exited the horizon. To mitigate these errors, we perform the integrals Eqs. (A32) to (A34) via quadrature of the numerically-integrated gauge field fluctuations. See Appendix B for full details of this procedure. Other than improving the gauge-field power spectra, our procedure for generating initial conditions is unchanged from Ref. [46]. We initialize the lattice simulations one to two e-folds (depending on the model; see Table II) before the end of inflation, where the inflaton is initialized with a mean value and velocity given by the background evolution. We seed the fluctuations in each field and its (conformal-time) velocity as independent, Gaussianrandom fields by drawing, for each Fourier mode on the lattice, an amplitude from a Rayleigh distribution and a uniform random phase. The variance of the Rayleigh distribution for each mode is set according to the power spectrum obtained from the numerical solution of the linearized equation of motion for the field fluctuations evolved through inflation. Finally, the chosen Lorenz gauge condition is satisfied on the initial slice by setting A 0 = 0 and projecting the polarization fields, A ± (k), onto their vector components A i (k), which automatically yields ∂ i A i = 0. The second difference in our numerical implementation is that, in contrast to the pseudospectral solver used in Ref. [46], we represent spatial derivatives with finite differencing, using fourth-order-accurate centered difference stencils. In particular, whereas in Ref. [46] we evolved the equations of motion for the tensor metric fluctuations h ij in Fourier space (computing and Fourier transforming the source term Eq. (A12) and projecting with Eq. (A10) at each timestep), we here evolve the equation of motion in position space without applying the transverse-traceless projection [120], u ij + 2Hu ij − ∂ k ∂ k u ij = 2 M 2 pl T ij .(18) Instead, we apply the transverse-traceless projection to u ij to obtain h ij = P il P jm − 1 2 P ij P lm u lm ,(19) only when computing the gravitational wave spectrum Ω gw (k) via Eq. (A14). The advantage of this procedure is that it requires no Fast Fourier Transforms at each timestep, which scale poorly to distributed-memory systems. We have verified that the axion-gauge-field dynamics and the gravitational wave spectra Ω gw (k) reproduce the results of the pseudospectral method extremely well. The software we developed for this purpose is pystella, 5 a Python-based, MPI-parallel and GPU-accelerated code making use of PyOpenCL [121] and Loo.py [122] for the generation of OpenCL code to run on GPUs. Because pystella may thus run on multiple GPUs (or any architecture with OpenCL support), we evolve lattices with N 3 = 384 3 points, enabling reliable simulations of larger couplings α/f than in Ref. [46]; we also checked a representative sample of our results against simulations with 512 3 points to show convergence. The inflationary models we consider exhibit different post-inflationary Hubble scales relative to the oscillation timescale about the minima of their respective potentials, m φ (defined by m 2 φ = ∂ 2 V /∂φ 2 evaluated at the minimum of the potential). For this reason, we choose different comoving box-lengths L for different models. In all cases, we have checked our results are insensitive to the precise choice of L, i.e., that the simulations have converged. We also tune the initialization time (relative to the end of inflation) based on the model, starting sufficiently early to capture any nonlinear effects at the end of inflation (but no earlier than necessary to save computational expense). We tabulate these choices, as well as other relevant parameters for each model, in Table II. We use a timestep ∆τ = ∆x/10 = L/N/10 in all cases and set the scale factor a = 1 at the end of inflation, when H ≡ −Ḣ/H 2 = 1. IV. ANALYTIC ESTIMATES To establish expectations for the effects of the inflationary model on preheating and gravitational wave production, we begin with some analytical estimates. While the linear theory reviewed in Appendix A is not valid during preheating, it can be used to gain intuition for the scaling of the backreaction on the expansion rate and on the motion of the inflaton during the initial phase of preheating which is typically the most violent. We additionally use a "rule of thumb" for stochastic gravitational wave production from cosmological processes [123] to make contact between the characteristics of the inflationary potential and the efficiency of gravitational wave production from preheating. A. Efficiency of preheating From the Friedmann equation, Eq. (A1), by making use of the approximation for the energy density in the gauge fields, Eq. (A35), we can derive the approximation H 2 M 2 pl = 1 (3 − φ ) V M 4 pl 1 + V M 4 pl 1.4 · 10 −4 e 2πξ ξ 3 (3 − φ ) 2 + O V M 4 pl 3 ,(20) where we introduce the slow roll parameters φ ≡φ 2 2M 2 pl H 2 , H ≡ −Ḣ H 2 .(21) In standard single-field slow-roll inflation, the Einstein equations imply that φ = H . However, in the regime of strong gauge field backreaction, this relation does not hold. Equation (20) quantifies the contribution of the gauge fields to the Hubble rate relative to the contribution from the inflaton. As such, it provides a rough estimate of the value of ξ (and therefore φ /H, once α/f is fixed) required in order for the gauge field contribution to the energy density to be comparable with the inflaton's. Beyond modifying the value of the Hubble parameter as in Eq. (20), the gauge fields induce a new friction term in the equation of motion of the inflaton. Substituting the spatially-averaged interaction, Eq. (A35), into the equation of motion for the inflaton, Eq. (A22), while making use of the lowest order approximation of Eq. (20) and assuming φ /H 2 M pl , we can derive the usual slowroll condition, now including the effect of the gauge-field backreaction, M 2 pl V ,φ V −φ H 1 + V M 4 pl 1.4 · 10 −4 e 2πξ ξ 3 (3 − φ ) 2 + 2.4 · 10 −4 α f V M 2 pl e 2πξ ξ 4 (3 − φ ) 2 .(22) Note that the gauge-field backreaction changes the relationship between the slope of the potential relative to its magnitude and so also the slow-roll parameter in Eq. (21). In this scenario, gauge-field-induced friction provides an additional mechanism to enforce the flatness of the potential required for slow-roll inflation [51,55,119]. There are two effects at play in Eq. (22). The first is that the backreaction changes the Hubble rate, increasing Hubble friction (on the inflaton), while the second is the backreaction of the E · B term. The impact of these two effects can be understood by comparing Eq. (22) with Eq. (20). As the same (V -dependent) term is compared with unity (for the backreaction on H) and with 2 φ (for the backreaction on the EOM of the inflaton) respectively, the gauge-field-induced modification of the Hubble parameter is higher order (in V /M 4 pl ) than the new friction term in the EOM of the inflaton. In addition, while both the gradient of the potential and the (lowest order expression of the) Hubble friction term in Eq. (22) do not depend explicitly on the scale of V , the gauge-field friction depends linearly on the inflationary energy scale. As a consequence, low-scale models are expected to require a larger value of ξ in order for the gauge fields to become important. Finally, while V is nearly constant (which is the case during inflation), the gauge-field-induced backreaction grows exponentially with ξ. However, if V is not constant (as is the case during preheating where it decreases as fast as, or faster than, e −2πξ ) then the gauge field induced backreaction may either never become relevant or be shut off. Such a possibility is realized in potentials without minima, such as a tanh(φ/M ) shape. We leave investigations of this class of potentials to future work. To set expectations for the range of couplings α/f relevant for preheating in a given model, we make use of Eq. (20) (to leading order in slow roll) to obtain scaling relations between α/f and the Hubble scale at the onset of preheating (and so the parameters of the model). We quantify efficiency in terms of the fraction of the Universe's energy residing in the gauge fields, Hilltop (p = 4, v = 4M pl ) 3.06 × 10 −6 20 -2 0.24 1.3 1.1 × 10 −3 0.951 1.4 × 10 −4 Hilltop (p = 4, v = 8M pl ) 5.33 × 10 −6 20 -2 0.33 0.95 1.7 × 10 −3 0.955 1.7 × 10 −3 D-brane (p = 2, v = M pl /2) 4.90 × 10 −5 40 -1 0.073 2.1 2.5 × 10 −3 0.975 2.2 × 10 −3 Natural (v = √8πMρ gauge ρ ≈ 1.4 · 10 −4 3 (H/M pl ) 2 ξ 3 e 2πξ .(23) In the regime of this approximation's validity, we may ask how much one must tune ξ (or α/f ) to compensate for a reduction in H/M pl in order to keep ρ gauge /ρ fixed. Considering two sets of parameters (H 1 , ξ 1 ) and (H 2 , ξ 2 ), this amounts to solving the nonlinear equation ρ gauge,1 ρ gauge,2 = H 1 H 2 2 ξ 2 ξ 1 3 e 2π(ξ1−ξ2) = 1(24) for ξ 2 in terms of H 1 /H 2 and ξ 1 . When ξ 1 and ξ 2 are both small (and so ξ 1 − ξ 2 is as well), we solve H 1 H 2 2 ξ 2 ξ 1 3 ≈ 1,(25) yielding ξ 2 = ξ 1 (H 2 /H 1 ) 2/3 . Alternatively, if ξ 1 and ξ 2 are both large, H 1 H 2 2 e 2π(ξ1−ξ2) ≈ 1,(26) in which case ξ 2 = ξ 1 + ln(H 1 /H 2 )/π. From this we read that the coupling must increase by an additive amount proportional to the logarithm of the ratio of Hubble scales. In our analysis below we demonstrate that this scaling relation is sufficiently accurate for our estimates. Refs. [40,41,46] demonstrate that in chaotic inflation models with m φ ≈ 10 −6 , the threshold value of α/f for which preheating is complete (i.e., max ρ gauge /ρ ∼ 80%) is 9 M −1 pl . With this as a baseline, we apply Eq. (24) to estimate how this threshold scales as we change the energy scale of inflation (and, later, the inflaton model itself). Consider the simplest case of a quadratic potential, an approximation useful for all of our models during the preheating phase, V (φ) = 1 2 m 2 φ φ 2 .(27) In this case, the Hubble parameter scales as H ∼ m φ , since at the end of inflation H 2 = ρ 3M 2 pl ≈ V (φ) 3M 2 pl ∼ m 2 φ .(28) In terms of the coupling, Eq. (26) tells us that, for some other mass m φ,2 , we require α 2 f = α 1 f + 1 π\|∂φ/∂N \| ln m φ,1 m φ,2(29) for comparable preheating efficiency. Since we are interested in the efficiency of preheating (rather than the inflationary production of gauge bosons), in this expression we evaluate ∂φ/∂N at the end of inflation, N = 0. At the end of chaotic inflation, ∂φ/∂N ≈ 1.4 M pl , in which case Eq. (29) reduces to α 2 f = α 1 f + 1.1 log 10 m φ,1 m φ,2 .(30) Thus, for comparable preheating efficiency we expect to need to increase the axion-gauge coupling α/f by ∼ M −1 pl for each order of magnitude we reduce m φ . B. Gravitational wave production and the "rule of thumb" While regimes of highly efficient preheating exist for all models (as we show in Section V B), the structure of resonance varies from model to model. To understand the relationship between the scales at which gravitational waves are produced and their resulting amplitude, we use the "rule of thumb" developed by Ref. [123] to estimate the stochastic gravitational wave production from cosmological processes. By approximating the source as a Gaussian of width σ peaked at wavenumber k * , Ref. [123] estimates the peak amplitude as Ω gw,0 ≈ 2.3 × 10 −4 α 2 βw 2 k * σ H * k * 2 ,(31) where α is the fraction of the energy in the gravitational wave source relative to the Universe's total energy density at that time, β encodes the anisotropy of the source, and w is the equation of state of the Universe at that time. Observe that Ω gw,0 h 2 decreases quadratically with the ratio of the peak wavenumber k * to the Hubble parameter at that time, H * . The oscillation frequency of the inflaton background (i.e., its effective mass) sets the scales of efficient resonance, while ξ ∼ α/f controls how far inside the horizon tachyonic resonance occurs. The former depends on the shape of the potential, while the latter can depend on the inflationary scale via the arguments of Section IV A: lower inflationary scales require larger couplings for complete preheating. To quantify the effect of the potential shape, note that the effective inflaton mass, defined by m 2 φ ≡ ∂ 2 V /∂φ 2 evaluated at the minimum of the potential, sets the scales of interest during preheating. As such, k * /H * ∼ m φ /H * ∼ M pl V /V -i.e. , the shape of the potential (about its minimum) determines the scales at which preheating occurs, which in turn affects the size of the gravitational wave signal according to Eq. (31). V. NUMERICAL RESULTS A. Dependence on inflationary scale In this section, we present simulations detailing the effect of the energy scale of inflation on the efficiency of preheating and the subsequent generation of gravitational waves. For this analysis we fix a chaotic inflationary potential while tuning m φ to study different energy scales of inflation (i.e., for the time being, we ignore the fact that m φ should be chosen to fit the normalization of the scalar power spectrum during inflation). This simplification allows us to separate the effect of the scale of the potential from that of its shape, which we consider in Section V B. Further, preheating studies are often restricted to chaotic inflationary models, a choice justified because preheating probes the inflaton's oscillation about the minimum of its potential, which is quadratic to leading order. We first seek to determine how the axion-gauge coupling α/f must be tuned in order to achieve complete preheating as we lower the inflationary scale and to evaluate the accuracy of the analytic estimates made in Section IV A. In Fig. 1 we depict the relationship between the efficiency of preheating and the net gravitational wave production across five decades of m φ . Because larger couplings result in power transfer to modes with larger momentum, for values of m φ < 10 −6 M pl we use a box-length L = 7.5 m −1 φ in order to ensure sufficient short-wavelength resolution in all cases. We immediately observe that Eq. (30) does indeed accurately predict the range of couplings for which preheating becomes efficient and even complete (ρ gauge /ρ 80%). It is reassuring that this scaling argument-derived from linearized approximationsis applicable and that it breaks down in the large-coupling limit when the approximations are least valid. To visualize this scaling, in the right panel of Fig. 1 we plot the preheating efficiency and net gravitational wave production as functions of an "adjusted" coupling, i.e., the value of α/f that would be required to obtain the same efficiency were m φ = 6.16 × 10 −6 M pl , as predicted by the analytic estimate Eq. (29). The efficiency of gauge field production becomes more complicated at larger couplings where nonlinear processes become important. After a regime of α/f in which preheating remains comparably efficient (which, for m φ = 6.16 × 10 −6 M pl , corresponds to 9 M −1 pl α/f 10.4 M −1 pl ), ρ gauge /ρ begins to decrease. At even stronger coupling, energy transfer to the gauge fields continues, gradually increasing ρ gauge /ρ to 100%. Regardless, the inflaton condensate is totally depleted, as shown by the dotted lines in Fig. 1 which indicate the (maximum) fraction of energy in either gauge fields or fluctuations of the axion. To examine the physics that realizes this trend, consider comparing the energy in inflaton fluctuations, ρ δφ ≡ ρ φ − ρ φ(32)= 1 2a 2 φ 2 + 1 2a 2 ∂ i φ∂ i φ + V (φ) − 1 2a 2 φ 2 + V ( φ ) ,(33) to the energy in the gauge fields, ρ gauge . In Fig. 2 we plot the ratio of these two quantities, ρ gauge /ρ δφ , as well as ρ δφ /ρ φ which measures the degree to which the axion has fragmented. In the top row, we observe that as preheating approaches ∼ 80% efficiency (at α/f ∼ 9 M −1 pl ), the inflaton also becomes more fragmented. Since the axion's equation of motion is linear for a quadratic potential, any inflaton particle production must occur through backscattering from produced gauge bosons, an inherently nonlinear process. As the coupling increases to 9 M −1 pl α/f 10.2 M −1 pl , the inflaton becomes totally fragmented and the energy in gauge fields relative to inflaton fluctuations does not increase. Preheating thus remains comparably efficient in this coupling regime. Evident in the middle row is that, as α/f is increased past 10.2 M −1 pl and inflaton fragmentation occurs more rapidly, the energy in fluctuations of the axion approach roughly half that of the gauge fields. The exponential amplification of the gauge fields likewise ends earlier and earlier, as the tachyonic resonance is driven by the motion of the (now-depleted) inflaton condensate. Thus, preheating becomes slightly less efficient, as apparent in Fig. 1. Increasing α/f past 11.6 M −1 pl , we see that while the initial phase of resonance continues to end earlier and earlier, the inflaton also becomes less fragmented during this phase. The backreaction of the gauge fields onto the axion's background dynamics now exerts a dramatic amount of friction before the axion even first crosses through the minimum of its potential. Tachyonic amplification of the gauge fields recommences and proceeds at a slower and slower rate as the coupling increases (as this increases the gauge-field friction). This process terminates later and later at values of ρ gauge /ρ increasingly close to 100%, as shown in Fig. 1, at which point the inflaton condensate is totally depleted. Indeed, preheating is 100% efficient in these cases because the gauge-field friction is strong enough to ensure that tachyonic amplification continues until preheating is complete (in contrast to the axion's oscillations allowing for backscattering effects to become important). Note that these results (which reproduce those originally obtained in Ref. [40]) are very similar to those presented more recently in Ref. [45] for the case of chaotic inflation, and our analyses of these regimes are likewise similar. The inflaton halts higher and higher up its potential as the gauge-field friction becomes more important, and eventually accelerated expansion (w < −1/3) recommences (for α/f 13.2 M −1 pl ). As observed in Ref. [41], the inflaton is momentarily "trapped," as depicted in Fig. 3. Its vanishing velocity momentarily shuts off the tachyonic instability, so that the gauge fields redshift and the inflaton starts to roll again (restarting the resonant enhancement of the gauge fields). At the highest couplings we simulate here (α/f ∼ 14.8 M −1 pl ), the inflaton is trapped before the background stops accelerating (i.e., before w ≥ −1/3), resembling models of inflation which achieve slow-roll via the same axion-gauge-field coupling [119]. Regardless of the value of m φ , each set of simulations exhibits the same qualitative features as the coupling increases. However, as m φ is decreased, the range of couplings α/f spanned by these features broadens in a manner that cannot be described by the analytic estimates above. The backscattering effects we posit as responsible for the dip in efficiency depend on the amplitude of axion fluctuations, which scales linearly with m φ . Thus, in order for backscattering to be important to a comparable degree, the coupling has to be increased by a greater relative amount to compensate for the smaller amplitude of vacuum fluctuations at lower values of m φ . From the analytical estimates presented at the beginning of this section, we expect the backreaction of the gauge fields onto the axion condensate to scale with the energy scale of the inflationary potential. Thus, models with lower m φ require a larger relative increase in α/f to enter the regime of slow tachyonic resonance due to the gauge fields' friction on the axion. The approximate universality we observe also extends to the gravitational waves produced by the dynamics of preheating. Of note in Fig. 1 is that any scenario which exhibits (near-) complete preheating (namely, ρ gauge /ρ ∼ 80%) results in a net gravitational wave production that could be detected (or ruled out) by CMB-S4, as discussed in Section II C. For the strongest couplings we simulate, the total integrated Ω gw,0 h 2 exceeds 10 −6 , which is already ruled out by Planck data [103]. To quantify this claim, in Fig. 4 we scatter-plot the fractional energy density in gravitational waves today versus the efficiency of preheating. The cluster of data points around and above max ρ gauge /ρ ∼ 0.8 mostly lies above the CMB-S4 bound, indicating that next-generation experiments will place bounds on the axion-gauge-field coupling α/f , regardless of m φ . In the upper panel of Fig. 4 we also observe that gravitational wave production is moderately less effective at lower inflationary scales. Returning to the arguments of Section IV B, the larger couplings required for comparable preheating at lower m φ push the resonance further inside the horizon. As the momenta undergoing the tachyonic instability are k/aH < ξ ∼ α/f , from the rule of thumb, Eq. (31), we expect Ω gw to decrease with (k * /H) 2 ∼ (α/f ) 2 . To verify this scaling, in the lower panel of Fig. 4 we scale Ω gw,0 h 2 by the squared ratio of each simulation's coupling α/f to its "adjusted" value from Eq. (29). With this multiplicative factor, the trends of Ω gw,0 h 2 versus max ρ gauge /ρ line up closely, confirming our hypothesis. The shape of the gravitational wave spectra (which is fairly generic and qualitatively comparable to that produced by tachyonic resonances) is similar for all values of m φ (when comparing couplings which yield comparable preheating efficiency). Plotting the signals that would be observed today in Fig. 5 demonstrates this observation, and also depicts the √ m φ scaling of the characteristic present-day frequencies at which these signals would be observed. 6 In general, Ω gw,0 h 2 increases exponentially with α/f , which is to be expected as the tachyonic resonance is exponential in ξ ∝ α/f . The actual rate at which Ω gw,0 h 2 increases as a function of exp(α/f ) falls into two regimes: the range of couplings where the actual efficiency of preheating increases exponentially with α/f , and those for which preheating is always complete (the latter of which is slower than the former). In the first regime the gravitational wave source (parameterized by α in Eq. (31)) is growing exponentially. In the second, while the simulations all completely transition to radiation domination, gravitational waves are continually sourced by the second phase of slow tachyonic resonance at an efficiency which still increases with the coupling strength. Lastly, we note that, despite the complicated trend of ρ gauge /ρ as α/f increases, the inflaton condensate always ends up depleted as the coupling increases past the critical value where efficient preheating is first achieved (e.g., α/f ∼ 9.6 M −1 pl for m φ = 6.16×10 −6 M pl ). Thus, in these cases the end state of the simulations is always radiation domination. However, the proportion of that radiation comprised of axion fluctuations varies with coupling, as depicted in Fig. 2. Ignoring any decay channels, the axion's fluctuations redshift after preheating until their physical momenta drop below their mass, at which point they become non-relativistic. From this point on the axion energy density redshifts like matter. If the axion's lifetime is sufficiently long, its energy eventually dominates over the gauge fields (which, being radiation-like, decay faster than matter). Any deviation from an equation of state of radiation, w ≡ p/ρ = 1/3, suppresses the gravitational wave density observed today, Ω gw,0 h 2 , relative to what the transfer function Eq. (A16) accounts for (which assumes the Universe was radiation dominated from the time of emission until matter-radiation equality). In Appendix C we demonstrate that Bose enhancement resulting from the larger occupation numbers from preheating ensures that perturbative decays happen quickly enough that the Universe remains radiation dominated. As such, we expect little to no suppression of Ω gw,0 h 2 relative to the values we report. B. Dependence on the shape of the potential We now explore the dependence of our results-in particular the amplitude of the resulting gravitational wave spectrum-on the shape of the potential during the reheating phase. We simulate preheating in the inflationary models detailed in Section II A and discuss the extent to which the results of Section V A are modified. In Fig. 6, we plot the efficiency of preheating and corresponding gravitational wave production over a range of axion-gauge-field couplings α/f . We first observe that the relationship between max ρ gauge /ρ and α/f follows the same general trend as presented in Section V A. That is, once α/f is large enough for preheating to be efficient, max ρ gauge /ρ remains roughly 90% until backscattering effects become important, at which point we observe a dip in efficiency. Finally, even larger couplings lead again to a regime of strong backreaction leading to slow tachyonic resonance, resulting in near-completely efficient preheating. Turning to the lower panel of Fig. 6, we see that i) depending on the coupling strength, preheating in all models can yield a net production of gravitational waves that would be probed by CMB-S4 measurements of N eff , and ii) models with r 10 −2 are already constrained by Planck data. While all models we study here reach a regime of gravitational wave production detectable by future experiments such as CMB-S4, for those with r 10 −2 the entire regime of efficient preheating could be ruled out by a null detection of ∆N eff . In Fig. 7 we display the energy density in gravitational waves today, Ω gw,0 h 2 , as a function of the efficiency of gauge preheating, which, as above, we quantify by the maximum fraction of energy in gauge field fluctuations during the simulation. These results exhibit the sensi- [103] are plotted in solid and dashed black, respectively, while the region between CMB-S4's 1σ and 2σ projections [102] is shaded grey. Note that, to save clutter, we only plot a subset of the hilltop parameter points listed in Table I. tivity of Ω gw,0 h 2 to the details of the potential. That is, while the general relationship between gravitational wave production and preheating efficiency follows a similar trend, the overall scaling of Ω gw,0 h 2 differs from model to model as observed in Section V A. Again, in general the larger coupling α/f required for complete preheating for a particular model, the less efficient the subsequent gravitational wave production. Referring to Table II, low-scale models require larger coupling for efficient preheating and present correspondingly weaker detection prospects. This observation, together with Figs. 6 and 7, leads us to make the broad claim that for models of inflation with tensor-to-scalar ratios observable by CMB-S4 experiments, preheating into gauge fields could be simultaneously probed via the contribution of gravitational waves to ∆N eff . A positive detection of r 10 −3 together with ∆N eff by CMB-S4 experiments could provide evidence for a pseudoscalar inflaton (axion) reheating the Universe through preheating to gauge fields. Alternatively, detection of r with a measurement of ∆N eff consistent with zero would provide stringent bounds on the axion-gauge coupling α/f -in particular the regime in which preheating is the sole mechanism by which the Universe was reheated would be ruled out. In the former case, a precise prediction of the end of inflation (relative to the pivot scale used to parameterize CMB observables) may require the methods of Appendix B to accurately model the backreaction of gauge fields onto the inflationary background toward the end of inflation. To evaluate the claim that the efficiency of gravitational wave production is correlated with r, we vary the free parameter v of the hilltop model, which allows us to tune the flatness of the potential during inflation, and so both the energy scale of inflation and r. In Fig. 8 we observe the exact trend we noted in our survey of different potentials: while the relationship between max ρ gauge /ρ and Ω gw,0 h 2 is clearly universal, as v increases, gravitational wave production is more efficient (as is preheating, which requires increasingly lower α/f to be efficient). Consulting Table II, increasing v decreases k * /H * , which (referring to Table I) correlates to larger tensor-to-scalar ratios, r. From Fig. 8 we extrapolate that, regardless of the choice of v, at sufficiently large coupling the hilltop models will exhibit complete preheating (entering the regime of slow tachyonic resonance sustained by gauge-field backreaction). However, the resulting gravitational wave production will be suppressed, reducing the constraining power of ∆N eff in this scenario. We note, however, that existing constraints on the instability parameter ξ during inflation (the strongest of which are due to primordial black hole production [53,[58][59][60][61][62]) still provide significantly weaker constraints on α/f for small-r (and so small-H ) models, as ∂φ/∂N ∼ √ H . At the same time, for larger values of v there is a regime of detectable gravitational waves from preheating (while still r 10 −3 , below the target for CMB-S4). As such, a detection of nonzero ∆N eff but not nonzero r would be consistent with axion inflation in models with r 10 −3 . Finally, we note that we have repeated this analysis for the D-brane model, finding the same relationship as for the hilltop model. VI. CONCLUSIONS In this paper, we have studied the effect of the inflationary potential on the dynamics of gauge preheating and gravitational wave production after axion inflation. Abelian gauge fields coupled via φFF are amplified via a tachyonic resonance from the post-inflationary axion condensate, which rapidly and efficiently reheats the Universe and copiously produces a stochastic background of gravitational waves. For strong enough coupling, these signals, first computed in Ref. [46], contribute so greatly to the radiation content of the Universe that Planck data already places bounds on the axion-gauge coupling α/f , which CMB-S4 will improve upon significantly. Here we have extended this study to consider a variety of inflationary models, finding that such a level of gravitational wave production typically occurs in scenarios where preheating completely reheats the Universe. While the results are qualitatively similar across models, the efficiency of preheating (at a given coupling α/f ) varies depending on both the energy scale of inflation and the shape of the potential. The scale of the potential (as well as the coupling parameter α/f ) controls the significance of backreaction of the gauge fields onto the axion's background dynamics. At strong enough couplings, this gauge-field friction supports an extended phase of slow tachyonic resonance which persists until 100% of the axion's energy has transferred to the gauge fields. As such, lower-scale inflation models require larger coupling to enter into this regime, which suppresses the resulting gravitational wave amplitude by moving the source further inside the horizon. The tensor-to-scalar ratio is sensitive to the flatness of the potential at CMB, measured by H , and (once the amplitude of scalar fluctuations is set to match CMB data) to the energy scale of inflation. As a result, gravitational wave production is comparatively less efficient in models with smaller tensor-to-scalar ratios. However, when preheating completely transitions the Universe to radiation domination, generally the level of gravitational wave production contributes to the effective number of radiation-like degrees of freedom N eff at a level that could be detected (or ruled out) by next-generation CMB experiments. Should CMB-S4 (which targets r 10 −3 ) detect nonzero r, a simultaneous detection of nonzero ∆N eff could be an indication that preheating occurred via strong coupling to gauge fields. Conversely, the lack of observed gravitational waves via ∆N eff would place severe constraints on α/f , in particular ruling out most of the parameter space in which preheating in this model is solely responsible for the transition to radiation domination. Notably, for these models CMB-S4 could rule out the entire regime of couplings for which preheating alone reheated the Universe. Planck data currently sets α/f 15 M −1 pl and 18.9 M −1 pl for the chaotic and monodromy potentials, respectively, which CMB-S4 will improve to α/f 9 M −1 pl and 13 M −1 pl . These results improve upon bounds from primordial black hole (over-) production, which constrain α/f 21.9M −1 pl −24.9 M −1 pl and α/f 35.9 M −1 pl , respectively [53,58]. Our results place similarly tight constraints for Starobinsky inflation (evident in Fig. 6) and natural inflation (similar to those for chaotic inflation). The significant gravitational response (of the tensor part of the metric) to the preheating dynamics studied here motivates a similar investigation into the metric's scalar degrees of freedom (i.e., curvature perturbations) and the associated potential for primordial black hole formation. Nonlinear gravitational effects could potentially alter our findings here, which may be studied in a framework similar to that employed by Ref. [124]. Finally, we have ignored the effects of the backreaction of any charged matter on the gauge preheating process. Charged particles are produced via the Schwinger effect, leading to a non-zero conductivity in the resulting plasma which could possibly damp gauge field production [71]. We leave a detailed investigation of these questions for future work. [125] through allocation TG-PHY180049, which is supported by National Science Foundation grant number ACI-1548562, and also made use of hardware purchased by the National Science Foundation, Kenyon College, and the Kenyon College Department of Physics. This work made use of the Illinois Campus Cluster, a computing resource that is operated by the Illinois Campus Cluster Program (ICCP) in conjunction with the National Center for Supercomputing Applications (NCSA) and which is supported by funds from the University of Illinois at Urbana-Champaign. P.A. acknowledges the hospitality of the Yukawa Institute for Theoretical Physics at Kyoto University, where some of this work was completed during the YITP-T-19-02 on "Resonant instabilities in cosmology". Appendix A: Equations of motion and linear analysis In this appendix, we write down the equations of motion for the system and collect some well-known results about the linear evolution of the system during inflation and the backreaction of the gauge field on the inflaton. Gravitational sector The evolution of the background FLRW spacetime (Eq. (3)) is governed by the Einstein equations H(τ ) 2 = a(τ ) 2 3M 2 pl ρ(τ ) (A1) H (τ ) = − a(τ ) 2 2M 2 pl (3ρ(τ ) + p(τ )) ,(A2) where τ is the conformal time and H = a /a is the conformal Hubble parameter related to the usual Hubble parameter via H = aH. For the action in Eq. (1), the spatially-averaged energy density and pressure are ρ(τ ) ≡ φ 2 2a 2 + (∂ i φ) 2 2a 2 + V (φ) + 1 2 E 2 + B 2 , (A3) p(τ ) ≡ φ 2 2a 2 − (∂ i φ) 2 6a 2 − V (φ) + 1 6 E 2 + B 2 ,(A4) where brackets · · · denote a spatial average and we have defined the electric and magnetic fields E i = 1 a 2 A i − ∂ i A 0 , B i = 1 a 2 ijk ∂ j A k . (A5) Finally, our Fourier convention is set by f (k) = d 3 x f (x)e ik·x ,(A6)f (x) = d 3 k (2π) 3 f (k)e −ik·x .(A7) Gravitational wave dynamics To study the production of gravitational waves, we consider an FLRW metric including tensor (but not scalar nor vector) perturbations, ds 2 = a(τ ) 2 −dτ 2 + (δ ij + h ij ) dx i dx j ,(A8) where ∂ i h ij = h ii = 0 is the transverse-traceless perturbation of the spatial metric. In this work we compute the dynamics of h ij passively-that is, we determine the amount of gravitational waves sourced by the axion and the gauge fields but we neglect their backreaction onto the system. From the linearized Einstein equations we obtain the equation of motion for h ij , h ij − ∂ k ∂ k h ij + 2Hh ij − 2 2H + H 2 h ij = 2 M 2 pl T TT ij .(A9) In this expression, T TT ij is the transverse-traceless component of the stress-energy tensor, T TT ij = P il P jm − 1 2 P ij P lm T lm ,(A10) where the transverse-traceless projector is P ij = δ ij − k i k j k 2 .(A11) The background Einstein equations show that 2 2H + H 2 is proportional to the background pressure p(τ ). Since usually all the modes in our simulations are subhorizon, this term in Eq. (A9) induces a negligible amount of dispersion and so we neglect it. Under the approximations discussed after Eq. (A8), the stress-energy tensor is expressed as T ij = ∂ i φ∂ j φ + F iα F jβḡ αβ −ḡ ij 1 2 ∂ µ φ∂ µ φ + V (φ) + 1 4 F µν F µν ,(A12) whereḡ µν denotes the unperturbed FLRW metric. Moreover, since the terms proportional toḡ ij are pure trace, they do not contribute to Eq. (A10) (they are projected out by P ij ). As a consequence, only the first two terms in Eq. (A12) are responsible for gravitational wave generation. The stress tensor of gravitational waves is [126] T gw µν = M 2 pl 4 h ij,µ h ij,ν ,(A13) with the sum over i and j implied. The corresponding fractional energy density in gravitational waves as Ω gw (k) ≡ 1 ρ dρ gw d ln k (A14) = 1 24π 2 L 3 k 3 H 2 i,j h ij (k, τ )) 2 .(A15) The spectrum redshifted to today is related to the spectrum at emission by the transfer function [14] Ω gw,0 (f )h 2 = Ω gw,e (f ) g 0 g * 1/3 Ω r,0 h 2 (A16) at frequencies f ≈ 2.7 × 10 10 k phys M pl H Hz,(A17) where k phys is the physical wave number and H is the Hubble parameter evaluated at the time when the spectrum is being computed. Above, g 0 /g * is the ratio of thermal degrees of freedom today to matter-radiation equality. Here we assume that the Universe was radiation dominated from the moment of emission until matter-radiation equality. Field equations The Euler-Lagrange equations for the action Eq. (1) yield the dynamics of the gauge fields, ∂ µ √ −gF µν + √ −g α fF µν = 0.(A18) In terms of the gauge potentials, the ν = i equations are the dynamical equations of motion 0 = A i − ∂ i A 0 − ∂ j ∂ j A i + ∂ i ∂ j A j − α f ikl φ ∂ k A l + α f ikl ∂ k φ (A l − ∂ l A 0 ) ,(A19) while the ν = 0 equation is the Gauss constraint ∂ i A i − ∂ j ∂ j A 0 = − α f ijk ∂ k φ∂ i A j .(A20) The four components of the gauge field are not physical: the theory is invariant under gauge transformations where A µ →Ã µ = A µ − ∂ µ β,(A21) where β is an arbitrary function. This freedom allows us to eliminate one degree of freedom by fixing a gauge. Together with Gauss's law, the gauge sector represents only two physical, dynamical degrees of freedom. Likewise, the Euler-Lagrange equation for the inflaton provides its equation of motion, φ − ∂ i ∂ i φ + 2Hφ + a 2 dV dφ = −a 2 α 4f F µνF µν . (A22) Note that F µνF µν /4 = a 4 E · B. Linear theory The system described by Eqs. (A18) and (A22) is difficult to solve analytically due to the nonlinear interactions between the axion and gauge field. However, provided the coupling α/f is not too large, a linear treatment provides some insight into the dynamics of this system during inflation and the early phase of preheating. We begin by linearizing the system of equations, expanding the axion about its homogeneous background φ =φ(τ ) + δφ and treating the gauge field as a first-order perturbation. We choose the temporal gauge, A 0 = 0. Note that, at linear order this gauge is equivalent to Coulomb gauge ∂ i A i = 0 via the gauge constraint; the gauge field may be taken to be purely transverse. Expanding into Fourier modes A i (τ, x) = λ=± d 3 k (2π) 3 A λ k (τ )ε λ (k)e −ik·x (A23) where the polarization vectors ε ± (k) form an orthogonal basis of polarizations transverse to the momentum k ijk k j ε ± k (k) = ∓ikε ± i (k),(A24) and satisfy the relations ε λ i (k)ε λ i (k) * = δ λλ ,(A25)k i ε ± i (k) = 0, (A26) ε ± i (−k) = ε ± i (k) * ,(A27)ε ± i (k) * = ε ∓ i (k).(A28) At linear order, the equation of motion, Eq. (A19), reduces to A ± (k) + k (k ∓ 2Hξ) A ± (k) = 0,(A29) where we have defined the instability parameter ξ ≡ α 2f φ H .(A30) Equation (A29) shows that the interaction between the inflaton and the gauge fields induces a tachyonic instability (for modes with k/H < 2\|ξ\|) for one of the two helicity states (A + if φ > 0, or conversely A − if φ < 0). The modes within this band experience (for values of ξ 2) exponential enhancement slightly before horizon crossing and then approach a constant value on superhorizon scales. In the limit of a nearly constant ξ, the solutions to this equation are Whittaker functions (see, e.g., [40]). Assuming φ > 0, near horizon crossing (k/aH ∼ 1) these solutions are well-approximated by A + 1 √ 2k k 2Hξ 1/4 e πξ−2 √ 2ξk/H .(A31) These results evince the importance of the instability parameter ξ which controls the exponential enhancement of the gauge field. Computing the backreaction of the gauge field onto the background dynamics requires the integrals 1 2 E 2 = 1 4π 2 a 4 λ=± dk k 2 \|A λ (k)\| 2 (A32) 1 2 B 2 = 1 4π 2 a 4 λ=± dk k 4 \|A λ (k)\| 2 ,(A33)E · B = − 1 4π 2 a 4 λ=± λ dk k 3 ∂ τ \|A λ (k)\| 2 , (A34) where A λ (k) denotes the helicity modes of the gauge potentials. (Note that Eqs. (A32) to (A34) assume the temporal gauge.) For ξ 4 [57], fairly accurate approximations of these quantities are given by [119] E · B 2.4 · 10 −4 H 4 a 4 ξ 4 e 2πξ , 1 2 E 2 + B 2 1.4 · 10 −4 H 4 a 4 ξ 3 e 2πξ .(A35) Substituting Eq. (A35) into Eq. (A22) demonstrates that gauge fields exert an additional friction term for the inflaton (see Eq. (22)). Since ξ is proportional to the inflaton background's velocity, it is expected to grow during slowroll inflation. As such, backreaction may significantly alter the evolution of the inflaton towards the end of inflation, even before preheating. In Appendix B we present our procedure to accurately account for these effects during inflation (as used to set initial conditions for the lattice simulations). Appendix B: Initial conditions As implemented in Refs. [40,41,46], in order to capture the tachyonic enhancement of (one polarization of) the gauge fields during inflation, we numerically integrate the linearized equations of motion for the fluctuations of the gauge-field helicity modes, Eq. (A29), during inflation. These fluctuations are integrated alongside the background dynamics, i.e., Eq. (A2) and the homogeneous part of the inflaton's equation of motion, Eq. (A22). To set initial conditions for the subsequent lattice simulation, we evaluate the background quantities and obtain power spectra from the integrated gauge field modes between one and two e-folds before H = 1, marking the end of inflation, as described in Section III. By beginning the lattice simulation sufficiently early before the end of inflation we capture nonlinear effects that become important, and thus only use the solutions to the linearized equations while they remain valid. To account for the gauge fields' backreaction onto the background dynamics during inflation (as described in Appendix A 3), Refs. [40,41,46] use the approximations Eqs. (A35) and (A36). The accuracy of these expressions is sensitive to two main assumptions. First, the solution of Eq. (A29) is well-approximated by the Whittaker function only in the limit that ξ is constant, and H = −1/τ (in near-de Sitter space). Second, Eqs. (A35) and (A36) themselves are approximations to the integrals Eqs. (A32) to (A34) over Eq. (A31), accurate for ξ 4 [57]. When ξ is nearly constant but small, the approximate integrals Eqs. (A35) and (A36) are inaccurate, but the Whittaker solution is still valid. However, if ξ is not approximately constant then the Whittaker solution itself is inaccurate, meaning the integrands of Eqs. (A32) to (A34) themselves are inaccurate, regardless of the size of ξ. At low couplings (which roughly corresponds to those which do not achieve complete preheating), the gauge fields, while amplified, have a negligible effect on the background evolution during inflation. Thus, while ξ is not large enough for Eqs. (A35) and (A36) to be valid, their effect is small enough that this error is unimportant. At larger couplings for which preheating is complete, the gauge fields have a more substantial impact on the background evolution during the final e-folds of inflation, but these effects can in principle be captured by linearized calculations (without further approximations). However, in this regime analytic approximations are in fact never simultaneously valid: during slow-roll when the Whittaker solution is approximately valid, \|ξ\| < 4, meaning the approximation to the analytic integral is not trustworthy [57]. In fact, by the time ξ is large enough to validate the approximation to the integral, the inflaton is no longer slowly rolling, so the Whittaker solution is no longer valid. Further, the backreaction of the gauge fields onto the background dynamics is no longer negligible (during the final e-folds of inflation), meaning the background evolution using these approximations may be inaccurate. At even stronger couplings, nonlinear dynamics have an important impact on the background evolution during the final e-folds of inflation. The friction exerted by the gauge fields on the inflaton background (beyond what is captured by a linear treatment) can even postpone the end of inflation. In this regime we might not trust any calculation using only the linear equations of motion toward the end of inflation, so we leave a detailed study of such strong coupling to future work. Here we improve upon this procedure by computing the gauge-field integrals Eqs. (A32) to (A34) via numerical quadrature of the numerically-evolved gauge fields. As chosen for the analytic results Eqs. (A35) and (A36), we take as a UV cutoff for these integrals k = 2aHξ, the upper end of the tachyonic instability band; we find that, in the regime where the gauge fields' contribution to the background evolution is non-negligible, the integration is insensitive to slight variation about this cutoff. With this method, the evolution of the background in tandem with the linearized fluctuations agrees very well with the background dynamics of the subsequent lattice calculation (for the overlapping final one to two e-folds of inflation.) In practice, it is prohibitively expensive to include the full dynamic range of fluctuations which are amplifiedthe band of tachyonic instability k 2aHξ is proportional to the comoving horizon H = aH, which increases exponentially during inflation. To address this, we only include modes which experience amplification during the last ∼ 20 e-folds of inflation, well before backreaction has any impact on the background dynamics of inflation (for the values of α/f we consider). We sample this range of modes (spanning 8-10 orders of magnitude) logarithmically, yielding good results from quadrature of Eqs. (A32) to (A34) over ln k. A final complication is that fluctuations deep inside the horizon oscillate extremely rapidly compared to the Hubble rate. However, this regime (k 2aHξ) is precisely the regime where the WKB approximation A ± (k, τ ) ≈ 1 2ω ± (k, τ ) e iω±(k,τ )τ (B1) with ω ± (k, τ ) 2 = k 2 ∓2aHξk, is extremely accurate. Thus, we only numerically integrate a given mode k when k ≤ 10 × 2aHξ, using the WKB solution until this point. Using an adaptive ODE integrator (in practice, SciPy's RK45 routine [127,128]) ensures that we take as large of timesteps as possible while maintaining a prescribed relative accuracy (typically to one part in 10 5 ). A similar strategy was employed in Ref. [129] to study this model's dynamics during both inflation and preheating; however, the validity of such a method applied to study preheating is unclear. In Fig. 9 we investigate the difference between the analytic approximations (Eqs. (A35) and (A36)) and the numerical computations described above, fixing a chaotic inflaton potential. We consider couplings where the background evolutions agree well (α/f = 8 M −1 pl ) and disagree (α/f = 14 M −1 pl ). We additionally compare to the numerical quadrature of Eqs. (A32) to (A34) using the actual Whittaker-function solutions, which demonstrates agreement with the approximations Eqs. (A35) and (A36) once \|ξ\| 3 − 4. Earlier in inflation, when exact slow-roll (and so the Whittaker-function solution) is a good approximation, we observe agreement between the exact integrals of the Whittaker function solutions and the integrals over numerically-evolved gauge field perturbations. However, in neither case do the analytic approximations agree with the results of numerical evolution and quadrature at any time. While these errors have no significant effect on the inflationary background deep in inflation, Figure 9 indicates that the analytic approximations greatly overestimate E 2 + B 2 and E · B during the final two e-folds of inflation, regardless of the coupling to the axion. In the strong-coupling case (where gauge-field backreaction is significant), the analytic result's overestimation of the energy in the gauge fields leads to inflation ending artificially early, as indicated by the vertical lines (marking when H = 1) in Fig. 9. In this case, with the analyticbased evolution we would end up initializing the lattice simulation with background values roughly half an e-fold earlier than intended (depending on the coupling). Note that α/f = 8 M −1 pl is roughly the largest coupling for which the background evolutions visibly agree. Appendix C: Post-preheating dynamics In this appendix we study the dynamics after preheating to evaluate whether the Universe remains radiationdominated. In particular, we seek to quantify the amount by which the fractional energy density in gravitational waves is suppressed due to any deviation from radiation domination between preheating and the end of reheating. As discussed in Section V A, at the end of all of our simulations a small of the total energy remains in axion fluctuations (O(1%) for the cases where preheating is most efficient). While these are typically relativistic at this point, because the axion is rather massive these fluctuations become non-relativistic within just a few efolds and gravitate like pressureless matter. Depending on their lifetime, these massive axions can come to dominate the energy density of the Universe, leading to a period of matter domination. Even if the Universe does not become matter-dominated, the Universe could still depart significantly from radiation domination. The resulting effect on the gravitational wave transfer function depends on i) how much energy remains in the axion, Ω φ , and ii) its decay rate into radiation, Γ. Noting that ρ GW ∝ a −4 for the modes generated inside the horizon during preheating, the fractional density in gravitational waves scales as Ω gw ≡ ρ gw ρ ∝ 1 (a/a 0 ) 4 (H/H 0 ) 2 ∝ a a 0 3w−1 ,(C1) where a subscript 0 denotes some reference time (which for our purposes is the end of preheating) and w ≡ p/ρ is the time-dependent equation of state. Thus, if the Universe remains radiation-dominated (w = 1/3), then Ω gw remains constant (as it redshifts at the same rate as the rest of the Universe). On the other hand, if the Universe is dominated by matter (w = 0) or by a mixture of matter and radiation (0 < w < 1/3), then Ω gw decreases with the scale factor. Thus, any suppression of gravitational waves after preheating depends on both the (evolution of the) equation of state w and the duration for which w < 1/3. In the standard reheating scenario (without a preheating phase), the equation of state of the Universe is that of matter due to the inflaton condensate's oscillation about the minimum of its potential. 7 Matter domination persists until the decay of the inflaton into relativistic species (in our case, bosons) becomes efficient, which occurs when the inflaton's decay rate Γ becomes comparable to present the Hubble scale H. Accounting for preheating affects this description in two significant ways: the equation of state is not (initially) w = 0 and the coupled sector is already (highly) occupied. The former would delay the possible onset of matter domination before reheating completes, while the latter significantly alters the inflaton decay rate due to Bose enhancement. Accounting for the 7 More precisely, the equation of state of a coherently oscillating scalar field time-averages to zero. non-trivial phase space distribution of the gauge fields f A (p) in our models enhances the decay rate by a factor ∼ 1 + 2f A (m φ /2) [130]. For simplicity, we study the dynamics after preheating with the standard Boltzmann equations describing threebody decay of inflaton particles at rest into two relativistic daughter particles [131], dρ φ dt + 3Hρ φ = −Γρ φ (C2) dρ γ dt + 4Hρ γ = Γρ φ .(C3) Rather than fixing the decay rate Γ and the initial (i.e., post-preheating) fractional energy in the axion Ω φ according to our model (and the final state of our simulations), we instead study a wide range of parameter space and use order-of-magnitude estimates to determine whether this period significantly affects our results. The amount by which Ωgw would decrease between preheating and the time the Universe fully reheats, as a function of the fraction of the Universe's energy remaining in the axion after preheating, Ω φ , and the effective decay rate Γ of the axion into a relativistic species (relative to the Hubble rate H at the end of preheating). In Fig. 10 we numerically integrate Eqs. (C2) and (C3) over a number of decades in Γ and Ω φ , plotting the amount by which Ω gw would reduce due to w being less than 1/3 during this epoch (using Eq. (C1)). In this figure, the vertical slice at Ω φ = 1 corresponds to reheating without any preheating phase. In our model, the zero-temperature decay rate of the axion into gauge fields is [132] Γ 0 H = 1 H α 2 m 3 φ 64πf 2 ∼ 10 −9 .(C4) To estimate f A (m φ /2) we extract the occupation number of the gauge fields from the simulations, which varies between ∼ 10 8 and 10 12 , in line with results from tachyonic resonance in scalar-field preheating [133]. Thus, the relevant portion of Fig. 10 is 10 −2 Ω φ 10 −1 and Γ 10 −1 , for which the fraction energy density in gravitational waves would drop by no more than ∼ 10%, and likely by less than 1%. Naturally, the regime for which Ω gw,0 h 2 exceeds the projected CMB-S4 bounds on ∆N eff corresponds to both the smallest Ω φ (approaching 1%) and the largest occupation numbers. Therefore, our results are most robust for the coupling regimes which would be probed or excluded by CMB-S4. In principle, the true dynamics depend on the timedependence of the entire phase-space distribution f (p), as well as other possible thermal effects due to sectors coupled to the daughter particles. We have further approximated that the axion particles are at rest (while at the end of the simulations all would have non-zero momentum). However, Fig. 10 makes clear that we are far from the regime where Ω gw is suppressed by any appreciable amount: the effective decay rate could drop by ∼ 2 − 8 orders of magnitude before Ω gw would redshift to half of its original value. As such, we conclude that the Universe would remain radiation dominated after preheating, and so our reported Ω gw,0 h 2 are robust. . The specific parameters chosen for each inflationary model under consideration: the effective inflaton mass, the simulation box-length, the simulation start-time in terms of the number of e-folds relative to the end of inflation, the Hubble parameter at the end of inflation He, the ratio of the lattice's infrared cutoff to the comoving Hubble scale at the end of inflation, equal to (2π/L)/He, and the energy scale at the end of inflation. = 6.16 × 10 −6 M pl m φ = 6.16 × 10 −7 M pl m φ = 6.16 × 10 −8 M pl m φ = 6.16 × 10 −9 M pl m φ = 6.16 × 10 −10 M pl Planck COrE + Euclid CMB-S4 1 − 2σ FIG. 1.Left panels: preheating efficiency, quantified by the maximum ρgauge/ρ over the simulation (top panels), and the total fractional energy in gravitational waves today, Ωgw,0h 2 (bottom panel), as functions of axion-gauge coupling α/f . Each curve fixes a quadratic potential and corresponds to various values of m φ , decreasing by factors of 10 from m φ = 6.16 × 10 −6 M pl (red) to m φ = 6.16 × 10 −10 M pl (purple). The dashed lines (which follow the same color scheme) correspond to the maximum total energy in fluctuations, i.e., the maximum amount of energy in the simulation outside of the inflaton condensate. Lines indicating the current ∆N eff bounds on Ωgw,0h 2 from Planck and CMB-S4 from Ref.[103] are plotted in solid and dashed black, respectively, while the region between CMB-S4's 1σ and 2σ projections[102] is shaded grey. Right panels: same, but with horizontal axis adjusted according to Eq.(29). That is, the horizontal axis corresponds to the actual coupling α/f shifted to the coupling that would be required for the same preheating efficiency for m φ = 6.16 × 10 −6 M pl (the value which fits the amplitude of the scalar power spectrum), as predicted by the analytic estimate Eq.(29). FIG. 2 . 2Left: ratio of the energy in gauge fields, ρgauge, to the energy in inflaton fluctuations, ρ δφ (defined in Eq. (32)), as a function of e-folds after inflation, N = ln a. Right: fragmentation of the inflaton, measured by ρ δφ /ρ φ . Each row depicts a different coupling regimes, with couplings α/f labeled in the legend of the left column (in units of M −1 pl ). The displayed simulations fix a chaotic inflationary potential with m φ = 6.16 × 10 −6 M pl to fit the amplitude of the scalar power spectrum. FIG. 3 . 3The equation of state, w = P / ρ and the mean value of the axion field φ . Couplings range from α/f = 11.6 M −1 pl to 14.8 M −1 pl as labeled in the legend. FIG. 4 . 4The total gravitational wave energy today versus the final preheating efficiency ρgauge/ρ, plotted for a chaotic potential with varying m φ . The bottom panel depicts the same data, but scaled by the square of the ratio of α/f (for a given simulation) to the coupling required for equal preheating efficiency for m φ = 6.16 × 10 −6 M pl , as estimated by Eq.(29). = 6.16 × 10 −6 M pl , α/f = 9.6 M −1 pl m φ = 6.16 × 10 −7 M pl , α/f = 10.8 M −1 pl m φ = 6.16 × 10 −8 M pl , α/f = 12.0 M −1 pl m φ = 6.16 × 10 −9 M pl , α/f = 13.0 M −1 pl m φ = 6.16 × 10 −10 M pl , α/f = 14.4 M −1 pl FIG. 5. Gravitational wave spectrum observed today for various inflationary energy scales. Each curve corresponds to a different simulation of preheating in chaotic inflation, with coupling α/f and inflaton mass m φ labeled in the legend. In each case, the total amount of gravitational wave production Ωgw,0 ∼ 10 −7 . FIG. 6 . 6Preheating efficiency, quantified by the maximum ρgauge/ρ over the simulation (top panel), and the total fractional energy in gravitational waves today, Ωgw,0h 2 (bottom panel), as functions of axion-gauge coupling α/f . Each color denotes a different inflationary potential, as labeled in the legend. The dashed lines (which follow the same color scheme) correspond to the maximum total energy in fluctuations, i.e., the maximum amount of energy in the simulation outside of the inflaton condensate. Lines indicating the current ∆N eff bounds on Ωgw,0h 2 from Planck and CMB-S4 from Ref. FIG. 7 . 7The total gravitational wave energy today versus the final preheating efficiency ρgauge/ρ for different inflationary potentials as indicated by the legend. FIG. 8 . 8The total gravitational wave energy today versus the final preheating efficiency ρgauge/ρ. Each color denotes the hilltop model with p = 4 and values of v equal to M pl , 2M pl , 4M pl , and 8M pl corresponding to colors red through purple. programme under the Marie Sk lodowska-Curie grant agreement No 713366. Z.J.W. is supported in part by the United States Department of Energy Computational Science Graduate Fellowship, provided under Grant No. DE-FG02-97ER25308. The development of pystella made use of the Extreme Science and Engineering Discovery Environment (XSEDE) FIG. 9 . 9Comparison of the analytic approximations (solid blue lines) to E 2 + B 2 (top panels) and E · B (bottom panels) (i.e., Eqs. (A35) and (A36)) to those obtained via direct numerical quadrature of gauge field fluctuations evolved with their linearized equations of motion (dashed orange lines). Both background evolutions are initialized identically, with N = 0 corresponding to the time inflation ends (i.e., H = 1) in the evolution using the analytic approximations. The vertical lines indicate the point at which inflation ends according to both evolutions. We additionally plot the result of direct numerical quadrature of Eqs. (A32) to (A34) using the Whittaker-function solutions to evaluate the accuracy of Eqs. (A35) and (A36) as approximations to this exact integral. FIG. 10. The amount by which Ωgw would decrease between preheating and the time the Universe fully reheats, as a function of the fraction of the Universe's energy remaining in the axion after preheating, Ω φ , and the effective decay rate Γ of the axion into a relativistic species (relative to the Hubble rate H at the end of preheating). TABLE I . IThe combinations of p and v we consider for the hilltop model, the corresponding value of V0 from the Planck normalization, and the corresponding predictions for the scalar tilt and tensor-to-scalar ratio. For other models of this class see also Refs.[86][87][88].3 Hilltop (p = 4, v = 2M pl ) 1.60 × 10 −6 20 -2 0.15 2.1 6.5 × 10 −4 0.949 9.8 × 10 −6Model m φ /M pl Lm φ N0 He/m φ kIR/He 4 √ ρe/M pl ns r Chaotic (n = 2) 6.16 × 10 −6 15 -2 0.51 0.82 2.3 × 10 −3 0.966 .13 Starobinsky (v = 10M pl /3) 1.06 × 10 −5 20 -2 0.37 0.85 2.6 × 10 −3 0.969 .016 Monodromy (φc = M pl /10) 4.66 × 10 −5 50 -2 0.15 0.84 3.5 × 10 −3 0.975 .067 Hilltop (p = 3, v = M pl ) 1.12 × 10 −7 20 -1 0.11 2.7 1.5 × 10 −4 0.932 2.0 × 10 −8 Hilltop (p = 4, v = M pl ) 8.39 × 10 −7 20 -1 0.088 3.6 3.6 × 10 −4 0.949 6.4 × 10 −7 In the context of supergravity this can be achieved by, e.g., using a Kähler potential which is shift-symmetric under the imaginary part of a chiral superfield[80]. In order to generate a potential for the inflaton the shift symmetry must be broken, which can be achieved, for example, by coupling the inflaton to a 3-form[81][82][83]. 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[ "The discovery of a very cool binary system", "The discovery of a very cool binary system" ]	[ "Ben Burningham \nCentre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield\n", "S K Leggett \nGemini Observatory\n670 N. A'ohoku Place96720HiloHIUSA\n", "P W Lucas \nCentre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield\n", "D J Pinfield \nCentre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield\n", "R L Smart \nIstituto Nazionale di Astrofisica\nOsservatorio Astronomico di Torino\nStrada Osservatrio 20, 10025 Pino TorineseItaly\n", "A C Day-Jones \nUniversidad de Chile\nCamino el Observatorio # 151536-DSantiagoCasillaChile\n", "H R A Jones ", "D Murray \nCentre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield\n", "E Nickson \nCentre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield\n\nUniversity of Southampton\nSouthamptonUK\n", "M Tamura \nCentre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield\n\nNational Astronomical Observatory\n181-8588MitakaTokyo\n", "Z Zhang \nCentre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield\n", "N Lodieu \nInstituto de Astrofísica de Canarias\n38200La LagunaSpain\n", "C G Tinney \nSchool of Physics\nUniversity of New South Wales\n2052Australia\n", "M R Zapatero Osorio \nCentro de Astrobiología (CSIC-INTA)\nE-28850Torrejón de Ardoz, MadridSpain\n" ]	[ "Centre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield", "Gemini Observatory\n670 N. A'ohoku Place96720HiloHIUSA", "Centre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield", "Centre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield", "Istituto Nazionale di Astrofisica\nOsservatorio Astronomico di Torino\nStrada Osservatrio 20, 10025 Pino TorineseItaly", "Universidad de Chile\nCamino el Observatorio # 151536-DSantiagoCasillaChile", "Centre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield", "Centre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield", "University of Southampton\nSouthamptonUK", "Centre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield", "National Astronomical Observatory\n181-8588MitakaTokyo", "Centre for Astrophysics Research\nScience and Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfield", "Instituto de Astrofísica de Canarias\n38200La LagunaSpain", "School of Physics\nUniversity of New South Wales\n2052Australia", "Centro de Astrobiología (CSIC-INTA)\nE-28850Torrejón de Ardoz, MadridSpain" ]	[ "Mon. Not. R. Astron. Soc" ]	We report the discovery of a very cool d/sdL7+T7.5p common proper motion binary system, SDSS J1416+13AB, found by cross-matching the UKIDSS Large Area Survey Data Release 5 against the Sloan Digital Sky Survey Data Release 7. The d/sdL7 is blue in J-H and H-K and has other features suggestive of low-metallicity and/or high gravity. The T7.5p displays spectral peculiarity seen before in earlier type dwarfs discovered in UKIDSS LAS DR4, and referred to as CH 4 -J-early peculiarity, where the CH 4 -J index, based on the absorption to the red side of the J-band peak, suggests an earlier spectral type than the H 2 O-J index, based on the blue side of the J-band peak, by ∼ 2 subtypes. We suggest that CH 4 -J-early peculiarity arises from low-metallicity and/or high-gravity, and speculate as to its use for classifying T dwarfs. UKIDSS and follow-up UKIRT/WFCAM photometry shows the T dwarf to have the bluest nearinfrared colours yet seen for such an object with H − K = −1.31 ± 0.17. Warm Spitzer IRAC photometry shows the T dwarf to have extremely red H − [4.5] = 4.86 ± 0.04, which is the reddest yet seen for a substellar object. The lack of parallax measurement for the pair limits our ability to estimate parameters for the system. However, applying a conservative distance estimate of 5-15 pc suggests a projected separation in range 45-135 AU. By comparing H − K : H − [4.5] colours of the T dwarf to spectral models we estimate that T eff = 500 K and [M/H]∼ −0.30, with log g ∼ 5.0. This suggests a mass of ∼30 M Jupiter for the T dwarf and an age of ∼10 Gyr for the system. The primary would then be a 75 M Jupiter object with log g ∼ 5.5 and a relatively dust-free T eff ∼ 1500K atmosphere. Given the unusual properties of the system we caution that these estimates are uncertain. We eagerly await parallax measurements and high-resolution imaging which will constrain the parameters further.	10.1111/j.1365-2966.2010.16411.x	[ "https://arxiv.org/pdf/1001.4393v1.pdf" ]	17,310,918	1001.4393	a432b42eb574ee619dfcdfc381489c24ffb481a8	The discovery of a very cool binary system 25 Jan 2010 25 January 2010 25 January 2010 Ben Burningham Centre for Astrophysics Research Science and Technology Research Institute University of Hertfordshire AL10 9ABHatfield S K Leggett Gemini Observatory 670 N. A'ohoku Place96720HiloHIUSA P W Lucas Centre for Astrophysics Research Science and Technology Research Institute University of Hertfordshire AL10 9ABHatfield D J Pinfield Centre for Astrophysics Research Science and Technology Research Institute University of Hertfordshire AL10 9ABHatfield R L Smart Istituto Nazionale di Astrofisica Osservatorio Astronomico di Torino Strada Osservatrio 20, 10025 Pino TorineseItaly A C Day-Jones Universidad de Chile Camino el Observatorio # 151536-DSantiagoCasillaChile H R A Jones D Murray Centre for Astrophysics Research Science and Technology Research Institute University of Hertfordshire AL10 9ABHatfield E Nickson Centre for Astrophysics Research Science and Technology Research Institute University of Hertfordshire AL10 9ABHatfield University of Southampton SouthamptonUK M Tamura Centre for Astrophysics Research Science and Technology Research Institute University of Hertfordshire AL10 9ABHatfield National Astronomical Observatory 181-8588MitakaTokyo Z Zhang Centre for Astrophysics Research Science and Technology Research Institute University of Hertfordshire AL10 9ABHatfield N Lodieu Instituto de Astrofísica de Canarias 38200La LagunaSpain C G Tinney School of Physics University of New South Wales 2052Australia M R Zapatero Osorio Centro de Astrobiología (CSIC-INTA) E-28850Torrejón de Ardoz, MadridSpain The discovery of a very cool binary system Mon. Not. R. Astron. Soc 000000025 Jan 2010 25 January 2010 25 January 2010Printed (MN L A T E X style file v2.2)surveys -stars: low-mass, brown dwarfs We report the discovery of a very cool d/sdL7+T7.5p common proper motion binary system, SDSS J1416+13AB, found by cross-matching the UKIDSS Large Area Survey Data Release 5 against the Sloan Digital Sky Survey Data Release 7. The d/sdL7 is blue in J-H and H-K and has other features suggestive of low-metallicity and/or high gravity. The T7.5p displays spectral peculiarity seen before in earlier type dwarfs discovered in UKIDSS LAS DR4, and referred to as CH 4 -J-early peculiarity, where the CH 4 -J index, based on the absorption to the red side of the J-band peak, suggests an earlier spectral type than the H 2 O-J index, based on the blue side of the J-band peak, by ∼ 2 subtypes. We suggest that CH 4 -J-early peculiarity arises from low-metallicity and/or high-gravity, and speculate as to its use for classifying T dwarfs. UKIDSS and follow-up UKIRT/WFCAM photometry shows the T dwarf to have the bluest nearinfrared colours yet seen for such an object with H − K = −1.31 ± 0.17. Warm Spitzer IRAC photometry shows the T dwarf to have extremely red H − [4.5] = 4.86 ± 0.04, which is the reddest yet seen for a substellar object. The lack of parallax measurement for the pair limits our ability to estimate parameters for the system. However, applying a conservative distance estimate of 5-15 pc suggests a projected separation in range 45-135 AU. By comparing H − K : H − [4.5] colours of the T dwarf to spectral models we estimate that T eff = 500 K and [M/H]∼ −0.30, with log g ∼ 5.0. This suggests a mass of ∼30 M Jupiter for the T dwarf and an age of ∼10 Gyr for the system. The primary would then be a 75 M Jupiter object with log g ∼ 5.5 and a relatively dust-free T eff ∼ 1500K atmosphere. Given the unusual properties of the system we caution that these estimates are uncertain. We eagerly await parallax measurements and high-resolution imaging which will constrain the parameters further. INTRODUCTION The current generation of wide-field surveys (e.g. UKIRT Infrared Deep Sky Survey, UKIDSS; Canada-France Brown Dwarf Survey, CFBDS; Lawrence et al. 2007;Delorme et al. 2008b) is significantly expanding the sam-⋆ E-mail: [email protected] ple of late type T dwarfs (e.g. Delorme et al. 2008a;Lodieu et al. 2007;Pinfield et al. 2008;Burningham et al. 2010). Recent discoveries of extremely cool T dwarfs probe new low-temperature extremes, with T eff as low as 500K Delorme et al. 2008a; Burningham et al. 2009; Leggett et al. 2009). In addition to probing new T eff regimes, we can expect the expanded sample to populate other hitherto unexplored regions of T dwarf parameter space. Of particular interest is the growing diversity seen in metallicity and gravity for late-T dwarfs (e.g. Leggett et al. 2010), and the potential for extending the lowmetallicity subdwarf sequence to very low temperatures. To date, the sample of ultracool subdwarfs (UCSDs) consists of just one proposed T subdwarf, 2MASS J09373487+2931409 (Burgasser et al. 2002(Burgasser et al. , 2006, along with a small number of L subdwarfs (e.g. 2MASS J1626+3925 -sdL4; SDSS J1256-0224 -sdL4; 2MASS J0616-6407 -sdL5; ULAS J1350+0815 -sdL5; 2MASS J0532+8246 -sdL7; Burgasser et al. 2004;Sivarani et al. 2009;Cushing et al. 2009;Lodieu et al. 2009;Burgasser et al. 2003, respectively). Recent parallax determinations and model comparisons by Schilbach et al. (2009) suggest that of these, only the earliest type objects (2MASS J1626+39 and SDSS J1256-02) have metallicities consistent with subdwarf classification on the scheme that Gizis (1997) defined for M subdwarfs. Based on this Schilbach et al. (2009) suggest that an intermediate d/sd classification should be applied to the two coolest objects (2MASS J0532+82 and 2MASS J0937+29). It is important to remember, however, that the subdwarf classification scheme is empirically based, and metallicities are associated with specific subdwarf classes only by model comparisons. That the model comparisons for the latest type UCSDs suggest higher metallicities than seen for earlier type objects should not be a sole basis for reclassification. The higher metallicity inferred from the colours of the coldest objects may actually highlight problems with the models to which they are compared. The spectral classification of subdwarfs should be based on observed spectral features that distinguish these objects from "normal" ultracool dwarfs (UCDs). As such, in this paper we adopt the position that the sdL objects described above are subdwarfs, since their spectra are clearly distinct from those of the bulk population of L dwarfs in a manner broadly consistent with subdwarf status. The more limited sample of T dwarfs, however, precludes such classification at this time, and we adopt the "peculiar" description for possible subdwarfs of this type (e.g. Burgasser et al. 2006;Burningham et al. 2010). However, in both cases the limited sample of "subdwarf" objects means that the current classification system may require significant revision as the true diversity of the spectra of low-metallicity UCDs becomes apparent in the era of larger, deeper surveys such as VISTA and WISE. We report here the discovery of a nearby d/sdL7+T7.5p common proper motion binary. The rest of the paper is laid out as follows. In Section 2 we describe the identification, photometric follow-up, spectral classification and proper motion determination for the two objects. In Section 3 we demonstrate their association as a common proper motion binary pair, and we provide initial estimates for some of their properties in Section 4. Our results and conclusions are summarised in Section 5. TWO NEW ULTRACOOL DWARFS Our searches of the UKIDSS Large Area Survey (LAS; see Lawrence et al. 2007) have been successful at identifying late-type T dwarfs (e.g. Lodieu et al. 2007; Warren et al. 2007;Pinfield et al. 2008;Burningham et al. 2008Burningham et al. , 2009Burningham et al. , 2010. Using the same search methodology as previously described in detail in Pinfield et al. (2008) and Burningham et al. (2010), we identified ULAS J141623.94+134836.30 (hereafter ULAS J1416+13) as a candidate late-T dwarf in Data Release 5 of the LAS with unusually blue H − K = −1.35. The subsequent photometric and spectroscopic follow-up, which resulted in its classification as a T7.5p dwarf, are described in the following sub-sections. Inspection of the surrounding field in SDSS, required to establish the red nature of ULAS J1416+13, revealed the presence of a nearby, very red object at a separation of 9 ′′ . Interrogation of SDSS DR7 revealed this object, SDSS J141624.08+134826.7 (hereafter SDSS J1416+13), to have an SDSS spectrum with L dwarf spectral morphology (see also Table 1 for SDSS photometry of this object). Since our initial identification of this L dwarf, its discovery has been published by Schmidt et al. (2009) andBowler et al. (2009), who have classified it as a blue L5 and L6pec ±2 dwarf respectively. In the following sub-sections, we also describe our follow-up photometry of this target, and describe our analysis of this source that was carried out independently prior to the Schmidt et al. (2009) andBowler et al. (2009) publications. Figure 1 shows a UKIDSS J band finding chart for both the L and T dwarf. Near-infrared photometry Near-infrared follow-up photometry was obtained using the Wide Field CAMera (WFCAM; Casali et al. 2007) on UKIRT on the night of 17 th June 2009, and the data were processed using the WFCAM science pipeline by the Cambridge Astronomical Surveys Unit (CASU) (Irwin et al. 2004), and archived at the WFCAM Science Archive (WSA; Hambly et al. 2008). Observations consisted of a three point jitter pattern in the Y and J bands, and five point jitter patterns in the H and K bands repeated twice, all with 2x2 microstepping and individual exposures of 10 seconds result-ing in total integration times of 120 seconds in Y and J and 400 seconds in H and K. The resulting photometry for both our targets is given in Table 2. The WFCAM filters are on the Mauna Kea Observatories (MKO) photometric system (Tokunaga et al. 2002) 2.2 Warm-Spitzer IRAC photometry The Spitzer General Observer program 60093 allowed us to obtain IRAC photometry of apparently very late-type T dwarfs discovered in the UKIDSS data. This Cycle 6 warm mission program provides only photometry at the shortest two wavelengths, [3.6] and [4.5]. Note that [3.6] and [4.5] are nominal filter wavelengths and, as the photometry is not colour-corrected for the dwarfs' spectral shapes, the results cannot be translated to a flux at the nominal wavelength (e.g. Cushing et al. 2008;Reach et al. 2005). Data were obtained for SDSS J1416+13 and ULAS J1416+13 on 23 rd August 2009. The telescope was pointed mid-way between the L and T dwarf; with a separation of 9 ′′ both dwarfs were near the centre of the 5.2 arcminute field of view. Individual frame times were 30 seconds, repeated three times, with a 16 position spiral dither pattern, for a total integration time of 24 minutes in each band. The post-basic-calibrated-data (pbcd) mosaics generated by the Spitzer pipeline were used to obtain aperture photometry. The photometry was derived using a 0.6-arcsecond pixel aperture radius, with separate (i.e. not annular) skies chosen to avoid the flaring due to the bright primary. The aperture correction was taken from the IRAC handbook 1 . The error is estimated by the variation with sky aperture, which is larger than that implied by the uncertainty images (noise pixel maps) provided by the Spitzer pipeline, and is much less than 1% for the A component in both bands, and 4% and 0.7% for the B component at [3.6] and [4.5] respectively. The description of the primary issues with early release warm IRAC data 2 indicates that the only significant concern is the uncertainty in the linearity correction for SDSS J1416+13; the total uncertainty due to this correction is estimated to be 5-7% at [3.6] and 4% at [4.5] for bright sources. Otherwise the photometry for both sources is uncertain by the usual 3% due to uncertainties in the absolute calibration and pipeline processing. Table 3 gives the photometry and the total uncertainties for both dwarfs. Near-infrared spectroscopy We used JH and HK grisms in the InfraRed Camera and Spectrograph (IRCS; Kobayashi et al. 2000) on the Subaru telescope on Mauna Kea to obtain a R∼ 100 JH and HK spectra for ULAS J1416+13 on 7 th May 2009 and 31 st December 2009 respectively. The observations were made up of a set of eight 300s sub-exposures for the JH spectrum and eighteen 200s sub-exposures in an ABBA jitter pattern to facilitate effective background subtraction, with a slit width of 1 arcsec. The length of the A-B jitter was 10 arcsecs. The spectrum was extracted using standard IRAF packages. The AB pairs were subtracted using generic IRAF tools, and median stacked. The data were found to be sufficiently uniform in the spatial axis for flat-fielding to be neglected. We used a comparison argon arc frame to obtain the dispersion solution, which was then applied to the pixel coordinates in the dispersion direction on the images. The resulting wavelength-calibrated subtracted pairs had a low-level of residual sky emission removed by fitting and subtracting this emission with a set of polynomial functions fit to each pixel row perpendicular to the dispersion direction, and considering pixel data on either side of the target spectrum only. The spectra were then extracted using a linear aperture, and cosmic rays and bad pixels removed using a sigma-clipping algorithm. Telluric correction was achieved by dividing each extracted target spectrum by that of the F4V star HIP 72303, which was observed just after the target and at a similar airmass. Prior to division, hydrogen lines were removed from the standard star spectrum by interpolating the stellar continuum. Relative flux calibration was then achieved by multiplying through by a blackbody spectrum of the appropriate T eff . The spectra were then normalised using the measured near-infrared photometry to place the spectra on an absolute flux scale, and rebinned by a factor of three to increase the signal-to-noise, whilst avoiding under-sampling of the spectral resolution. Spectral types As noted in the Section 1, the discovery SDSS J1416+13 has recently been published by Schmidt et al. (2009) and Bowler et al. (2009). Schmidt et al. (2009) find an optical spectral type of L5 and an infrared type of L5-6 (using the Geballe et al. 2002, indices). Bowler et al. (2009) similarly find an optical type of L6±0.5 and an infrared type of L7-7.5. The template fits carried out in both papers show some discrepancies beyond 9000Å however, and here we use the SDSS spectrum of the source to produce an alternative classification as follows. The top two panels of Figure 2 show the SDSS DR7 spectrum of SDSS J1416+13 along with the optical spectra of the L6 and L7 spectral templates 2MASS J0103+19 and DENIS J0205-11. Whilst the SDSS J1416+13 is good match over much of the range to the L6 template, they disagree significantly beyond 9000Å. On the other hand, the slope of the pseudo-continuum is very similar to that of an L7 across the entire 6000-9200Å range, although the prominent TiO, FeH and CrH features are considerably stronger in the spectrum of SDSS J1416+13. This behaviour is more typical of low-metallicity objects, where it has been speculated that that the low-metallicity atmosphere inhibits the formation of the condensate dust clouds, allowing the opacity due to alkali and hydride species to become more apparent (e.g. Burgasser et al. 2003;Reiners & Basri 2006). Hence, we do not classify this object following the system for L dwarfs defined by Kirkpatrick et al. (1999), and instead rely on comparison to other metal-poor L dwarfs. The lower panel of Figure 2 shows the close similarity between the spectrum of SDSS J1416+13 and that of the metal-poor L dwarf 2MASS J0532+8246. Burgasser et al. (2003) demonstrated that this object not only displays features characteristic of a low-metallicity atmosphere, but also has kinematics consistent with halo membership, and classify it as sdL7. Whilst the general agreement between the spectrum of SDSS J1416+13 and the sdL7 spectrum is good across the entire range considered, there are specific areas of disagreement that should be noted. In particular the CsI and NaI absorption features are somewhat deeper than in the sdL7 template, and more suggestive of dwarf classification than that of a subdwarf. This suggests that SDSS J1416+13 may be less metal poor than 2MASS J0532+82. Given the apparent intermediate nature of SDSS J1416+13 between the L7 and sdL7 spectra, we classify it as d/sdL7 (optical). We note that Bowler et al. (2009) suggest that SDSS J1416+13 is unlikely to have significantly reduced metallicity based on the optical TiO and CaH features. Burgasser et al. (2008b) and Stephens et al. (2009) discuss various mechanisms which may lead to unusually blue L dwarfs including low metallicity, high gravity and thin condensate cloud decks. We explore the physical properties of the L dwarf further in Section 4. The IRCS spectrum of the T dwarf, ULAS J1416+13, is shown in Figure 3, along with spectra of the T7 and T8 spectral standards (Burgasser et al. 2006). With the exception of the poor match to both templates on the red side of the J-band peak and the heavily suppressed K band peak, the spectrum appears intermediate between the two. This is reflected in the spectral typing ratios (see Table 4), and we classify this object as T7.5p. The early type suggested by the CH4-K index clearly reflects the small amount of flux in the K band peak. The type of peculiarity seen here in the red side of the J band peak, and reflected in the spectral typing ratios, has been described for at least three other T dwarfs in Burningham et al. (2010), and has been suggested as a possible tracer of low-metallicity and/or high-gravity. The significance of this feature is discussed in more detail in Section 4. Object u ′ g ′ r ′ i ′ z ′ g ′ − r ′ r ′ − i ′ i ′ − z ′ SDSS Index Ratio Value Type H 2 O-J 1.165 1.14 f (λ)dλ 0.29 ± 0.02 T4 Table 4. The spectral flux ratios for ULAS J1416+13. Those used for spectral typing are indicated on Figure 3.The NH 3 index is not used for assigning a type (see Burningham et al. 2008, and Burningham et al 2010 for a discussion of this), but is included for completeness and to permit future comparison with other late T dwarfs. Proper motions The photometric follow-up observations that were carried out provided a second epoch of imaging data, showing the position of the two sources of interest 1.1 years after the LAS image was measured. We used the IRAF task GEOMAP to derive spatial transformations from the WFCAM follow-up J-band image into the original UKIDSS LAS J-band image based on the positions of 18 reference stars. The transform allowed for linear shifts and rotation, although the rotation that was required was negligible. We then transformed the WFCAM follow-up pixel coordinates of the targets into the Figure 2. The SDSS optical spectrum for SDSS J1416+13 displayed with the template spectra for the L6 dwarf 2MASS J0103+19 (Kirkpatrick et al. 2000), L7 dwarf DENIS J0205-1159 (Delfosse et al. 1997) and sdL7 dwarf 2MASS J0532+82 (Burgasser et al. 2003). The spectra are normalised to unity at 8100Å. LAS images using GEOXYTRAN, and calculated their change in position (relative to the reference stars) between the two epochs. The uncertainties associated with our proper motion measurement primarily come from the spatial transformations, and the accuracy with which we have been able to measure the position of the targets (by centroiding) in the image data. Centroiding uncertainties for the targets should be small, since the seeing and signal-to-noise of the sources was good in both epochs, so this latter source of uncertainty will be neglected. For the LAS image the seeing was ∼0.9 ′′ in the J-band, whilst for the WFCAM image it was ∼ 1.1 ′′ . The root-mean-square (rms) scatter in the difference between the transformed positions of the reference stars and their actual measured positions was ±0.24 pixels in declina-tion and ±0.18 pixels in right ascension, corresponding to 0.048 and 0.036 ′′ in the J-band LAS image. We thus estimate proper motion uncertainties of ±45 mas/yr and ±33 mas/yr in declination and right ascension respectively. The final, relative, proper motion measurements are µ αcosδ = 248 ± 33mas/yr, µ δ = 100 ± 45mas/yr for SDSS J1416+13 and µ αcosδ = 221 ± 33mas/yr, µ δ = 115 ± 45mas/yr for ULAS J1416+13. It should be noted that the relative proper motions calculated here disagree with the absolute values found for the primary by Schmidt there is likely to considerable geometric distortion across the field of view, this poor distribution of reference stars will likely result in an unreliable absolute fit to the coordinates. Secondly, they do not take into account the parallax of the targets. The first and second epoch data were taken on 12 th May 2008 and 17 th June 2009 respectively, which would suggest the influence of parallax should be small. However, given that the distance for both objects may be as low as 5 pc (see Section 4), we do not rule this out as a significant effect. These concerns should not effect the reliability of these proper motions as relative values, but we caution that they include systematic effects that prevent their use in any absolute manner. A WIDE LOW-MASS BINARY The close agreement of the proper-motions for these two objects, and their 9 ′′ proximity on the sky suggests that they represent a common proper motion binary pair. To estimate the probability that the proper motions are aligned by chance, rather than because of a bona-fide association, we have considered the proper motions of objects in the Su-perCosmos Sky Survey (Hambly et al. 2001) in the direction of our targets. Since we do not have a parallax for either object, we instead estimate a liberal range of distances based on their spectral types and apparent magnitudes for the purposes of placing broad limits on their shared volume. In Section 4 we refine this distance estimate based on subsequent analysis of these objects. If we apply the MJ vs. spectral type relations of Liu et al. (2006) we find that an L7 and a T7.5 dwarf with the apparent magnitudes of our objects can be expected to lie at distances ranging from 5 pc to 25 pc. Of the ∼ 50 SuperCosmos objects with apparent distances (based on colour-magnitude relation for field stars) similar to those of our targets, none shared a common motion to within 2σ. We thus conclude that the likelihood of a com-mon proper motion occurring by chance in this direction is less than 1/50. Since the statistics for the properties of the ultracool subdwarf population are not currently known, we will use the space density of "normal" L dwarfs to estimate a conservative probability that this pair are unrelated, and are found in close proximity by chance. Using our liberal distance range of 5-25 pc, and given the separation of 9 ′′ , we can thus estimate that two objects likely share a volume of 0.01 pc 3 . The space density for field L dwarfs was determined by Cruz et al. (2007) to be 0.0038 pc −3 . The probability of finding an L dwarf within the same 0.01 pc 3 as our T7.5 dwarf is thus 3.8 × 10 −5 . It is reasonable to surmise that the probability of finding two ultracool subdwarfs within this volume would be considerably smaller. These combined arguments suggest the probability of a chance alignment in space and motion for these two objects is less than 10 −6 . If we apply these arguments to the total UKIDSS LAS T dwarf sample up to DR4 we find that we would need a sample of approximately 1000 times larger before we would expect to identify one chance alignment such as this. It is worth stressing that our estimate for this probability is somewhat conservative. Given the apparently unusual nature of the objects discussed here, it is likely that true probability for chance alignment is considerably lower. We thus conclude that SDSS J1416+13 and ULAS J1416+13 represent a binary pair, which we shall henceforth refer to as SDSS J1416+13AB. THE PROPERTIES OF SDSS J1416+13AB The optical spectral classification of SDSS J1416+13A as a dwarf/subdwarf implies that we could reasonably classify the secondary as a dwarf/subdwarf also, given that most binary systems are expected to be coevally formed in the same cloud core. Figure 4 shows near-infrared colours Knapp et al. (2004) with T spectral types updated to the Burgasser et al. (2006) system. Additional data for late-T dwarfs taken from Burningham et al. (2010). Known binary systems are shown as green dots, whilst known metal poor objects discussed in the text are shown as red dots, and labelled in the lower plot. The only other known T dwarf with K band photometry that displays CH 4 -J-early peculiarity is shown as an orange dot, whilst SDSS J1416+13AB are shown as red asterisks. With the exception of SDSS J1416+13A and 2MASS J0532+82, all spectral types are near-infrared types. 2MASS photometry for 2MASS0532+82 has been converted to the MKO system using the Stephens & Leggett (2004) relationships, which give consistent results with synthetic colours calculated from the object's nearinfrared spectrum. as a function of spectral type for L and T dwarfs, with SDSS J1416+13AB indicated with red asterisks. Blue H −K near-infrared colours for mid-to late T dwarfs have typically been interpreted as indicative of low-metallicity and/or high-gravity (e.g. Burgasser et al. 2002;Knapp et al. 2004;Liu et al. 2007), caused by K band suppression by pressure sensitive collisionally induced absorption by hydrogen (CIA H2; Saumon et al. 1994). Blue J − H colours in metal poor L dwarfs have also been interpreted in terms of H band suppression by CIA H2 (e.g. Burgasser et al. 2003). The blue J − H colour of SDSS J1416+13A, and the blue H − K colour of SDSS J1416+13B, therefore, support the interpretation that both objects have low-metallicity and/or high-gravity, and we interpret the peculiar spectral shape of SDSS J1416+13B in this context. The spectral morphology in the J band peak of SDSS J1416+13B is reminiscent of a number of T dwarfs recently discovered that have been classified as peculiar . These also show a J band peak that appears earlier in type on the red side (as indicated by the CH4-J index) compared to the blue side (as indicated by the H20-J and WJ indices). This morphology was referred to by Burningham et al. (2010) as CH4-J-early peculiarity, and we continue this convention here. Only one of the objects already found with CH4-J-early peculiarity, ULAS J1233+1219, currently has K band photometry. It also appears very blue, with H − K = −0.75 (indicated by an orange filled circle in Figure 4), and is as notable an outlier in H − K for its type as SDSS J1416+13AB. It thus seems plausible that CH4-J-early peculiarity is indicative of low-metallicity and/or high gravity. There is some theoretical basis for preferring a lowmetallicity interpretation of CH4-J-early peculiarity. =-0.5 metallicity, and also for solar metallicity with log g = 5.0 and log g = 5.5. Enhancement of the red side of the J band peak is apparent in both the low-metallicity and high-gravity cases, but is most pronounced in the former. We speculate that CH4-J-early peculiarity may represent a useful tracer of low-metallicity atmospheres, although its presence in a system with fiducial metallicity and age constraints will need to be observed before a robust interpretation will be possible. It is interesting to note that the spectral shape of SDSS J1416+13B also deviates from that of the spectral templates blueward of 1.1µm, in a manner similar to that seen in Figure 5 for the low-metallicity case. The same behaviour is not predicted for the high-gravity case. A spectrum with better coverage in the Y band may provide a useful means of breaking the gravity-metallicity degeneracy. The need for a more complex spectral classification scheme to take account of spectral variations that result from changes in metallicity and gravity in addition to T eff has been highlighted by Kirkpatrick (2005). As more objects that exhibit CH4-J-early spectral peculiarity are identified, its behaviour may provide a convenient method for more detailed classification of T dwarf spectra. The lack of a known parallax for this binary pair precludes a detailed assessment of their properties, since spectroscopic distances are not well constrained for ultracool T dwarfs. In the case of the one previously identified sdL7, 2MASS J0532+82, the determined absolute magnitude (MJ = 13.00) is 1-2 mags brighter than might otherwise be expected for a field dwarf of type L7 (Burgasser et al. 2008a). If we assume that SDSS J1416+13A has MJ = 13.0 as was the case for for 2MASS J0532+82 we arrive at a distance estimate of 10 pc. However, SDSS J1416+13A is considerably less blue in J − H than 2MASS J0532+82 (J − H = 0.55 vs J − H = 0.08 respectively, see Figure 4) and, as previously discussed, may have rather different properties. Assuming spectral types L7 and T7.5, however, suggests distances of 5 pc and 20 pc for the objects respectively by applying the MJ vs spectral type relations of Liu et al. (2006). In the case of the metal poor T dwarf 2MASS J0937+29 this method would overestimate the distance by ∼ 30%. Using this as a correction suggests a distance for SDSS J1416+13AB of 14 pc. This would represent a significant discrepancy in the distances of the primary and secondary members of SDSS J1416+13AB, implying that the primary could be an unresolved binary. However, the K band suppression in SDSS J1416+13B is greater than in the case of 2MASS J0937+29 and, as discussed below, it appears to be considerably cooler. It thus seems likely that SDSS J1416+13B is fainter still, and a distance as close as 10 pc seems plausible. A distance of ∼ 10pc is also in broad agreement with that estimated by Schmidt et al. (2009) andBowler et al. (2009) for the primary. We thus conservatively estimate the distance to SDSS J1416+13AB to lie in the 5-15 pc range. The implied projected separation of the binary pair at this range of distances is 45 -135 AU. It is thus possible that this pair also represents a rare very low-mass wide binary system (e.g. Figure 9 in Lafrenière et al. 2008). The longer baseline provided by our Spitzer IRAC photometry offers the opportunity to estimate parameters of the system through comparison to predictions of model spectra. The IRAC colours of SDSS J1416+13A are normal for a Figure 6. Spitzer IRAC colours as a function of spectral type for L and T dwarfs. Data for L and T dwarfs on the MKO system taken from Knapp et al. (2004) with T spectral types updated to the Burgasser et al. (2006) system. Additional data for late-T dwarfs taken from Leggett et al. (2010). Known binary systems are shown as green dots, whilst known metal poor objects discussed in the text are shown as red dots, and labelled in the upper plot. SDSS J1416+13AB are shown as red asterisks. late-type L dwarf, although the colours of these objects show significant scatter (see Figure 6). All of the low-metallicity late-T dwarfs plotted in Figure 6 display H − [4.5] that is at least 0.5 magnitudes redder than would otherwise be expected for a "normal" T dwarf of their subtype. However, the HMKO -[4.5] colour of SDSS J1416+13B is the reddest yet measured. In addition to apparently indicating low-metallicity atmospheres, this colour is a good indicator of T eff (e.g. Warren et al. 2007;Stephens et al. 2009;Leggett et al. 2010) and so SDSS J1416+13B appears to be very cool. of 30-40 MJupiter for SDSS J1416+13B and an age around 10 Gyr or older for the system using the evolutionary models of . The near-infrared indices of Geballe et al. (2002) for SDSS 1416+13A suggest a near-infrared spectral type of L7-7.5 (Bowler et al. 2009). The near-infrared spectral type-T eff relations of Stephens et al. (2009) suggest that blue L7 dwarfs have T eff ∼1500 K. If the system is aged at ∼ 10Gyr as implied by the secondary then the evolutionary models of suggest that the primary is a ∼75MJupiter dwarf with log g ∼ 5.5. Hence the L dwarf is at the stellar/substellar boundary as also suggested by Bowler et al. (2009). Stephens et al. (2009) have shown that the atmospheres of blue L dwarfs are less dusty than the bulk population, deriving a high value of f sed ∼ 3 using the Marley et al. (2002) models. These authors also find that there is an indication that dust clearing may occur at higher temperatures for higher gravity systems. The blue colours and almost dustfree atmosphere of this relatively warm ∼1500 K L dwarf is therefore consistent with a high gravity and relatively old age for the system. 1.4, −27±9) kms −1 respectively 3 , and interpret its kinematics as indicative of thin disk membership. This is consistent with the age of 10 Gyr and slight metal-paucity that we find here (Robin et al. 2003;Haywood et al. 1997). It is intriguing that SDSS J1416+13B appears to be ∼250 K and ∼100 K cooler than 2MASS J1114-26 2MASS J0939-24 respectively, which are of similar spectral type (T7.5 and T8). It is plausible that the nearinfrared spectral type vs. T eff relation for very late T dwarfs shows significant dependence on metallicity and gravity, with lower-metallicity dwarfs of a given subtype having lower T eff than similar type objects of higher metallicity. The cool nature of 2MASS J0939-24 compared to other T8 dwarfs spectral type would tend to support this assertion, although interpretation of this object is complicated by its probable binarity (Burgasser et al. 2008c). If SDSS J1416+13B had significantly lower metallicity and/or higher gravity than 2MASS J1114-26 2MASS J0939-24, such an effect might account for their similar types but diverse T eff s. However, the same model predictions seen in Figure 7 that suggest such different T eff also suggest fairly similar metallicities and gravities for the three objects. Finally, it should be noted that we cannot currently rule out the presence of a cooler unresolved companion to SDSS J1416+13B, which might explain its extremely red H −[4.5] colour coupled with is T7.5p near-infrared morphology. Unfortunately, the lack of parallax and high-resolution imaging for this target prevent us from adequately exploring this issue here. 3 U positive towards the Galactic centre. SUMMARY We have identified what appears to be the coolest binary system yet found. The association of the T7.5p component with the d/sdL7 primary allows us to now extend the highgravity and low-metallicity sequence to the lowest observed temperatures, and we suggest that CH4-J-early peculiarity ) may in future prove to be a useful discriminator for this type of object. The likely close proximity of the system to the Sun should facilitate the determination of the trigonometric parallax in the near-future, which will allow a more robust determination of the properties for this exciting system. Marley et al. (2002) and , after Figure 11 from Leggett et al. (2010). The location of SDSS J1416+13B is indicated with an asterisk. The small dots represent the comparison sample of late-T dwarfs used in Leggett et al. (2010), with binaries shown as ringed symbols. Larger symbols are T dwarfs examined in detail in that work, where symbol size indicates gravity and colour indicates metallicity. Largest to smallest filled circles correspond to log g ≈ 5.4 (5.2-5.5), log g ≈ 5.2 (5.0 -5.4), log g ≈ 5.0 (4.8 -5.1), and log g ≈ 4.3 (4.0 -4.5). Open circles are unconstrained gravity. Red symbols indicate metal-rich, green solar, and cyan metal-poor dwarfs; black are dwarfs with unconstrained metallicity. Model sequences with a range of gravity and metallicity are shown, with log g = 4.0, 4.5 and 5.0 and [m/H]=0 shown as solid lines, whilst log g = 4.5 with [m/H]= -0.3 and +0.3 are shown as dashed lines. The T eff values for the log g = 4.5 [m/H]= 0 model are indicated on the top axis, and crosses along the sequences indicate the 800K and 600K points for each model set. Figure 1 . 1A 1'×1' J band finding chart for ULAS J1416+13 and SDSS J1416+13 taken from the UKIDSS database. et al. (2009) and Bowler et al. (2009) at the 4σ level. This discrepancy likely arises as a result of two factors. Firstly, in the first epoch images both targets lie within 30 ′′ of the detector edge. As a result, the distribution of reference stars is not even about the targets. Since Figure 3 . 3The IRCS JHK spectrum for ULAS J1416+13 plotted with T7 and T8 spectral standards 2MASS J072718.24+171001.2 and 2MASS J04151954-093506.6. Overlaid are the spectral ranges for the numerators and denominators (D) of the spectral typing indices ofBurgasser et al. (2006) andBurningham et al. (2008). Figure 4 . 4J − H and H − K colour as a function of spectral type for L and T dwarfs. Data for L and T dwarfs on the MKO system are taken from Figure 5 shows comparisons of Burrows et al. (2006) model spectra for log g = 5.0, T eff = 700K T dwarfs with solar and [Fe/H] Figure 5 . 5Burrows et al. (2006) models for log g = 5.0 700K T dwarfs with solar and [Fe/H]0=-0.5 metallicity, and for solar metallicity combined with log g = 5.0 and log g = 5.5. Figure 7 reproduces 7Figure 11ofLeggett et al. (2010) with the location of SDSS J1416+13B indicated. It can be seen that this T dwarf forms a sequence with the other known metal-poor ([m/H]∼ -0.3) high-gravity (log g ∼ 5.0 − 5.3) dwarfs: 2MASS J0937347+293142, 2MASS J12373919+6526148, 2MASS J11145133-2618235, 2MASS J09393548-2448279. The dwarfs have T eff ∼ 950, 825, 750 and 600 K respectively (Geballe et al. 2009; Liebert & Burgasser 2007; Leggett et al. 2007, 2009; Burgasser et al. 2008c). Extrapolating these values using Figure 7 and the models (Marley et al. 2002; Saumon & Marley 2008) shown in the figure, implies that SDSS J1416+13B has [m/H]∼-0.3, log g ∼ 5.0 to 5.3 and T eff ∼ 500 K. This indicates a mass Both Schmidt et al. (2009) and Bowler et al. (2009) have estimated (U, V, W )LSR for SDSS J1416+13A, finding (−7.9±2.1, 10.2±1.2, −31.4±4.7) kms −1 and (−6±4, 10.2± Figure 7 . 7A plot of H − K vs H M KO − [4.5] for a selection of late-T dwarfs and model predictions of Table 1. SDSS DR7 AB system photometry for SDSS J1416+13.Table 2. UKIDSS LAS DR5 and WFCAM follow-up photometry of the newly discovered ultracool dwarfs presented here. In the case of each object the first row is the UKIDSS survey photometry, and the second is the results of the WFCAM follow-up. All near-infrared photometry presented here is on the Vega system and uses the MKO photometric system.Table 3. Spitzer IRAC photometry for the d/sdL7 and T7.5p dwarfs presented here.J1416+13 23.55 ± 0.57 23.08 ± 0.18 20.69 ± 0.04 18.38 ± 0.01 15.92 ± 0.01 2.39 ± 0.19 2.31 ± 0.04 2.46 ± 0.01 Object Y J H K Y − J J − H H − K ULAS J1416+13 18.16 ± 0.02 17.26 ± 0.02 17.58 ± 0.03 18.93 ± 0.24 0.90 ± 0.03 −0.32 ± 0.03 −1.35 ± 0.25 18.13 ± 0.02 17.35 ± 0.02 17.62 ± 0.02 18.93 ± 0.17 0.78 ± 0.03 −0.27 ± 0.03 −1.31 ± 0.17 SDSS J1416+13 14.25 ± 0.01 12.99 ± 0.01 12.47 ± 0.01 12.05 ± 0.01 1.26 ± 0.01 0.52 ± 0.01 0.42 ± 0.01 14.28 ± 0.01 13.04 ± 0.01 12.49 ± 0.01 12.08 ± 0.01 1.24 ± 0.01 0.55 ± 0.01 0.41 ± 0.01 Name [3.6] [4.5] SDSS J1416+13 10.99 ± 0.07 10.98 ± 0.05 ULAS J1416+13 14.69 ± 0.05 12.76 ± 0.03 http://ssc.spitzer.caltech.edu/irac/dh/ 2 http://ssc.spitzer.caltech.edu/irac/documents/iracwarmdatamemo.txt ACKNOWLEDGEMENTSSKL is supported by the Gemini Observatory, which is operated by AURA, on behalf of the international Gemini partnership of Argentina, Australia, Brazil, Canada, Chile, the United Kingdom, and the United States of America. EN received support from a Royal Astronomical Society small grant. NL was funded by the Ramón y Cajal fellowship number 08-303-01-02. CGT is supported by ARC grant DP0774000. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and has benefited from the SpeX Prism Spectral Libraries, maintained by Adam Burgasser at http://www.browndwarfs.org/spexprism. . B P Bowler, M C Liu, T J Dupuy, ArXiv 0912.3796ApJ in pressBowler B. P., Liu M. C., Dupuy T. J., 2009, ApJ in press, ArXiv 0912.3796 . A J Burgasser, T R Geballe, S K Leggett, J D Kirkpatrick, D A Golimowski, ApJ. 6371067Burgasser A. J., Geballe T. R., Leggett S. K., Kirkpatrick J. D., Golimowski D. A., 2006, ApJ, 637, 1067 . A J Burgasser, J D Kirkpatrick, M E Brown, I N Reid, A Burrows, J Liebert, K Matthews, J E Gizis, C C Dahn, D G Monet, R M Cutri, M F Skrutskie, ApJ. 564421Burgasser A. J., Kirkpatrick J. D., Brown M. E., Reid I. N., Burrows A., Liebert J., Matthews K., Gizis J. E., Dahn C. 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[ "Families index for Boutet de Monvel operators", "Families index for Boutet de Monvel operators" ]	[ "Severino T Melo ", "Thomas Schick ", "Elmar Schrohe " ]	[]	[ "Münster Journal of Mathematics c Münster J. of Math" ]	We define the analytical and the topological indices for continuous families of operators in the C * -closure of the Boutet de Monvel algebra. Using techniques of C * -algebra K-theory and the Atiyah-Singer theorem for families of elliptic operators on a closed manifold, we prove that these two indices coincide.	null	[ "https://arxiv.org/pdf/1203.0482v2.pdf" ]	118,499,698	1203.0482	83b3fce3dbf133478d83bced1a15dc15cdf62cb0	Families index for Boutet de Monvel operators 2008. 2008 2010 Mathematics Subject Classification: 19K56 Severino T Melo Thomas Schick Elmar Schrohe Families index for Boutet de Monvel operators Münster Journal of Mathematics c Münster J. of Math 12008. 2008 2010 Mathematics Subject Classification: 19K56Münster We define the analytical and the topological indices for continuous families of operators in the C * -closure of the Boutet de Monvel algebra. Using techniques of C * -algebra K-theory and the Atiyah-Singer theorem for families of elliptic operators on a closed manifold, we prove that these two indices coincide. Introduction Boutet de Monvel's calculus [5] provides a pseudodifferential framework which encompasses the classical differential boundary value problems. In an extension of the concept of Lopatinski and Shapiro, it associates to each operator two symbols: a pseudodifferential principal symbol, which is a bundle homomorphism, and an operator-valued boundary symbol. Ellipticity requires the invertibility of both. In this case, the calculus allows the construction of a parametrix. If the underlying manifold is compact, elliptic elements define Fredholm operators, and the parametrices are Fredholm inverses. Boutet de Monvel showed how then the index can be computed in topological terms. The crucial observation is that elliptic operators can be mapped to compactly supported K-theory classes on the cotangent bundle over the interior of the manifold. The topological index map, applied to this class, then furnishes an integer which is equal to the index of the operator. For the construction of the above map, Boutet de Monvel combined operator homotopies and classical (vector bundle) K-theory in a very refined way. It therefore came as a surprise that this map -which is neither obvious nor trivial -can also be obtained as a composition of various standard maps in K-theory for C * -algebras -which was not yet available when [5] was written. In fact, it turns out to be basically sufficient to have a precise understanding of the short exact sequence induced by the boundary symbol map, [17], see also [16]. In the spirit of the classical result of Atiyah and Singer [3] we introduce and consider in this article families of operators in Boutet de Monvel's calculus, an issue that has not been addressed in [5]. More specifically, we consider a compact manifold X with boundary and then a fiber bundle Z → Y with fiber X over a compact Hausdorff space Y . We are then studying fiberwise (elliptic) Boutet de Monvel operators, depending continuously on y ∈ Y . In order to be able to use the powerful tools of C * -algebra K-theory we define such an operator family A over Y as a continuous section of a bundle of C * -algebras over Y , a concept which is slightly more general than that of Atiyah and Singer, who equip the set of operators with a Fréchet-space topology. In fact, restricted to the case without boundary, our algebra of continuous families A contains that of [3] as a dense subalgebra. While the analytic index ind a (A) of such an elliptic family A as an element of K(Y ) is easily defined following Atiyah [2] and Jänich [11], cf. Definition 15 below, it is less obvious how to obtain the topological description. Similar to Boutet de Monvel's approach, the essential step is the construction of a map which associates to an elliptic family an element of the compactly supported K-theory of the total space of the bundle of cotangent spaces over the interior of the underlying manifolds. We regard this map as a homomorphism defined on K 1 (A/K), where K denotes the ideal of continuous families which have values in compact operators. In its definition, we use a fact which builds upon an observation of Boutet de Monvel: There exists a natural subalgebra A † of A for which K * (A † /K) ∼ = K * (A/K) so that each elliptic family A in A can be represented by a class a ∈ K 1 (A † /K). Moreover, A † /K is commutative which allows us to make the connection to classical (vector bundle) K-theory. Then ind t (A) is defined by applying the classical construction of the topological index to a, compare Definition 16. Our main result is then that these two indices are equal. To prove this, we reduce to the classical families index theorem of Atiyah and Singer [3]. We assign in a canonical way to A an index problem on a bundle of closed manifolds, namely the double of our original bundle of manifolds with boundary. We then show that this associated family has the same analytic as well as topological index as A. In this step we make once more use of the isomorphism K 1 (A/K) ∼ = K 1 (A † /K). It is perhaps worth stressing that our index theorem does not use the Boutet de Monvel index theorem for boundary value problems, which can actually be obtained from ours by taking Y equal to one point. Taking the families index theorem for granted, Albin and Melrose derived a more refined formula for the Chern character of the index bundle in terms of symbolic data [1,Theorem 3.8]. The paper is structured as follows: Section 1 starts with a review of the Boutet de Monvel calculus for a single manifold. We introduce the C * -algebra A of Boutet de Monvel operators of order and class zero and the boundary symbol map γ. Section 2 gives the technical introduction of operator families in Boutet de Monvel's calculus over a compact Hausdorff space Y . We define them as the continuous sections into a bundle of operator algebras whose typical fiber is the C * -algebra A. In order to keep the exposition simple, we first treat the case where E is trivial one-dimensional and F = 0. We introduce γ as the fiberwise symbol map and extend the results on the kernel and image of γ to the family situation. While in the single operator case this was sufficient to compute the K-theory of A/K, the situation is more complicated in the families case. In fact, an important ingredient in [17] is that fact that whenever X is connected and ∂X = ∅ there exists a continuous section of S * X • . This is no longer true in the families case. Instead, we prove in Theorem 12 the fact alluded to above: For F = 0 we define A † as the C * -algebra generated by all sections whose pseudodifferential part is independent of the co-variable at the boundary and whose singular Green part vanishes. Then A † /K is commutative. Moreover, we use a Mayer-Vietoris argument to show that the inclusion map induces an isomorphism (1) K * (A † /K) ∼ = K * (A/K). In Section 3 we study the index problem. Again, we confine ourselves first to the case of trivial onedimensional bundles. We introduce the analytic and topological index and, as our main result, prove that the analytic and the topological index are equal. To achieve this, we reduce with the help of a doubling procedure to the case of families of closed manifolds. This reduction is based on the fact that we can use the isomorphism in (1) to represent any element of K 1 (A/K) as a K 1 -class of A † /K. In Section 4 we finish by explaining the arguments needed for the general situation. Two appendices give technical details about the structure group of our families and about the Künneth theorem we are using. Boutet de Monvel calculus for a single manifold In this section, we introduce notation and recall the case of single operators. Details can be found in the monographs of Rempel and Schulze [20] and Grubb [8] as well as in the short introduction [22]. Let X be a compact manifold of dimension n with boundary ∂X and interior X • . We equip X with a collar (i.e, a neighborhood U of the boundary and a diffeomorphism δ : U → ∂X × [0, 1)) which then induces the boundary defining function x n = pr [0,1) • δ The variables of ∂X will be denoted x ′ . The collar is used to provide the double 2X of X with a (noncanonical) smooth structure. Recall that 2X is the union of two copies X + and X − of X quotiented by identification of the two copies of ∂X. An element in Boutet de Monvel's calculus is a matrix of operators A = P + + G K T S : C ∞ (X, E 1 ) C ∞ (X, E 2 ) ⊕ −→ ⊕ C ∞ (∂X, F 1 ) C ∞ (∂X, F 2 ) ,(2) acting between sections of vector bundles E 1 , E 2 over X and F 1 , F 2 over ∂X. In this article we shall focus on the case of endomorphisms, where E 1 = E 2 = E and F 1 = F 2 = F . For convenience, we choose a Riemannian metric g on M and Hermitean metrics on E, F to later obtain fixed Hilbert spaces structures, although the results do not depend on these choices. The operator P + in the upper left corner is a truncated pseudodifferential operator, derived from a (classical) pseudodifferential operator P on 2X. Given u ∈ C ∞ (X, E), P + u is defined as the composition r + P e + u. Here e + extends u by zero to a function on 2X, to which P is applied. The result then is restricted (via r + ) to X. In general it is not true that P + u ∈ C ∞ (X, E). In order to ensure this, P is required to satisfy the transmission condition: If p ∼ p j is the asymptotic expansion of the local symbol p of P into terms p j (x, ξ), which are positively homogeneous of degree j in ξ one requires that, for x n = 0 and ξ = (0, ±1) one has D β x D α ξ p j (x ′ , 0, 0, 1) = (−1) j−\|α\| D β x D α ξ p j (x ′ , 0, 0, −1). As for the remaining entries, G is a singular Green operator, T a trace operator, K a potential operator, and S a pseudodifferential operator on the boundary. Operators in Boutet de Monvel's calculus have an order and a class or type. There are invertible elements in the calculus which allow us to reduce both, order and class, to zero. The operators then form a * -subalgebra of the bounded operators on the Hilbert space H := L 2 (X, E) ⊕ L 2 (∂X, F ). Definition 1. Let A • (E, F ) denote the algebra of the (polyhomogeneous) Boutet de Monvel operators of order and class zero on H = L 2 (X, E) ⊕ L 2 (∂X, F ), endowed with its natural Fréchet topology, and A its C * -closure in the algebra of all bounded operators on H. We write A • and A if E = X × C is trivial one-dimensional and F = 0. Let A ∈ A • (E, F ) be given as in (2). For each entry P, S, G, T, K we have a symbol. This is the usual one for P and S, while G, T , and K can be considered as operator-valued pseudodifferential operators on ∂X with classical symbols in the sense of Schulze [23]. These are defined as follows, see [22]: The principal pseudodifferential symbol σ(A) of A is the restriction of the principal symbol of P to the cosphere bundle over X. In order to define the boundary principal symbol γ(A) we first denote by p 0 , g 0 , t 0 , k 0 , and s 0 the principal symbols of P , G, T , K, and S, respectively. We let E 0 x ′ ,ξ ′ be the pullback of E\| {xn=0} to the normal bundle of X, lifted to (x ′ , ξ ′ ) ∈ S * ∂X. For fixed (x ′ , ξ ′ ) ∈ S * ∂X, ξ n → p 0 (x ′ , 0, ξ ′ , ξ n ) is a function on the conormal line in (x ′ , ξ ′ ), acting on E 0 x ′ ,ξ ′ . It induces a truncated pseudodifferential operator p 0 (x ′ , 0, ξ ′ , D n ) + = r + p 0 (x ′ , 0, ξ ′ , D n )e + : L 2 (R ≥0 , E 0 x ′ ,ξ ′ ) → L 2 (R ≥0 , E 0 x ′ ,ξ ′ ) . In local coordinates near the boundary we then define the boundary principal symbol γ(A)(x ′ , ξ ′ ) : L 2 (R ≥0 , E 0 x ′ ,ξ ′ )⊕ F x ′ ,ξ ′ → L 2 (R ≥0 , E 0 x ′ ,ξ ′ ) ⊕ F x ′ ,ξ ′ by (3) γ(A)(x ′ , ξ ′ ) := p 0 (x ′ , 0, ξ ′ , D n ) + + g 0 (x ′ , ξ ′ , D n ) k 0 (x ′ , ξ ′ , D n ) t 0 (x ′ , ξ ′ , D n ) s 0 (x ′ , ξ ′ ) , with D n indicating that we let the symbol act as an operator with respect to the variable x n only. Note that the operator g 0 (x ′ , ξ ′ , D n ) is compact and that k 0 (x ′ , ξ ′ , D n ), t 0 (x ′ , ξ ′ , D n ) and s 0 (x ′ , ξ ′ ) even have finite rank. The operator p 0 (x ′ , 0, ξ ′ , D n ) + on the other hand is a Toeplitz type operator; it will not be compact unless p 0 = 0. Denoting by K = K(H) the ideal of compact operators on L(H), one has the following important estimate based on work by Gohberg [7], Seeley [24] and Grubb-Geymonat [9], see [20, 2.3.4.4, Theorem 1] for a proof: inf K∈K A + K = max{ σ(A) sup , γ(A) sup },(4) where the sup-norms on the right hand side are over the cosphere bundles in X and ∂X, respectively. This estimate implies, in particular, that both symbols extend continuously to C * -algebra homomorphisms defined on A(E, F ). For fixed (x ′ , ξ ′ ) the range {γ(A)(x ′ , ξ ′ ) \| A ∈ A} forms an algebra of Wiener-Hopf type operators. It also follows from this estimate that γ vanishes on K. Since the entries of γ(A)(x ′ , ξ ′ ) induced by g 0 , k 0 , t 0 and s 0 are (pointwise) compact while that induced by p 0 is not (unless p 0 = 0), we conclude that a Boutet de Monvel operator A belongs to ker γ if and only if σ(A) vanishes at the boundary. Based on this observation (see [16,Section 2] for details) one can show that σ induces an isomorphism (5) ker γ/K ∼ = C 0 (S * X • ). The K-theory of the range of γ was described in [16,Section 3]. Let b : C(∂X) → Im γ denote the C * -homomorphism that maps g to γ(m(f )), where m(f ) is the operator of multiplication by a function f ∈ C(X) whose restriction to ∂X equals g. Then b induces a K-theory isomorphism. K-Theory of the families C * -algebra To simplify the exposition, we shall assume in this section that E = X ×C is the trivial one-dimensional line bundle and F = 0. Let Diff(X) denote the group of diffeomorphisms of X, equipped with its usual Fréchet topology. Recall that δ : U → ∂X × [0, 1) is the collar fixed at the beginning of Section 1. Let G denote the subgroup of Diff(X) consisting of those φ such that δ • φ • δ −1 : ∂X × [0, 1/2) → ∂X × [0, 1) is of the form (x ′ , x n ) → (ϕ(x ′ ), x n ) for some diffeomorphism ϕ : ∂X → ∂X. We are going to use two properties that each φ ∈ G satisfies: the boundary defining function is preserved (x n • φ = x n for 0 ≤ x n ≤ 1/2), and the canonical map 2φ : 2X → 2X, defined by 2φ • i ± = i ± • φ, where i ± : X ± → 2X are the two canonical embeddings of X in 2X, is a diffeomorphism of 2X. Throughout this paper, π : Z → Y will denote a fiber bundle over the compact Hausdorff space Y with fiber X and structure group G. Note, however, that this choice of structure group is just for convenience and can always be (essentially uniquely) arranged for a general bundle with typical fiber X, see the Appendix A for details. We denote Z y := π −1 (y). Each Z y is a smooth manifold with boundary, non-canonically diffeomorphic to X. The restriction of π to ∂Z = ∪ y ∂Z y is a fiber bundle π ∂ : ∂Z → Y with fiber ∂X and structure group Diff(∂X). Next we define a bundle of Hilbert spaces, and later a C * -algebra which will act on its space of sections. This is a bit delicate, as it depends on some further choices; therefore we give the details. We choose a continuous family of Riemannian metrics (g y ) y∈Y with corresponding measures µ y on Z y and define H y := L 2 (Z y , µ y ). Recall that such a family (g y ) exists: we can patch them together using trivializations of the bundle and a partition of unity on Y , as the space of Riemannian metrics on X is convex. The union H = y∈Y H y is a fiber bundle of topological vector spaces over Y , canonically associated to π : Z → Y , with trivializations induced from the trivializations of π in the obvious way. The structure group is the group of invertible bounded operators on H, equipped with the strong topology. Remark 2. That we obtain here the strong topology and not the norm topology comes from the fact that the changes of trivialization are implemented by pullback with the diffeomorphisms of G, and this is continuous in the strong, but not the norm topology. This makes our considerations about bundles of operators later quite cumbersome and requires to use the fact that we deal with pseudodifferential operators. Moreover, the choice (g y ) y∈Y gives rise to a continuous family of inner products on H inducing the given topology of the fibers H y . Let A y be the Boutet de Monvel algebra of order and class zero on L 2 (Z y ). We want to define the bundle of Boutet de Monvel algebras ℵ = y∈Y A y as locally trivial bundle with structure group the automorphism group of the C * -algebra A with the norm topology, associated to Z → Y . To achieve this, we need the diffeomorphism invariance of the Boutet de Monvel algebra in a precise form. Definition 3. Given φ ∈ G, let T φ denote the bounded operator on L 2 (X) defined by f → f • φ −1 . Proposition 4. We have a well defined continuous action (for the Fréchet topology on G and the norm topology on A) G × A ∋ (φ, A) → T φ AT −1 φ ∈ A. Moreover, by restriction we get an action G × A • → A • . Proof. This corresponds to [3,Proposition 1.3]. In fact, even if X is closed, Atiyah and Singer consider a slightly different situation in that they close A • with respect to the operator norm of the action on all Sobolev spaces, while we only use the operator norm on L 2 . Their argument still applies verbatim, since they treat the action on each Sobolev space separately. Indeed, the proof of [3, Proposition 1.3] uses only a number of formal properties of the algebra of pseudodifferential operators which are also satisfied by the Boutet de Monvel algebra, and therefore applies in the same way to our general situation. To be more specific, let us list these properties: (1) the Boutet de Monvel algebra A • is diffeomorphism invariant, i.e. in particular T φ AT −1 φ ∈ A • for A ∈ A • and φ ∈ G. (2) Each T φ is a bounded operator on L 2 (X) and the map G → L(L 2 (X)) is strongly continuous. Moreover, for a sufficiently small open neighborhood of 1, the image has uniformly bounded norm. The proof of this fact as given in [3] works for compact manifolds with boundary exactly the same way as for closed manifolds. (3) Let V G denote the space of vector fields on X which, in the collar, pull back from vector fields on ∂X. The exponential map, defined with the help of Riemannian metrics which respect the collar structure, gives a local diffeomorphism (of Fréchet manifolds) between V G and G. (4) If V ∈ V G and A ∈ A • then the commutator [A, V ] belongs to A • by the rules of the calculus, cf. [8, Theorem 2.7.6]. All these properties are either well known or easy to establish. Corollary 5. We obtain the bundle ℵ = y∈Y A y of topological algebras with bundle of subalgebras ℵ • = y∈Y A • y , modelled on (A, A • ) with structure group the automorphism group of A with its norm topology and the automorphism group of A • its Fréchet topology. The local trivializations are induced by the local trivializations of π : Z → Y , where a diffeomorphisms α y : Z y → X obtained from the trivialization map A y to A by conjugation with T αy . Moreover, the choice of metrics (g y ) y∈Y induces a continuous family of norms on the fibers of ℵ inducing the topology. With these norms the bundle becomes a bundle of C * -algebras. Proof. The statement about the bundle of topological algebras follows immediately from Proposition 4. Moreover, it is well known that each A y is closed under taking adjoints in L(L 2 (Z y )). We now check that with this structure, we obtain a locally trivial bundle of C * -algebras. Fix a local trivialization with diffeomorphisms α y : Z y → X. If we pull back the inner products on H y to H = L 2 (X) with the induced maps, then the corresponding Gram operator G y , expressing this pullback inner product in terms of the original one on L 2 (X), is the multiplication with a smooth positive function m y which depends continuously on y: the density of α * y µ y with respect to a chosen measure µ on X. Note that G y belongs to A and its norm, which is just the supremum, depends continuously on y. Now compose the original trivialization of A y with conjugation by G y and the resulting trivialization will respect the C * -algebra structures, but inherit the norm continuity of transition maps. To summarize: with a canonical modification (given in terms of the inner products) we have obtained trivializations of our bundle ℵ as a bundle of C * -algebras, as claimed. Definition 6. We denote by A the set of continuous sections of the bundle ℵ of C * -algebras. With the pointwise operations and the supremum norm, this becomes a C * -algebra. The underlying topological algebra is canonically associated to π : Z → Y , the norm and the * -operation depend on the choice of the family of metrics (g y ) y∈Y . The principal symbol and the boundary principal symbol extend continuously to two families of C algebra homomorphisms σ y : A y → C(S Z y ) and γ y : A y → C(S * ∂Z y , L(L 2 (R ≥0 ))), where S * denotes cosphere bundle and L bounded operators. Here γ y is well defined, since the structure group of the bundle π : Z → Y leaves the boundary defining function invariant, see [8,Theorem 2.4.11]. Let us denote by S * Z the disjoint union of all S * Z y . This can canonically be viewed as the total space of a fiber bundle over Y with structure group G. One analogously defines S * ∂Z = ∪ y S * ∂Z y and S * Z • = ∪S * Z • y . Definition 7. Given A ∈ A, let σ A be the function on S * Z defined by piecing together all the σ y 's. Then A → σ A defines a C * -algebra homomorphism σ : A −→ C(S * Z). One also gets, analogously, γ : A −→ C(S * ∂Z, L(L 2 (R ≥0 ))). Let K denote the subalgebra of A consisting of the sections (A y ) y∈Y such that A y is compact for every y ∈ Y . It follows immediately from the corresponding statement for a single manifold that ker σ ∩ ker γ = K. It is also straightforward to generalize the description of ker γ for a single manifold (5): Theorem 8. The principal symbol restricted to ker γ induces a C * -algebra isomorphism (6) ker γ/K ≃ C 0 (S * Z • ). Here C 0 (S * Z • ) consists of the elements of C(S * Z) which, for every y ∈ Y , vanish on all points of S * Z y with base point belonging to ∂Z y . Regarding each f ∈ C(Z) as a family of multiplication operators on (H y ) y∈Y , furnishes an embedding of C(Z) in A, which we denote m : C(Z) → A. Mapping a g ∈ C(∂Z) to the boundary principal symbol of m(f ), where f ∈ C(Z) is such that its restriction to ∂Z is g, defines the C * -algebra homomorphism b : C(∂Z) → Im γ. Theorem 9. The homomorphisms b * : K i (C(∂Z)) → K i (Im γ), i = 0, 1, induced by b are isomorphisms. Proof: Given an open set U ⊆ Y , let us denote by π U : Z U = π −1 (U ) → U the restriction of π to U , by A U the algebra of sections in A which vanish outside U and by γ U the restriction of γ to A U . Moreover we let C 0 (∂Z U ) = {f ∈ C(∂Z) : supp f ⊆ π −1 ∂ (U )} and write b U for the restriction of b to C 0 (∂Z U ). If the bundle π is trivial over U , then A U is isomorphic to C 0 (U, A) and, with respect to this isomorphism, b U corresponds to the tensor product of the identity on C 0 (U ) with the corresponding map for a single manifold, also denoted by b on [16,17]. It is the content of [16, Corollary 8] that b induces a K-theory isomorphism onto the image of γ. It then follows from the Künneth formula for C * -algebras [21] that b U induces isomorphisms b U * : K i (C 0 (∂Z U )) −→ K i (Im γ U ), i = 0, 1, see Proposition 21 in Appendix B. Now let (Im γ) U denote the subset of Im γ consisting of those functions which vanish outside ∪ y∈U S * ∂Z y . It is obvious that Im γ U ⊆ (Im γ) U . Since both Im γ U and (Im γ) U are closed in C(S * ∂Z, L(L 2 (R ≥0 ))), to show that they are equal it suffices to show that the former is dense in the latter. This follows from the fact that multiplication by a complex continuous function with support contained in U maps (Im γ) U to Im γ U . This simple observation implies that, for open sets U and V , we have a canonical C * -algebra isomorphism (7) Im γ U∩V ∼ = {(f, g) ∈ Im γ U ⊕ Im γ V ; f = g}. Now suppose that we have shown b U * to be an isomorphism for some open U and that V is open and π trivial over V , and so in particular also over U ∩ V . We then consider the two -thanks to (7) - diagrams C 0 (∂Z U∩V ) → C 0 (∂Z U ) ↓ ↓ C 0 (∂Z V ) → C 0 (∂Z U∪V ) and Im γ U∩V → Im γ U ↓ ↓ Im γ V → Im γ U∪V . Because they are cartesian, we may extract from both diagrams cyclic exact Mayer-Vietoris sequences (see [4, 21.2.2] or [15, 7.2.1]), and we may use the K-theory maps induced by b U , b V , b U∩V and b U∪V to map the first cyclic sequence to the second. By assumption and the case of trivial bundles, the maps induced by b U , b V and b U∩V are isomorphisms. It then follows from the five-lemma that also b U∪V induces a K-theory isomorphism. Since Y has a finite cover by open sets over which π is trivial, induction shows that b induces K-theory isomorphisms. ✷ Using Theorem 8, we obtain the following commutative diagram of C * -algebra homomorphisms, whose horizontal lines are exact: 0 −→ C 0 (S * Z • ) −→ A/K γ −→ Imγ −→ 0  m •   m  b 0 −→ C 0 (Z • ) −→ C(Z) r −→ C(∂Z) −→ 0 . We have denoted by r the map that pieces together all restrictions r y : C(Z y ) → C(∂Z y ), y ∈ Y , and by Z • the union ∪ y Z • y . Since the isomorphism (6) is induced by the principal symbol, and the principal symbol of an operator of multiplication by a function is the function itself, the map m • in the diagram above is actually the map of composition with the canonical projection S * Z • → Z • . We may apply the cone-mapping functor [17, Lemma 9] to the above diagram and get (using the same arguments that prove (11) in [17]) the following commutative diagram of cyclic exact sequences (8) K 0 (C 0 (Z • )) −→ K 0 (C(Z)) ↓ m • * ↓ m * K 0 (C 0 (S * Z • )) −→ K 0 (A/K) ↓ ↓ K 1 (Cm • ) ∼ = −→ K 1 (Cm) ↓ ↓ K 1 (C 0 (Z • )) −→ K 1 (C(Z)) ↓ m • * ↓ m • * K 1 (C 0 (S * Z • )) −→ K 1 (A/K) ↓ ↓ K 0 (Cm • ) ∼ = −→ K 0 (Cm) ↓ ↓ K 0 (C 0 (Z • )) −→ K 0 (C(Z)) , where ∼ = denotes isomorphism. Up to this point, everything goes exactly as in the case of a single manifold, but here comes a difference: The homomorphism m 0 does not necessarily have a left inverse (in the case of a single manifold X, such a left inverse is defined by composition with a section of S * X), and hence the cyclic exact sequences above do not have to split into short exact ones. To proceed we now introduce the subalgebra A † of A and an associated subalgebra B of C(S * Z) with the properties outlined in the introduction: For each y ∈ Y , let B y denote the subalgebra of C(S * Z y ) consisting of the functions which do not depend on the co-variable over the boundary, that is, an f ∈ C(S * Z y ) belongs to B y if and only if the restriction of f to the points of S * Z y over ∂Z y equals g • p y , for some g ∈ C(∂Z y ), where p y : S * Z y → Z y is the canonical projection. We then define A † y as the C * -subalgebra of A y generated by {P + ; P is a pseudodifferential operator with the transmission property and σ y (P + ) ∈ B y }. Definition 10. Let B denote the subalgebra of C(S * Z) consisting of the functions whose restriction to each S * Z y belongs to B y . We let then A † be the C * -subalgebra of A consisting of the sections (A y ) y∈Y such that A y ∈ A † y for every y ∈ Y . Proposition 11. The C * -algebra A † /K is commutative, and the map A † /K ∋ [A]σ −→σ(A) ∈ B is a C * -algebra isomorphism. Proof. Let P = (P y ) be a family of pseudodifferential operators with symbol independent of the covariable over the boundary, i.e. a generator of A † . According to (3), γ(P ) can be considered as a function on ∂Z, acting for z ∈ ∂Z on L 2 (R ≥0 ) by multiplication with γ(P )(z). Moreover, for z ∈ ∂Z we have γ(z) = σ(z) independent of the co-variable by assumption. It follows that the composed algebra homomorphism σ : A † σ⊕γ − −− → C(S * Z) ⊕ C(S * ∂Z, L(L 2 (R ≥0 ))) pr − → C(S * Z) has the same kernel as σ ⊕ γ, namely K and so the map we consider is injective and in particular A † /K is commutative. By the very definition of A † , σ : A † → B has dense image, as a morphism of C * -algebras it is therefore also surjective. This allows us to describe the K-theory of A/K: Theorem 12. The composition K i (A/K) ι −1 * −→K i (A † /K)σ * −→K i (B) is an isomorphism, i = 0, 1. The proof makes use of the following proposition, which is easily established by a diagram chase, compare [10, Exercise 38 of Section 2.2]: Proposition 13. Let there be given a commutative diagram of abelian groups with exact rows, · · · → A ′ i f ′ i −→ B ′ i g ′ i −→ C ′ i h ′ i −→ A ′ i+1 → · · · ↑ ai ↑ bi ↑ ci ↑ ai+1 · · · → A i fi −→ B i gi −→ C i hi −→ A i+1 → · · · , where each c i is an isomorphism. Then the sequence · · · −→ A i (ai,−fi) −→ A ′ i ⊕ B i f ′ i ,bi −→ B ′ i hic −1 i g ′ i −→ A i+1 −→ · · · is exact, where f ′ i , b i is the map defined by f ′ i , b i (α, β) = f ′ i (α) + b i (β) . We are now ready to prove Theorem 12. Applying Proposition 13 to the diagram (8), we get the exact sequence (9) K 0 (C 0 (Z • )) → K 0 (C(Z)) ⊕ K 0 (C 0 (S * Z • )) → K 0 (A/K) ↑ ↓ K 1 (A/K) ← K 1 (C(Z)) ⊕ K 1 (C 0 (S * Z • )) ← K 1 (C 0 (Z • )) . We next consider the following diagram of commutative C * -algebras (10) C 0 (Z • ) m • −→ C 0 (S * Z • ) ↓ ↓ p2 C(Z) p1 −→ B . As C 0 (Z • ) is canonically isomorphic to {(f, g) ∈ C(Z) ⊕ C 0 (S * Z • ); p 1 (f ) = p 2 (g)}, the Mayer-Vietoris exact sequence associated to (10) is the exact sequence (11) K 0 (C 0 (Z • )) → K 0 (C(Z)) ⊕ K 0 (C 0 (S * Z • )) → K 0 (B) ↑ ↓ K 1 (B) ← K 1 (C(Z)) ⊕ K 1 (C 0 (S * Z • )) ← K 1 (C 0 (Z • )) . The map ι : B ∼ = A † /K ֒→ A/K and the identity on the other K-theory groups furnish morphisms from the cyclic sequence (11) to the cyclic sequence (9). The five lemma then shows that the induced maps in K-theory are isomorphisms. Together with Proposition 11 we obtain the assertion. ✷ The Boutet de Monvel family index theorem The index of a continuous function with values in Fredholm operators was defined by Jänich [11] and Atiyah [2]. Using the following Proposition 14, their definition can be extended to sections of our ℵ. Proposition 14. Let H and A be as above, k ∈ N and let (A y ) y∈Y ∈ M k (A) be such that, for each y, A y is a Fredholm operator, where we interpret M k (A) as the sections of the bundle with fiber M k (A y ). Then there are continuous sections s 1 , · · · , s q of H k such that the maps A y : H k y ⊕ C q −→ H k y ⊕ C q (v, λ) −→ (A y v + q j=1 λ j s j (y),(A) = [kerÃ] − [Y × C q ] ∈ K(Y ) . This is independent of the choices of q and of s 1 , · · · , s q and we call it the analytical index of A. If A = (A y ) y∈Y ∈ M k (A) is a section such that each A y is a Fredholm operator on H k y then the projection to M k (A/K) is invertible and hence defines an element of K 1 (A/K). Since ind a (A) is invariant under stabilization, homotopies and perturbations by compact operator valued sections, we get a homomorphism (12) ind a : K 1 (A/K) −→ K(Y ). Next we define the topological index, also as a homomorphism ind t : K 1 (A/K) −→ K(Y ). Let T * Z denote the union of all T * Z y , and B * Z the union of all B * Z y , equipped with their canonical topologies, where B * Z y denotes the bundle of closed unit balls of T * Z y . One may regard B * Z as a compactification of T * Z and identify the "points at infinity" with S * Z. Let ∼ denote the equivalence relation that identifies, for each y ∈ Y , all points of each ball of B * Z y which lies over a point of ∂Z y . The C * -algebra B of Theorem 12 is isomorphic to the algebra of continuous functions on the quotient space S * Z/∼. Let β : K 1 (C(S * Z/∼)) → K 0 (C 0 (T * Z • )) denote the index map associated to the short exact sequence 0 −→ C 0 (T * Z • ) −→ C(B * Z/∼) −→ C(S * Z/∼) −→ 0, where T * Z • is the union over y ∈ Y of all points of T * Z y which lie over interior points of Z y and the map from C(B * Z/∼) to C(S * Z/∼) is induced by restriction. Let 2Z denote the union ∪ y 2Z y , where each 2Z y is the double of Z y , and π d : 2Z → Y the canonical projection. This can be given the structure of a Diff(2X)-bundle, with trivializations obtained by "doubling" (as explained at the beginning of Section 2) the trivializations of the bundle π : Z → Y . Each fiber 2Z y is then equipped with the smooth structure induced by the trivializations of π d : 2Z → Y and we can form the bundles T * 2Z and S * 2Z as the unions, respectively, of all cotangent bundles T * (2Z y ) and of all cosphere bundles S * (2Z y ), y ∈ Y . We denote by as-ind t : K 0 (C 0 (T * 2Z)) → K(Y ) the composition of Atiyah and Singer's [3] topological families-index for the bundle of closed manifolds 2Z with the canonical isomorphism K(T * 2Z) ≃ K 0 (C 0 (T * 2Z)). Theorem 12 allows us to define the topological index: Definition 16. The topological index ind t is the following composition of maps ind t : K 1 (A/K)σ * •ι −1 * −→ K 1 (C(S * Z/∼)) β −→K 0 (C 0 (T * Z • )) e * −→ K 0 (C 0 (T * 2Z)) ↓ as−indt K(Y ), where e : C 0 (T * Z • ) → C 0 (T * 2Z) denotes the map which extends by zero. If A = (A y ) y∈Y ∈ A is a family of Fredholm operators we denote by ind t (A) the topological index evaluated at the element of K 1 (A/K) that A defines. Theorem 17. Let A = (A y ) y∈Y ∈ A be a continuous family of Fredholm operators in the closure of the Boutet de Monvel algebra for each y. Then (13) ind a (A) = ind t (A). Proof: Our strategy is to derive the equality of the indices from the classical Atiyah-Singer index theorem for families [3,Theorem (3.1)]. To this end we define an operator familyÂ acting on a vector bundle over the double of Z by a gluing technique involving the principal symbol family of A. We proceed in several steps. Step 1 consists of a few preliminary remarks on the choice of the representative of the K-theory class of A. In Step 2 we describe the construction of the bundle. We then define the operator familyÂ over 2Z in Step 3. Its topological index coincides with that of A as we shall see in Step 4. The equality of the analytic indices of A andÂ is the content of Step 5. Step 1. We need to prove that ind t and ind a coincide on K 1 (A/K). Using that K 1 (A/K) = K 1 (A † /K) by Theorem 12, an arbitrary element of K 1 (A/K) is a class [[A]] 1 (the inner brackets denoting a class in the quotient by the compacts), for some operator family A = (A y ) y∈Y ∈ M k (A † ), k ∈ N, such that, for each y, A y : H k y → H k y is a Fredholm operator with symbol in B. It will be convenient to pick a representative with special properties. We denote by C ∞ (S * X/ ∼) the subset of C ∞ (S * X) of functions which factor through S * X/∼, i.e. are independent of the co-variable at the boundary. The algebraic tensor product C 0 (U ) ⊗ C ∞ (S * X/∼) is dense in C(U × S * X/∼) for every open subset U of Y . Furthermore, the inclusion of the space of all elements in C ∞ (S * X/∼) which are independent of the co-variable even in a neighborhood of ∂Z into C ∞ (S * X/ ∼) is a homotopy equivalence. We can therefore assume that the symbol family (σ y (A y )) y∈Y is given as a finite sum of elements supported in open subsets U of Y over which Z is trivial, and each of these is a pure tensor in C 0 (U ) ⊗ C ∞ (S * X) which is independent of the co-variable near the boundary. Hence it suffices to prove equality for such an A. Step 2. For each y ∈ Y , let Z + y and Z − y denote the two copies of Z y which are glued together at ∂Z y to form 2Z y . The map i y : ∂Z + y → ∂Z − y identifies the two copies of ∂Z y . We define E y as the quotient of the disjoint union Z + y × C k ∪ Z − y × C k by the equivalence relation that identifies the pairs (x, v) and (x ′ , w) if and only if they are equal or x ′ = i y (x), x ∈ ∂Z + y , and w = σ y (A y )(x)v (remembering that at points of S * Z y over ∂Z y , σ y (A y ) is independent of the co-vector variable). This set E y naturally becomes a smooth vector bundle over Z y . Let E denote the union of all E y , which in the same way becomes a vector bundle over Y . When defining families of smooth manifolds with smooth vector bundles, Atiyah and Singer make the technical assumption that the fiberwise vector bundles are isomorphic to a fixed vector bundle on the typical fiber. If Y is not connected, this is not necessarily satisfied. However, the isomorphism type of E y depends only on the homotopy type of the map σ y , in particular only on the component of the space of all continuous maps from ∂Z y to M k (C) in which it lies. By the compactness of Y , the latter decomposes into finitely many open and closed subsets over each of which the isomorphism type of E y is constant. As the K-theory of Y as well as A/K split as direct sums under such disjoint union decompositions of Y , and as ind a , ind t respect this, we can restrict to one such subset of Y . Then we are canonically in the situation of [3, Definition 1.2], i.e. E is a smooth vector bundle over the family of smooth manifolds 2Z. Step 3. Let π s : S * 2Z → 2Z denote the canonical projection and S * Z + and S * Z − , respectively, the union of all S * Z + y and S * Z − y , y ∈ Y . The bundle π * s E can be seen as the disjoint union of S * Z + ×C k and S * Z − × C k quotiented by the equivalence relation that identifies a boundary point ( s, v) in S * Z + × C k with (s, σ A (s) · v) in S * Z − × C k . Similarly, the bundle S * 2Z × C k can be seen as the disjoint union of S * Z + × C k and S * Z − × C k quotiented by the equivalence relation that identifies a boundary point (s, v) in S * Z + × C k with (s, v) in S * Z − × C k . We then defineâ ∈ Hom(π * s E, S * 2Z × C k ) by (14)â(s, v) = σ A (s) · v, if (s, v) ∈ S * Z + × C k , v, if (s, v) ∈ S * Z − × C k . We want to show thatâ is the symbol of a continuous family of pseudodifferential operators. As any element of Hom(π * s E, S * 2Z × C k ), ourâ can be regarded as a family (â y ) y∈Y ,â y ∈ Hom(π * s E y , S * 2Z y × C k ). It is easily checked that our definition ofâ indeed mends continuously at boundary points. But more is true. Since σ y (A y ) is smooth and independent of the co-variable near the boundary, eachâ y is smooth. Moreover, since we assumed in Step 1 that a is a finite sum of local elementary tensors, we see thatâ is the symbol of an Atiyah-Singer family of pseudodifferential operators on 2Z 1 . Step 4. Let ι : K 0 (C 0 (T * 2Z)) → K(B * 2Z, S * 2Z) ≃ K(T * 2Z) denote the canonical isomorphism (we refer to [4] and mainly [12] for topological K-theory definitions and notation). By Definition 16, it is enough to show that ι(e * (β([σ A ] 1 ))) is equal to the element of K(B * 2Z, S * 2Z) defined by the triple (π * b E, B * 2Z × C k ,â), where π b : B * 2Z → 2Z denotes the canonical projection. The main step here is to understand β([σ A ] 1 ). Now, σ A can and will be considered as a function on S * Z/ ∼ with values in Gl k (C), representing an element in K 1 (C(S * Z/ ∼)) and at the same time the corresponding element of the topological K-theory K 1 (S * Z/ ∼), [12, 3.2]. Recall from [12, 3.21] that for the pair of compact topological spaces S * Z/ ∼ ⊂ B * Z/ ∼, the boundary map in topological K-theory assigns to σ A the relative K-class ((B * Z/∼) × C k , (B * Z/∼) × C k , σ A ), corresponding under the excision isomorphism K((B * Z/∼), (S * Z/∼)) ∼ = K(B * Z, S * Z) to (B * Z × C k , B * Z × C k , σ A ), compare [12, 2.35]. Moreover, this corresponds to β under the isomorphism with C * -algebra K-theory. We next have to compute the map e top : K(B * Z, S * Z) → K(B * 2Z, S * 2Z) in topological K-theory, representing e * : K 0 (C 0 (T * Z)) → K 0 (C 0 (T * 2Z)). Recall, however, that e top (V, W, τ ) is given by any extensionṼ of V ,W of W to B * 2Z and an extension of τ to an isomorphismτ betweenṼ andW on all of (B * 2Z \ B * Z) ∪ S * Z,τ finally restricted to S * 2Z. Finally, observe that (π * b E, B * 2Z × C k ,â) provides exactly such an extension (asâ extends as id over all of B * 2Z \B * Z) and therefore represents ιe * (β([σ A ])), as we had to prove. Step 5. In order to show that the analytic indices coincide, we will introduce yet another operator family. Since σ(A) is independent of the co-variable near the boundary, there is an open set U ⊆ 2Z containing Z − = ∪ y Z − y and a bundle isomorphism Φ : E\| U −→ U × C k such that the restriction ofâ to π −1 s (U ) is equal to the pullback of Φ by π s . Let (χ + y ) y∈Y and (χ − y ) y∈Y be continuous families of smooth functions on 2Z with 0 ≤ χ ± y ≤ 1, (χ + y ) 2 + (χ − y ) 2 = 1. Moreover, let the support of each χ + y be contained in the interior of Z + y and χ + y ≡ 1 outside a neighborhood of ∂Z + y in U . ThenB y = χ + yÂy χ + y + χ − y Φ y χ − y , defines a family of pseudodifferential operators in the sense of Atiyah and Singer which has the same principal symbol -and hence the same analytic index -asÂ. For each y ∈ Y , we canonically identify the space L 2 (E y ) of L 2 -sections of E y with the direct sum L 2 (Z + y ; C k ) ⊕ L 2 (Z − y ; C k ) and denote by e ± y and r ± y the maps of extension by zero and restriction, e ± y : L 2 (Z ± y ; C k ) → L 2 (E y ) and r ± y : L 2 (2Z y ; C k ) → L 2 (Z ± y ; C k ). Then B y = r + yBy e + y defines a continuous family B = (B y ) y∈Y in M k (A). As σ(A) = σ(B) (and hence γ(A) = γ(B)), it suffices to prove that the analytic indices of B andB are equal. Proposition (2.2) of [3], applied to the familyB provides us with sections s j y ∈ C ∞ (2Z y ; C k ), y ∈ Y , 1 ≤ j ≤ q, such thatQ y : C ∞ (2Z y ; E y ) ⊕ C q −→ C ∞ (2Z y ; C k ) (u; λ 1 , · · · , λ q ) −→B y (u) + q j=1 λ j s j y is onto, kerQ = (kerQ y ) y∈Y is a vector bundle and the analytic index ofB is equal to [kerQ] − [Y × C q ]. Now let t j y = r + y s j y ∈ C ∞ (Z y ; C k ). The continuity with respect to y that we get from [3, Proposition (2. 2)] is enough to ensure that (t j y ) y∈Y is a continuous section of our bundle of Hilbert spaces y∈Y L 2 (Z y ; C k ). We then define Q y : L 2 (Z y ; C k ) ⊕ C q −→ L 2 (Z y ; C k ) (u; λ 1 , · · · , λ q ) −→ B y (u) + q j=1 λ j t j y Since B y is elliptic, ker Q y ⊂ C ∞ (Z y ; C k ). Using that Φ y is local, it is straightforward to check that B y = e + y r + yBy e + y r + y + e − y r − yBy e − y r − y = e + y B y r + y + e − y r − y Φ y e − y r − y and, hence, ker Q y and kerQ y are isomorphic for each y (because Φ is an isomorphism). Moreover, Q y is also surjective: Given v ∈ L 2 (Z y ; C k ), if u ∈ L 2 (2Z y ; E y ) is a preimage of e + y v underQ y , then r + y u is a preimage of v under Q y . Hence the analytic index of B is given by [ker Q] − [Y × C q ]. The bundles ker Q = (ker Q y ) y∈Y and kerQ are isomorphic and then ind a (B) = [ker Q] − [Y × C q ] = [kerQ] − [Y × C q ] = ind a (B), as we wanted. ✷ Nontrivial bundles In this section we discuss families of Boutet de Monvel operators acting between vector bundles. The case considered in the first two sections correspond to the case of trivial bundles over the manifolds and the zero bundle over the boundary. In addition to the data assumed up to this point (a bundle of manifolds π : Z → Y with fiber X), we take smooth vector bundles E and F over X and ∂X, respectively. Let Diff(∂X, F ) denote the group of diffeomorphisms of F which map fibers to fibers linearly, and let G E denote the group of diffeomorphisms of E which map fibers to fibers linearly and whose restrictions to the base belong to the group G defined on page 100001. We equip Diff(∂X, F ) with its canonical topology [3, page 123] and do a similar construction for G E . Note that there are homomorphisms "forget the action in the fiber" h ∂ : Diff(∂X, F ) → Diff(∂X) and h : G E → G. Define the fiber product group G r := {(φ, ψ) ∈ Diff(∂X, F ) × G E \| h ∂ (φ) = h(ψ)}. Let (p :Ẽ → Z; q :F → ∂Z) be maps such that (π • p :Ẽ → Y ; π ∂ • q :F → Y ) are bundles with, respectively, fibers E and F and structure group G r . It follows that, for each pair of local trivializations (α, β) of (π • p :Ẽ → Y ; F → Y ) there are local trivialization α 0 of π : Z → Y and β 0 of ∂Z → Y such that the diagram (15) (π • p) −1 (U ) α −→ U × E   p   π −1 (U ) α0 −→ U × X commutes, where the right vertical arrow is the identity on U times the bundle projection on E. This defines a vector bundle structure for p :Ẽ → Z. Moreover, for each y ∈ Y , the restriction of p tõ E y = (π • p) −1 (y) defines a smooth vector bundle p y :Ẽ y → Z y , isomorphic to E → X. We obtain the corresponding result for the the map q and get a vector bundle q :F → ∂Z and, for each y ∈ Y , a smooth vector bundle q y :F y → ∂Z y isomorphic to F → ∂X. Choose now, in addition to the family of Riemannian metrics (g y ) y∈Y families of Hermitean metrics on E y and F y which depend continuously on y ∈ Y . Using them, we get families of Hilbert spaces H y := L 2 (Z y ; E y )⊕L 2 (∂Z y ; F y ) which patch together to a bundle of Hilbert spaces. Let A(E, F ) y denote the C * -subalgebra of the algebra of all bounded operators on H y generated by the polyhomogeneous Boutet de Monvel operators of order and class zero. Exactly as [3, Proposition 1.3] our Proposition 4 generalizes to the case of non-trivial bundles and their diffeomorphisms and is the basis for the generalization of Corollary 5 to the case of non-trivial bundles: the A(E, F ) y form in a canonical way a continuous bundle of C * -algebras, which we continue to call ℵ by abuse of notation. Let A denote the set of continuous sections of the bundle ℵ, forming again a C * -algebra with pointwise operations and supremum norm. The K-theory results of Section 2 can be extended to this more general setting using arguments similar to those used in [17]. In particular, the analytic and topological index given in Section 3 can also be defined as maps K 1 (A) → K(Y ). Theorem 17 then extends to this more general setting. Remark 18. Variants of Theorem 17, the family index theorem for the Boutet de Monvel algebra for real K-theory or for equivariant K-theory should hold as well, and one should be able to derive them along the lines used in the present article. In this appendix, we prove that, for any bundle (over a paracompact space) with structure group Diff(X) we have a unique (up to isomorphism) reduction to the structure group G. In other words, the functor from bundles (over a given paracompact base) with structure group G to bundles with structure group Diff(X) which "forgets the collar" is an equivalence of categories. [This is similar to the (unique up to isomorphism) choice of a Riemannian metric on a given finite dimensional vector bundle: reduction of the structure group from Gl(n) to O(n).] It is well known that we get this unique reduction of structure group if the inclusion G → Diff(X) is a homotopy equivalence, compare [6] for a rather refined version of this fact. We therefore show Theorem 19. The inclusion G → Diff(X) (and therefore the corresponding map BG → BDiff(X)) are homotopy equivalences. Proof. Observe first that G and Diff(X) as well as BG and BDiff(X) are paracompact Fréchet manifolds by [14,Sections 41,42,44.21] (the reference is for Diff(X), but the proofs easily generalize to G). Therefore it suffices by [19,Theorem 15] to show that G → Diff(X) is a weak homotopy equivalence and it follows automatically that it is a homotopy equivalence. To show that the map is a weak homotopy equivalence, we have for a continuous map f : K → Diff(X), where K is a compact CW-complex, to construct a homotopy f s from f 0 = f to an f 1 which takes values in G. Moreover, the homotopy should be constant on every CW-subcomplex K 0 of K where f already maps to G. Note that K 0 is a deformation retract of a neighborhood U , i.e. there is a homotopy h : K × [0, 1] → K from the identiy to h 1 such that h 1 (U ) = K 0 and such that h t is the identity on K 0 . Be precomposing with h 1 we can therefore assume that f maps the neighbourhood U of K 0 to G. Let us now construct the family f t . Choose η ∈ (0, 1] such thatf (k) = δ • f (k) • δ −1 maps ∂X × [0, η) to ∂X × [0, 1) for all k ∈ K and writef (k)(x ′ , t) = (ϕ(x ′ , t; k), τ (x ′ , t; k)). In two steps we shall now first deform τ to a functionτ which equals t for small t and then ϕ to a function which depends only on x ′ for small t. Observe that, as f (k) is a diffeomorphism of a manifold with boundary, ∂τ ∂t > 0 and therefore, by the compactness of K, if we choose η small enough, C > ∂τ ∂t > c > 0 for some C > c > 0 on all of K × ∂X × [0, η). Pick a smooth function a : [0, η) → [0, 1] such that a(t) ≡ 0 for t close to zero, a(t) ≡ 1 for t close to η and such thatτ (x ′ , t; k) = (1 − a(t))t + a(t)τ (x ′ , t; k), (x ′ , t) ∈ ∂X × [0, η), satisfies ∂τ (x ′ , t; k)/∂t ≥ c/2 for every x ′ ∈ ∂X end every k ∈ K. To construct such an a, we use the uniform growth of τ : Choose, for some given ε > 0, the function a so that (1 − a)t is monotonely increasing on the interval [0, 4ε] with (1 − a)t = t on [0, ε] and (1 − a)t = 2ε on [3ε, 4ε]. Then a is necessarily increasing with a ≡ 0 near 0 and a(4ε) = 1/2. Moreover,τ is strictly increasing as τ is. Finally choose a on [4ε, η] such that (1 − a)t monotonely decreases to 0 and equals zero on [η − ε, η]. Moreover, we arrange for the derivative ∂ t ((1 − a)t) to be always ≥ −2 2ε η−5ε . Again, a is necessarily increasing with a ≡ 1 near η. The derivative ∂ t (aτ ) can therefore be estimated from below by c/2. For ε sufficiently small, we will therefore have 2 2ε η−5ε < c and thus ∂ tτ (x ′ , t; k) > 0 for all x ′ , t, k. Note that thenτ (x ′ , t; k) = t for t close to zero, andτ (x ′ , t; k) = τ (x ′ , t; k) for t close to η, uniformly in k. We then let τ s = sτ + (1 − s)τ, 0 ≤ s ≤ 1. Then ∂τs ∂t ≥ c/2 on K × ∂X × [0, η). For the second step fix a smooth function ρ : [0, 1) → [0, 1) with ρ(t) = 0 for t < ε and ρ(t) = t for t > 1 − ε. Next choose a smooth family of smooth functions ρ s , 0 ≤ s ≤ 1 such that ρ 0 is the identity and ρ 1 = ρ. By compactness, we have a uniform bound \|dρ s (t)/dt\| ≤ R. For a given η > 0, define ρ η s (t) : [0, η) → [0, η); t → ηρ s (η −1 t). Then still \|dρ η s /dt\| ≤ R, even independently of η. Let ϕ s (x ′ , t) := ϕ(x ′ , ρ η s (t)) andf s (k)(x ′ , t) = (ϕ s (x ′ , t), τ s (t)). Thenf s equals the givenf for t close to η. Therefore f s = δ −1 •f s • δ extends (independently of s) to a self-map of X. Moreover, \| ∂τs ∂x ′ \| ≤ \| ∂τ ∂x ′ \| for all s. And for t = 0 we have ∂τ ∂x ′ = 0. On the other hand, ∂ρs ∂x ′ \| (x ′ ,t) = ∂ρ ∂x ′ \| (x ′ ,ρs(t)) is, for η small enough, invertible on [0, η] with uniform bound on the norm of the inverse (and with better bounds if we choose η smaller), and \| ∂φs ∂t (x ′ , t)\| = \| ∂φ ∂t (x ′ ,ρs(t)) \| · \|dρ η s /dt(t)\| which is uniformly bounded, independent of η. By choosing η small enough, therefore ∂τ s will be linearly independent from ∂ϕ(x ′ , ρ s (t)) and so f s (k) is a submersion for all s, k. We check that we actually constructed diffeomorphisms. We made our construction such that all the maps f s (k) are submersions which map the boundary to itself, therefore the image is an open subset of X. As X is compact, the image is also closed, and the map being a local diffeomorphism, is a covering map. Because it is homotopic to the diffeomorphism f (k), it is a trivial covering map and therefore a diffeomorphism. It is obvious that f 0 = f and f 1 (k) lies in the variant of G where 1/2 is replaced by η − ǫ. Next, we compose with a family of reparametrizations of the collar [0, 1) which stretches [0, η − ǫ) to [0, 1/2) such that in the end we really map to G. Note that our construction is carried out in such a way that for k ∈ U , where f (k) was already in G, f s (k) ∈ G for all s, although, because of the last reparametrization step, not necessarily f s (k) = f (k). Therefore, finally, we choose a function β : K → [0, 1] which is 1 outside U and 0 on K 0 and replace the homotopy f s (k) with f β(k)s (k). This yields the desired homotopy from f 0 = f to an f 1 taking values in G. Moreover, the mapping is constant on K 0 . Appendix B. The Künneth formula By the "Künneth formula", we mean the following theorem of Schochet [21]: Theorem 20. Let A and B be C * -algebras with A in the smallest subcategory of the category of separable nuclear C * -algebras which contains the separable Type I algebras and is closed under the operations of taking ideals, quotients, extensions, inductive limits, stable isomorphism, and crossed product by Z and by R. Then there is a natural Z/2-graded exact sequence We use this Theorem to prove a statement made in the proof of Theorem 9: Proposition 21. b U * : K i (C 0 (∂Z U )) → K i (Im γ U ) is an isomorphism, i = 0, 1. Proof: Let A = C 0 (U ) and B = C(∂X). Then Im γ U is equal to A ⊗ C, where C is the image of the boundary principal symbol for the single manifold X. As explained in the Introduction of [16], C can be regarded as a C * -subalgebra of C(S * ∂X) ⊗ T , where T denotes the Toeplitz algebra. Since T belongs to the category defined in the statement of Theorem 20 (see Examples 5.6.4 and 6.5.1 in [18]), we may apply Schochet's theorem for A ⊗ B and for A ⊗ C. Now let b : C(∂X) → C be the map analogous to the map b defined right before the statement of Theorem 9. In [16,Section 3], it is proven that b induces a K-theory isomorphism (b was denoted b in [16,17]). Using that the exact sequence of Theorem 20 is natural, we can map (16) to the corresponding sequence obtained by replacing B with C. Since the maps induced by b are isomorphisms, it follows from the five-lemma that the maps induced by b U = id A ⊗ b are also isomorphisms. ✷ K * (A) ⊗ K * (B) → K * (A ⊗ B) → Tor(K * (A), K * (B)) → 0. Münster Journal of MathematicsVol. 1 (2008), 99999-99999 Recall that they use a slightly stricter definition of operator families: While we here require continuity of the family with respect to the L 2 (X)-operator norm, they take into account the norms on the whole range of Sobolev spaces.Münster Journal of Mathematics Vol. 1 (2008), 99999-99999 AcknowledgementsWe greatly benefited from numerous discussions with our friends Johannes Aastrup and Daniel Tausk. We thank them for their generosity and for the great time we had talking Math to them. We are also grateful to Jochen Ditsche for pointing out Proposition 13 to us. Severino Melo was partially supported by a grant from the Brazilian agency CNPq (Processo 304783/2009-9). Thomas Schick was partially supported by the Courant Center "Higher order structures of mathematics" within the Excellence initiative's Institutional strategy of Georg-August-Universität Göttingen.Appendix A. Reduction of the structure group Let, as in the main body of the text, X be a compact smooth manifold with boundary ∂X, and fix a collar diffeomorphism δ : U → ∂X × [0, 1) with collar coordinate x n . Recall that G was defined as the subgroup of the diffeomorphism group Diff(X) of those diffeomorphisms which respect the product structure and collar coordinate for x n ∈ [0, 1/2). For convenience, in the text we were working with bundles of manifolds modelled on X and with structure group G, i.e. with a canonically defined collar of the boundary in each fiber of the bundle. Relative Chern character, boundaries and index formulas. P Albin & R. Melrose, J. Topol. Anal. 13P. Albin & R. Melrose. Relative Chern character, boundaries and index formulas. J. Topol. Anal. 1 (2009), no. 3, 207-250. . M F Atiyah, K-Theory, D. W. Anderson. W. A. Benjamin, Inc.New York-AmsterdamM. F. Atiyah. K-Theory, Lecture notes by D. W. Anderson. W. A. Benjamin, Inc., New York-Amsterdam, 1967. The index of elliptic operators IV. M F M Atiyah & I, Singer, Ann. of Math. 2M. F. Atiyah & I. M. Singer. The index of elliptic operators IV. Ann. of Math. (2) 93 (1971), 119-138. K-Theory for Operator Algebras. B Blackadar, Cambridge University PressCambridgeB. Blackadar. K-Theory for Operator Algebras. Cambridge University Press, Cambridge, 1998. Boundary problems for pseudo-differential operators. L Boutet De Monvel, Acta Math. 1261-2L. Boutet de Monvel. Boundary problems for pseudo-differential operators. Acta Math. 126 (1971), no. 1-2, 11-51. Partitions of unity in the theory of fibrations. A Dold, Ann. of Math. 2A. Dold. Partitions of unity in the theory of fibrations. Ann. of Math. (2) 78 (1963), 223-255. On the theory of multidimensional singular integral operators. I C Gohberg, Dokl. Akad. Nauk SSSR. 133I. C. Gohberg. On the theory of multidimensional singular integral operators. Dokl. Akad. Nauk SSSR 133 (1960), 1279-1282. Functional Calculus of Pseudodifferential Boundary Problems. G Grubb, BostonSecond Edition. BirkhäuserG. Grubb. Functional Calculus of Pseudodifferential Boundary Problems, Second Edition. Birkhäuser, Boston, 1996. The essential spectrum of elliptic systems of mixed order. G Grubb, & G Geymonat, Math. Ann. 227G. Grubb & G. Geymonat. The essential spectrum of elliptic systems of mixed order. Math. Ann. 227 (1977), 247-276. Algebraic topology. A Hatcher, Cambridge University PressCambridgeA. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. Vektorraumbündel und der Raum der Fredholm-Operatoren. K Jänich, Math. Ann. 161K. Jänich. Vektorraumbündel und der Raum der Fredholm-Operatoren. Math. Ann. 161 (1965), 129-142. K-theory. An introduction. Grundlehren der Mathematischen Wissenschaften. M Karoubi, Springer-Verlag226Berlin-New YorkM. Karoubi. K-theory. An introduction. Grundlehren der Mathematischen Wissenschaften, Band 226. Springer-Verlag, Berlin-New York, 1978. An algebra of pseudo-differential operators. J J Kohn, & L Nirenberg, Comm. Pure Appl. Math. 18J. J. Kohn & L. Nirenberg. An algebra of pseudo-differential operators. Comm. Pure Appl. Math. 18 (1965), 269-305. The convenient setting of global analysis. Mathematical Surveys and Monographs 53. A W Kriegl & P, Michor, American Mathematical SocietyProvidence, RIA. Kriegl & P. W. Michor. The convenient setting of global analysis. Mathematical Surveys and Monographs 53. American Mathematical Society, Providence, RI, 1997. Lecture Notes on the K-Theory of Operator Algebras. R Matthes, & W Szymanski, R. Matthes & W. Szymanski. Lecture Notes on the K-Theory of Operator Algebras, www.impan.gov.pl/Manuals/K theory.pdf, 2007. Schrohe. C * -structure and K-theory of Boutet de Monvel's algebra. S T Melo, R Nest, E , J. reine angew. Math. 561S. T. Melo, R. Nest, E. Schrohe. C * -structure and K-theory of Boutet de Monvel's algebra. J. reine angew. Math. 561 (2003), 145-175. A K-theoretic proof of Boutet de Monvel's index theorem for boundary value problems. S T Melo, T Schick, E Schrohe, J. reine angew. Math. 599S. T. Melo, T. Schick, E. Schrohe. A K-theoretic proof of Boutet de Monvel's index theorem for boundary value problems. J. reine angew. Math. 599 (2006), 217-233. C * -Algebras and Operator Theory. G J Murphy, Academic PressBostonG. J. Murphy. C * -Algebras and Operator Theory. Academic Press, Boston, 1990. Homotopy theory of infinite dimensional manifolds. R S Palais, Topology. 5R. S. Palais. Homotopy theory of infinite dimensional manifolds. Topology 5 (1966), 1-16. Index Theory of Elliptic Boundary Problems. S Rempel, & B.-W Schulze, Akademie VerlagBerlinS. Rempel & B.-W. Schulze. Index Theory of Elliptic Boundary Problems. Akademie Verlag, Berlin, 1982. Topological methods for C * -algebras II: geometric resolutions and the Künneth formula. C Schochet, Pacific J. Math. 982C. Schochet. Topological methods for C * -algebras II: geometric resolutions and the Künneth formula. Pacific J. Math. 98 (1982), no. 2, 443-458. A short introduction to Boutet de Monvel's calculus; Approaches to Singular Analysis. E Schrohe, BerlinE. Schrohe. A short introduction to Boutet de Monvel's calculus; Approaches to Singular Analysis (Berlin, 1999). . Oper. Theory Adv. Appl. 125Oper. Theory Adv. Appl. 125, 85-116, Birkhäuser, Basel, 2001. B.-W Schulze, Pseudo-Differential Operators on Manifolds with Singularities. AmsterdamNorth HollandB.-W. Schulze. Pseudo-Differential Operators on Manifolds with Singularities. North Holland, Amsterdam, 1991. Integro-differential operators on vector bundles. R T Seeley, Trans. Amer. Math. Soc. 117R. T. Seeley. Integro-differential operators on vector bundles. Trans. Amer. Math. Soc. 117 (1965), 167-204. Universidade de São Paulo Rua do Matão 1010, 05508-090 São Paulo, Brazil-E-mail: [email protected]. T Severino, Melo Instituto De Matemática E Estatística, Severino T. Melo Instituto de Matemática e Estatística, Universidade de São Paulo Rua do Matão 1010, 05508-090 São Paulo, Brazil- E-mail: [email protected] URL: http://www.ime.usp.br/~toscano [email protected] URL. Göttingen, Germany E-mail37073Thomas Schick Mathematisches Institut, Georg-August-Universität Göttingen BunsenstrThomas Schick Mathematisches Institut, Georg-August-Universität Göttingen Bunsenstr. 3-5, 37073 Göttingen, Germany E-mail: [email protected] URL: http://www.uni-math.gwdg.de/schick	[]
[ "Computational approaches to many-body dynamics of unstable nuclear systems", "Computational approaches to many-body dynamics of unstable nuclear systems" ]	[ "Alexander Volya \nDepartment of Physics\nFlorida State University\n32306-4350TallahasseeFLUSA\n" ]	[ "Department of Physics\nFlorida State University\n32306-4350TallahasseeFLUSA" ]	[]	The goal of this presentation is to highlight various computational techniques used to study dynamics of quantum many-body systems. We examine the projection and variable phase methods being applied to multi-channel problems of scattering and tunneling; here the virtual, energy-forbidden channels and their treatment are of particular importance. The direct time-dependent solutions using Trotter-Suzuki propagator expansion provide yet another approach to exploring the complex dynamics of unstable systems. While presenting computational tools, we briefly revisit the general theory of the quantum decay of unstable states. The list of questions here includes those of the internal dynamics in decaying systems, formation and evolution of the radiating state, and lowenergy background that dominates at remote times. Mathematical formulations and numerical approaches to time-dependent problems are discussed using the quasi-stationary methods involving effective Non-Hermitian Hamiltonian formulation.	null	[ "https://arxiv.org/pdf/1412.6335v1.pdf" ]	118,504,520	1412.6335	fe06f1b46dfe71fe21f3b47d883752cd4ba150ef	Computational approaches to many-body dynamics of unstable nuclear systems Alexander Volya Department of Physics Florida State University 32306-4350TallahasseeFLUSA Computational approaches to many-body dynamics of unstable nuclear systems Quantum many-body dynamics, Time Dependent Continuum Shell ModelVariable Phase MethodTrotter-Suzuki propagator expansion The goal of this presentation is to highlight various computational techniques used to study dynamics of quantum many-body systems. We examine the projection and variable phase methods being applied to multi-channel problems of scattering and tunneling; here the virtual, energy-forbidden channels and their treatment are of particular importance. The direct time-dependent solutions using Trotter-Suzuki propagator expansion provide yet another approach to exploring the complex dynamics of unstable systems. While presenting computational tools, we briefly revisit the general theory of the quantum decay of unstable states. The list of questions here includes those of the internal dynamics in decaying systems, formation and evolution of the radiating state, and lowenergy background that dominates at remote times. Mathematical formulations and numerical approaches to time-dependent problems are discussed using the quasi-stationary methods involving effective Non-Hermitian Hamiltonian formulation. Introduction There is no physical system that is truly isolated from the rest of the world, the closed system idealization may be convenient but becomes poor or completely invalid for many questions of modern-day science. In nuclear physics, as interests shift towards weakly bound, unbound or even dynamically evolving reaction states, the theoretical approaches for dealing with unstable dynamics of open quantum systems with multiple degrees of freedom must be revisited. The availability of advanced computational technologies calls forth innovative thinking and new philosophies in addressing these types of quantum many-body problems. In this presentation, using different models and realistic examples from the world of nuclear physics, we discuss computational strategies and techniques for dealing with dynamically unstable many-body systems. The Nuclear Theory in the Supercomputing Era venue is especially timely and allows us to put emphasis on some of the techniques, that due to their computational nature, remained behind the curtains in a number of recent investigations [1][2][3]. see also Refs. [2,4]. Consider a model of scattering illustrated in Fig. 1. In this one-dimensional problem two particles with masses µ 1 and µ 2 comprise a composite system of unit mass µ 1 +µ 2 = 1. The system can be described with the center-of-mass and relative coordinates, X = µ 1 x 1 + µ 2 x 2 and x = x 1 − x 2 , respectively. The two particles are confined by a potential v(x). The intrinsic Hamiltonian h = − 1 µ ∂ 2 ∂x 2 + v(x)(1) is assumed to have a complete set of discrete eigenstates ψ n (x) with corresponding intrinsic energies n : hψ n (x) = n ψ n (x), n = 0, 1, 2, ... Here the reduced mass is µ = µ 1 µ 2 and we select our units so that 2 /2 = 1. We assume that this system scatters off an infinite wall and the wall interacts only with the second particle. Therefore the full Hamiltonian is H = − ∂ 2 ∂X 2 + U (x 2 ) + h, where U (x 2 ) = ∞ if x 2 ≥ 0 0 if x 2 < 0 .(2) As illustrated in Fig. 1, we assume that the incident beam is traveling from the left and contains the projectiles in an intrinsic state (channel) n. A complete set of reflected waves is characterized by the amplitudes R nm defined here so that \|R mn \| 2 represents the probability for the initial beam in channel n to reflect in channel m; R nm = R mn due to time-reversal invariance. The scattering wave function is Φ(X, x) = e iKnX \|K n \| ψ n (x) + ∞ m=0 R mn \|K m \| e −iKmX ψ m (x),(3) where K n (E) = (E − n )(4) is the center-of-mass momentum of the two-particle system while in the n th intrinsic state, and E is the total energy. Figure 1: Schematic picture of scattering. A composite system of two particles bound by a harmonic oscillator potential scatters off an infinite wall. One of the particles does not interact with the wall, at the same time the wall is impenetrable for the second particle. A channel n is considered to be open if E ≥ n and the corresponding momentum K n is real. The conservation of particle-number in all the open channels necessitates m∈open \|R mn \| 2 = 1. The channel is closed if E < n , in which case K n is purely imaginary. We stress that the principal value of the square root is implied in Eq. (4). The boundary condition set by an impenetrable wall Φ(X, x) = 0 at x 2 = 0(5) is to be used for determining the set of coefficients R mn . Since at x 2 = 0 the center-ofmass coordinate X = µ 1 x, the boundary condition can be expressed in the intrinsic coordinate x only Φ(µ 1 x, x) = 0. Therefore we can project the reaction problem onto a complete set of intrinsic basis states, which leads to the following linear equation m D n m [−iµ 1 (K n + K m )] \|K m \| R mn = − δ n n \|K n \| ,(6) where the matrix D is defined as D mn (κ) = ψ m \| exp(κx)\|ψ n .(7) Eq. (6) represents a typical mathematical challenge associated with the formulation of reaction problems where reaction states are projected onto the intrinsic states; see also Sec. 3. It is a linear algebra problem where the construction of the scattering matrix amounts to matrix inversion in the projected space. The scattering energy E is a running parameter here, and studies of scattering at different energies is therefore time consuming. And, finally, the underlying matrix is highly singular and there are issues with convergence. The latter difficulty is the one that we would like to illustrate using this example. If the two particles forming a composite system are bound by a harmonic oscillator confinement, v(x) = µω 2 x 2 /2 in Eq. (1), the D-matrix is then known analytically [2]. Then to solve the problem we truncate the channel space at some large number N of oscillator quanta, and solve Eq. (6) using standard numerical techniques. This turns out to be a difficult task; the matrix element D mn (κ) for virtual channels, where κ is real, are exponentially large, making the process of matrix inversion difficult and numerically unstable [2,5,6]. As shown in Fig. 2, left panel, the absolute values of the reflection amplitudes, R n ≡ R 0n exponentially diverge for increasingly remote virtual channels. While it is possible to overcome the numerical issues, further examination shows that the approach has fundamental flaws. In Fig. 2, right panel, the phase shift, defined as e 2iδ = −R 00 , is shown as a function of N . While satisfactory and seemingly convergent results can be easily found for the cases where the mass of the noninteracting particle is small, in general, as N increases, the results start oscillating; situations where the non-interacting particle is heavy and therefore deeply penetrates the wall are particularly difficult to handle. It was emphasized in Refs. [2,4] that there is no numerical convergence with increasing N . Variable Phase Method The above example shows that reaction problems call for new techniques. One approach, based on the Variable Phase Method (VPM), see Ref. [7], is proposed in Ref. [2]. The VPM is an effective technique for solving the coupled-channel problem of the form ∂ 2 ∂X 2 + K 2 n Ψ n (X) − n V nn (X)Ψ n (X) = 0,(8) where scattering observables are to be expressed relative to free-space solutions normalized to unit current Ξ ± nn (X) = e ±iKnX √ −2iK n δ nn ;(9) the ± sign corresponds to a wave moving in the right/left direction. In the VPM approach the coupled-channel Schödinger's equation (8) is reformulated as a set of first order differential equations for dynamic reflection and transmission amplitude matrices R nn (X ) and T nn (X ). These amplitudes correspond to a potential that is cut at X , namely to V nn (X)θ(X − X ) : , is plotted as a function of truncation N. The problems with the approach are highlighted by an unstable and oscillatory behavior of the phase shifts. The problem is particularly severe when the non-interacting particle of mass µ 1 is heavy. Different curves show phase-shifts for different mass ratios µ 1 /µ 2 = 1, 2, 3, 5, 10, as labeled; the exact values obtained with Variable Phase Method (see Sec. 2.2) are shown by the horizontal grid lines with the tic-marks on the right. Inset shows the case when µ 1 /µ 2 = 3 extending the study to considerably large values of N and emphasizing that for any choice of parameters the approach fails at some point. dR(X) dX = Ξ + + R(X) Ξ − V Ξ + + Ξ − R(X) , R nn (∞) = 0,(10)dT (X) dX = T (X) Ξ − V Ξ + + Ξ − R(X) , T nn (∞) = δ nn .(11) These equations, being solved from X = +∞ towards X → −∞, recover the reflection and transmission amplitudes R nn (−∞) = R nn and T nn (−∞) = T nn . Using factorization of the form Φ(X, x) = n Ψ n (X)ψ n (x) the Schrödinger's equation for the scattering problem described in Fig. 1 can be transformed into a coupled-channel equation (8) for the center-of-mass wave-functions Ψ n (X) where the folded potentials are V nn (X) = ∞ −∞ ψ * n (x) U (X, x) ψ n (x)dx .(12) Some representative results for the scattering problem where an oscillator-bound system interacts with an infinite wall are shown in Figs Probability density X µ 1 , E _____ 0.5, 0.5 0.9, 0.5 0.5, 7.5 0.9, 7.5 Figure 4: The density of probability for the center of mass of the projectile to be at a location X when it is reflected from an infinite wall at X = 0. energy. The kinetic energy is expressed in units of oscillator's ω and therefore for each integer value thereof a new channel opens. One can notice typical cusps at thresholds associated with the loss of flux into newly opened channels. In Fig. 4 the probability distribution for the center of mass is shown. The four curves show four of the most representative situations; low and high incident kinetic energies E = 0.5 ω and E = 7.5 ω, respectively, and two different mass-ratios µ 1 = 0.5 and 0.9. Time-dependent approach Turning to a time-dependent approach is a natural strategy for dealing with nonstationary systems. There are various computational techniques; see Ref. [8] for some recent tests and comparisons of methods being applied to one dimensional Schrödinger's equation. In time-dependent techniques preservation of unitarity is often at the core of computational difficulties: lack of unitarity could lead to exponential amplification of numerical noise even for single channel, while in multi-channel problems discontinuities near thresholds are particularly challenging. Here we propose and demonstrate another approach that is computationally efficient, even in multi-variable cases, and preserves unitarity exactly. The time propagation Φ(x, t) = exp − i Ht Φ(x, 0)(13) can be performed by considering, separately, the potential and kinetic parts of the hamiltonian H = K + V. In the discretized space of generalized coordinates x = {x 1 , x 2 , . . . } the potential V (x) is diagonal, so that the exponential operator exp(−iV t/ ) can be readily applied. Similarly, in the conjugate momentum space p = {p 1 , p 2 . . . } the propagation with kinetic energy operator, which is diagonal, is also easy to perform. While the operators K and V do not commute, the time evolution (13) with the combined Hamiltonian can be done efficiently with the Trotter-Suzuki approach [9,10]. In this approach the propagation is done in small time steps ∆T ; for each of these steps the evolution operator is approximated as exp − i H∆t = exp − i 2 V ∆t exp − i K∆t exp − i 2 V ∆t + O(∆t 3 ).(14) The Fast Fourier Transform allows for an efficient transition between coordinate and momentum representations so that exponentials of operators are always applied in the diagonal form. Even with the finite time steps the unitarity is fully retained; the method is applicable to time-dependent Hamiltonians. The computational cost of two back and forth Fourier transforms involved in each step is N log(N ) assuming the coordinate space is discretized into N points. While this at first appears to be higher than the typical O(N ) scaling of the traditional methods, in practice the cost cN of any high quality method involves a constant factor c that often exceeds log(N ). Moreover, modern computer hardware often comes with signal processing tools which are optimized at hardware and software level to perform Fast Fourier Transform with incredible efficiency. Let us return to the problem of scattering illustrated in Fig. 1. The time dependent picture of the scattering process is shown in Fig. 5 with a series of four plots showing the two-dimensional wave function using a density plot at four different times. The plot of the density projection onto the center of mass coordinate X, which is the time dependent analog of Fig. 4, is shown below each of the four snapshots. The initial wave function at t = 0, shown on the first panel, is selected as the ground state wave function for the intrinsic potential, and as a moving Gaussian wave packet for the center of mass coordinate, Φ(X, x) = 1 σ 0 √ π exp 1 2σ 2 0 (X − X 0 ) 2 + iK 0 X ψ 0 (x).(15) In this example σ 0 = 2, X 0 = −5, and initial momentum K 0 = 1, all quantities are being expressed here in dimensionless units of distance as defined earlier. While this time dependent consideration is different from the stationary state formulation studied above, the series of snapshots for different times shown in Fig. 5 highlights some similar features. At high energies the dynamics of virtual excitations is complex; this is illustrated in Fig. 6, where the initial wave packet is selected to have K 0 = 5. Some semiclassical interpretation can be given to the stages of the process. Initial compression at t = 1 is followed by two particles bouncing apart at t = 2. Having equal masses, their center of mass remains at the origin but the relative separation x becomes large so that the particles are positioned roughly symmetrically on the opposite sides of the wall. Next at t = 3 the center of mass moves into X < 0 region pressing the interacting particle A.Volya against the wall. Finally, the system is reflected at t = 4 with the initial wave packet being considerably distorted. In comparison to the projection and VPM techniques discussed earlier, the timedependent approach is substantially faster numerically; moreover, any potential U (x 1 , x 2 ) can be considered with ease in this approach. One has to keep in mind, however, that it is not always easy to provide quantitative answers to stationary state questions, such as determination of scattering phase shifts in this example, using time-dependent techniques. The exact choice of the initial state as well as the energy uncertainty of the initial state can be important for some stages of time evolution. The physics of decay of unstable states represents a particularly important class of time-dependent process. The familiar exponential decay law is only an incomplete picture, requiring some subtle approximations, and being valid only within certain time limits. The complex intrinsic dynamics that can occur in the decaying manybody system further complicates the time evolution. The non-exponential decay laws in quantum mechanics have been studied and revisited by many authors ( see Ref. [3] and references therein). The presence of three regimes, namely, initial, exponential, and long-time power law, appears to be a universal feature of the decay processes. The transitions from one regime to another are accompanied by the interference of corresponding quantum amplitudes that is seen as oscillations on the decay curve. As another demonstration of the time-dependent technique based on the Trotter-Suzuki expansion and as an introduction to the section that follows, we demonstrate in Fig. 7 the decay process in Winter's model [11], which has been a very popular tool for exploring non-exponential features in decays. In this model a particle is confined to a region x ≥ 0 by an impenetrable wall at x = 0 and is held by a delta barrier at x = 1. The initial state at t = 0 is taken as Ψ(x, 0) = √ 2 sin(πx). The survival probability shown in the solid red line on the left panel of Fig. 7 illustrates the three general regimes: pre-exponential, exponential, and post-exponential. Oscillations can be seen in transitional regions. The snapshots of the wave function at different times are shown on the right. The pre-exponential behavior at very early times is influenced by the memory of how the state was created and, in particular, by the high energy components in the state. Later in time the internal structure and transitions between the intrinsic states become relevant. Short times correspond to remote energy components where the presence of other resonant states is to be considered. The high energy components have much shorter lifetimes and decay quickly leading to an exponential decay phase. This phase is dominated by a single resonant component, the radiating state, so that the wave function retains its shape while decreasing in amplitude. This can be seen on the right panel of Fig. 7. In the same figure one can also trace a moving away background component. The background contains very low energy particles; being far off-resonance, they essentially do not interact but move slowly away from the interaction region. Near the decay threshold the number of such particles with a certain energy is determined by the available phase space, which for neutral particles scales with energy following a power-law E l+1/2 where l is the angular momentum quantum number. This type of scaling leads to non-resonant components that follow a power-law decay S(t) ∼ 1/t 2 +3 . While the non-resonant component can be very small in the initial state, eventually it becomes dominant due to its slower-thanexponential decay. Further discussion of decay processes in quantum mechanics and other examples can be found in Ref. [3]. The near-threshold phase space scaling with energy which leads to power-law decay at remote times is an important consideration in the Time Dependent Continuum Shell Model approach that is discussed in the following section, see also Refs. [1,12,13], as well as in more complicated sequential decay processes [14]. 3 Time dependent continuum shell model 3 .1 Continuum Shell Model A seamless transition between structure physics and reactions is one of the central present-day theoretical problems. The computational aspect associated with transitions from discrete levels to a continuum of reaction states is especially challenging. The Continuum Shell Model approach [12,13] and its time-dependent version, in particular, is one among several theoretical tools confronting these issues. In the Continuum Shell Model the Feshbach projection formalism [15,16] is used to express the exact dynamics in the full Hilbert space using an effective Hamiltonian in the projected intrinsic subspace of interest, Q : H(E) = H QQ +H(E) whereH(E) = H QP 1 E − H PP H PQ .(16) Here the effective Hamiltonian contains H QQ which is the part of the original Hamiltonian that acts in the space Q, and the energy-dependent non-Hermitian termH(E), that emerges from the coupling of the space Q to an external space containing a continuum of reaction states, P. In practical applications the intrinsic space Q is assumed to represent the configuration space of the traditional shell model, built from states \|1 that are Slater determinants constructed from bound-state single-particle wave functions. The space P contains continua of reaction states \|c, E characterized by the channel index, c, and the continuous energy parameter E. There is a certain threshold energy E (c) thr for each channel c. The energy-dependent non-Hermitian effective Hamiltonian (16) is then represented by a matrix H 12 (E) ≡ 1\|H(E)\|2 , H 12 (E) = H 12 + ∆ 12 (E) − i 2 W 12 (E) , where(17)∆ 12 (E) = c PV ∞ E (c) thr. dE A c 1 (E )A c 2 * (E ) E − E , W 12 (E) = 2π c(open) A c 1 (E)A c 2 * (E), and the channel amplitudes are the matrix elements A c 1 (E) = 1\|H\|c, E . The traditional shell model Hamiltonian is recovered when the internal space Q is isolated and thus is decoupled, A c 1 (E) = 0. The computational challenges of the traditional shell model approach are well known, they are mainly associated with the need to find some selected eigenvalues and eigenvectors of the Hamiltonian matrix H 12 . The matrix is generally sparse, thanks to few-body nature of the underlying nucleon-nucleon interactions which inhibits mixing of very remote configurations, thus iterative techniques such as Lanczos approach are commonly used. The physics of weakly-bound and unstable nuclear systems is much more rich as questions of interest span from properties of bound states to features in scattering cross sections. Narrow resonances are well characterized by the usual properties of bound states with the decay width being an additional characteristic. This requires the non-Hermitian eigenvalue problem H(E)\|I = E\|I to be solved. The resulting complex energies E represent positions of resonances, E = Re(E), and their widths, Γ = −2 Im(E). The most practical technique here is to start with the perturbative treatment and evaluate the termH(E) associated with continuum, using the wave functions of the traditional shell model Hamiltonian H QQ . As coupling to the continuum increases the states become broad and one is forced to treat the non-Hermitian energy-dependent eigenvalue problem as an iterative non-Hermitian diagonalization process. In this limit a major problem is associated with the physical interpretation of the resonances and their widths. Formally, the energy-dependent non-Herminitan Hamiltonian provides an exact propagator for the intrinsic space and therefore the scattering matrix is S cc (E) = exp(iξ c + iξ c ) [δ cc − 2πiT cc (E)] , where T cc (E) = 12 A c 1 (E) 1 E − H 12 A c 2 (E).(18) Here ξ c is a potential (direct-reaction) phase. The matrix is unitary (see Ref [1]) and the unitarity is related to a factorized form of the imaginary W 12 (E) in Eq. (17). The eigenvalues of the non-Hermitian Hamiltonian are therefore poles of the scattering matrix. In the limit of broad resonances one has to address the reaction problem where obtaining a reaction cross section is the main goal. There are several numerical challenges associated with Eq. (18), many of these challenges being similar to the ones discussed in Sec. 2.1. First, the size of the Hamiltonian matrix and the complex arithmetic involved are not making this problem simpler as compared to matrix diagonalization. Second, the scattering energy E represents a running parameter so that the procedure should be repeated for all energies of interest. Finally, the problem is numerically unstable: bound states, as well as resonances with widths ranging by many orders of magnitude, may be encountered and should be treated consistently. All of these technical issues are resolved by the Time-dependent Continuum Shell Model approach which we discuss next. Time-dependent many-body evolution operator The many-body wave function follows the time evolution which is a Fourier image of the retarded propagator involved in the scattering matrix (18): G(E) = 1 E − H = −i ∞ 0 dt exp(iEt) exp(−iHt).(19) Here H is an arbitrary Hamiltonian, but as discussed below, it is advantageous to include a factorized imaginary part W using a different procedure described in Sec. 3.3. Thus, we view H as being a Hermitian Hamiltonian of the traditional shell model in which case it is set to have an infinitesimal negative-definite imaginary part. The time-dependent evolution operator can be factorized using a Chebyshev polynomial expansion method, see Ref. [1,17,18]. exp(−iHt) = ∞ n=0 (−i) n (2 − δ n0 ) J n (t) T n (H),(20) where J n is the Bessel function of the first kind and T n represents Chebyshev polynomials. The Chebyshev polynomials, defined as T n [cos(θ)] = cos(nθ) or, in explicit form, T n (x) = n 2 k≤n/2 k=0,1,... (−1) k n − k n − k k (2x) n−2k ,(21) provide a complete set of orthogonal functions covering uniformly the interval [-1, 1]. In contrast, Taylor expansion relies on power functions which favor the edges of the interval and thus are more sensitive to extreme eigenvalues. The "angular addition" identity 2T n (x)T m (x) = T n+m (x) + T n−m (x) , n ≥ m(22) which follows from the definition, allows one to obtain these polynomials using the recurrence relation T 0 (x) = 1, T 1 (x) = x, and T n+1 (x) = 2xT n (x) − T n−1 (x).(23) Therefore, the process of evaluation of Chebyshev polynomials of the Hamiltonian operator is an iterative procedure, similar to the one in Lanczos approach. For a given initial state \|λ ≡ \|λ 0 , a sequence \|λ n = T n (H)\|λ can be constructed as \|λ 0 = \|λ , \|λ 1 = H\|λ , and \|λ n+1 = 2H\|λ n − \|λ n−1 . For overlap functions, assuming Hermitian H, one can also use the following identity (20) needed for convergence is denoted as n max (τ ). The asymptotic of Bessel functions J n (x) ≈ 1/(2πn)[ex/(2n)] n suggests n max (τ ) ≈ eπτ /2 ≈ 4τ. At fixed values of n but for large times the convergence remains stable due to J n (t) ≈ 2/(πt) cos(t − πn/2 − π/4) in this limit. For the desired energy resolution ∆E/N the propagation in time has to be extended up to ≈ τ N which requires n max ≈ 4N ; therefore 2N matrix-vector multiplications are required if one also uses Eq. (25). λ \|T n+m (H)\|λ = 2 λ m \|λ n − λ \|λ n−m , n ≥ m.(25) The time-dependent approach provides the Green's function for all energies at once; it is also exceptionally stable numerically when dealing with very narrow resonances or with stable states. Indeed, the time-dependent behavior of stationary states is regular and the corresponding delta function in energy is well handled by Fourier transform, which at the desired energy resolution properly conservers the integrated strength. In order to illustrate the approach, let us consider strength and integrated strength functions defined for a given state \|λ as F λ (E) = λ\|δ(E − H)\|λ = − 1 π Im λ\|G(E)\|λ , I λ (E) = E −∞ F λ (E )dE .(27) In Fig. 8 both strength (left) and integrated strength (right) functions are shown for 15 N for neutron channels where \|λ corresponds to different angular momentum channels constructed from 1 + ground state in 14 N coupled to a single nucleon on either d 5/2 (top panels) or d 3/2 (bottom panels) single-particle states. This theoretical study follows recent experimental work in Ref. [19]. The full p-sd valence space is used with the Hamiltonian from Ref. [20]. With about 10 7 m-scheme basis states, obtaining and computing strength functions in energy regions around 20 MeV of excitation is impractical; the time-dependent method provides an excellent alternative. Sherman-Morrison-Woodbury relations It is certainly possible to implement the Chebyshev polynomial expansion procedure for a full non-Hermitian Hamiltonian using Eq. (20); however the factorized structure ofH offers a different alternative which is much more computationally advantageous. The two propagators corresponding to Eq. (16) G(E) = 1 E − H QQ and G(E) = 1 E − H(E)(28) can be related through Dyson's equation G(E) = G(E) + G(E)H(E)G(E). Since the contribution from the continuum emerges in the factorized form H(E) = cc \|c H cc (E) c \|,(29) the expression for the full propagator can be found in a closed form in the space spanned by the channel states G = G 1 −HG −1 = 1 − GH −1 G.(30) The operators here are represented by matrices in the channel subspace G ab = a\|G(E)\|b and G ab = a\|G(E)\|b . In computer science these relations are known as Sherman-Morrison-Woodbury matrix inversion equations [21]. The unitarity of the scattering matrix immediately follows from these relations, see [1]. We illustrate the TDCSM approach in its complete form in Figs. 9 and 10 where the resonances in 24 O are considered. The system is treated in the sd valence space using the USD shell model Hamiltonian [22]. In Fig. 9 the norm of the survival amplitude is shown as a function of time for the following set of most representative states 2 + 1 (4180, 2.7), 1 + 1 (5291, 195.1), 4 + 1 (6947, 0.0), 2 + 3 (8107, 92.5), and 2 + 4 (9673, 17.5). The states are listed here with their excitation energies followed by the decay widths, both in keV. The initial wave functions at t = 0 are taken as eigenstates of the traditional shell model. For the states such as 4 + 1 , which cannot decay in this model due to high angular momentum, the norm of the survival amplitude remains constant. Narrow states, exhibit a nearly exponential decay, for the state 2 + 4 the survival amplitude expected in exponential decay is shown. The decay is non-exponential for broad states such as 1 + 1 and 2 + 3 . In Fig. 10 . They are eigenstates of the traditional USD SM but are non-stationary resonances in the TDCSM, except for the 4 + 1 state which due to its high spin does not decay within the sd valence space. To emphasize the non-exponentiality in the decay law the unmarked solid line shows the exp(−Γ α t/2) function with parameters for the 2 + 4 state. The time-dependent approach provides an effective computational strategy for treating many-body systems that feature both bound and unbound states. In contrast to the stationary state formalism, the time dependent approach addresses the evolution of states in a natural way, thus providing a computationally robust and stable strategy, where experimental observables are easily recovered and fundamental principles of quantum mechanics, such as linearity and unitarity, are followed. From the computational perspective, the matrix-vector multiplication, the most efficient operation available, is utilized in building the time evolution operator with full con-Computational approaches to many-body dynamics of unstable nuclear systems 15 trol of the desired energy and time resolution. The specifics of the terms that emerge due to coupling to continuum in Feshbach projection formalism can be used to build the full evolution operator using Sherman-Morrison-Woodbury relations. TDCSM found broad practical applications, see Refs. [23][24][25] for example. Conclusions As our interests shift towards open, reacting, decaying, and otherwise evolving quantum many-body systems, new theoretical and computational techniques must be developed to address multiple new challenges that emerge. The goal of this presentation is to highlight some of the methods used in the recent scientific projects. We use a simple model to demonstrate three distinctly different techniques. The most straightforward method involves projecting the dynamics onto a set of basis states, allowing subsequently for the well-developed methods of linear algebra to be used; in certain reaction problems this method appears to have significant drawbacks associated with numerical instabilities and poor convergence. We demonstrate the Variable Phase Method that can treat reaction problems efficiently in a discretized coordinate space. Finally, we consider explicitly time-dependent techniques that are perhaps most adequate for the time-dependent dynamics associated with decay. We put forward the Time Dependent Continuum Shell Model approach, as a practical tool and demonstrate its application to realistic problems in nuclear physics. start by illustrating the difficulties that one faces while trying to reformulate reaction problems using the basis projection methods typical for structure physics; Proceedings International Conference 'Nuclear Theory in the Supercomputing Era -2014' (NTSE-2014), Khabarovsk, Russia, June 23-27, 2014. Eds. A. M. Shirokov and A. I. Mazur. Pacific National University, Khabarovsk, Russia, 2014, p. 1. Figure 2 : 2This figure refers to a system of two particles, bound by a harmonic oscillator confinement, which collides with an infinite wall. The incident kinetic energy is exactly half of the oscillator quantum so that only the ground state channel is open. Left panel: For a system where µ 1 = µ 2 the absolute values of amplitudes \|R n \| ≡ \|R 0n \| in virtual channels are shown as a function of n assuming different truncations N . The asymptotic dependence is illustrated with the straight line "exp(n)." Right panel: The phase shift, defined for a single open channel as e 2iδ = −R 00 Figure 3 : 3. 3 and 4. The reflection probabilities for different channels are shown inFig. 3as functions of Reflection probabilities in different channels as a function of incident kinetic energy. The incident beam contains a composite projectile in the ground state. Equal masses µ 1 = µ 2 are assumed for both interacting and non-interacting particles. Figure 5 : 5Four panels show the wave function \|Φ(X, x)\| 2 as a density plot for different times t = 0, 5, 10 and 15, as labeled. For each of the time snapshots the lower plot shows the density distribution over the center of mass coordinate computed as \|Φ(X, x)\| 2 dx. The initial wave function at t = 0 is given the Gaussian wave packet, Eq. 15. For this system µ 1 = µ 2 , the border of inaccessible area x 2 > 0 is shown with a solid line. Figure 6 : 6Four panes, similar to those inFig. 5, show the wave function \|Φ(X, x)\| 2 at most representative moments of time t = 1, 2, 3 and 4, during the high energy collision with the impenetrable wall. Here K 0 = 5, the remaining parameters being the same as inFig. 5. Figure 7 : 7Left: survival probability S(t) = \| Ψ(0)\|Ψ(t) \| 2 is shown as a function of time (solid red line). The exponential decay law, where mean lifetime τ = 0.65 is known from the poles of the scattering matrix, is shown with a double-dotted black line, the background component that decays following a power law is shown with a dot-dash blue line. The survival probability at very early times is shown in inset. Right: wave function of a decaying state is shown at times t = 0.8, 1.2, and 1.6: upper panel shows the probability distribution \|Ψ(x)\| 2 , middle panel displays current j(x, t), and the wave function in momentum space is shown in the lower panel. Here the strength of the delta function G = 6, in units where = 2m = 1. ( Well controlled energy resolution is one advantage of the method. In applications of the method the energy interval [E min , E max ], which should contain all eigenvalues of H, is mapped onto [−1, 1] by rescaling the Hamiltonian as H → (H −E)/∆E where E = (E max + E min )/2 and ∆E = (E max − E min )/2. For a desired energy resolution ∆E/N where N is some even integer number, the discrete Fourier transform allows one to evaluate Green's function in the corresponding energy points of the rescaled interval E p = p/N with p = −N/2 . . . −i) n (2 − δ n0 )J n (πτ ) λ \|T n (Hthe evaluation of the evolution operator at times t = πτ, where τ = 0 . . . N −1. For each desired time point τ the number of terms in expansion Figure 8 : 8Single-particle strength function (left) and cumulative or integrated strength function (right) are shown as functions of excitation energy (in units of MeV) for 15 N. Figure 9 : 9Time evolution of several low-lying states in 24 O. The absolute value of the survival overlap \| α\| exp(−iHt)\|α \| is shown as a function of time. Different lines, as marked, correspond to states α(E α , Γ α ) Figure 10: Scattering cross section for 23 O(n, n) 23 O reaction showing resonances in 24 O. the scattering cross section is shown for elastic neutron scattering on the ground state of 23 O, where the same resonant states can be observed. . A Volya, Phys. Rev. C. 7944308A. Volya, Phys. Rev. C 79, 044308 (2009). . N Ahsan, A Volya, Phys. Rev. C. 8264607N. Ahsan and A. Volya, Phys. Rev. C 82, 064607 (2010). . M Peshkin, A Volya, V Zelevinsky, EPL. 10740001M. Peshkin, A. Volya, and V. Zelevinsky, EPL 107, 40001 (2014). . 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[ "Pseudo-Spin Symmetry and its Applications 1", "Pseudo-Spin Symmetry and its Applications 1" ]	[ "Octavio Castaños \nInstituto de Ciencias Nucleares\nUNAM; A. Postal\n70-54304510\n", "Jorge G Hirsch \nDepartamento de Física\nCINVESTAV; A. Postal\n14-74007000MéxicoD. F\n", "Peter O Hess \nInstituto de Ciencias Nucleares\nUNAM; A. Postal\n70-54304510\n" ]	[ "Instituto de Ciencias Nucleares\nUNAM; A. Postal\n70-54304510", "Departamento de Física\nCINVESTAV; A. Postal\n14-74007000MéxicoD. F", "Instituto de Ciencias Nucleares\nUNAM; A. Postal\n70-54304510" ]	[]	The pseudo-spin symmetry is reviewed. A mapping that produces the separation of the total angular momentum into pseudoorbital and pseudo-spin degrees of freedom is discussed, together with the analytic transformations that take us from the normal parity space to the eigenstates of a pseudo-oscillator with one quanta less. The manyparticle version of the unitary transformation to the pseudo-SU(3) space is established. As an example, these symmetries are used to describe the double beta decay phenomenon in heavy deformed nuclei.	10.1063/1.48794	[ "https://arxiv.org/pdf/nucl-th/9501028v1.pdf" ]	15,442,155	nucl-th/9501028	46ccbe68fdcfad39a710d0efd7b9274cb14f62da	Pseudo-Spin Symmetry and its Applications 1 26 Jan 1995 Octavio Castaños Instituto de Ciencias Nucleares UNAM; A. Postal 70-54304510 Jorge G Hirsch Departamento de Física CINVESTAV; A. Postal 14-74007000MéxicoD. F Peter O Hess Instituto de Ciencias Nucleares UNAM; A. Postal 70-54304510 Pseudo-Spin Symmetry and its Applications 1 26 Jan 1995 The pseudo-spin symmetry is reviewed. A mapping that produces the separation of the total angular momentum into pseudoorbital and pseudo-spin degrees of freedom is discussed, together with the analytic transformations that take us from the normal parity space to the eigenstates of a pseudo-oscillator with one quanta less. The manyparticle version of the unitary transformation to the pseudo-SU(3) space is established. As an example, these symmetries are used to describe the double beta decay phenomenon in heavy deformed nuclei. INTRODUCTION The pseudo-spin symmetry was introduced in [1,2], where a separation of the single particle Hilbert space into pseudo orbital and pseudo-spin degrees of freedom was proposed. This new coupling scheme provides a simple interpretation of many striking features of the Nilsson orbits: i) The observed approximate degeneracy of the single particle levels of the type (N, l, s)j and (N, l + 2, s)j + 1 for the spherical case or the one associated to the levels with asymptotic quantum numbers [N, N z , Λ] Ω = Λ + 1/2 and [N, N z , Λ+2] Ω = Λ+3/2 for the large deformation case. These energy orbitals can be considered as pseudo-spin orbit doublets, implying that the strength of the pseudo spin-orbit interaction is small. ii) The expectation value of the pseudo-spin operator, iii) the calculation of the decoupling parameters, and iv) the matrix elements of the Coriolis interaction [1]. More recently the pseudospin symmetry has been considered to be useful to characterize identical bands [3] and as a signature of superdeformation [4]. For heavy nuclei the nuclear shells can be divided into two parts: the associated to the single particle orbits of the same parity, which are called normal parity levels and the intruder that comes from the shell above of opposite parity and thus it has been called unique or abnormal parity orbital. Thus for a given shell, N α one has j N α = {1/2, 3/2, · · · , N α − 1/2 }, j A α = N α + 3/2, where α = π or ν for protons and neutrons, respectively. This separation of the space for the single particle states of Nilsson or Wood Saxon potentials is reasonable because it has been proved that the intruder orbitals are nearly unmixed for standard deformations [5]. When relabelled in the pseudo spin scheme, the normal parity levels form a major shell for a pseudo-oscillator potential with N α = N α − 1 quanta. Since the symmetry of this oscillator is of course SU(3), one may consider the pseudo-SU(3) coupling scheme. In the Second Section, we review the transformation to the pseudo space introduced by Bohr, Mottelson and Hamamoto (BHM) [6] and the mappings from oscillators to pseudo-oscillators wave functions for the spherical and asymptotic cases [7]. We emphasize the difference between the pseudo-spin and the pseudo-oscillator symmetries. The first one implies the separation of the normal parity degrees of freedom into pseudo spin and pseudo orbital spaces (called pseudo-space) while the second one for the one particle case is related with the use of oscillator wavefunctions with spin to describe the pseudo-space. At the end of this section, the many-particle unitary transformation to the pseudo-SU(3) space is introduced and applied to one body operators. In the final section, theoretical calculations for the 2ν double beta decays of 150 Nd and 238 U are presented [8]. UNITARY TRANSFORMATIONS The BHM unitary transformation acts on the angular-spin parts of the wave functions of the particle, without affecting the radial motion. It is given by the scalar product between the spin operator S andr a unit vector in the direction of the position of the particle, U BHM = 2 ir · S. It is straightforward, to calculate the action on the single-particle wave function of the three dimensional harmonic oscillator with quantum numbers ψ N (l,s)jm ( r) = R N l (r) µσ lµ, sσ\|jm Y lµ (θ, φ)χ s σ ,(1) which can be written as a product of the radial part times the superposition of two spinor wave functions [6]. This transformation gives rise to the results l = j ± 1/2 →l = j ∓ 1/2 . Thus it produces the breaking of the normal degrees of freedom into pseudo-spin and pseudo-orbital spaces, but it does not give the wave functions of a pseudo-oscillator with one quanta less. In the set of wave functions (2) corresponding to a given N value, we disregard those with j = N + 1 2 , which define the unique or abnormal space, and only consider the remaining wave functions of the normal parity orbitals. These states can be mapped onto the eigenstates \|Ñ(ls)jm of a pseudo-oscillator via the following unitary operator [7] U = 2 ( ξ · S) (N − 2 L · S) − 1 2 ,(2) where ξ is the annihilation harmonic oscillator quantum. The action of U onto eigenstates of a three dimensional harmonic oscillator gives the result U\|N(l, s)jm = \|Ñ(ls)jm ,(3) with the relations between the labelsÑ = N − 1,s = s,j = j,m = m, and l = l ± 1 according to whether j = l ± 1/2 . Although the previous mapping connecting the normal parity harmonic oscillator eigenstates with the full set of eigenstates of a pseudo-oscillator was proposed long ago [1,2], the explicit form of the unitary operator (3) was constructed for the first time in [7]. More recently, the transformation to the pseudo-oscillator space has been carried out by using the Algebraic Generator Coordinate Method [9]. Now we consider the asymptotic wave functions of the Nilsson Hamiltonian [7] associated with large deformations. Then the Nilsson orbitals are described by the states characterized by \| [N, N z , Λ] Ω , the cylindrical harmonic oscillator states with spin. The degeneracy mentioned above can be explained through the unitary transformation U ∞ = 2 (ξ + S + + ξ − S − ) (N ρ − 2 L z · S z ) − 1 2 ,(4) where the operators ξ ± and η ± are the spherical components of the creation η and annihilation ξ harmonic oscillator operators, the spin operators are expressed in terms of the Pauli matrices S m = σ m /2 andN ρ is the number operator in the plane. The U ∞ and their corresponding hermitean conjugated operator U † ∞ are unitary, if they act onto states belonging to the normal space orbitals, that is if we disregard the levels of a given N shell that satisfy Ω = N ρ + 1/2. Thus, the unitary transformations for the spherical and asymptotic Nilsson orbitals to the pseudo-oscillator wave functions are not equivalent. This fact which has not been previously emphasized, is reflected in the separation itself into normal and abnormal parity spaces. However for standard deformations the exact wave functions of the Nilsson Hamiltonian can be evaluated and it is observed that there is no mixing of the unique (spherical) parity orbital with the remaining levels of the shell [10], which suggest that for these deformations the spherical transformation is the most appropriate. The division of the shell model space into normal and unique as proposed in the asymptotic limit may be useful in applications to superdeformation phenomena. For a system of n particles the unitary transformation to the pseudooscillator space is defined by U = n s U s .(5) In the Fock space, if we denote byÛ the corresponding unitary operator associated to U, then a general one body operator F is mapped to the pseudo-SU(3) space byF =ÛFÛ † . If two single particle state vectors \|α and \|α are related by a unitary transformation U (see Eq. 2), then the fermion operators are related byÛ a † αÛ † = a † α . Then, F in the pseudo-space is given bŷ F = N, l, j, µ ′ N ′ , l ′ , j ′ , µ ′ ′ N ′ (l ′ , 1 2 ) j ′ µ ′ \|F 1 \|N (l, 1 2 ) j µ a †Ñ ′ ,l ′ ,j ′ ,μ ′ aÑ ,l,j,μ ,(6) where the primes on the sum indicate that the unique parity orbitals are excluded. A similar result is obtained for the two body operators, that is only the labels of the fermion creation and annihilation operators are changed by its pseudo quantum numbers. The previous results give formal support to the recipe indicated in Refs. [1,11]. The many body expressions in the pseudo space of the SU(3) generators are given by the series expansionŝ L q = k LLq + · · · ,Q µ = k QQ µ + · · · ,(7) where only the leading terms of the series are indicated. These operatorsL q ,Q µ have the same form as the SU(3) generators but of a shell with one quanta less, but we want to emphasize that they are not the transformed operatorŝ L q ,Q µ to the pseudo space. DOUBLE BETA DECAY Double beta decay is a rare transition between two nuclei with the same mass number A involving change of the nuclear charge number Z by two units. This exotic phenomenon is a useful tool to test the lepton number conservation, neutrino properties and models of nuclear structure [12] . It can be classified into various modes according to the light particles besides the electrons associated with the decay. The two neutrino mode (ββ 2ν ), in which two electrons and two neutrinos are emitted, takes place independently of the neutrino properties, and conserves the electric charge and lepton number. The 0ν mode violates lepton number conservation and therefore it is forbidden in the standard electroweak theory. To proceed the 0ν decay, the virtual neutrino must be emitted in one vertex and absorbed in the other one, thus it is required that: i) the exchanged neutrino is a Majorana particle (ν =ν) and ii) both neutrinos have a common helicity component [13]. Next, we restrict to describe, within the pseudo-SU(3) formalism, the 2ν double beta decays of 150 Nd and 238 U. The decay rate of the 2ν-mode can be calculated through the formulae (τ 1/2 2ν ) −1 = G 2ν \| M GT 2ν \| 2 ,(8) where G 2ν is a kinematic factor and M GT 2ν = M 2ν is a nuclear matrix element strongly dependent on the considered nuclear model, that is M 2ν = N 1 E 0 + E N − E i 0 + f \|\| Γ \|\| 1 + N 1 + N \|\| Γ \|\| 0 + i .(9) The Γ is denoting the Gamow-Teller operator Γ m = s σ ms t − s and the E 0 = 1 2 Q ββ + m e c 2 is the half of the total energy released. The E N gives the energy of the intermediate state \|1 + N . The E i is the energy of the ground state of the initial nucleus \|0 + i and the ket \|0 + f describes the ground state of the final nucleus. In order to compute M 2ν , we have to perform a sum over all the intermediate states . Fortunately, an alternative form of calculate this matrix element has been developed [14], i.e., M 2ν = 1 E 0 0 + f \| m (−1) m Γ −m F m \|0 + i ,(10) where the operator F m is defined by: F m = ∞ λ (−1) λ E λ 0 [H, [H, . . . , [H, Γ m ] . . .] (λ−times) .(11) A reasonable model for describing spectra and BE2 transitions of heavy deformed nuclei is [8]: H = α H α − 1 2 χ Q a · Q a + ζ 1 K 2 + ζ 2 L 2 ,(12) where H α denotes the spherical Nilsson Hamiltonian for neutrons or protons plus a constant term V α , which represent the depth of the potential well. The quadrupole-quadrupole interaction in a given shell and K 2 is a linear combination of L 2 , X 3 and X 4 , which are rotational scalar operators built with generators of the algebra of SU(3) [15]. Notice that the quadrupole-quadrupole force, L 2 and the K 2 interaction are independent of the spin degrees of freedom and symmetric in the neutron and proton components. To evaluate (13), we express H α and Γ m in the second quantization formalism, H α =hω η l j m ǫ α (η, l, j) a † ηl 1 2 jmα a ηl 1 2 jmα ,(13)Γ m = π ν σ(π, ν)A(π, ν, m) ,(14) with A(π, ν, m) = [a † ηπ lπ 1 2 ;jπ ã ην lν 1 2 ;jν ] 1 m denoting the angular coupling of proton creation and neutron annihilation operators, ǫ α the single particle energies and σ(π, ν) ≡ π ν 2j π + 1 3 η π l π 1 2 ; j π \|\|σ\|\|η ν l ν 1 2 ; j ν . Afterwards some algebraic manipulations, M 2ν is rewritten by M 2ν = π ν π ′ ν ′ σ(π ′ , ν ′ ) σ(π, ν) 0 + f \| A (π ′ , ν ′ )· A (π, ν) \| 0 + i E 0 + ǫ (ηπ , lπ , jπ) − ǫ (ην , lν , jν )(16) In the pseudo SU(3) Hilberty space the above sum is restricted to η π = η ν , l π = l ν , η π ′ = η ν ′ , l π ′ = l ν ′ , j π = j ν + 1. Following [8], the single particle energy difference in the denominator takes the form ǫ(η, l, j π ) − ǫ(η, l, j ν ) = −hωk π 2j π + ∆ C . The constants k α are well known [16] and ∆ C is used to determine the value V ν − V π . The ∆ C is the difference Coulomb energy and it is evaluated by the expression ∆ C = 0.70 A 1/3 [2Z + 1 − 0.76((Z + 1) 4/3 − Z 4/3 )]MeV .(18) The description of the correlated deformed ground states is done using the pseudo SU(3) scheme for the normal parity space and seniority zero configurations, that is all the nucleons coupled by pairs to angular momentum zero, for the unique part [11,15]. The occupancies of these spaces are determined from the corresponding Nilsson diagrams by selecting a reasonable deformation and filling each level with a pair of particles in order of increasing energy. These numbers fix the totally antisymmetric irreducible representations (irreps) of the unitary groups associated to the normal U((N α + 1)(N α + 2)) and unique U(2N α + 4) spaces. For the normal parity space, one separates the degrees of freedom in pseudo-orbital U(Ω N α ) and pseudo-spin U α (2) parts, with Ω N α = (N α + 1)(N α + 2)/2 and their irreps {f α } are characterized by the partitions of the number of particles in the normal part, n N α . Then of the pseudo SU(3) irreps, (λ α , µ α ), contained in {f α }, one considers those with maximum eigenvalue of the Casimir operator, (C 2 ) α = (λ α + µ α + 3) (λ α + µ α ) − λ α µ α . In Table 1, the results found for the participant nuclei in the double beta decays of 150 Nd and 238 U are presented. Finally we used the strong coupled limit [11], and from the Kronecker product (λ π , µ π ) × (λ ν , µ ν ), the (λ π + λ ν , µ π + µ ν ) irrep will dominate the low-lying energy structure. In Table 2, we display the calculated Gamow-Teller matrix elements, energy denominators, predicted and experimentally determined double beta half lives of 150 Nd and 238 U. They are given in the last column of Table 2 and these half lives are lower bounds because the nuclear matrix elements could be reduced by considering, for example the inclusion of the pairing interaction, and therefore giving longer (never shorter) ββ half lives. Table 1 . 1Ground states in the Pseudo-SU(3) coupling scheme. NUCLEUS n Nπ n A π n N ν n A ν U(Ω N π ) U(Ω N ν ) SU π (3) SU ν (3) 150 Nd 6 4 6 2 {2 3 } {2 3 } (12, 0) (18, 0) 150 Sm 6 6 4 2 {2 3 } {2 2 } (12, 0) (12, 2) 238 U 6 4 12 8 {2 3 } {2 6 } (18, 0) (36, 0) 238 Pu 6 6 10 8 {2 3 } {2 5 } (18, 0) (30, 4) Table 2 . 2Theoretical estimates for the half-life ββ-decay in the 2ν mode.T ransition < 0 + f \|Γ 2 \|0 + i > E[MeV ] τ 1/2 theo [y] τ 1/2 exp [y][13] 150 Nd → 150 Sm 1.31 12.2 6.0 × 10 18 9-17 × 10 18 238 U → 238 P u 1.51 16.8 1.4 × 10 21 2 × 10 21 Work supported in part by project UNAM-CONACYT 3513-E9310. . K T Hecht, A Adler, Nucl. Phys. A. 137129K. T. Hecht and A. Adler, Nucl. Phys. A 137, 129 (1969); . R D Raju, J P Draayer, K T Hecht, Nucl. Phys. A. 202433R.D. Ratna Raju, J.P. Draayer and K.T. Hecht, Nucl. Phys. A 202, 433 (1973). . A Arima, M Harvey, K Shimizu, Phys. Lett. B. 30517A. Arima, M. Harvey and K. Shimizu, Phys. Lett. B 30, 517 (1969). . A J Kreiner, Rev. Mex. Fís. 40122A. J. Kreiner, Rev. Mex. Fís. 40 Supl. 1, 22 (1994). . F S Stephens, Phys. Rev. Lett. 642623F. S. Stephens et al., Phys. Rev. Lett. 64, 2623 (1990). . R Bergtsson, J Dudek, W Nazarewics, P Olanders, Physica Scripta. 39196R. Bergtsson, J. Dudek, W. Nazarewics and P. Olanders, Physica Scripta 39, 196 (1989). . A Bohr, I Hamamoto, B R Mottelson, Physica Scripta. 26267A. Bohr, I. Hamamoto and B. R. Mottelson, Physica Scripta 26, 267 (1982). . O Castaños, M Moshinsky, C Quesne, Phys. Lett. B. 277238O. Castaños , M. Moshinsky and C. Quesne, Phys. Lett. B 277, 238 (1992); . O Castaños, V Velázquez, P O Hess, J G Hirsch, Phys. Lett. B. 321303O. Castaños, V. Velázquez, P.O. Hess and J.G. Hirsch, Phys. Lett. B 321, 303 (1994). . O Castaños, J G Hirsch, O Civitarese, P O Hess, Nucl. Phys. 571276O. Castaños , J. G. Hirsch, O. Civitarese and P. O. Hess, Nucl. Phys. A571, 276 (1994). . A Góźdź, A Staszczak, K Zajac, Acta Physica Polonica. 25665A. Góźdź, A. Staszczak and K. Zajac, Acta Physica Polonica. B25, 665 (1994). . D Troltenier, W Nazarewics, Z Szymanski, J P Draayer, Nucl. Phys. 567591D. Troltenier, W. Nazarewics, Z. Szymanski and J. P. Draayer, Nucl. Phys. A567, 591 (1994). . J P Draayer, K J Weeks, Ann. Phys. 15641J. P. Draayer and K. J. Weeks, Ann. Phys. 156, 41 (1984). . M Doi, T Kotani, E Takasugi, Progr. Theo. Phys. Suppl. 831M. Doi, T.Kotani and E. Takasugi, Progr. Theo. Phys. Suppl. 83(1985) 1. . M K Moe, P Vogel, Ann. Rev. Nuc. Part. Sci. to be publishedM. K. Moe, P. Vogel, Ann. Rev. Nuc. Part. Sci. (to be published). . O Civitarese, J Suhonen, Phys. Rev. 472410O. Civitarese and J. Suhonen, Phys. Rev. C47 (1993) 2410. . O Castaños, J P Draayer, Y Leschber, Z. Phys. A. 32933O. Castaños, J.P. Draayer and Y. Leschber, Z. Phys. A 329 (1988) 33; . H A Naqvi, J P Draayer, Nucl. Phys. 516351H.A.Naqvi and J.P. Draayer, Nucl. Phys. A516 (1990) 351. The Nuclear Many Body Problem. P Ring, P Schuck, Springer VerlagNew YorkP. Ring, P. 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[ "Absence of Wavepacket Diffusion in Disordered Nonlinear Systems", "Absence of Wavepacket Diffusion in Disordered Nonlinear Systems" ]	[ "G Kopidakis \nMax Planck Institute for the Physics of Complex Systems\nNöthnitzer Str. 38D-01187DresdenGermany\n\nDepartment of Materials Science and Technology\nUniversity of Crete\n71003HeraklionGreece\n", "S Komineas \nMax Planck Institute for the Physics of Complex Systems\nNöthnitzer Str. 38D-01187DresdenGermany\n", "S Flach \nMax Planck Institute for the Physics of Complex Systems\nNöthnitzer Str. 38D-01187DresdenGermany\n", "S Aubry \nMax Planck Institute for the Physics of Complex Systems\nNöthnitzer Str. 38D-01187DresdenGermany\n\nLaboratoire Léon Brillouin (CEA-CNRS)\nCEA Saclay\n91191Gif-sur-YvetteFrance\n" ]	[ "Max Planck Institute for the Physics of Complex Systems\nNöthnitzer Str. 38D-01187DresdenGermany", "Department of Materials Science and Technology\nUniversity of Crete\n71003HeraklionGreece", "Max Planck Institute for the Physics of Complex Systems\nNöthnitzer Str. 38D-01187DresdenGermany", "Max Planck Institute for the Physics of Complex Systems\nNöthnitzer Str. 38D-01187DresdenGermany", "Max Planck Institute for the Physics of Complex Systems\nNöthnitzer Str. 38D-01187DresdenGermany", "Laboratoire Léon Brillouin (CEA-CNRS)\nCEA Saclay\n91191Gif-sur-YvetteFrance" ]	[]	We study the spreading of an initially localized wavepacket in two nonlinear chains (discrete nonlinear Schrödinger and quartic Klein-Gordon) with disorder. Previous studies suggest that there are many initial conditions such that the second moment of the norm and energy density distributions diverge as a function of time. We find that the participation number of a wavepacket does not diverge simultaneously. We prove this result analytically for norm-conserving models and strong enough nonlinearity. After long times the dynamical state consists of a distribution of nondecaying yet interacting normal modes. The Fourier spectrum shows quasiperiodic dynamics. Assuming this result holds for any initially localized wavepacket, a limit profile for the norm/energy distribution with infinite second moment should exist in all cases which rules out the possibility of slow energy diffusion (subdiffusion). This limit profile could be a quasiperiodic solution (KAM torus).PACS numbers: Valid PACS appear hereIt is well-known that Anderson localization occurs for a one-dimensional linear system with uncorrelated random potential. Since all the linear eigenmodes -Anderson modes (AMs) -are localized, any wavepacket which is initially localized remains localized for all time. Therefore there is no energy diffusion [1]. When nonlinearities are added to such models, AMs interact with each other, giving rise to more complex situations[2]. Numerical studies of wavepacket propagation in several models showed that the second moment of the norm/energy distribution growths subdiffusively in time as t α [3, 4, 5], with α in the range 0.3 − 0.4, though not being accurately determined. The conclusion was that the initial excitation will completely delocalize for infinite times.Recently, experiments were performed on light propagation in spatially random nonlinear optical media[6].Spatially periodic nonlinear systems will support Discrete Breathers (DBs), which are spatially localized time periodic solutions [7] with frequencies outside the frequency spectrum of the linear system. The temporal evolution of a localized wavepacket leads to the formation of a DB, while a part of the energy of the wavepacket is radiated ballistically to infinity (in the form of weakly nonlinear plane waves)[8]. In that case, the second moment of the energy density distribution diverges as t 2 , falsly suggesting complete delocalization. The participation number P of the norm/energy distribution (or similar quantities) is a well-known measure of the degree of localization. In the case of a periodic nonlinear lattice, P will saturate at a finite value, correctly indicating the formation of a DB.For nonlinear random systems it was proven rigorously that AMs survive in the presence of nonlinearities as spatially localized and time-periodic solutions [9] with frequencies which depend on the amplitude of the mode. The allowed frequencies form a fat Cantor set (with finite measure) whose density becomes unity for weak nonlin-earity. They are located inside the frequency spectrum of the linear system. Numerical techniques for obtaining these (dynamically stable) intraband DB solutions at computer accuracy were developed[10]. When they are chosen as an initial wavepacket, they persist for infinite time and there is no diffusion at all.Here we analyse carefully the evolution of the participation number of wavepackets as a function of time, in situations where subdiffusion is claimed to exist [3, 4, 5]. We study two models. The Hamiltonian of the disordered discrete nonlinear Schrödinger equation (DNLS)(1) with complex variables ψ n . The random on-site energies ǫ n are chosen uniformly from the interval − W 2 , W 2 . The equations of motion are generated byψ n = ∂H D /∂(iψ ⋆ n ). We choose β = 1 and V = −1 here [4] and note that varying the norm of the initial wavepacket is strictly equivalent to varying β.The Hamiltonian of the quartic Klein-Gordon chain (KG)The equations of motion areü n = −∂H K /∂u n ,ǫ n = 1 + ǫ n (W = 1), and g = 1.For β = g = 0 both models are reduced to the linear eigenvalue problem λA n = ǫ n A n − V (A n+1 + A n−1 ). The eigenvectors A ν n are the AMs, and the eigenvalues λ ν are the frequencies of the AMs for the DNLS, while the KG modes have frequencies ω ν = √ λ ν + 1 + 2V . Hamiltonian (1) (unlike(2)), in addition to conserving the energy, also conserves the total norm S = n \|ψ n \| 2 = ψ\|ψ . We use this norm conservation for proving rigorously that initially localized wavepackets with a large	10.1103/physrevlett.100.084103	[ "https://arxiv.org/pdf/0710.2621v1.pdf" ]	887,541	0710.2621	44dbc1a1d3610ad705a7b452df3aeb2a008c3917	Absence of Wavepacket Diffusion in Disordered Nonlinear Systems 13 Oct 2007 (Dated: February 2, 2008) G Kopidakis Max Planck Institute for the Physics of Complex Systems Nöthnitzer Str. 38D-01187DresdenGermany Department of Materials Science and Technology University of Crete 71003HeraklionGreece S Komineas Max Planck Institute for the Physics of Complex Systems Nöthnitzer Str. 38D-01187DresdenGermany S Flach Max Planck Institute for the Physics of Complex Systems Nöthnitzer Str. 38D-01187DresdenGermany S Aubry Max Planck Institute for the Physics of Complex Systems Nöthnitzer Str. 38D-01187DresdenGermany Laboratoire Léon Brillouin (CEA-CNRS) CEA Saclay 91191Gif-sur-YvetteFrance Absence of Wavepacket Diffusion in Disordered Nonlinear Systems 13 Oct 2007 (Dated: February 2, 2008) We study the spreading of an initially localized wavepacket in two nonlinear chains (discrete nonlinear Schrödinger and quartic Klein-Gordon) with disorder. Previous studies suggest that there are many initial conditions such that the second moment of the norm and energy density distributions diverge as a function of time. We find that the participation number of a wavepacket does not diverge simultaneously. We prove this result analytically for norm-conserving models and strong enough nonlinearity. After long times the dynamical state consists of a distribution of nondecaying yet interacting normal modes. The Fourier spectrum shows quasiperiodic dynamics. Assuming this result holds for any initially localized wavepacket, a limit profile for the norm/energy distribution with infinite second moment should exist in all cases which rules out the possibility of slow energy diffusion (subdiffusion). This limit profile could be a quasiperiodic solution (KAM torus).PACS numbers: Valid PACS appear hereIt is well-known that Anderson localization occurs for a one-dimensional linear system with uncorrelated random potential. Since all the linear eigenmodes -Anderson modes (AMs) -are localized, any wavepacket which is initially localized remains localized for all time. Therefore there is no energy diffusion [1]. When nonlinearities are added to such models, AMs interact with each other, giving rise to more complex situations[2]. Numerical studies of wavepacket propagation in several models showed that the second moment of the norm/energy distribution growths subdiffusively in time as t α [3, 4, 5], with α in the range 0.3 − 0.4, though not being accurately determined. The conclusion was that the initial excitation will completely delocalize for infinite times.Recently, experiments were performed on light propagation in spatially random nonlinear optical media[6].Spatially periodic nonlinear systems will support Discrete Breathers (DBs), which are spatially localized time periodic solutions [7] with frequencies outside the frequency spectrum of the linear system. The temporal evolution of a localized wavepacket leads to the formation of a DB, while a part of the energy of the wavepacket is radiated ballistically to infinity (in the form of weakly nonlinear plane waves)[8]. In that case, the second moment of the energy density distribution diverges as t 2 , falsly suggesting complete delocalization. The participation number P of the norm/energy distribution (or similar quantities) is a well-known measure of the degree of localization. In the case of a periodic nonlinear lattice, P will saturate at a finite value, correctly indicating the formation of a DB.For nonlinear random systems it was proven rigorously that AMs survive in the presence of nonlinearities as spatially localized and time-periodic solutions [9] with frequencies which depend on the amplitude of the mode. The allowed frequencies form a fat Cantor set (with finite measure) whose density becomes unity for weak nonlin-earity. They are located inside the frequency spectrum of the linear system. Numerical techniques for obtaining these (dynamically stable) intraband DB solutions at computer accuracy were developed[10]. When they are chosen as an initial wavepacket, they persist for infinite time and there is no diffusion at all.Here we analyse carefully the evolution of the participation number of wavepackets as a function of time, in situations where subdiffusion is claimed to exist [3, 4, 5]. We study two models. The Hamiltonian of the disordered discrete nonlinear Schrödinger equation (DNLS)(1) with complex variables ψ n . The random on-site energies ǫ n are chosen uniformly from the interval − W 2 , W 2 . The equations of motion are generated byψ n = ∂H D /∂(iψ ⋆ n ). We choose β = 1 and V = −1 here [4] and note that varying the norm of the initial wavepacket is strictly equivalent to varying β.The Hamiltonian of the quartic Klein-Gordon chain (KG)The equations of motion areü n = −∂H K /∂u n ,ǫ n = 1 + ǫ n (W = 1), and g = 1.For β = g = 0 both models are reduced to the linear eigenvalue problem λA n = ǫ n A n − V (A n+1 + A n−1 ). The eigenvectors A ν n are the AMs, and the eigenvalues λ ν are the frequencies of the AMs for the DNLS, while the KG modes have frequencies ω ν = √ λ ν + 1 + 2V . Hamiltonian (1) (unlike(2)), in addition to conserving the energy, also conserves the total norm S = n \|ψ n \| 2 = ψ\|ψ . We use this norm conservation for proving rigorously that initially localized wavepackets with a large We study the spreading of an initially localized wavepacket in two nonlinear chains (discrete nonlinear Schrödinger and quartic Klein-Gordon) with disorder. Previous studies suggest that there are many initial conditions such that the second moment of the norm and energy density distributions diverge as a function of time. We find that the participation number of a wavepacket does not diverge simultaneously. We prove this result analytically for norm-conserving models and strong enough nonlinearity. After long times the dynamical state consists of a distribution of nondecaying yet interacting normal modes. The Fourier spectrum shows quasiperiodic dynamics. Assuming this result holds for any initially localized wavepacket, a limit profile for the norm/energy distribution with infinite second moment should exist in all cases which rules out the possibility of slow energy diffusion (subdiffusion). This limit profile could be a quasiperiodic solution (KAM torus). PACS numbers: Valid PACS appear here It is well-known that Anderson localization occurs for a one-dimensional linear system with uncorrelated random potential. Since all the linear eigenmodes -Anderson modes (AMs) -are localized, any wavepacket which is initially localized remains localized for all time. Therefore there is no energy diffusion [1]. When nonlinearities are added to such models, AMs interact with each other, giving rise to more complex situations [2]. Numerical studies of wavepacket propagation in several models showed that the second moment of the norm/energy distribution growths subdiffusively in time as t α [3,4,5], with α in the range 0.3 − 0.4, though not being accurately determined. The conclusion was that the initial excitation will completely delocalize for infinite times. Recently, experiments were performed on light propagation in spatially random nonlinear optical media [6]. Spatially periodic nonlinear systems will support Discrete Breathers (DBs), which are spatially localized time periodic solutions [7] with frequencies outside the frequency spectrum of the linear system. The temporal evolution of a localized wavepacket leads to the formation of a DB, while a part of the energy of the wavepacket is radiated ballistically to infinity (in the form of weakly nonlinear plane waves) [8]. In that case, the second moment of the energy density distribution diverges as t 2 , falsly suggesting complete delocalization. The participation number P of the norm/energy distribution (or similar quantities) is a well-known measure of the degree of localization. In the case of a periodic nonlinear lattice, P will saturate at a finite value, correctly indicating the formation of a DB. For nonlinear random systems it was proven rigorously that AMs survive in the presence of nonlinearities as spatially localized and time-periodic solutions [9] with frequencies which depend on the amplitude of the mode. The allowed frequencies form a fat Cantor set (with finite measure) whose density becomes unity for weak nonlin-earity. They are located inside the frequency spectrum of the linear system. Numerical techniques for obtaining these (dynamically stable) intraband DB solutions at computer accuracy were developed [10]. When they are chosen as an initial wavepacket, they persist for infinite time and there is no diffusion at all. Here we analyse carefully the evolution of the participation number of wavepackets as a function of time, in situations where subdiffusion is claimed to exist [3,4,5]. We study two models. The Hamiltonian of the disordered discrete nonlinear Schrödinger equation (DNLS) H D = n ǫ n \|ψ n \| 2 − 1 2 β\|ψ n \| 4 − V (ψ n+1 ψ ⋆ n + ψ ⋆ n+1 ψ n ) (1) with complex variables ψ n . The random on-site energies ǫ n are chosen uniformly from the interval − W 2 , W 2 . The equations of motion are generated byψ n = ∂H D /∂(iψ ⋆ n ). We choose β = 1 and V = −1 here [4] and note that varying the norm of the initial wavepacket is strictly equivalent to varying β. The Hamiltonian of the quartic Klein-Gordon chain (KG) H K = n p 2 n 2 +ǫ n 2 u 2 n + 1 4 gu 4 n + V 2 (u n+1 − u n ) 2 . (2) The equations of motion areü n = −∂H K /∂u n ,ǫ n = 1 + ǫ n (W = 1), and g = 1. For β = g = 0 both models are reduced to the linear eigenvalue problem λA n = ǫ n A n − V (A n+1 + A n−1 ). The eigenvectors A ν n are the AMs, and the eigenvalues λ ν are the frequencies of the AMs for the DNLS, while the KG modes have frequencies ω ν = √ λ ν + 1 + 2V . Hamiltonian (1) (unlike (2)), in addition to conserving the energy, also conserves the total norm S = n \|ψ n \| 2 = ψ\|ψ . We use this norm conservation for proving rigorously that initially localized wavepackets with a large enough amplitude cannot spread to arbitrarily small amplitudes. The consequence is that a part of the initial energy must remain well-focused at all times. This proof is inspired by [11]. We split the total energy H D = ψ\|L\|ψ + H N L into the sum of its quadratic term of order 2 and its nonlinear terms of order strictly higher than 2. Then, L is a linear operator which is bounded from below (and above). In our specific example, we have ψ\|L\|ψ ≥ ω m ψ\|ψ = ω m S where ω m ≥ −2 − W 2 is the lowest eigenvalue of L. Otherwise, the higher order nonlinear terms have to be strictly negative. If we assume that the wavepacket amplitudes spread to zero at infinite time, we have lim t→+∞ (sup n \|ψ n \|) = 0. Then lim t→+∞ ( n \|ψ n \| 4 ) < lim t→+∞ (sup n \|ψ 2 n \|)( n \|ψ n \| 2 ) = 0 since S = n ψ 2 n is time invariant. Consequently , for t → +∞ we have H N L = 0 and H D ≥ ω m n \|ψ n \| 2 = ω m S. Since H D and S are both time invariant, this inequality should be fulfilled at all times. However when the initial amplitude A of the wavepacket is large enough, it cannot be initially fulfilled because the nonlinear energy diverges as −A 4 while the total norm diverges as A 2 only. For example, a wavepacket initially at a site 0 ( ψ n = 0 for n = 0 and ψ 0 = √ A ) has energy H D = ǫ 0 A 2 − 1 2 A 4 . Consequently, the above inequality is not fulfilled when A 2 > −2(ω m − ǫ 0 ) > 0. Thus such an initial wavepacket cannot spread to zero amplitudes at infinite time. This proof is valid for DNLS models with any W (including the periodic case) and any lattice dimension and can be easily extended to larger classes of DNLS models where the nonlinear terms are either strictly negative, or strictly positive. Note that the large amplitude regime where we prove that complete energy diffusion is impossible in DNLS models, is precisely the one where subdiffusion is claimed to completely delocalize the wavepacket [5]. Thus we disprove these claims. We performed extensive numerical simulations, and characterized the wavepacket spreading both in real space for DNLS and normal mode space (Anderson space or AS) for KG. We used initial wavepackets with all the energy localized on a single site n 0 , or single AM, or combinations, close to n 0 . Nonlinearity induces diffusion in Anderson space, where each AM is characterized by a amplitude a ν and momentumȧ ν . We analyze distributions z l ≥ 0 using the second moment m 2 = l (l − l 0 ) 2 z l and the participation number P = ( l z l ) 2 / l z 2 l , which measures the number of the strongest excited sites in z l . We order the AMs in space by increasing value of the center-of-norm coordinate X ν = n nA 2 n . In the results presented here, for the DNLS z n = \|ψ n \| 2 is the norm density in real space, and for the KG z ν =ȧ 2 ν /2 + ω 2 ν a 2 ν /2 is the (harmonic) energy density in AS. The system size was N = 1000 for KG, and N = 2000 for DNLS. Excitations did not reach the boundaries during the integration time, and results are unchanged when further increasing N . We show in Fig.1 V = 0.25 at times t = 6 × 10 7 , 1.2 × 10 8 . Two rather strongly excited modes are surviving almost unchanged on these time scales. The insets show their eigenvectors, which are well localized, and practically do not overlap. The same distributions on a logarithmic scale (KG and DNLS) show a chapeau of weaker excited AMs, with exponential tails due to its finite width (Fig.2). This chapeau is perhaps slowly growing. The subdiffusive growth of the second moment at these times (see Fig.3) is mainly due to weak excitation of tail modes. The participation number P (t) is plotted in Fig.3 for the same runs. We observe no growth. P fluctuates around a value of 7-10, confirming the results in Fig.1, that we observe a localized state, similar to a DB. Assume that the rest of the weakly excited modes continues to subdiffuse in the chapeau. We use a modified distribution z ν for the KG run, where the 10 strongest mode contributions are zeroed (top panel, green curve). The weak mode participation number is now fluctuating around 70, but again does not grow. Therefore the chapeau appears not to diffuse, and the observed growth of m 2 ∼ t 0.3...0.4 is not related to a delocalization process. Instead, we find that the packet does not delocalize. Indeed, assuming that the chapeau homogeneously spreads in a subdiffusive way as claimed, it follows that P (t) ∼ t α/2 , which clearly contradicts our observations. We repeated these runs with various initial conditions and disorder realizations with similar results. However the localization pattern (Fig.1), and the observed averaged participation number P , fluctuate. Performing an averaging of the final distribution over several realizations [3,4,5] will there- fore completely smear out the sharp localization patterns in the distributions. Closer inspection of the evolution of m 2 shows, that the exponent α is strongly depending on the time intervals of study, and also on the given disorder realization. There are some indications suggesting that α might decay at long time and even that m 2 (t) may saturate, but further clarification may call for very extensive numerical investigations. Finally we calculated the Fourier transform I(ω) of P (t) (after t = 2 × 10 7 , over an interval of ∆t = 2000), see Fig.4. We find a quasiperiodic spectrum, which is close to periodic, with no hints of a chaos-induced continuous part. For the KG case the energy densities are quadratic forms of the AM coordinates, thus the main peak position ω ≈ 3 corresponds to a frequency ω ≈ 1.5 for the AM coordinate dependence, which coincides with the frequencies of the strong excited Anderson modes in Fig.1. Our main result is, that in both models (1) and (2), whatever the initial wavepacket is (even if it is not fulfilling the conditions for our theorem), and irrespective of the model parameters and the disorder realization, the participation number does not diverge as a function of time as it should in case of subdiffusion (as t α/2 ) but instead fluctuates between finite upper and lower bounds. Let us now propose an interpretation of our observations. First, it is useful to recall the wavepacket behavior in the absence of disorder. When its amplitude is large enough for generating a DB, there is a transient dynamical state which is more or less chaotic, with a broad band time-Fourier spectrum overlapping the spectrum of the linear system. Because of that, a part of the energy of the wavepacket is radiated to infinity. With that, the remaining DB like excitation becomes quasiperiodic first, and finally, approaches an equidistant spectrum of periodic motion, which completely stops further radiation. The energy which has been emitted spreads towards infinity. Therefore there is a limit profile which is a localized time-periodic solution -an exact DB. This is the only possibility for the limit profile, in order to avoid ra-diation. This is an example where the initial wavepacket self-organizes in order to stop radiation. When the system is both random and nonlinear, radiation into the linear spectrum is impossible due to Anderson localization. Nevertheless, the same process starts as before, but the energy emitted by the initial wavepacket cannot spread towards infinity since the participation number (full and partial) does not diverge. The following cascading scenario may be true. The core of the wavepacket emits a part of its energy which remains within the linear localization length nearby the initial wavepacket (due to the nonlinearity-induced coupling between the AMs). The same process should repeat for the emitted energy. A part of it remains localized while another part is reemitted a bit farther from the central site within the localization length and so on. This process of reemission repeats forever and generates a tail for the wavepacket which will become much more extended than the localization length. The central amplitude of the wavepacket does not tend to zero. The process of energy reemission slows down when the amplitude at the edge of the tail becomes small which explains the very slow numerical convergence. The final result is that at infinite time, the energy (or norm) distribution should converge to a nonvanishing limit profile which is summable since energy (or norm) is conserved. However, it may or may not have a finite second moment, which makes the question of the evolution of the second moment secondary. Unlike the standard DB case in spatially periodic systems, the limit profile is not a time periodic solution. It was proven rigorously ( [12,13]) that stable spatially localized quasiperiodic solutions with finite energy exist in similar nonlinear models with infinitely many degrees of freedom without or with degenerate linear spectrum. These KAM tori are quasiperiodic DBs which in some sense are linear combination of Anderson modes surviving in the presence of nonlinearity. Indeed, we find that the Fourier spectrum of the wavepacket dynamics becomes quasiperiodic, with narrow peaks and a small background as time grows suggesting the motion tends to become quasiperiodic (Fig.4). If the limit profile becomes a KAM torus, we should also observe that the largest Lyapunov exponent tends to 0 as t → +∞. Indeed, we find that this Lyapunov exponent drops rapidly during the first expansion part of the wavepacket, and slowly further decays, with characteristic values of 10 −4 at the end of our simulations. The corresponding time scale is 10 4 , and four orders of magnitude smaller than the simulation times. No chaotic dynamics is observable, and we think that the convergence to the final KAM torus is very slow because the surrounding KAM tori are expected to become dense. We should even expect to enter the regime of Arnol'd diffusion which is expected to be very slow and difficult to investigate both numerically and analytically. Note that this convergence to a quasiperiodic limit profile can only occur in infinite systems because if the system is finite, the regularization process of the initially chaotic trajectories ends when the packet tails reach the edge of the box. Then, we should expect to get equipartition of the energy after a sufficiently large time and a trajectory which remains chaotic with a nonzero largest Lyapunov exponent. In summary, we have proved by a rigorous analytical argument, and completed by numerical investigations of the participation number, that a wavepacket in a random nonlinear system does not spread ad infinitum. A limiting quasiperiodic profile is approached, and the slow increase of the second moment of the energy/norm distribution does not violate these findings. It is an open question whether the limiting profile will have a finite or infinite second moment. Thus, we observe absence of diffusion in nonlinear disordered systems. Note that this conclusion can be equally well applied to higher dimensional systems, provided all AMs are localized. FIG the KG energy distribution in AS for a single site excitation with energy E = 1. 1: (color online) KG: Energy distribution at t = 6 × 10 7 (black solid) and t = 1.2 × 10 8 (red dashed) in AS. Initial single site excitation with energy E = 1, V = 0.25. Insets: profiles of the strongest excited AMs in real space. FIG. 2 : 2(color online) Same as in Fig.1 but on a logarithmic scale. Top panel: KG, t = 1.2 × 10 8 , AS. Bottom panel: DNLS, W = 4, t = 1.2 × 10 8 , real space. FIG. 3 : 3(color online) P and m2 versus time, on logarithmic scale. Parameters as in Fig.2. Top panel: KG, AS. Bottom panel: DNLS, real space. FIG. 4 : 4(color online) I(ω) for KG and DNLS. Parameters as inFig.3. Insert: magnification of the main peak. AcknowledgmentsThis work was performed within the program of the Advanced Study Group 2007 at the MPIPKS Dresden http://www.mpipks-dresden.mpg.de/ asg2007/. 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[ "Cluster Variables on Double Bruhat Cells G u,e of Classical Groups and Monomial Realizations of Demazure Crystals", "Cluster Variables on Double Bruhat Cells G u,e of Classical Groups and Monomial Realizations of Demazure Crystals" ]	[ "Yuki Kanakubo ", "Toshiki Nakashima " ]	[]	[]	Let G be a simply connected simple algebraic group over C, B and B− its two opposite Borel subgroups, and W the associated Weyl group. It is shown that the coordinate ring C [G u,v1,8]). In the case that a classical group G is of type Br, Cr or Dr, we shall describe the non-trivial last r initial cluster variables {∆(k; i)} (m−2)r<k≤(m−1)r (m is given below) of the cluster algebra C[L u,e ] ([6,8]) in terms of monomial realization of Demazure crystals, where L u,e is the reduced double Bruhat cell of type (u, e). The relation between ∆(k; i) on G u,e and on L u,e is described in Proposition 6.3 below. We also present the corresponding results for type Ar though the results for all initial cluster variables have been obtained in[9].	10.1093/imrn/rnw344	[ "https://arxiv.org/pdf/1604.05956v2.pdf" ]	119,122,445	1604.05956	fc123b40ae1d209b068c2b9fc0137cf31594853e	Cluster Variables on Double Bruhat Cells G u,e of Classical Groups and Monomial Realizations of Demazure Crystals 17 Jan 2017 Yuki Kanakubo Toshiki Nakashima Cluster Variables on Double Bruhat Cells G u,e of Classical Groups and Monomial Realizations of Demazure Crystals 17 Jan 2017 Let G be a simply connected simple algebraic group over C, B and B− its two opposite Borel subgroups, and W the associated Weyl group. It is shown that the coordinate ring C [G u,v1,8]). In the case that a classical group G is of type Br, Cr or Dr, we shall describe the non-trivial last r initial cluster variables {∆(k; i)} (m−2)r<k≤(m−1)r (m is given below) of the cluster algebra C[L u,e ] ([6,8]) in terms of monomial realization of Demazure crystals, where L u,e is the reduced double Bruhat cell of type (u, e). The relation between ∆(k; i) on G u,e and on L u,e is described in Proposition 6.3 below. We also present the corresponding results for type Ar though the results for all initial cluster variables have been obtained in[9]. Introduction In [4,5] Fomin and Zelevinsky initiated the theory of cluster algebras, which is a commutative algebra generated by so-called "cluster variables". Let G be a simply connected complex simple algebraic group of rank r, B, B − ⊂ G the opposite Borel subgroups, H := B ∩ B − the maximal torus, N ⊂ B, N − ⊂ B − the unipotent radical and W = s i \|1 ≤ i ≤ r the associated Weyl group generated by the simple reflections {s i } 1≤i≤r . For u, v ∈ W , define the (reduced) double Bruhat cell G u,v := (BuB) ∩ (B − vB − ) (resp. L u,v := (N uN ) ∩ (B − vB − )). In [1], it is revealed that the coordinate ring C[G u,v ] (u, v ∈ W ) of double Bruhat cell G u,v is isomorphic to the upper cluster algebra A(i) C . Recently, in [8], Goodear and Yakimov have shown that the algebra C[G u,v ] (u, v ∈ W ) is isomorphic to the cluster algebra A(i) C (see Subsection 4.2 and 4.4): φ : A(i) C ∼ −→ C[G u,v ], x k −→ ∆(k; i), where i is a reduced word for the shuffle (u, v) and ∆(k; i) is certain generalized minor on G u,v . It means that the k-th initial cluster variable of the cluster algebra C[G u,v ] is given as certain generalized minors ∆(k; i) on G u,v . In [9], for type A r and a pair (u, e) with the specific reduced word i we gave the explicit forms of these initial cluster variables {∆(k; i)} and described them by using monomial realization of certain Demazure crystals. In this article, we shall get the partial results for the classical groups of type B r , C r and D r . Indeed, we find that the non-trivial last r initial cluster variables are expressed by using monomial realizations of Demazure crystals. Let us explain what we obtain here in more details. As the reduced longest word let us take i 0 = (12 · · · r) r ′ (r ′ = r for B r , C r , and r ′ = r − 1 for D r ), and set i a left factor of i 0 whose length is in [(m − 1)r + 1, mr] (m ≤ r ′ ), u the corresponding Weyl group element and (m − 2)r < k ≤ (m − 1)r. Let x L i (t) be as in (3.6) and ∆ L (k; i) as in Definition 6.1. For example, in Example 6.6 for type C 2 we have ∆ L (2; i)(Y ) = Y 2 1,1 Y 1,2 + 2 Y 1,1 Y 2,1 + Y 1,2 Y 2 2,1 + 1 Y 2,2 (1.1) where i = (1, 2, 1, 2) and Y = (Y 1,1 , Y 1,2 , Y 2,1 , Y 2,2 ). On the other hand, we have the crystal graph B(Λ 2 ) of type C 2 as in Example 5.3: Y 0,2 2 −→ Y 2 1,1 Y 1,2 1 −→ Y 1,1 Y 2,1 1 −→ Y 1,2 Y 2 2,1 2 −→ 1 Y 2,2 . (1.2) Comparing (1.1) and (1.2), it is not difficult to find their relations, that is, ∆ L (2; i) is expressed as a summation of all vertices but Y 0,2 of the graph, which is what we want to clarify in this article. Here we also observe the feature different from the one for type A r . It is that coefficients greater than 1 appear in ∆ L (k; i), which is possible to occur for types B r , C r and D r . In the case k ≤ (m − 2)r, indeed, the generalized minors ∆(k; i) seem to be expressed by monomial realizations of certain subset of crystals, nevertheless, not necessarily Demazure crystals. This is the reason that here we only treat the non-trivial last r generalized minors. So we expect to find a new characterization to abstract a subset of crystals matching the generalized minors, which is a further task for us. Let us see the organization of this article. In Sect.2, we review the explicit forms of fundamental representations of classical groups. In Sect.3, we introduce double Bruhat cells and in Sect.4, we shall review the notion of cluster algebras and generalized minors. Sect.5 is devoted to recall the theory of crystals and their monomial realizations. In Sect.6, we state Theorem 6.5 the main results of the article, which claims as follows: suppose that u is a Weyl group element corresponding to a left factor of i 0 with m(r − 2) < k ≤ m(r − 1) < l(u) ≤ mr and v = e. In the setting, ∆ L (k; i) is expressed by a summation of monomials in some Demazure crystals (see Subsection 5.2). In Sect.7, the explicit forms of ∆ L (k; i) are described using the results in [9,10,11]. In the last section, the proof of the main theorem is presented. Fundamental representations First, we review the fundamental representations of the complex simple Lie algebras g of type A r , B r , C r , and D r [16,18]. We shall use them in calculations of generalized minors (see Subsection 4.3). Let I := {1, · · · , r} be a finite index set, A = (a ij ) i,j∈I be the Cartan matrix of g, and (h, {α i } i∈I , {h i } i∈I ) be the associated root data satisfying α j (h i ) = a ij where α i ∈ h * is a simple root and h i ∈ h is a simple co-root. Let {Λ i } i∈I be the set of the fundamental weights satisfying Λ i (h j ) = δ i,j , P = i∈I ZΛ i the weight lattice and P * = i∈I Zh i the dual weight lattice. Type A r Let g = sl(r + 1, C) be the simple Lie algebra of type A r . The Cartan matrix A = (a i,j ) i,j∈I of g is as follows: a i,j =      2 if i = j, −1 if \|i − j\| = 1, 0 otherwise. For g = h, e i , f i (i ∈ I) , let us describe the vector representation V (Λ 1 ). Set B (r) := {v i \| i = 1, 2, · · · , r + 1} and define V (Λ 1 ) := v∈B (r) Cv. The weights of v i (i = 1, · · · , r + 1) are given by wt(v i ) = Λ i − Λ i−1 , where Λ 0 = Λ r+1 = 0. We define the g-action on V (Λ 1 ) as follows: hv j = h, wt(v j ) v j (h ∈ P * , j ∈ J), (2.1) f i v i = v i+1 , e i v i+1 = v i (1 ≤ i ≤ r),(2.2) and the other actions are trivial. Let Λ i be the i-th fundamental weight of type A r . As is well-known that the fundamental representation V (Λ i ) (1 ≤ i ≤ r) is embedded in ∧ i V (Λ 1 ) with multiplicity free. The explicit form of the highest (resp. lowest) weight vector u Λi (resp. v Λi ) of V (Λ i ) is realized in ∧ i V (Λ 1 ) as follows: u Λi = v 1 ∧ v 2 ∧ · · · ∧ v i , v Λi = v i+1 ∧ v i+2 ∧ · · · ∧ v r+1 . (2.3) Type C r Let g = sp(2r, C) be the simple Lie algebra of type C r . The Cartan matrix A = (a i,j ) i,j∈I of g is given by a i,j =          2 if i = j, −1 if \|i − j\| = 1 and (i, j) = (r − 1, r), −2 if (i, j) = (r − 1, r), 0 otherwise. Note that α i (i = r) are short roots and α r is the long simple root. Define the total order on the set J C := {i, i\|1 ≤ i ≤ r} by 1 < 2 < · · · < r − 1 < r < r < r − 1 < · · · < 2 < 1. (2.4) For g = h, e i , f i (i ∈ I) , let us describe the vector representation V (Λ 1 ). Set B (r) := {v i , v i \|i = 1, 2, · · · , r} and define V (Λ 1 ) := v∈B (r) Cv. The weights of v i , v i (i = 1, · · · , r) are given by wt(v i ) = Λ i − Λ i−1 and wt(v i ) = Λ i−1 − Λ i , where Λ 0 = 0. We define the g-action on V (Λ 1 ) as follows: hv j = h, wt(v j ) v j (h ∈ P * , j ∈ J C ), (2.5) f i v i = v i+1 , f i v i+1 = v i , e i v i+1 = v i , e i v i = v i+1 (1 ≤ i < r), (2.6) f r v r = v r , e r v r = v r ,(2.7) and the other actions are trivial. Let Λ i be the i-th fundamental weight of type C r . As is well-known that the fundamental representation V (Λ i ) (1 ≤ i ≤ r) is embedded in ∧ i V (Λ 1 ) with multiplicity free. The explicit form of the highest (resp. lowest) weight vector u Λi (resp. v Λi ) of V (Λ i ) is realized in ∧ i V (Λ 1 ) as follows: u Λi = v 1 ∧ v 2 ∧ · · · ∧ v i , v Λi = v 1 ∧ v 2 ∧ · · · ∧ v i . (2.8) Type B r Let g = so(2r + 1, C) be the simple Lie algebra of type B r . The Cartan matrix A = (a i,j ) i,j∈I of g is given by a i,j =          2 if i = j, −1 if \|i − j\| = 1 and (i, j) = (r, r − 1), −2 if (i, j) = (r, r − 1), 0 otherwise. Note that α i (i = r) are long roots and α r is the short simple root. Define the total order on the set J B := {i, i\|1 ≤ i ≤ r} ∪ {0} by 1 < 2 < · · · < r − 1 < r < 0 < r < r − 1 < · · · < 2 < 1. (2.9) For g = h, e i , f i (i ∈ I) , let us describe the vector representation V (Λ 1 ). Set B (r) := {v i , v i \|i = 1, 2, · · · , r} ∪ {v 0 } and define V (Λ 1 ) := v∈B (r) Cv. The weights of v i , v i (i = 1, · · · , r) and v 0 are as follows: wt(v i ) = Λ i − Λ i−1 , wt(v i ) = Λ i−1 − Λ i (1 ≤ i ≤ r − 1), (2.10) wt(v r ) = 2Λ r − Λ r−1 , wt(v r ) = Λ r−1 − 2Λ r , wt(v 0 ) = 0, where Λ 0 = 0. We define the g-action on V (Λ 1 ) as follows: 13) and the other actions are trivial. Let Λ i (1 ≤ i ≤ r − 1) be the i-th fundamental weight of type B r . Similar to the C r case, the fundamental representation V (Λ i ) is embedded in ∧ i V (Λ 1 ) with multiplicity free. In ∧ i V (Λ 1 ), the highest (resp. lowest) weight vector u Λi (resp. v Λi ) of V (Λ i ) is realized as the same form as in (2.8). hv j = h, wt(v j ) v j (h ∈ P * , j ∈ J B ), (2.11) f i v i = v i+1 , f i v i+1 = v i , e i v i+1 = v i , e i v i = v i+1 (1 ≤ i < r), (2.12) f r v r = v 0 , e r v r = v 0 , f r v 0 = 2v r , e r v 0 = 2v r ,(2. The fundamental representation V (Λ r ) is called the spin representation. It can be realized as follows: Set B (r) sp := {(ǫ 1 , · · · , ǫ r )\| ǫ i ∈ {+, −} (i = 1, 2, · · · , r)}, V (r) sp := v∈B (r) sp Cv, and define the g-action on V (r) sp as follows: h i (ǫ 1 , · · · , ǫ r ) = ǫi·1−ǫi+1·1 2 (ǫ 1 , · · · , ǫ r ) if i < r, ǫ r (ǫ 1 , · · · , ǫ r ) if i = r,(2. 14) f i (ǫ 1 , · · · , ǫ r ) =        (ǫ 1 , · · · , i −, i+1 + , · · · , ǫ r ) if ǫ i = +, ǫ i+1 = −, i = r, (ǫ 1 , · · · , ǫ r−1 , r −) if ǫ r = +, i = r, 0 otherwise, (2.15) e i (ǫ 1 , · · · , ǫ r ) =        (ǫ 1 , · · · , i +, i+1 − , · · · , ǫ r ) if ǫ i = −, ǫ i+1 = +, i = r, (ǫ 1 , · · · , ǫ r−1 , r +) if ǫ r = −, i = r, 0 otherwise. (2.16) Then the module V (r) sp is isomorphic to V (Λ r ) as a g-module. Type D r Let g = so(2r, C) be the simple Lie algebra of type D r . The Cartan matrix A = (a i,j ) i,j∈I of g is given by a i,j =      2 if i = j, −1 if \|i − j\| = 1 and (i, j) = (r, r − 1), (r − 1, r), or (i, j) = (r − 2, r), (r, r − 2), 0 otherwise. Define the partial order on the set J D := {i, i\|1 ≤ i ≤ r} by 1 < 2 < · · · < r − 1 < r r < r − 1 < · · · < 2 < 1. (2.17) Note that there is no order between r and r. For g = h, e i , f i (i ∈ I) , let us describe the vector representation V (Λ 1 ). Set B (r) := {v i , v i \|i = 1, 2, · · · , r} and define V (Λ 1 ) := v∈B (r) Cv. The weights of v i , v i (i = 1, · · · , r) are as follows: wt(v i ) = Λ i − Λ i−1 , wt(v i ) = Λ i−1 − Λ i (1 ≤ i ≤ r − 2, i = r), (2.18) wt(v r−1 ) = Λ r + Λ r−1 − Λ r−2 , wt(v r−1 ) = Λ r−2 − Λ r−1 − Λ r , where Λ 0 = 0. We define the g-action on V (Λ 1 ) as follows: hv j = h, wt(v j ) v j (h ∈ P * , j ∈ J D ), (2.19) f i v i = v i+1 , f i v i+1 = v i , e i v i+1 = v i , e i v i = v i+1 (1 ≤ i < r), (2.20) f r v r = v r−1 , f r v r−1 = v r , e r v r = v r−1 , e r v r−1 = v r ,(2.21) and the other actions are trivial. Let Λ i (1 ≤ i ≤ r − 2) be the i-th fundamental weight of type D r . Similar to the B r and C r cases, the fundamental representation V (Λ i ) is embedded in ∧ i V (Λ 1 ) with multiplicity free. In ∧ i V (Λ 1 ), the highest (resp. lowest) weight vector u Λi (resp. v Λi ) of V (Λ i ) is realized as the formula (2.8). The fundamental representations V (Λ r−1 ) and V (Λ r ) are also called the spin representations. They can be realized as follows: Set B (+,r) sp (resp. B (−,r) sp ) := {(ǫ 1 , · · · , ǫ r )\| ǫ i ∈ {+, −}, ǫ 1 · · · ǫ r = + (resp. −)}, V (+,r) sp (resp. V (−,r) sp ) := v∈B (+,r) sp (resp. B (−,r) sp ) Cv, and define the g-action on V (±,r) sp as follows: h i (ǫ 1 , · · · , ǫ r ) = ǫi·1−ǫi+1·1 2 (ǫ 1 , · · · , ǫ r ) if i < r, ǫr−1·1+ǫr·1 2 ǫ r (ǫ 1 , · · · , ǫ r ) if i = r, (2.22) f i (ǫ 1 , · · · , ǫ r ) =        (ǫ 1 , · · · , i −, i+1 + , · · · , ǫ r ) if ǫ i = +, ǫ i+1 = −, i = r, (ǫ 1 , · · · , r−1 − , r −) if ǫ r−1 = +, ǫ r = +, i = r, 0 otherwise, (2.23) e i (ǫ 1 , · · · , ǫ r ) =        (ǫ 1 , · · · , i +, i+1 − , · · · , ǫ r ) if ǫ i = −, ǫ i+1 = +, i = r, (ǫ 1 , · · · , r−1 + , r +) if ǫ r−1 = −, ǫ r = −, i = r, 0 otherwise. (2.24) Then the module V (+,r) sp (resp. V (−,r) sp ) is isomorphic to V (Λ r ) (resp. V (Λ r−1 )) as a g-module. Factorization theorem In this section, we shall introduce (reduced) double Bruhat cells G u,v , L u,v , and their properties for v = e and some special u ∈ W . In [2,3], these properties have been proven in more general setting. We shall state a relation between certain functions generalized minors on double Bruhat cells and crystal bases, which is our main result (Theorem 6.5). For l ∈ Z >0 , we set [1, l] := {1, 2, · · · , l}. Double Bruhat cells Let G be the simple complex algebraic group of classical type, B and B − be two opposite Borel subgroups in G, N ⊂ B and N − ⊂ B − be their unipotent radicals, H := B ∩ B − a maximal torus. We set g := Lie(G) with the triangular decomposition g = n − ⊕ h ⊕ n. Let e i , f i (i ∈ [1, r]) be the generators of n, n − . For i ∈ [1, r] and t ∈ C, we set x i (t) := exp(te i ), y i (t) := exp(tf i ). (3.1) Let W := s i \|i = 1, · · · , r be the Weyl group of g, where {s i } are the simple reflections. We identify the Weyl group W with Norm G (H)/H. An element s i := x i (−1)y i (1)x i (−1) (3.2) is in Norm G (H), which is representative of s i ∈ W = Norm G (H)/H [18]. For u ∈ W , let u = s i1 · · · s in be its reduced expression. Then we write u = s i1 · · · s in , call l(u) := n the length of u. We have two kinds of Bruhat decompositions of G as follows: G = u∈W BuB = u∈W B − uB − . Then, for u, v ∈ W , we define the double Bruhat cell G u,v as follows: G u,v := BuB ∩ B − vB − . This is biregularly isomorphic to a Zariski open subset of an affine space of dimension r + l(u) + l(v) [3, Theorem 1.1]. We also define the reduced double Bruhat cell L u,v as follows: L u,v := N uN ∩ B − vB − ⊂ G u,v . As is similar to the case G u,v , L u,v is biregularly isomorphic to a Zariski open subset of an affine space of dimension l(u) + l(v) [2,Proposition 4.4]. Definition 3.1. Let u = s i1 · · · s in be a reduced expression of u ∈ W (i 1 , · · · , i n ∈ [1, r]). Then the finite sequence i := (i 1 , · · · , i n ) is called a reduced word for u. For example, the sequence (1, 2, 3, 1, 2, 1) is a reduced word of the longest element s 1 s 2 s 3 s 1 s 2 s 1 of the Weyl group of type A 3 . For all the cases A r , B r , C r and D r , we fix the reduced word i 0 of the longest element as follows: i 0 =      (1, 2, · · · , r, 1, 2, · · · , r − 1, · · · , 1, 2, 3, 1, 2, 1) for A r , (1, 2, · · · , r − 1, r) r for B r , C r , (1, 2, · · · , r − 1, r) r−1 for D r . In this paper, we mainly treat (reduced) Double Bruhat cells of the form G u,e := BuB ∩ B − , L u,e := N uN ∩ B − , and the element u ∈ W whose reduced word can be written as a left factor of i 0 . Factorization theorem In this subsection, we shall introduce the isomorphisms between double Bruhat cell G u,e and H × (C × ) l(u) , and between L u,e and (C × ) l(u) . As in the previous subsection, let G be a complex classical algebraic group of type A r , B r , C r and D r . For a reduced word i = (i 1 , · · · , i n ) (i 1 , · · · , i n ∈ [1, r]), we define a map x G i : H × C n → G as x G i (a; t 1 , · · · , t n ) := a · y i1 (t 1 ) · · · y in (t n ). Next, for i ∈ [1, r] and t ∈ C × , we define as follows: α ∨ i (t) := t hi , x −i (t) := y i (t)α ∨ i (t −1 ). (3.5) For i = (i 1 , · · · , i n ) (i 1 , · · · , i n ∈ [1, r]), we define a map x L i : C n → G as x L i (t 1 , · · · , t n ) := x −i1 (t 1 ) · · · x −in (t n ). (3.6) We have the following theorem which is similar to the previous one. We define a map x G i : H × (C × ) n → G u,e as x G i (a; t 1 , · · · , t n ) = ax L i (t 1 , · · · , t n ), where a ∈ H and (t 1 , · · · , t n ) ∈ (C × ) n . In [9,10,11], we have proven the following proposition in the case G is of type A r , B r and C r . Similarly, we can prove it in the case G is type D r . Proposition 3.4. In the above setting, the map x G i is a biregular isomorphism between H × (C × ) n and a Zariski open subset of G u,e . Cluster algebras and generalized minors Following [1,3,4,7], we review the definitions of cluster algebras and their generators called cluster variables. It is known that the coordinate rings of double Bruhat cells have cluster algebra structures, and generalized minors are their cluster variables [1,8]. We will refer to a relation between certain cluster variables on double Bruhat cells and crystal bases in Sect.6. We set [1, l] := {1, 2, · · · , l} and [−1, −l] := {−1, −2, · · · , −l} for l ∈ Z >0 . For n, m ∈ Z >0 , let x 1 , · · · , x n , x n+1 , · · · , x n+m be commuting variables and P be a free multiplicative abelian group generated by x n+1 , · · · , x n+m . We set ZP := Z[x ±1 n+1 , · · · , x ±1 n+m ]. Let F := C(x 1 , · · · , x n , x n+1 , · · · , x n+m ) be the field of rational functions. Cluster algebras of geometric type In this subsection, we recall the definitions of cluster algebras. LetB = (b ij ) 1≤i≤n+m, 1≤j≤n be an (n + m) × n integer matrix. The principal part B ofB is obtained from B by deleting the last m columns. ForB and k ∈ [1, n], the new (n + m) × n integer matrix µ k (B) = (b ′ ij ) is defined by b ′ ij := −b ij if i = k or j = k, b ij + \|b ik \|b kj +b ik \|b kj \| 2 otherwise. One calls µ k (B) the matrix mutation in direction k ofB. If there exists a positive integer diagonal matrix D such that DB is skew symmetric, we say B is skew symmetrizable. It is easily verified that ifB has a skew symmetrizable principal part then µ k (B) also has a skew symmetrizable principal part [7,Proposition 3.6]. We can also verify that µ k µ k (B) =B. Define x := (x 1 , · · · , x n+m ) and we call the pair (x,B) initial seed. For 1 ≤ k ≤ n, a new cluster variable x ′ k is defined by x k x ′ k = 1≤i≤n+m, b ik >0 x b ik i + 1≤i≤n+m, b ik <0 x −b ik i . Let µ k (x) be the set of variables obtained from x by replacing x k by x ′ k . Ones call the pair (µ k (x), µ k (B)) the mutation in direction k of the seed (x,B). Now, we can repeat this process of mutation and obtain a set of seeds inductively. Hence, each seed consists of an (n + m)-tuple of variables and a matrix. Ones call this (n + m)-tuple and matrix cluster and exchange matrix respectively. Variables in cluster are called cluster variables. Definition 4.1. [3,7] LetB be an integer matrix whose principal part is skew symmetrizable and Σ = (x,B) a seed. We set A := ZP. The cluster algebra (of geometric type) A = A(Σ) over A associated with seed Σ is defined as an A-subalgebra of F generated by all cluster variables in all seeds which can be obtained from Σ by sequences of mutations. Cluster algebra A(i) Let G be a simple classical algebraic group, g := Lie(G) and A = (a i,j ) be its Cartan matrix. In Definition 3.1, we define a reduced word i = (i 1 , · · · , i l(u) ) for an element u of Weyl group W . In this subsection, we define the cluster algebra A(i), which obtained from i. It satisfies that A(i) ⊗ C is isomorphic to the coordinate ring C[G u,e ] of the double Bruhat cell [1,8]. Indeed, in [1], it is shown that C[G u,e ] holds the structure of an upper cluster algebra and in [8] it possesses the structure of a cluster algebra. Let i k (k ∈ [1, l(u)]) be the k-th index of i from the left. For t ∈ [−1, −r], we set i t := t. For k ∈ [−1, −r] ∪ [1, l(u)], we denote by k + the smallest index l such that k < l and \|i l \| = \|i k \|. For example, if i = (1, 2, 3, 1, 2) then, 1 + = 4, 2 + = 5 and 3 + is not defined. We define a set e(i) as e(i) := {k ∈ [1, l(u)]\|k + is well − defined}. Following [1], we define a quiver Γ i as follows. The vertices of Γ i are the numbers [−1, −r] ∪ [1, l(u)]. For two vertices k ∈ [−1, −r] ∪ [1, l(u)] and l ∈ [1, l(u)] with k < l, there exists an arrow k → l (resp. l → k) if and only if l = k + (resp. l < k + < l + and a i k ,i l < 0). For k, l ∈ [−1, −r], there exists an arrow k → l if and only if l < l + < k + and a i k ,i l < 0. Next, let us define a matrixB =B(i). If there exists an arrow k → l (resp. l → k) in Γ i , then b kl := 1 (resp. − 1) if \|i k \| = \|i l \|, −a \|i k \|\|i l \| (resp. a \|i k \|\|i l \| ) if \|i k \| = \|i l \|. If there exist no arrows between k and l, we set b kl = 0. Definition 4.4. By Proposition 4.3, we can construct a cluster algebra from the matrixB(i) by applying mutations. We denote this cluster algebra by A(i). Generalized minors and bilinear form Set A(i) C := A(i)⊗C. It is known that the coordinate ring C[G u,e ] of the double Bruhat cell is isomorphic to A(i) C (Theorem 4.6). To describe this isomorphism explicitly, we need generalized minors. We set G 0 := N − HN , and let x = [x] − [x] 0 [x] + with [x] − ∈ N − , [x] 0 ∈ H, [x] + ∈ N be the corresponding decomposition. Definition 4.5. For i ∈ [1, r] and u ′ ∈ W , the generalized minor ∆ uΛi,Λi is a regular function on G whose restriction to the open set uG 0 is given by ∆ uΛi,Λi (x) = ([u −1 x] 0 ) Λi . Here, Λ i is the i-th fundamental weight, for a = t h ∈ H (h ∈ P * ) and λ ∈ P , we set a λ := t λ(h) . In particular, we write ∆ Λi := ∆ Λi,Λi and call it a principal minor. We can calculate generalized minors by using a bilinear form in the fundamental representation of g = Lie(G) (Sect.2). Let x i (t), y i (t) be the ones in Sect.3 (3.1), and ω : g → g be the anti-involution ω(e i ) = f i , ω(f i ) = e i , ω(h) = h, and extend it to G by setting ω(x i (c)) = y i (c), ω(y i (c)) = x i (c) and ω(t) = t (t ∈ H). There exists a g (or G)-invariant bilinear form on the fundamental representation V (Λ i ) of g such that au, v = u, ω(a)v , (u, v ∈ V (λ), a ∈ g (or G)). For g ∈ G, we have the following simple fact: ∆ Λi (g) = gu Λi , u Λi , where u Λi is a properly normalized highest weight vector in V (Λ i ). Hence, for w ∈ W , we have ∆ wΛi,Λi (g) = ∆ Λi (w −1 g) = w −1 g · u Λi , u Λi = g · u Λi , w · u Λi , (4.1) where w is the one we defined in Sect.3 (3.2), and note that ω(s ± i ) = s ∓ i . Cluster algebras on Double Bruhat cells For u = s i1 s i2 · · · s in and k ∈ [1, l(u)], we set u ≤k = u ≤k (i) := s i1 s i2 · · · s i k . (4.2) For k ∈ [−1, −r], we set u ≤k := e. For k ∈ [−1, −r] ∪ [1, l(u)], we define ∆(k; i)(x) := ∆ u ≤k Λ \|i k \| ,Λ \|i k \| (x). We set F (i) := {∆(k; i)(x)\|k ∈ [−1, −r] ∪ [1, l(u)]}. It is known that the set F (i) is an algebraically independent generating set for the field of rational functions C(G u,e ) [3, Theorem 1.12]. Then, we have the following. Theorem 4.6. [1, 8] The isomorphism of fields ϕ : F → C(G u,e ) defined by ϕ(x k ) = ∆(k; i) (k ∈ [−1, −r] ∪ [1, l(u)]) restricts to an isomorphism of algebras A(i) C → C[G u,e ]. Monomial realizations and Demazure crystals As mentioned in the beginnings of Sect.3 and 4, our findings are relations between generalized minors on double Bruhat cells and crystal bases. More precisely, we shall describe generalized minors in terms of the monomial realizations of Demazure crystals. Let us recall these notion in this section. Let g be a complex simple Lie algebra and we will use the same notation as in Sect. 2. Monomial realizations of crystals In this subsection, we review the monomial realizations of crystals [13,15,17]. First, let us recall the crystals. Definition 5.1. [12] A crystal associated with the Cartan matrix A is a set B together with the maps wt : B → P ,ẽ i ,f i : B ∪ {0} → B ∪ {0} and ε i , ϕ i : B → Z ∪ {−∞}, i ∈ I, satisfying some properties. We call {ẽ i ,f i } i∈I the Kashiwara operators. Let U q (g) be the quantum enveloping algebra [12] associated with the Cartan matrix A, that is, U q (g) has generators {e i , f i , h i \| i ∈ I} over C(q) satisfying some relations, where q is an indeterminate. Let V (λ) (λ ∈ P + = ⊕ i∈I Z ≥0 Λ i ) be the finite dimensional irreducible representation of U q (g) which has the highest weight vector u λ , and (L(λ), B(λ)) be the crystal base of V (λ). The crystal B(λ) has a crystal structure. Let us introduce monomial realizations of crystals which realize each element of crystals as a certain Laurent monomial. We define a set of integers p = (p j,i ) j,i∈I, j =i such that p j,i = 1 if j < i, 0 if i < j. For doubly-indexed variables {Y s,i \| i ∈ I, s ∈ Z}, we define the set of monomials Y :=    Y = s∈Z, i∈I Y ζs,i s,i ζ s,i ∈ Z, ζ s,i = 0 except for finitely many (s, i)    . Finally, we define maps wt : Y → P , ε i , ϕ i : Y → Z, i ∈ I. For Y = s∈Z, i∈I Y ζs,i s,i ∈ Y, let wt(Y ) := i,s ζ s,i Λ i , ϕ i (Y ) := max    k≤s ζ k,i \| s ∈ Z    , ε i (Y ) := ϕ i (Y ) − wt(Y )(h i ). (5.1) We set A s,i := Y s,i Y s+1,i j =i Y aj,i s+pj,i,j (5.2) and define the Kashiwara operators as follows f i Y = A −1 n f i ,i Y if ϕ i (Y ) > 0, 0 if ϕ i (Y ) = 0,ẽ i Y = A ne i ,i Y if ε i (Y ) > 0, 0 if ε i (Y ) = 0, where n fi := min    n ϕ i (Y ) = k≤n ζ k,i    , n ei := max    n ϕ i (Y ) = k≤n ζ k,i    . Then the following theorem holds: 15,17]). Theorem 5.2 ([(i) For the set p = (p j,i ) as above, (Y, wt, ϕ i , ε i ,f i ,ẽ i ) i∈I is a crystal. When we emphasize p, we write Y as Y(p). (ii) If a monomial Y ∈ Y(p) satisfies ε i (Y ) = 0 (resp. ϕ i (Y ) = 0) for all i ∈ I, then the connected component containing Y is isomorphic to B(wt(Y )) (resp. B(w 0 · wt(Y )), where w 0 is the longest element of W ). Example 5.3. Let us consider the case of type C 2 . By (5.2), we get A s,1 = Y s,1 Y s+1,1 Y s,2 , A s,2 = Y s,2 Y s+1,2 Y 2 s+1,1 (s ∈ Z). We set Y := Y 0,2 ∈ Y. Following the definitions, we obtain wt(Y ) = Λ 2 , ϕ 2 (Y ) = 1, ε 2 (Y ) = ϕ 2 (Y ) − Λ 2 (h 2 ) = 0 and ϕ 1 (Y ) = ε 1 (Y ) = 0. It follows from n f2 = 0 thatf 2 Y = A −1 0,2 Y 0,2 = Y 2 1,1 Y 1,2 ,f 1 Y = 0. Similar to this, we also obtaiñ f 1f2 Y = A −1 1,1 Y 2 1,1 Y 1,2 = Y 1,1 Y 2,1 ,f 2f2 Y = 0. Repeating this argument, we see that the connected component containing Y = Y 0,2 is as follows: Y 0,2 2 −→ Y 2 1,1 Y 1,2 1 −→ Y 1,1 Y 2,1 1 −→ Y 1,2 Y 2 2,1 2 −→ 1 Y 2,2 . (5.3) The graph (5.3) is the monomial realization of the crystal base B(Λ 2 ) = B(wt(Y 0,2 )). Example 5.4. Let us consider the case of type B 3 . By (5.2), we get A s,1 = Y s,1 Y s+1,1 Y s,2 , A s,2 = Y s,2 Y s+1,2 Y s+1,1 Y 2 s,3 , A s,3 = Y s,3 Y s+1,3 Y s+1,2 , , for s ∈ Z. We set Y := Y 0,3 ∈ Y. Following the definitions, we obtain wt(Y ) = Λ 3 , ϕ 3 (Y ) = 1, ε 3 (Y ) = ϕ 3 (Y ) − Λ 3 (h 3 ) = 0 and ϕ i (Y ) = ε i (Y ) = 0 (i = 1, 2). It follows from n f3 = 0 that f 3 Y = A −1 0,3 Y 0,3 = Y 1,2 Y 1,3 ,f i Y = 0 (i = 1, 2). Similar to this, we also obtaiñ f 2f3 Y = Y 1,3 Y 2,1 Y 2,2 ,f 1f2f3 Y = Y 1,3 Y 3,1 ,f 3f2f3 Y = Y 2,1 Y 2,3 . Repeating this argument, we see that the connected component containing Y = Y 0,3 is as follows: Y 0,3 3 −→ Y 1,2 Y 1,3 2 −→ Y 1,3 Y 2,1 Y 2,2 1 −→ Y 1,3 Y 3,1 (5.4) ↓ 3 ↓ 3 Y 2,1 Y 2,3 1 −→ Y 2,2 Y 2,3 Y 3,1 2 −→ Y 2,3 Y 3,2 3 −→ 1 Y 3,3 . The graph (5.4) is the monomial realization of the crystal base B(Λ 3 ) = B(wt(Y 0,3 )). Demazure crystals For w ∈ W and λ ∈ P + , an upper Demazure crystal B + (λ) w ⊂ B(λ) is inductively defined as follows. Definition 5.5. Let u λ be the highest weight vector of B(λ). For the identity element e of W , we set B + (λ) e := {u λ }. For w ∈ W , if s i w < w, B + (λ) w := {f k i b \| k ≥ 0, b ∈ B + (λ) si w ,ẽ i b = 0} \ {0}. Similarly, we define a lower Demazure crystal B − (λ) w inductively. Definition 5.6. Let v λ be the lowest weight vector of B(λ). We set B − (λ) e := {v λ }. For w ∈ W , if s i w < w, B − (λ) w := {ẽ k i b \| k ≥ 0, b ∈ B − (λ) siw ,f i b = 0} \ {0}. Theorem 5.7. [14] For w ∈ W , let w = s i1 · · · s in be an arbitrary reduced expression. Let u λ (resp. v λ ) be the highest (resp. lowest) weight vector of B(λ). Then B + (λ) w = {f a(1) i1 · · ·f a(n) in u λ \|a(1), · · · , a(n) ∈ Z ≥0 } \ {0}, B − (λ) w = {ẽ a(1) i1 · · ·ẽ a(n) in v λ \|a(1), · · · , a(n) ∈ Z ≥0 } \ {0}. Generalized minors and crystals Let G be a complex classical algebraic group. In this section, we describe certain initial cluster variables on G u,e by monomial realizations of Demazure crystals. In the rest of paper, we only treat elements u ∈ W (l(u) = n) whose reduced word i can be written as a left factor of i 0 in (3.3), which means that i is defined by          (1, 2, · · · , r 1st cycle , 1, 2, · · · , r − 1 2nd cycle , · · · , 1, 2, · · · , r − m + 1 m−1th cycle , 1, 2, · · · , i n mth cycle ) for A r , (1, 2, · · · , r 1st cycle , 1, 2, · · · , r 2nd cycle · · · 1, 2, · · · , r m−1th cycle , 1, 2, · · · , i n mth cycle ) for B r , C r , D r , (6.1) where m ∈ [1, r] (for type A r , B r , C r ) and m ∈ [1, r − 1] (for type D r ), i n ∈ [1, r − m] (for type A r ) and i n ∈ [1, r] (for type B r , C r , D r ) . Let i k be the k-th index of i from the left, and belong to m ′ -th cycle (m ′ ≤ m). As we shall show in Lemma 6.4, we may assume i n = i k . By Theorem 4.6, we can regard C[G u,e ] as a cluster algebra and {∆(k; i)} as its cluster variables. Each ∆(k; i) is a regular function on G u,e . On the other hand, by Proposition 3.4 (resp. Theorem 3.3), ∆(k; i) can be seen as a function on H × (C × ) l(u) (resp. (C × ) l(u) ). Here, we change the variables of {∆(k; i)} as follows: Definition 6.1. Along (6.1), we set a variable Y ∈ (C × ) n as Y :=          (Y 1,1 , Y 1,2 , · · · , Y 1,r , Y 2,1 , Y 2,2 , · · · , Y 2,r−1 , · · · , Y m−1,1 , · · · , Y m−1,r−m+1 , Y m,1 , · · · , Y m,in ), (A r ) (Y 1,1 , Y 1,2 , · · · , Y 1,r , Y 2,1 , Y 2,2 , · · · , Y 2,r , · · · , Y m−1,1 , · · · , Y m−1,r , Y m,1 , · · · , Y m,in ), (B r , C r D r ) (6.2) and for a ∈ H, define ∆ G (k; i)(a, Y) := (∆(k; i) • x G i )(a, Y), ∆ L (k; i)(Y) := (∆(k; i) • x L i )(Y), where x G i and x L i are as in Subsection 3.2. Remark 6.2. If we see the variables Y s,0 , Y s,r+1 (1 ≤ s ≤ m) then we under- stand Y s,0 = Y s,r+1 = 1. For example, if i = 1 then Y s,i−1 = 1. Next proposition implies that ∆ G (k; i)(a, Y) is immediately obtained from ∆ L (k; i)(Y): Proposition 6.3. [9, 10, 11] We set d := i k . For a ∈ H, we have ∆ G (k; i)(a, Y) = (a u ≤k Λ d )∆ L (k; i)(Y) In the rest of the paper, we will treat ∆ L (k; i)(Y) only due to this proposition. Lemma 6.4. [9,10,11] Let i, Y be as in (6.1), (6.2), and u ∈ W be an element whose reduced word is i. Let i n+1 ∈ [1, r] be an index such that u ′ := us in+1 ∈ W satisfies l(u ′ ) > l(u). We set the reduced word i ′ for u ′ as i ′ = (1, · · · , r 1 st cycle , 1, · · · , r 2 nd cycle , · · · , 1, · · · , r m−1 th cycle , 1, · · · , i n m th cycle , i n+1 ), and set Y ′ ∈ (C × ) n+1 as Y ′ := (Y 1,1 , · · · , Y 1,r , · · · , Y m−1,1 , · · · , Y m−1,r , Y m,1 , · · · , Y m,in , Y ). For an integer k (1 ≤ k ≤ n), if d := i k = i n+1 , then ∆ L (k; i ′ )(Y ′ ) does not depend on Y . So, we can regard it as a function on (C × ) n . Furthermore, we have ∆ L (k; i)(Y) = ∆ L (k; i ′ )(Y ′ ). By this lemma, when we calculate ∆ L (k; i)(Y), we may assume that i n = i k without loss of generality. Let B(Λ d ) (1 ≤ d ≤ r) be the crystal base of the same type as G and v Λ d ∈ B(Λ d ) be its lowest weight vector. The following theorem is our main result. Theorem 6.5. Let i be the reduced word of u ∈ W in (6.1) and Y ∈ (C × ) n be the variables as in (6.2). We suppose that the index i k belongs to (m − 1)th cycle. Setting d : = i k = i n , there exist positive integers {a b \| b ∈ B − (Λ d ) u ≤k } such that ∆ L (k; i)(Y) = b∈B − (Λ d )u ≤k a b µ(b), (6.3) where B − (Λ d ) u ≤k is the lower Demazure crystal of B(Λ d ) and µ : B(Λ d ) → Y is the monomial realization of the crystal base B(Λ d ) such that µ(v Λ d ) = 1 Y m,d . Example 6.6. Let us consider the case of type C 2 . Take u = s 1 s 2 s 1 s 2 ∈ W and let i = (1, 2, 1, 2) be its reduced word. Following Subsection 2.2 and 4.3, we shall calculate ∆ L (2; i)(Y). First, it follows from α i (t) = t −hi and y i (t) = exp(tf i ) = 1 + tf i on V (Λ 2 ) that x L i (Y)(v 1 ∧ v 2 ) = y 1 (Y 1,1 )α ∨ 1 (Y −1 1,1 )y 2 (Y 1,2 )α ∨ 2 (Y −1 1,2 )y 1 (Y 2,1 )α ∨ 1 (Y −1 2,1 )y 2 (Y 2,2 )α ∨ 2 (Y −1 2,2 )(v 1 ∧ v 2 ) = y 1 (Y 1,1 )α ∨ 1 (Y −1 1,1 )y 2 (Y 1,2 )α ∨ 2 (Y −1 1,2 ) 1 Y 2,1 v 1 + v 2 ∧ Y 2,1 Y 2,2 v 2 + 1 Y 2,1 v 2 + v 1 = 1 Y 1,1 Y 2,1 v 1 + 1 Y 2,1 + Y 1,1 Y 1,2 v 2 + 1 Y 1,1 v 2 + v 1 ∧ Y 1,1 Y 2,1 Y 1,2 Y 2,2 v 2 + Y 2,1 Y 1,1 Y 2,2 + Y 1,2 Y 2,1 Y 1,1 v 2 + Y 2,1 Y 2,2 + Y 1,2 Y 2,1 + Y 1,1 v 1 . Since u ≤2 = s 1 s 2 , we obtain ∆ L (2; i)(Y) = x L i (Y)(v 1 ∧ v 2 ), s 1 s 2 (v 1 ∧ v 2 ) = x L i (Y)(v 1 ∧ v 2 ), v 2 ∧ v 1 = 1 Y 2,1 + Y 1,1 Y 1,2 · Y 2,1 Y 2,2 + Y 1,2 Y 2,1 + Y 1,1 − Y 1,1 Y 2,1 Y 1,2 Y 2,2 = Y 2 1,1 Y 1,2 + 2 Y 1,1 Y 2,1 + Y 1,2 Y 2 2,1 + 1 Y 2,2 . (6.4) Note that these four terms coincide with the monomials in Example 5. 3 (5.3) except for Y 0,2 . The set { Y 2 1,1 Y1,2 , Y1,1 Y2,1 , Y1,2 Y 2 2,1 , 1 Y2,2 } is a monomial realization of Demazure crystal B − (Λ 2 ) s1s2 . Example 6.7. Let us consider the case of type B 3 . Take the element u = s 1 s 2 s 3 s 1 s 2 s 3 s 1 s 2 s 3 ∈ W and let i = (1, 2, 3, 1, 2, 3, 1, 2, 3) be its reduced word. Following Subsection 2.3 and 4.3, we shall calculate ∆ L (6; i)(Y): ∆ L (6; i)(Y) = Y 1,2 Y 1,3 + Y 1,3 Y 2,1 Y 2,2 + Y 1,3 Y 3,1 + Y 2,1 Y 2,3 + Y 2,2 Y 2,3 Y 3,1 + Y 2,3 Y 3,2 + 1 Y 3,3 . Note that these seven terms coincide with the monomials in Example 5. 4 (5.4) except for Y 0,3 . The set { Y1,2 Y1,3 , Y1,3Y2,1 Y2,2 , Y1,3 Y3,1 , Y2,1 Y2,3 , Y2,2 Y2,3Y3,1 , Y2,3 Y3,2 , 1 Y3,3 } is a monomial realization of Demazure crystal B − (Λ 3 ) s1s2s3s1s2s3 . 7 The explicit formula of ∆ L (k; i) of type D r In [9,10,11], we have given the explicit formula of minors ∆ L (k; i) of type A r , B r and C r . In this section, we shall give the one for type D r in some special cases, which is needed in the proof of Theorem 6.5. Let i be a reduced word in (3.3) and i k be its k-th the index from the left. We suppose that i k belongs to (m − 1) th cycle as in the previous section. We shall prove the following theorems: The explicit formula of ∆ L (k; i) For l ∈ [1, m] and k ∈ J D , we define the Laurent monomials D(l, k) by D(l, k) :=                  Y l,k−1 Y l,k if 1 ≤ k ≤ r − 2, Y l,r−2 Y l,r−1 Y l,r if k = r − 1, Y l,r−1 Y l+1,r if k = r, Y l,r Y l,r−1 Y l+1,r−2 if k = r − 1, Y l,\|k\| Y l+1,\|k\|−1 if r − 2 ≤ k ≤ 1, or k = r. Theorem 7.1. In the setting of Theorem 6.5, suppose that i k = d < r − 1. We have ∆ L (k; i)(Y) = ( * ) D(l 1 , k 1 )D(l 2 , k 2 ) · · · D(l d , k d ), (7.3) l i := m − k i + i if 1 ≤ k i ≤ r − 1, m − r + i if k i ∈ {r, · · · , 1} ∪ {r}, where ( * ) is the condition for k i (1 ≤ i ≤ d) : k 1 k 2 · · · k d , (7.4) i ≤ k i ≤ m − 1 + i (1 ≤ i ≤ r − m), (7.5) i ≤ k i ≤ d − i + 1 (r − m < i ≤ d). (7.6) Theorem 7.2. In the setting of Theorem 6.5, suppose that i k = r (resp. i k = r − 1). We have ∆ L (k; i)(Y) = ( * ) D(m − 1, k 1 )D(m − 2, k 2 ) · · · D(m − s, k s )D(m − s, r + 1), where ( * ) is the condition for s ∈ Z ≥0 and k i (1 ≤ i ≤ s) : 0 ≤ s ≤ m − 1 and s is even (resp. odd), 1 ≤ k 1 < k 2 < · · · < k s ≤ r. (7.7) 7.2 The set X d (m, m − 1) of paths In this subsection, we shall introduce a set X d (m, m − 1) of paths which will correspond to the set of the monomials in ∆ L (k; i)(Y). Let m and d be the positive integers as in Sect.6. Let J D := {j, j\| 1 ≤ j ≤ r} be as in Subsection 2.4 and for l ∈ {1, 2, · · · , r}, set \|l\| = \|l\| = l. Definition 7.3. Let us define the directed graph (V d , E d ) as follows: The set V d = V d (m) of vertices is defined by V d (m) := {vt(m − s; a (s) )\| 0 ≤ s ≤ m, a (s) = (a (s) 1 , a (s) 2 , · · · , a (s) d ) ∈ J d D }. And we define the set E d = E d (m) of directed edges as E d (m) := {vt(m − s; a (s) ) → vt(m − s − 1; a (s+1) ) \| 0 ≤ s ≤ m − 1, vt(m − s; a (s) ), vt(m − s − 1; a (s+1) ) ∈ V d (m)}. Definition 7.4. Let X d (m, m − 1) be the set of directed paths p p = vt(m; a (0) 1 , · · · , a (0) d ) → vt(m−1; a (1) 1 , · · · , a (1) d ) → vt(m−2; a (2) 1 , · · · , a (2) d ) → · · · → vt(1; a (m−1) 1 , · · · , a (m−1) d ) → vt(0; a (m) 1 , · · · , a (m) d ), which satisfy the following conditions (i)-(v) : For s ∈ Z (0 ≤ s ≤ m), , · · · , a (s+1) d ) in V d (m) are connected if the conditions (i), (ii) and (iv) in Definition 7.4 are satisfied. And for a path p as above, we call the sequence a (i) a (s) 1 < a (s) 2 < · · · < a (s) d in the order (2.17), (ii) If 1 ≤ a (s) ζ ≤ r − 2, then a (s+1) ζ = a (s) ζ or a (s) ζ + 1. If r − 1 ≤ a (s) ζ ≤ 1, then a (s) ζ ≤ a (s+1) ζ ≤ 1, (iii) (a (0) 1 , a (0) 2 , · · · , a (0) d ) = (1, 2, · · · , d) and (a (m) 1 , · · · , a (m) d ) = (m, m + 1, · · · , r − 1, d − r + m, · · · , 2, 1) if m + d > r, (m, m + 1, · · · , m + d − 1) if m + d ≤ r,(iv)(0) i → a (1) i → a (2) i → · · · → a (m) i the i-sequence of p. We define Laurent monomials associated with edges in E d (m). Definition 7.5. (i) For each s (0 ≤ s ≤ m) and i, j ∈ J D with i ≤ j, we set Q (s) (i → j) := Ym−s,i−1 Ym−s,i if j = i, i ∈ [1, r − 2], 1 if j > i, i ∈ [1, r − 2], Q (s) (r − 1 → j) :=        Ym−s,r−2 Ym−s,r−1Ym−s,r if j = r − 1, 1 Ym−s,r if j = r, 1 Y m−s,\|j\|−1 if j ∈ {r, r − 1, · · · , 1}, Q (s) (r → j) := Ym−s,r−1 Ym−s,r if j = r, Ym−s,r−1 Y m−s,\|j\|−1 if j ∈ {r − 1, · · · , 1}, Q (s) (i → j) :=        Ym−s,r Y m−s,\|j\|−1 if i = r, Ym−s,r Ym−s,r−1 Y m−s,\|j\|−1 if i = r − 1, Y m−s,\|i\| Y m−s,\|j\|−1 if i ∈ {r − 2, · · · , 2 For an edge e i1,··· ,i d j1,··· , j d = vt(m − s; i 1 , · · · , i d ) → vt(m − s − 1; j 1 , · · · , j d ) in E d (m) such that i k ≤ j k (k = 1, 2, · · · , d), we define the label Q (s) (e i1,··· ,i d j1,··· ,j d ) of the edge e i1,··· ,i d j1,··· ,j d as Q (s) (e i1,··· ,i d j1,··· ,j d ) := d k=1 Q (s) (i k → j k ). (ii) Let p ∈ X d (m, m − 1) be a path: p = vt(m; a (0) 1 , · · · , a (0) d ) → vt(m−1; a(1)1 , · · · , a(1)d ) → vt(m−2; a(2) 1 , · · · , a d ) → · · · → vt(1; a (m−1) 1 , · · · , a (m−1) d ) → vt(0; a (m) 1 , · · · , a (m) d ).(2) For each s (0 ≤ s ≤ m − 1), we denote the label of the (m − s) th edge vt(m − s; a (s) 1 , · · · , a (s) d ) → vt(m − s − 1; a (s+1) 1 , · · · , a (s+1) d ) of p by Q (s) (p). And we define the label Q(p) of the path p as the total product: Q(p) := m−1 s=0 Q (s) (p). (iii) For a subpath p ′ p ′ = vt(m − s ′ ; a (s ′ ) ) → vt(m − s ′ − 1; a (s ′ +1) ) → · · · → vt(m − s ′′ ; a (s ′′ ) ) of p (0 ≤ s ′ < s ′′ ≤ m), we define the label of the subpath p ′ as Q(p ′ ) := s ′′ −1 s=s ′ Q (s) (p). One-to-one correspondence between paths in X d (m, m− 1) and monomials in ∆ L (k; i)(Y) We define x −i (Y ) = y i (Y ) · Y −hi in (3.5) (Y ∈ C × ). For 1 ≤ s ≤ m and i 1 , · · · , i d ∈ J D = {i, i\|1 ≤ i ≤ r}, let us define x (s) −[1,r] := x −1 (Y s,1 ) · · · x −r (Y s,r ). (7.8) and (s; i 1 , i 2 , · · · , i d ) := x (1) −[1,r] x (2) −[1,r] · · · x (s) −[1,r] (v i1 ∧ · · · ∧ v i d ), u ≤k (v 1 ∧ · · · ∧ v d ) . (7.9) Since i k belongs to (m − 1) th cycle of i, by (3.2) we have u ≤k (v 1 ∧ · · · ∧ v d ) = v m ∧ · · · ∧ v m+d−1 if d ≤ r − m, v m ∧ · · · ∧ v r−1 ∧ v d−r+m ∧ · · · ∧ v 1 if d > r − m. (7.10) We can verify ∆ L (k; i)(Y) = (m; 1, · · · , d). Following Subsection 2.4, we get the following: Lemma 7.6. We obtain x (s) −[1,r] v j = Y s,j−1 Y s,j v j + v j+1 , x (s) −[1,r] v j = j i=1 Y s,j Y s,i−1 v i (1 ≤ j ≤ r − 2) x (s) −[1,r] v r−1 = Y s,r−2 Y s,r−1 Y s,r v r−1 + 1 Y s,r v r + r i=1 1 Y s,i−1 v i , x (s) −[1,r] v r = Y s,r−1 Y s,r v r + r−1 i=1 Y s,r−1 Y s,i−1 v i x (s) −[1,r] v r = r i=1 Y s,r Y s,i−1 v i , x (s) −[1,r] v r−1 = r−1 i=1 Y s,r−1 Y s,r Y s,i−1 v i . Lemma 7.7. Let (s; i 1 , i 2 , · · · , i d ) (1 ≤ s ≤ m) be as in (7.9), (i) if i 1 · · · i d and there is no number j ∈ [1, d − 1] such that i j = r, i j+1 = r or i j = r, i j+1 = r then (s; i 1 , i 2 , · · · , i d ) = V d t=1 Q (m−s) (i t → j t )(s − 1; j 1 , · · · , j d ), (7.11) where V := {(j 1 , · · · , j d )\| i t ≤ j t , if i t ≤ r −2 then j t = i t or i t +1, if i t ≥ r − 1 then j t i t+1 }. (ii) We suppose that s ≤ m − r + j − 1 and i 1 < · · · < i j−1 for some 1 ≤ j ≤ d. If i j−1 ≤ r − 1, r − 1 ≤ i j+1 and either i j = r or i j = r, then (s; i 1 , i 2 , · · · , i d ) = 0. (iii) We suppose that i 1 · · · i d and s ≤ m − r + j − 1 for some 1 ≤ j ≤ d. If i j−2 ≤ r − 1, r − 1 ≤ i j+1 and either i j−1 = r, i j = r or i j−1 = r, i j = r then (s; i 1 , i 2 , · · · , i d ) = 0. Proof. (i) Let us calculate x (s) −[1,r] (v i1 ∧ · · · ∧ v i d ). It follows from Lemma 7.6 and Definition 7.5 (i) that x (s) −[1,r] (v i1 ∧ · · · ∧ v i d ) = V ′ d t=1 Q (m−s) (i t → j t )v j1 ∧ · · · ∧ v j d ,(7.12) where (j 1 , · · · , j d ) runs over the set V ′ := {(j 1 , · · · , j d )\| i t ≤ j t , if i t ≤ r − 2 then j t = i t or i t +1}. Now we set V ′′ := {(j 1 , · · · , j d ) ∈ V ′ \| ∃l such that \|j l \| ≤ \|i l+1 \|}. We define the map τ : V ′′ → V ′′ as follows: Take (j 1 , · · · , j d ) ∈ V ′′ . Let l (1 ≤ l ≤ d − 1) be the index such that \|j 1 \| > \|i 2 \|, \|j 2 \| > \|i 3 \|, · · · , \|j l−1 \| > \|i l \| and \|j l \| ≤ \|i l+1 \|. Since \|j l+1 \| ≤ \|i l+1 \| by the definition of V ′ , we have (j 1 , · · · , j l+1 , j l , · · · , j d ) ∈ V ′′ . So, we define τ (· · · , j l , j l+1 , · · · ) := (· · · , j l+1 , j l , · · · ). We can easily see that τ 2 = id V ′′ . In (7.12), (v j1 ∧ · · · ∧ j l ∧ j l+1 , · · · , j d ) and (v j1 ∧ · · · ∧ j l+1 ∧ j l ∧ · · · ∧ j d ) have the same coefficient and then they are cancelled, which yields that x (s) −[1,r] (v i1 ∧ · · · ∧ v i d ) = V d t=1 Q (m−s) (i t → j t )v j1 ∧ · · · ∧ v j d , since V = V ′ \ V ′′ . The definition (7.9) implies our desired result. (ii) We use induction on s. We suppose that i j−1 ≤ r−1, i j = r. By the same argument as in (i), (s; i 1 , i 2 , · · · , i d ) is a linear combination of {(s−1; l 1 , · · · , l d )} such that l j−2 ≤ r − 1 and l j−1 ≤ r − 1 or l j−1 = r. In the case l j−1 = r, (s − 1; l 1 , · · · , l d ) = 0 by the assumption of induction. Hence, we may assume that l j−1 ≤ r−1. Then (s−1; l 1 , · · · , l d ) is a linear combination of (0; ζ 1 , · · · , ζ d ) such that #{1 ≤ α ≤ d\| r ≤ ζ α ≤ 1} ≤ (s − 2) + (d − j + 2) = s + d − j ≤ m − r + d − 1, which implies that (0; ζ 1 , · · · , ζ d ) = 0 by (7.9) and (7.10). Hence we obtain (s; i 1 , i 2 , · · · , i d ) = 0. Similarly, we can verify that if i j−1 ≤ r − 1, i j = r then (s; i 1 , i 2 , · · · , i d ) = 0. (iii) Similar to (ii), we use induction on s. We suppose that i j−1 = r, i j = r. By the same argument as in (i), (s; i 1 , i 2 , · · · , i d ) is a linear combination of {(s − 1; l 1 , · · · , l d )} such that i t+1 l t and either l j−1 = r or l j = r. If l j−1 = r then we may assume that l j−2 ≤ r − 1 by the assumption of induction. In this case, however, (s−1; l 1 , · · · , l d ) = 0 by (ii), which implies that (s; i 1 , i 2 , · · · , i d ) = 0. If l j = r then we also get l j−2 ≤ r − 1, and (s − 1; l 1 , · · · , l d ) is a linear combination of {(s − 2; m 1 , · · · , m d )} such that m j−2 = r − 1 or r. Using (ii), we have (s − 2; m 1 , · · · , m d ) = 0, which implies (s; i 1 , i 2 , · · · , i d ) = 0. Supposing s ≤ m−r +j −1 and writing (m; 1, · · · , d) as a linear combination of {(s; i 1 , · · · , i d )\| 1 ≤ i 1 , · · · , i d ≤ 1}, then the coefficient of (s; i 1 , · · · , i d ) such that either i 1 < · · · < i j−1 = r, i j = r or i 1 < · · · < i j−1 = r, i j = r is 0. Proposition 7.8. We have ∆ L (k; i)(Y) = p∈X d (m,m−1) Q(p). Proof. We suppose that 1 ≤ i 1 < · · · < i d ≤ 1 and 1 ≤ s ≤ m. By the definition of V in Lemma 7.7, we see that (j 1 , · · · , j d ) ∈ V if and only if the vertices vt(s − 1; j 1 , · · · , j d ) and vt(s; i 1 , · · · , i d ) are connected (Definition 7.4). Further, the coefficient of (s − 1; j 1 , · · · , j d ) in (7.11) coincides with the label of the edge e i1,··· ,i d j1,··· ,j d between vt(s; i 1 , · · · , i d ) and vt(s − 1; j 1 , · · · , j d ) (Definition 7.5 (i)). Hence, we get (s; i 1 , · · · , i d ) = (j1,··· ,j d ) Q (m−s) (e i1,··· ,i d j1,··· ,j d ) · (s − 1; j 1 , · · · , j d ),(7.13) where (j 1 , · · · , j d ) runs over the set {(j 1 , · · · , j d )\| vt(s−1; j 1 , · · · , j d ) and vt(s; i 1 , · · · , i d ) are connected}. Since we know that if j l ∈ {r, · · · , 1} ∪ {0} then j l ≤ i l+1 , if i l ≤ r − 1 then j l = i l or i l + 1, and i l+1 ≤ j l+1 in V , one gets j l < j l+1 , and then j 1 < j 2 < · · · < j d . The followings is obtained in the same way as (7.13): (s − 1; j 1 , · · · , j d ) = (k1,··· ,k d ) Q (m−s+1) (e j1,··· ,j d k1,··· ,k d ) · (s − 2; k 1 , · · · , k d ), (7.14) where (k 1 , · · · , k d ) runs over the set {(k 1 , · · · , k d )\| vt(s−2; k 1 , · · · , k d ) and vt(s− 1; j 1 , · · · , j d ) are connected} and e j1,··· ,j d k1,··· ,k d is the edge between vertices vt(s − 1; j 1 , · · · , j d ) and vt(s−2; k 1 , · · · , k d ). By (7.13), (7.14), (s; i 1 , · · · , i d ) is a linear combination of {(s − 2; k 1 , · · · , k d )}, and the coefficient of (s − 2; k 1 , · · · , k d ) is as follows: (j1,··· ,j d ) Q (m−s) (e i1,··· ,i d j1,··· ,j d ) · Q (m−s+1) (e j1,··· ,j d k1,··· ,k d ) · (s − 2; k 1 , · · · , k d ), where (j 1 , · · · , j d ) runs over the set {(j 1 , · · · , j d )\| vt(s−1; j 1 , · · · , j d ) is connected to the vertices vt(s; i 1 , · · · , i d ) and vt(s − 2; k 1 , · · · , k d )}. The coefficient of (s − 2; k 1 , · · · , k d ) coincides with the label of subpath (Definition 7.5 (iii)) vt(s; i 1 , · · · , i d ) → vt(s − 1; j 1 , · · · , j d ) → vt(s − 2; k 1 , · · · , k d ). Repeating this argument, we see that (s; i 1 , · · · , i d ) is a linear combination of {(0; l 1 , · · · , l d )} (1 ≤ l 1 < · · · < l d ≤ 1). The coefficient of (0; l 1 , · · · , l d ) is equal to the sum of labels of all subpaths from vt(s; i 1 , · · · , i d ) to vt(0; l 1 , · · · , l d ). In the case m ′ + d > r (resp. m ′ + d ≤ r), for 1 ≤ l 1 < · · · < l d ≤ 1, if (l 1 , · · · , l d ) = (m ′ + 1, m ′ + 2, · · · , r, d − r + m ′ , · · · , 2, 1) (resp. = (m ′ + 1, m ′ + 2, · · · , m ′ + d)), then we obtain (0; l 1 , · · · , l d ) = 1 by (7.9). If (l 1 , · · · , l d ) is not as above, we obtain (0; l 1 , · · · , l d ) = 0. Therefore, we see that (s; i 1 , · · · , i d ) is equal to the sum of labels of subpaths from vt(s; i 1 , · · · , i d ) to vt(0; m, m + 1, · · · , r − 1, d − r + m, · · · , 2, 1) (resp. vt(0; m, m + 1, · · · , m + d − 1)). In particular, ∆ L (k; i)(Y) = (m; 1, 2, · · · , d) is equal to the sum of labels of paths in X d (m, m − 1), which shows ∆ L (k; i)(Y) = p∈X d (m,m−1) Q(p). (i) For i with r − m + 1 ≤ i ≤ d, we obtain a (m) i = a (m−1) i = · · · = a (r−i+1) i = d − i + 1. (7.16) (ii) Let a (0) i → a (1) i → a (2) i → · · · → a (m) i be the i-sequence of the path p (Def inition 7.4). (1) In the case i ≤ r − m, one has #{0 ≤ s ≤ m − 1\| 1 ≤ a (s) i ≤ r − 1, and a (s) i = a (s+1) i } = 1. (2) In the case i > r − m, one has #{0 ≤ s ≤ r−i+1\| 1 ≤ a (s) i ≤ r−1 and a (s) i = a (s+1) i }+#{0 ≤ s ≤ r−i+1\| a (s) i ∈ {r, r, · · · , 1}} = 1. Proof. (i) By Definition 7.4 (iii), we get a (m) r−m+1 = d − r + m. Using Definition 7.4 (iv) repeatedly, we obtain d − r + m = a (m) r−m+1 < a (m−1) r−m+2 < a (m−2) r−m+3 < · · · < a (r−d+1) d ≤ 1, which means a (r−i+1) i = d − i + 1 (r − m + 1 ≤ i ≤ d). It follows from Definition 7.4 (ii) and (iii) that d − i + 1 = a (r−i+1) i ≤ a (r−i+2) i ≤ · · · ≤ a (m−1) i ≤ a (m) i = d − i + 1, which yields (7.16). (ii) (1) In the case i ≤ r − m, Definition 7.4 (ii) and (iii) show that i = a (0) i ≤ a (1) i ≤ · · · ≤ a (m) i = m + i − 1, a (s+1) i = a (s) i or a (s) i + 1. (7.17) In particular, we get 1 ≤ a (s) i ≤ r for 1 ≤ s ≤ m. By (7.17), we obtain #{0 ≤ s ≤ m − 1\| a (s+1) i = a (s) i + 1} = m − 1, which implies #{0 ≤ s ≤ m − 1\| a (s) i = a (s+1) i } = 1. (2) In the case i > r − m, by Definition 7.4 (ii), we have i = a (0) i ≤ a (1) i ≤ · · · ≤ a (r−i+1) i ≤ 1. We suppose that i = a (0) i ≤ a (1) i ≤ · · · ≤ a (l) i ≤ r − 1, and r − 1 < a (l+1) i ≤ · · · ≤ a (r−i) i ≤ 1, (7.18) for some l (1 ≤ l ≤ r − i). Definition 7.4 (ii) implies that a (s+1) i = a (s) i or a (s) i + 1 (0 ≤ s ≤ l − 1) and a (l) i = r − 1. Therefore, i = a (0) i ≤ a (1) i ≤ · · · ≤ a (l) i = r − 1, a (s+1) i = a (s) i or a (s) i + 1. So, we have #{0 ≤ s ≤ l − 1\| a (s+1) i = a (s) i } = l − (r − 1 − i) = l − r + i + 1 in the same way as (2)(i). On the other hand, the assumption r−1 < a (l+1) i ≤ · · · ≤ a (r−i) i ≤ 1 in (7.18) means that {l+1 ≤ s ≤ r−i\| a (s) i ∈ {r, r, · · · , 1}} = {l+1, l+2, · · · , r−i}. Hence, #{0 ≤ s ≤ l − 1\| a (s+1) i = a (s) i } + #{l + 1 ≤ s ≤ r − i\| a (s) i ∈ {r, r, · · · , 1}} = (l − r + i + 1) + (r − i − l) = 1. By this lemma, we define l i ∈ {0, 1, · · · , m} (1 ≤ i ≤ d) for the path p ∈ X d (m, m − 1) in (7.15) as follows: For i ≤ r − m, we define 0 ≤ l i ≤ m as the unique number which satisfies a (li) i = a (li+1) i ≤ r − 1 (Lemma 7.9(ii) (1)). For i > r − m, we define l i (0 ≤ l i ≤ r − i + 1) as the unique number which satisfies either a (li) i = a (li+1) i ≤ r − 1 or a (li) i ∈ {r, r, · · · , 1} (Lemma 7.9(ii) (2)). We also set k i ∈ {j, j\| 1 ≤ j ≤ r} (1 ≤ i ≤ d) as k i := a (li) i . Lemma 7.10. For a path p as in the paragraph above, we have the following. (i) For 1 ≤ i ≤ d, l i = k i − i if k i ∈ {1, 2, · · · , r − 1}, r − i if k i ∈ {r, r, r − 1, · · · , 1}. (ii) For 1 ≤ i ≤ d − 1, if k i ∈ {1, 2, · · · , r − 1}, then k i < k i+1 , l i ≤ l i+1 . For 1 ≤ i ≤ d − 1, if k i ∈ {r, r, r − 1, · · · , 1}, then k i k i+1 , l i = l i+1 + 1. (iii) One has Q(p) = D(m − l 1 , k 1 )D(m − l 2 , k 2 ) · · · D(m − l d , k d ). Proof. (i) First, we suppose that k i ∈ {1, 2, · · · , r − 1}. The definition of l i says a (0) i = i, a(1)i = i + 1, a(2)i = i + 2, · · · , a (li) i = l i + i. Hence, k i = a (li) i = l i + i, which means that l i = k i − i. Next, we suppose that k i ∈ {r, r, r − 1, · · · , 1}. It follows from Lemma 7.9 (i) that l i = r − i. (ii) Since a (s) i < a (s) i+1 for 0 ≤ s ≤ m, if k i ∈ {1, 2, · · · , r − 1} then k i < k i+1 . The inequality l i ≤ l i+1 is obtained by (i). If k i ∈ {r, r, r − 1, · · · , 1} then k i k i+1 by Definition 7.4 (iv). It is easily to see l i = l i+1 + 1 by using (i). (iii) By Definition 7.5 (i) and Lemma 7.9 (ii), one gets Q (0) (a (0) i → a (1) i )Q (1) (a (1) i → a (2) i ) · · · Q (m−1) (a (m−1) i → a (m) i ) = D(m − l i , k i ) if i ≤ r − m, D(m − l i , k i ) · 1 Y m−r+i,d−i m s=r−i+1 Y m−s,d−i+1 Y m−s,d−i if i > r − m. Therefore, we get Q(p) = d i=1 m−1 s=0 Q (s) (a (s) i → a (s+1) i ) = D(m − l 1 , k 1 ) · · · D(m − l d , k d ). The proof of Theorem 7.1 We shall prove that ∆ L (k; i)(Y) = ( * ) D(m − l 1 , k 1 )D(m − l 2 , k 2 ) · · · D(m − l d , k d ), (7.19) l i := k i − i if 1 ≤ k i ≤ r − 1, r − i if k i ∈ {r, · · · , 1} ∪ {r}, which is equivalent to Theorem 7.1. [Proof of Theorem 7.1.] As we have seen in Lemma 7.10, the monomial Q(p) (p ∈ X d (m, m − 1)) is described as Q(p) = D(m − l 1 , k 1 )D(m − l 2 , k 2 ) · · · D(m − l d , k d ) with k 1 k 2 · · · k d . Definition 7.4 (iii) and the definition of l i imply that a (0) i = i ≤ a (li) i = k i ≤ m + i − 1 = a (m) i (i ≤ r − m) and a (0) i = i ≤ a (li) i = k i ≤ d − i + 1 = a (m) i (i > r − m) . Therefore, these {k i } 1≤i≤d satisfy the conditions (7.4), (7.5) and (7.6) in Theorem 7.1. Conversely, let {K i } 1≤i≤d be the sequence in J D which satisfies the conditions (7.4), (7.5) and (7.6) in Theorem 7.1: K 1 K 2 · · · K d , (7.20) i ≤ K i ≤ m − 1 + i (1 ≤ i ≤ r − m), (7.21) i ≤ K i ≤ d − i + 1 (r − m < i ≤ dQ(p) = D(m − L 1 , K 1 )D(m − L 2 , K 2 ) · · · D(m − L d , K d ) (7.23) and L i := K i − i if K i ∈ {1, · · · , r − 1}, r − i if K i ∈ {r, r, · · · , 1}, for 1 ≤ i ≤ d. Since we supposed K i K i+1 , we can easily verify if K i ∈ {1, · · · , r − 1} then L i ≤ L i+1 and if K i ∈ {r, r, · · · , 1} then L i = L i+1 + 1. We define a path p = vt(m; a (0) 1 , · · · , a (0) d ) → · · · → vt(0; a (m) 1 , · · · , a (m) d ) ∈ X d (m, m − 1) as follows: For i (1 ≤ i ≤ r − m), we define the i-sequence of p as a (0) i = i, a (1) i = i + 1, · · · , a (Li) i = L i + i, a (Li+1) i = L i + i, a (Li+2) i = L i + i + 1, a (Li+3) i = L i + i + 2, · · · , a (m) i = m + i − 1. For i (r − m < i ≤ d), if K i ∈ {1, 2, · · · , r − 1}, we define a (0) i = i, a (1) i = i + 1, · · · , a (Li) i = L i + i, a (Li+1) i = L i + i, a (Li+2) i = L i + i + 1, a (Li+3) i = L i +i+2, · · · , a (r−i) i = r−1, a (r−i+1) i = a (r−i+2) i = · · · = a (m) i = d − i + 1, if K i ∈ {r, r, · · · , 1}, define a (0) i = i, a (1) i = i+1, · · · , a (r−i−1) i = r−1, a (r−i) i = K i , a (r−i+1) i = a (r−i+2) i = · · · = a (m) i = d − i + 1. These mean that a 1 , · · · , k (s) t )\|0 ≤ t ≤ s ≤ m, 1 ≤ k (s) 1 < k (s) 2 < · · · < k (s) t ≤ r}. And we define the set E sp = E sp (m) of directed edges as E sp (m) := {vt(s; k (s) 1 , · · · , k (s) t ) → vt(s − 1; k (s+1) 1 , · · · , k (s+1) t ′ ) \| 0 ≤ s ≤ m − 1, vt(s; k (s) 1 , · · · , k (s) t ), vt(s − 1; k (s−1) 1 , · · · , k (s−1) t ′ ) ∈ V sp (m)}. Definition 7.12. Let X r (m, m − 1) (resp. X r−1 (m, m − 1)) be the set of directed paths p in (V sp , E sp ) which satisfy the following conditions: For s ∈ Z (1 ≤ s ≤ m), if vt(s; k (s) 1 , · · · , k (s) t ) → vt(s − 1; k (s−1) 1 , · · · , k (s−1) t ′ ) is an edge included in p, then (i) t ′ = t or t ′ = t + 2, (ii) if t ′ = t + 2, then k (s−1) t+2 = r, (iii) k (s−1) i ≤ k (s) i < k (s−1) i+1 , (iv) the starting vertex of p is vt(m; φ) (resp. vt(m; r)), (v) if m is odd (resp. even) then the ending vertex of p is vt(0; 1, 2, · · · , m−1), if m is even (resp. odd) then one of p is vt(0; 1, 2, · · · , m − 1, r). Define a Laurent monomial associated with each edge of paths in X r (m, m− 1) and X r−1 (m, m − 1). Definition 7.13. Let p be a path in X r (m, m − 1) or X r−1 (m, m − 1). For each s (1 ≤ s ≤ m), we define the label of an edge vt(s; i 1 , i 2 , · · · , i t ) → vt(s − 1; j 1 , j 2 , · · · , j t ′ ) in p as the Laurent monomial which is given as follows and denote it Q (s) (p) or Q(vt(s; i 1 , i 2 , · · · , i t ) → vt(s − 1; j 1 , j 2 , · · · , j t ′ )): Q (s) (i p → j p ) :=        Ym−s,i p Ym−s,j p −1 if i p ≤ r − 2, Ym−s,r−1Ym−s,r Ym−s,j p −1 if i p = r − 1, Ym−s,r Ym−s,j p −1 if i p = r, Q (s) (p) := 1 Ym−s,r t p=1 Q (s) (i p → j p ) if t ′ = t, 1 Ym−s,j t+1 −1 t p=1 Q (s) (i p → j p ) if t ′ = t + 2. And we define the label Q(p) of the path p as the total product: Q(p) := m−1 s=0 Q (s) (p). Proposition 7.14. In the setting of Theorem 7.2, we suppose that i k = r − 1 or r. ∆ L (k; i)(Y) = p∈X i k (m,m−1) Q(p). (7.24) Proof. We shall see the case i k = r. Using the bilinear form (4.1), we obtain ∆ L (k; i)(Y) = ∆ u ≤k Λr ,Λr (x L i (Y)) = x L i (Y) · u Λr , u ≤k · u Λr , where u Λr is the highest vector in the spin representation V (Λ r ). As we have seen in Subsection 2.4, we can describe basis vectors of V (Λ r ) as (ǫ 1 , · · · , ǫ r ) (ǫ i ∈ {+, −}). In particular, the highest weight vector u Λr is equal to (+, +, · · · , +). Using (2.23) and (2.24), we have u ≤k · u Λr = (s 1 s 2 · · · s r ) m−1 (+, +, +, · · · , +) = (s 1 s 2 · · · s r ) m−2 (−, +, +, · · · , −) . . . =          (−, −, · · · , − m−1 , +, · · · , +, −) if m is even, (−, −, · · · , − m−1 , +, · · · , +, +) if m is odd. Now, we define the following notation: For (ǫ 1 , · · · , ǫ r ) ∈ B (r) sp , let {k 1 , · · · , k t } be the set of indices such that {1 ≤ i ≤ r\| ǫ i = −} = {k 1 , · · · , k t }. Then we de- note [k 1 , · · · , k t ] := (ǫ 1 , · · · , ǫ r ). Applying x −1 (Y s,1 ) · · · x −r (Y s,r ) on [k 1 , · · · , k t ] (1 ≤ s ≤ m), it becomes a linear combination of {[j 1 , · · · , j t ]\| k l−1 < j l ≤ k l (1 ≤ l ≤ t)} ∪ {[j 1 , · · · , j t , j t+1 , r]\|k l−1 < j l ≤ k l (1 ≤ l ≤ t + 1)}. It follows from (2.14) and (2.15) that x −k l−1 (Y s,k l−1 ) · · · x −(k l −1) (Y s,k l −1 )[· · · , k l , · · · ] = · · ·+ 1 Y s,k l −1 1 2 Y s,k l −1 Y s,k l −2 1 2 · · · Y s,j l Y s,j l −1 1 2 Y s,j l −2 Y s,j l −1 1 2 Y s,j l −3 Y s,j l −2 1 2 · · · Y s,k l−1 +1 Y s,k l−1 +2 1 2 Y s,k l−1 (Y s,k l−1 +1 ) 1 2 [· · · , j l , · · · ]+· · · = · · ·+ Y s,k l−1 Y s,j l −1 [· · · , j l , · · · ]+· · · . Thus, the coefficient of [j 1 , · · · , j t ] in x −1 (Y s,1 ) · · · x −r (Y s,r )[k 1 , · · · , k t ] is Y s,k 1 Ys,j 1 −1 Y s,k 2 Ys,j 2 −1 · · · Y s,k t Ys,j t −1 · 1 Ys,r if k t−1 , k t = r − 1, Y s,k 1 Ys,j 1 −1 Y s,k 2 Ys,j 2 −1 · · · Y s,k t Ys,j t −1 if k t−1 or k t = r − 1. and the one of [j 1 , · · · , j t , j t+1 , r] is Y s,k1 Y s,j1−1 Y s,k2 Y s,j2−1 · · · Y s,kt Y s,jt−1 1 Y s,jt+1−1 , which coincide with Q(vt(m−s; k 1 , · · · , k t ) → vt(m−s+1; j 1 , · · · , j t ′ )) in Definition 7.13. In our notation, we can write [φ] := (+, +, · · · , +), [1, 2, · · · , m − 1] = (−, −, · · · , − m−1 , +, · · · , +) and [1, 2, · · · , m − 1, r] = (−, −, · · · , − m−1 , +, · · · , +−). Therefore, we get (7.24). We can similarly verify the case i k = r − 1 by the same way as the case i k = r. [Proof of Theorem 7. 2.] We take a path in X r (m, m − 1) in the form p = vt(m; φ) → vt(m − 1; k (m−1) 1 , · · · , k (m−1) tm−1 ) → vt(m−2; k (m−2) 1 , · · · , k (m−2) tm−2 ) → · · · → vt(1; k (1) 1 , · · · , k (1) t1 ) → vt(0; k (0) 1 , · · · , k (0) t0 ), where t m−s−1 = t m−s or t m−s +2 (0 ≤ s ≤ m−1, t m := 0). There exists a unique even number s (0 ≤ s ≤ m − 1) such that t m = 0, t m−1 = t m−2 = 2, t m−3 = t m−4 = 4, · · · , t m−s = s, t m−s−1 = s. We also see that k (m−s−1) s = · · · = k (0) s = s (1 ≤ s ≤ m − 1) by Definition 7.12 (ii), (iii) and (v). Then, similar to Lemma 7.10, we can verify Q(p) = D(m − 1, k (m−1) 1 )D(m − 2, k (m−2) 2 ) · · · D(m − s, k (m−s) s )D(m − s, r + 1) and we obtain Theorem 7.2 in the same way as the proof of Theorem 7.1. 8 The proof of Theorem 6.5 In this section, we shall give the proof of Theorem 6.5. We have already proven Theorem 6.5 for the simple algebraic group of type A r in [9]. So, we shall prove the theorem for other type B r , C r and D r . First, let us review the theorem for type A r . For k ∈ Z (1 ≤ k ≤ r), we set \|k\| = \|k\| = k and [1, k] := {1, 2, · · · , k}. Type A r For 1 ≤ l ≤ m and 1 ≤ k ≤ r, we set the Laurent monomial A(l, k) := Y l,k−1 Y l,k . Lemma 8.1. [9] In the setting of Theorem 6.5, we have ∆ L (k; i) = ( * ) A(m − k 1 + 1, k 1 )A(m − k 2 + 2, k 2 ) · · · A(m − k d + d, k d ), (8.1) where ( * ) is the condition for {k i } 1≤i≤d : 1 ≤ k 1 < · · · < k d ≤ m − 1 + d. Theorem 8.2. [9] ∆ L (k; i)(Y) = b∈B − (Λ d )u ≤k µ(b), where B − (Λ d ) u ≤k is the lower Demazure crystal of type A r , and µ : B(Λ d ) → Y is the monomial realization of the crystal base B(Λ d ) such that µ(v Λ d ) = 1 Y m,d , where v Λ d is the lowest weight vector in B(Λ d ). In Theorem 6.5 (6.3), we have used the notation a b ∈ Z >0 . In the case of type A r , we find that a b ≡ 1 for all b ∈ B − (Λ d ) u ≤k . Type C r We treat the type C r . In [10], we calculate the explicit formula of the generalized minors ∆ L (k; i) on the simple algebraic group of type C r and i in (6.1). For l ∈ [1, m] and k ∈ J C = {i, i\|i = 1, 2, · · · , r}, we set the Laurent monomials C(l, k) := Y l,k−1 Y l,k if 1 ≤ k ≤ r, Y l,\|k\| Y l+1,\|k\|−1 if r ≤ k ≤ 1. (8.2) Lemma 8.3. [10] In the setting of Theorem 6.5, we have ∆ L (k; i) = ( * ) C(l 1 , k 1 )C(l 2 , k 2 ) · · · C(l d , k d ), (8.3) l i := m − k i + i if 1 ≤ k i ≤ r, m − r − 1 + i if r ≤ k i ≤ 1. (8.4) where ( * ) is the condition for k i (1 ≤ i ≤ d) : 1 ≤ k 1 < k 2 < · · · < k d ≤ 1, (8.5) k i ≤ m − 1 + i (if 1 ≤ i ≤ r − m + 1), (8.6) k i ≤ d − i + 1 (if r − m + 2 ≤ i ≤ d). (8.7) Example 8.4. Let us consider the case of type C 2 , we set u = s 1 s 2 s 1 s 2 ∈ W and its reduced word i = (1, 2, 1, 2), which is the same setting in Example 6.6. Then m = 2, i 1 = i 3 = 1, i 2 = i 4 = 2. Following Lemma 8.3, we can calculate the generalized minor ∆ L (2; i)(Y): ∆ L (2; i)(Y) = C(2, 1)C(2, 2) + C(2, 1)C(1, 2) + C(2, 1)C(1, 1) + C(1, 2)C(1, 2) + C(1, 2)C(1, 1) = 1 Y 2,2 + Y 1,2 Y 2 2,1 + 2 Y 1,1 Y 2,1 + Y 2 1,1 Y 1,2 , which coincides with the one in (6.4). Unlike the case of type A r , there exists a term with coefficient 2. In the cases of type B r , C r , and D r , some terms in ∆ L (k; i)(Y) have coefficients greater than 1. Y l+1,k in [10], which corresponds to C(l, k) and C(l, k + 1) in (8.2). (ii) In the case d ≤ r − m+ 1, the generalized minor ∆ L (k; i) in (8.3) coincides with the one in (8.1) for type A r . Further, in this case, we can verify Theorem 6.5 by the same way as in [9]. Therefore, in the rest of paper, we shall show for d > r − m + 1. Set the subset B ⊂ Y as B := {C(l 1 , k 1 )C(l 2 , k 2 ) · · · C(l d , k d )\|{k i } satisfies (8.5), (8.6) and (8.7)}, (8.8) where above {l i } is as in (8.4). Remark 8.6. The definition (8.8) implies that X = C(l 1 , k 1 )C(l 2 , k 2 ) · · · C(l d , k d ) ∈ B if and only if (i) 1 ≤ k 1 < · · · < k d ≤ 1 (⇔ (8.5)),(ii) The monomial X has at most d − r + m − 1 factors in the form C(s, b) (s ∈ Z, b ∈ {r, · · · , 1}). Lemma 8.7. We take X := C(l 1 , k 1 )C(l 2 , k 2 ) · · · C(l d , k d ) ∈ B. The conditions k i ≤ m − 1 + i (1 ≤ i ≤ r − m + 1) and k i ≤ d − i + 1 (r − m + 2 ≤ i ≤ d) (i) For k ∈ [1, r − 1], we haveẽ k C(l, k) = A l−1,k · C(l, k) = C(l − 1, k + 1) and e k C(l, k + 1) = A l,k · C(l, k + 1) = C(l, k) in Y. Furthermore,ẽ r C(l, r) = A l−1.r · C(l, r) = C(l − 1, r). (ii) If k ∈ [1, r − 1] andẽ k X = 0, then there exists j ∈ [1, d] such that the following (a) or (b) holds: (a) k j = k, k j+1 > k + 1 and e k X = C(l 1 , k 1 ) · · · C(l j−1 , k j−1 )C(l j − 1, k + 1)C(l j+1 , k j+1 ) · · · C(l d , k d ). (8.9) (b) k j = k + 1, k j+1 > k and e k X = C(l 1 , k 1 ) · · · C(l j−1 , k j−1 )C(l j , k)C(l j+1 , k j+1 ) · · · C(l d , k d ). (8.10) (iii) Ifẽ r X = 0, then there exists j (1 ≤ j ≤ d) such that k j = r, k j+1 > r and e r X = C(l 1 , k 1 ) · · · C(l j−1 , k j−1 )C(l j − 1, r)C(l j+1 , k j+1 ) · · · C(l d , k d ). (8.11) Furthermore, we haveẽ 2 r X = 0. (iv) For k ∈ [1, r], suppose thatẽ k X = 0. Then, the monomialẽ k X satisfies the condition in Remark 8.6 (i), that is, we can writeẽ k X = C(l ′ 1 , k ′ 1 ) · · · C(l ′ d , k ′ d ) with some 1 ≤ k ′ 1 < · · · < k ′ d ≤ 1. Proof. (i) If 1 ≤ k ≤ r − 1, it follows from the definition ofẽ k in Y that e k C(l, k) = A l−1,k · C(l, k) = Y l−1,k Y l,k Y l−1,k+1 Y l,k−1 · Y l,k−1 Y l,k = C(l − 1, k + 1) . We can verify the remained cases in the similar way. (ii), (iii) For k ∈ [1, r − 1], we suppose thatẽ k X = 0. If the Laurent monomial X does not include factors in the form Y −1 * ,k , then we get ε k (X) = 0 andẽ k X = 0, which is absurd. Therefore, since k 1 < · · · < k d , the monomial X includes either one or two factors in the form Y −1 * ,k , which means that X has the factors in the form either C(l j , k) or C(l j ′ , k + 1), or both (see (8.2)). If X does not have the factor Y −1 lj ,k then X has the factor Y −1 l j ′ +1,k , k j ′ = k + 1, and k j ′ +1 > k. Sinceẽ k X = 0, we have n e k = l j ′ and henceẽ k X = A l j ′ ,k X = C(l 1 , k 1 ) · · · C(l j−1 , k j−1 )C(l j , k)C(l j+1 , k j+1 ) · · · C(l d , k d ). Similarly, if X does not have the factor Y −1 l j ′ +1,k then we have k j = k, k j+1 > k + 1, andẽ k X = A lj −1,k X = C(l 1 , k 1 ) · · · C(l j−1 , k j−1 )C(l j − 1, k + 1)C(l j+1 , k j+1 ) · · · C(l d , k d ). If X includes both factors Y −1 lj ,k and Y −1 l j ′ +1,k , then n e k = l j ′ or n e k = l j − 1, which implies thatẽ k X is in the form (8.9) or (8.10). Arguing similarly, we see that ifẽ r X = 0 thenẽ r X is in the form (8.11). Then, since k 1 < k 2 < · · · < k j = r < k j+1 < · · · < k d , the Laurent monomialẽ r X has no factors in the form C( * , r), which means thatẽ r X does not have factors in the form Y −1 * ,r . Hence, it follows from ε r (ẽ r X) = 0 thatẽ 2 r X = 0. (iv) The explicit forms (8.9), (8.10) and (8.11) with each condition for k j+1 imply that the monomialẽ k X satisfies the condition in Remark 8.6 (i). Lemma 8.8. For X = C(l 1 , k 1 )C(l 2 , k 2 ) · · · C(l d , k d ) ∈ B and s ∈ {0, 1, · · · , r − m + 1}, we suppose that 1 ≤ k 1 < · · · < k d−s ≤ r and r ≤ k d−s+1 < · · · < k d−1 < k d ≤ 1, that is, s := #{i ∈ [1, d]\|k i ∈ {r, · · · , 1}}. Then there exist non-negative integers {a(i, j)} 1≤i≤r−d+s, 1≤j≤r such that X =ẽ a(r−d+s,1) 1 · · ·ẽ a(r−d+s,r) r · · ·ẽ a(2,1) 1 · · ·ẽ a(2,r) rẽ a(1,1) 1 · · ·ẽ a(1,d) d 1 Y m,d . (8.12) Proof. We use induction on s. In the case s = 0, we obtain 1 ≤ k 1 < · · · < k d ≤ r. Let us put a(1, i) = a(2, i + 1) = · · · = a(k i − i, k i − 1) = 1 for 1 ≤ i ≤ d and a(i, j) = 0 for other (i, j). Using Lemma 8.7 (i), we see that e ki−1 · · ·ẽ i+2ẽi+1ẽi C(m, i) = C(m − k i + i, k i ) = C(l i .k i ) (i = 1, 2, · · · , d). Therefore, by the action ofẽ a(r−d,1) 1 · · ·ẽ a(r−d,r) r · · ·ẽ a(2,1) 1 · · ·ẽ a(2,r) rẽ a(1,1) 1 · · ·ẽ a(1,d) d , each factor C(m, i) in C(m, 1)C(m, 2) · · · C(m, d) = 1 Y m,d becomes C(m − k i + i, k i ) = C(l i .k i ), which means e a(r−d,1) 1 · · ·ẽ a(r−d,r) r · · ·ẽ a(2,1) 1 · · ·ẽ a(2,r) rẽ a(1,1) 1 · · ·ẽ a(1,d) d 1 Y m,d = C(l 1 , k 1 )C(l 2 , k 2 ) · · · C(l d , k d ) = X. Next, we consider the case s > 0. The explicit form of C(l, k) in (8.2) implies that k1−1 i=1 C(m − i, i) k2−1 i=k1+1 C(m − i + 1, i) k3−1 i=k2+1 C(m − i + 2, i) · · · · k d−s −1 i=k d−s−1 +1 C(m − i + d − s − 1, i) r i=k d−s +1 C(m − i + d − s, i) = Y m−k1+1,k1−1 Y m−k1+1,k1 · Y m−k2+2,k2−1 Y m−k2+2,k2 · · · Y m−k d−s +d−s,k d−s −1 Y m−k d−s +d−s,k d−s · Y m−r+d−s,r = C(l 1 , k 1 )C(l 2 , k 2 ) · · · C(l d−s , k d−s ) · Y m−r+d−s,r . Hence, we obtain C(l 1 , k 1 )C(l 2 , k 2 ) · · · C(l d−s , k d−s ) = C(m − 1, k ′ 1 )C(m − 2, k ′ 2 ) · · · C(m − r + d − s, k ′ r−d+s ) · 1 Y m−r+d−s,r , (8.13) where {k ′ 1 , k ′ 2 , · · · , k ′ r−d+s } := {1, 2, · · · , r} \ {k 1 , k 2 , · · · , k d−s } and k ′ 1 < k ′ 2 < · · · < k ′ r−d+s . Thus, the Laurent monomial X can be written as follows: X = C(m − 1, k ′ 1 )C(m − 2, k ′ 2 ) · · · C(m − r + d − s, k ′ r−d+s ) · 1 Y m−r+d−s,r · C(m − r + d − s, k d−s+1 ) · · · C(m − r + d − 2, k d−1 )C(m − r + d − 1, k d ). (8.14) Setting κ = \|k d−s+1 \|, we suppose that k ′ j−1 < κ ≤ k ′ j for some j (1 ≤ j ≤ r − d + s). Then the Laurent monomial X has the factor Y m−r+d−s,κ and does not have a factor in the form Y −1 a, * (a ≤ m − r + d − s). It follows from the definition (5.1) of ϕ κ that ϕ κ (X) > 0. Furthermore, as we have seen in Lemma 8.7 (i), if κ < k ′ j (resp. κ = k ′ j ), then the factor C(m − r + d − s, κ) in X is changed to C(m − r + d − s, κ + 1) by the action off κ (resp.f 2 κ ). Repeating this argument, we see that by the action of F 0 = (f rf 2 r−1 · · ·f 2 k ′ r−d+sf k ′ r−d+s −1 )(f 2 k ′ r−d+s −2 · · ·f 2 k ′ r−d+s−1f k ′ r−d+s−1 −1 ) · · · (f 2 k ′ j+2 −2 · · ·f 2 k ′ j+1f k ′ j+1 −1 )(f 2 k ′ j+1 −2 · · ·f 2 k ′ j +1f 2 k ′ jf k ′ j −1 · · ·f κ+1fκ ), each factor C(m−i, k ′ i ) is sent to C(m−i, k ′ i+1 − 1) (j ≤ i ≤ r−d+s−1) . And we find that C(m−r+d−s, k ′ r−d+s ) and C(m−r+d−s, k d−s+1 ) = C(m−r+d−s, κ) become C(m − r + d − s, r) and C(m − r + d − s + 1, r) respectively. Hence, F 0 · X = C(m − 1, k ′ 1 ) · · · C(m − j + 1, k ′ j−1 )C(m − j, k ′ j+1 − 1) · · · C(m − r + d − s + 1, k ′ r−d+s − 1)C(m − r + d − s, r) · 1 Y m−r+d−s,r ·C(m−r+d−s+1, r)C(m−r+d−s+1, k d−s+2 ) · · · C(m−r+d−2, k d−1 )C(m−r+d−1, k d ). Similar to (8.13) , setting {k ′′ 1 , k ′′ 2 , · · · , k ′′ d−s } := {1, 2, · · · , r}\{k ′ 1 , · · · , k ′ j−1 , k ′ j+1 − 1, k ′ j+2 − 1, · · · , k ′ r−d+s − 1, r} with 1 ≤ k ′′ 1 < · · · < k ′′ d−s < r, we have F 0 · X = C(l ′′ 1 , k ′′ 1 )C(l ′′ 2 , k ′′ 2 ) · · · C(l ′′ d−s , k ′′ d−s )C(m − r + d − s + 1, r) · C(m − r + d − s + 1, k d−s+2 ) · · · C(m − r + d − 2, k d−1 )C(m − r + d − 1, k d ), where l ′′ i := m − k ′′ i − i. By 1 ≤ k ′′ 1 < · · · < k ′′ d−s < r < r ≤ k d−s+2 < · · · < k d−1 < k d ≤ 1 , the monomial F 0 · X has the s − 1 factors in the form C( * , b) (b ∈ {r, · · · , 1}) and belongs to B (see Remark 8.6). Hence using the assumption of induction for F 0 · X, we get F 0 · X =ẽ a(r−d+s−1,1) 1 · · ·ẽ a(r−d+s−1,r) r · · ·ẽ a(1,1) 1 · · ·ẽ a(1,d) d 1 Y m,d , for some non-negative integers {a(i, j)}. In general, for each operator F :=f i1fi2 · · ·f in on Y, we denote the operator F * by F * :=ẽ in · · ·ẽ i2ẽi1 , (8.15) which satisfies F * · F · X = X if F · X = 0. Following (8.15), we set F * 0 := e κẽκ+1ẽk ′ j −1ẽ 2 k ′ j · · ·ẽ 2 r−1ẽr . Then X = F * 0ẽ a(r−d+s−1,1) 1 · · ·ẽ a(r−d+s−1,r) r · · ·ẽ a(1,1) 1 · · ·ẽ a(1,d) d 1 Y m,d , which is the same form as in (8.12). Similar to this, we can verify (8.12) in the case k ′ r−d+s < k d−s+1 = κ. We put κ ′ = k ′ r−d+s and suppose that \|k j \| < κ ′ ≤ \|k j−1 \| for some d − s + 2 ≤ j ≤ d. Using the operator F ′ 0 = (f rf 2 r−1 · · ·f 2 \|k d−s+1 \|f\|kd−s+1\|−1 )(f 2 \|k d−s+1 \|−2 · · ·f 2 \|k d−s+2 \|f\|kd−s+2\|−1 ) · · · (f 2 \|kj−3\|−2 · · ·f 2 \|kj−2\|f\|kj−2\|−1 )(f 2 \|kj−2\|−2 · · ·f 2 \|kj−1\|+1f 2 \|kj−1\|f\|kj−1\|−1 · · ·f κ ′ +1fκ ′ ), instead of F 0 , we get (8.12) by the same way as above. [Proof of Theorem 6.5 for type C r ] All we have to show is µ(B − (Λ d ) u ≤k ) = B, where B is as in (8.8). First, let us prove the inclusion µ(B − (Λ d ) u ≤k ) ⊂ B. Recall that the crystal B(Λ d ) has the lowest weight vector v Λ d , and (1, 2, · · · , r 1st cycle , 1, 2, · · · , r 2nd cycle · · · 1, 2, · · · , r m−2 th cycle , 1, 2, · · · , d m−1 th cycle ) is a reduced word of u ≤k (see (6.1) in Sect.6). Since we have µ(v Λ d ) = C(m, 1)C(m, 2) · · · C(m, d) = E m−1 1 Y m,d =ẽ a(m−1,1) 1 · · ·ẽ a(m−1,d) d 1 Y m,d is either a product of {C( * , b)\|1 ≤ b ≤ d + 1, * is an integer} belonging to B or 0. Similarly, E m−2 E m−1 1 Y m,d is either a product of {C( * , b)\|1 ≤ b ≤ d + 2} or 0. Repeating this argument, we see that E m−r+d · · · E m−2 E m−1 1 Y m,d is either a product of {C( * , b)\|1 ≤ b ≤ r} belonging to B or 0. Hence, E m−r+d · · · E m−2 E m−1 1 Y m,d = C(l 1 , k 1 )C(l 2 , k 2 ) · · · C(l d , k d ), for some 1 ≤ k 1 < k 2 < · · · < k d ≤ r. Using Lemma 8.7 (ii), (iii) and (iv), we see that E m−r+d−1 E m−r+d · · · E m−2 E m−1 1 Y m,d is either 0 or a product of {C( * , b)\|b ∈ J C } such that the number of product of {C( * , b ′ )\|b ′ ∈ {r, r − 1, · · · , 1}} is at most 1, and it satisfies the condition in Remark 8.6 (i). Since we have supposed 1 ≤ d − r + m − 1 in Remark 8.5 (ii), it also satisfies the condition in Remark 8.6 (ii), which means that it belongs to B ∪ {0}. Repeating this argument, we can verify that E 1 E 2 · · · E m−2 E m−1 1 Y m,d is either 0 or a product of {C( * , b)\|b ∈ J C } such that the number of product of {C( * , b ′ )\|b ′ ∈ {r, r − 1, · · · , 1}} is at most d − r + m − 1 , and it satisfies the condition in Remark 8.6 (i). Hence, E 1 E 2 · · · E m−2 E m−1 1 Y m,d ∈ B ∪ {0}, which means (8.16). Finally, we shall prove the inverse inclusion µ( B − (Λ d ) u ≤k ) ⊃ B. Let X = C(l 1 , k 1 )C(l 2 , k 2 ) · · · C(l d , k d ) ∈ B Type B r Next, we treat the case of type B r . For l ∈ [1, m] and k ∈ J B , we set the Laurent monomials B(l, k) :=                  Y l,k−1 Y l,k if 1 ≤ k ≤ r − 1, Y l,r−1 Y 2 l,r if k = r, Y l,r Y l+1,r if k = 0, Y 2 l,r Y l+1,r−1 if k = r, Y l,\|k\| Y l+1,\|k\|−1 if r − 1 ≤ k ≤ 1. (8.18) In the case i k = d < r, the minors ∆ L (k; i) are explicitly described as follows. Lemma 8.9. [11] In the setting of Theorem 6.5, if i k = d < r, we have ∆ L (k; i)(Y) = ( * ) a k1,··· ,k d B(l 1 , k 1 )B(l 2 , k 2 ) · · · B(l d , k d ), l i := m − k i + i if 1 ≤ k i ≤ r, m − r − 1 + i if k i ∈ {r, · · · , 1} ∪ {0}, a k1,··· ,k d := 2 if k i = 0 for some i ∈ [1, d], 1 if k i = 0 for any i ∈ [1, d], where ( * ) is the condition for k i (1 ≤ i ≤ d) : (ii) For X := B(l 1 , k 1 )B(l 2 , k 2 ) · · · B(l d , k d ) ∈ B and k ∈ [1, r − 1], ifẽ k X = 0, then there exists j (1 ≤ j ≤ d) such that either the following (a) or (b) holds: 1 ≤ k 1 ≤ k 2 ≤ · · · ≤ k d ≤ 1, k i = k i+1 if and only if k i = k i+1 = 0, (8.19) i ≤ k i ≤ m − 1 + i (1 ≤ i ≤ r − m + 1), (8.20) i ≤ k i ≤ d − i + 1 (r − m + 2 ≤ i ≤ d). (a) k j = k, k j+1 > k + 1 and e k X = B(l 1 , k 1 ) · · · B(l j−1 , k j−1 )B(l j − 1, k + 1)B(l j+1 , k j+1 ) · · · B(l d , k d ). (b) k j = k + 1, k j+1 > k and e k X = B(l 1 , k 1 ) · · · B(l j−1 , k j−1 )B(l j , k)B(l j+1 , k j+1 ) · · · B(l d , k d ). In particular, we haveẽ k X ∈ B. (iii) Ifẽ r X = 0, then there exists j (1 ≤ j ≤ d) such that either the following (a) or (b) holds: (a) k j = r, k j+1 = r and e r X = B(l 1 , k 1 ) · · · B(l j−1 , k j−1 )B(l j − 1, 0)B(l j+1 , k j+1 ) · · · B(l d , k d ). (b) k j = 0, k j+1 > r and e r X = B(l 1 , k 1 ) · · · B(l j−1 , k j−1 )B(l j , r)B(l j+1 , k j+1 ) · · · B(l d , k d ). Lemma 8.11. For X = B(l 1 , k 1 )B(l 2 , k 2 ) · · · B(l d , k d ) ∈ B and s ∈ Z (0 ≤ s ≤ r−m+1), we suppose that 1 ≤ k 1 < · · · < k d−s ≤ r and k d−s+1 , · · · , k d−1 , k d ∈ {0, r, · · · ,B(l 1 , k 1 )B(l 2 , k 2 ) · · · B(l d−s , k d−s ) = B(m − 1, k ′ 1 )B(m − 2, k ′ 2 ) · · · B(m − r + d − s, k ′ r−d+s ) · 1 Y 2 m−r+d−s,r , (8.23) where {k ′ 1 , k ′ 2 , · · · , k ′ r−d+s } := {1, 2, · · · , r} \ {k 1 , k 2 , · · · , k d−s } and k ′ 1 < k ′ 2 < · · · < k ′ r−d+s . Thus, the Laurent monomial X can be written as follows: X = B(m − 1, k ′ 1 )B(m − 2, k ′ 2 ) · · · B(m − r + d − s, k ′ r−d+s ) · 1 Y 2 m−r+d−s,r · B(m + d − r − s, k d−s+1 ) · · · B(m + d − r − 2, k d−1 )B(m + d − r − 1, k d ). (8.24) If k j = 0 for any j ∈ [1, d], then the monomial (8.24) is similar to (8.14). Therefore, in this case, we can show (8.22) in the same way as Lemma 8.8. Hence, we may assume that k d−s = k d−s+1 = · · · = k ζ = 0 and r ≤ k ζ+1 < k ζ+1 < · · · < k d ≤ 1 for some integer ζ (1 ≤ ζ ≤ s). Then we can write as X = B(m − 1, k ′ 1 )B(m − 2, k ′ 2 ) · · · B(m + d − r − s, k ′ r−d+s ) · 1 Y 2 m−r+d−s,r × B(m + d − r − s, 0)B(m + d − r − s + 1, 0) · · · B(m + d − r − s + ζ − 1, 0) ×B(m+d−r −s+ζ, k ζ+1 )B(m+d−r −s+ζ +1, k ζ+2 ) · · · B(m+d−r −1, k d ). We put κ = k ′ r−d+s and suppose that there exists some integer j (ζ + 2 ≤ j ≤ d + 1) such that \|k j \| < κ ≤ \|k j−1 \|, where \|k d+1 \| := 0. By the action of F 0 = (f 2 rf 2 r−1 · · ·f 2 \|k ζ+1 \|f\|kζ+1\|−1 )(f 2 \|k ζ+1 \|−2 · · ·f 2 \|kj−3\|f\|kj−3\|−1f 2 \|kj−3\|−2 · · ·f 2 \|kj−2\|f\|kj−2\|−1 )(f 2 \|kj−2\|−2 · · ·f 2 \|kj−1\|+1f 2 \|kj−1\|f\|kj−1\|−1 · · ·f κ+1fκ ), each factor B(m−d−r−s+i−1, k i ) of X becomes B(m−d−r−s+i−1, k i−1 − 1) (ζ + 2 ≤ i ≤ j − 1). Similarly, B(m − d − r − s + ζ, k ζ+1 ) is changed to B(m − d − r − s + ζ, 0) . And we also find that B(m + d − r − s, k ′ r−d+s ) = B(m + d − r − s, κ) and B(m + d − r − s, 0) are sent to B(m + d − r − s, r) and B(m + d − r − s + 1, r) respectively. Hence, it follows from (8.23) and the same argument as in the proof of Lemma 8.8, the monomial F 0 · X has s − 1 factors in the form B( * , b) (b ∈ {0, r, · · · , 1}). Thus, using the assumption of induction on s − 1, we get F 0 · X =ẽ a(r−d+s−1,1) 1 · · ·ẽ a(r−d+s−1,r) r · · ·ẽ a(1,1) 1 · · ·ẽ a(1,d) d 1 Y m,d , for some non-negative integers {a(i, j)}. And we can write X = F * 0ẽ a(r−d+s−1,1) 1 · · ·ẽ a(r−d+s−1,r) r · · ·ẽ a(1,1) 1 · · ·ẽ a(1,d) d 1 Y m,d , where F * 0 is as in (8.15). Therefore, X is in the desired form in (8.22). [Proof of Theorem 6.5 for type B r in the case i k = d < r] All we have to show is µ(B − (Λ d ) u ≤k ) = B. Using Lemma 8.10, we can prove µ(B − (Λ d ) u ≤k ) ⊂ B in the same way as the case of type C r . The inverse inclusion µ(B − (Λ d ) u ≤k ) ⊃ B is followed from Theorem 5.7 and Lemma 8.11 . Next, let us prove Theorem 6.5 for type B r in the case i k = d = r. We use the notation (8.18) and B(l, r + 1) := 1 Y l,r . Lemma 8.12. [11] In the setting of Theorem 6.5, suppose that i k = r. Then ∆ L (k; i) = ( * ) B(m − 1, k 1 )B(m − 2, k 2 ) · · · B(m − s, k s )B(m − s, r + 1), where ( * ) is the condition for s ∈ Z ≥0 and k i (1 ≤ i ≤ s) : 0 ≤ s ≤ m − 1, 1 ≤ k 1 < k 2 < · · · < k s ≤ r. (ii) For X := B(m − 1, k 1 ) · · · B(m − s, k s )B(m − s, r + 1) ∈ B sp and k ∈ [1, r − 1], ifẽ k X = 0 then there exists j (1 ≤ j ≤ s) such that k j = k + 1, k < k j+1 and e k X = B(m − 1, k 1 ) · · · B(m − j, k) · · · B(m − s, k s )B(m − s, r + 1). In particular, we haveẽ k X ∈ B sp . Ifẽ r X = 0 then k s = r and Proof. (i) We obtainẽ r B(l, r + 1) = A l−1,r 1 Y l,r = B(l − 1, r)B(l − 1, r + 1). (ii) For k ∈ [1, r−1], we suppose thatẽ k (X) = 0. If the Laurent monomial X does not have factors in the form 1 Y * ,k , then ε k (X) = 0, which impliesẽ k (X) = 0. So, X includes a factor in the form 1 Y * ,k . The explicit form (8.18) means that X has the factor B(m − j, k + 1) for some j ∈ [1, s] and does not have the factors in the form B( * , k). Then, we have n e k = m − j, and e k X = B(m − 1, k 1 ) · · · B(m − j, k) · · · B(m − s, k s )B(m − s, r + 1), by Lemma 8.10 (i). Similarly, we see that ifẽ r (X) = 0, then k s = r andẽ r X is described as in (8.26). [Proof of Theorem 6.5 for type B r in the case i k = r] All we have to show is µ(B − (Λ r ) u ≤k ) = B sp . Similar to the cases of type C r and type B r (i k < r), we can prove µ(B − (Λ r ) u ≤k ) ⊂ B sp by using Lemma 8.13 (ii). Let us prove µ(B − (Λ r ) u ≤k ) ⊃ B sp . Take an arbitrary monomial X = B(m− 1, k 1 ) · · · B(m − s, k s )B(m − s, r + 1) ∈ B sp . It follows from Lemma 8.13 that e k1+1 · · ·ẽ r−1ẽr B(m, r + 1) = B(m − 1, k 1 )B(m − 1, r + 1). Using Lemma 8.13 repeatedly, we obtain (ẽ ks+1 · · ·ẽ r−1ẽr ) · · · (ẽ k2+1 · · ·ẽ r−1ẽr )(ẽ k1+1 · · ·ẽ r−1ẽr )B(m, r + 1) = B(m − 1, k 1 ) · · · B(m − s, k s )B(m − s, r + 1) = X. Recall that B(m, r+1) = 1 Ym,r = µ(v Λr ), where v Λr is the lowest weight vector of the crystal base B(Λ r ). Therefore, by Theorem 5.7, we get X ∈ µ(B − (Λ r ) u ≤k ). Hence, we obtain µ(B − (Λ r ) u ≤k ) ⊃ B sp . Type D r Finally, we treat the case type D r . We shall use the notation (7.1). First, let us prove the theorem in the case i k = d < r − 1. Set the subset B ⊂ Y as B := {D(l 1 , k 1 )D(l 2 , k 2 ) · · · D(l d , k d )\|{k i } satisfy (7.4), (7.5) and (7.6)}. Similar to Lemma 8.7, we can verify the following lemma. Lemma 8.14. (i) For k ∈ [1, r − 1], we haveẽ k D(l, k) = D(l − 1, k + 1) and e k D(l, k + 1) = D(l, k) in Y. Furthermore,ẽ r D(l, r − 1) = D(l − 1, r) and e r D(l, r) = D(l, r − 1). (ii) For X := D(l 1 , k 1 )D(l 2 , k 2 ) · · · D(l d , k d ) ∈ B and k ∈ [1, r −1], ifẽ k X = 0, then there exists j (1 ≤ j ≤ d) such that either the following (a) or (b) holds: (a) k j = k, k j+1 > k + 1 and e k X = D(l 1 , k 1 ) · · · D(l j−1 , k j−1 )D(l j − 1, k + 1)D(l j+1 , k j+1 ) · · · D(l d , k d ). (b) k j = k + 1, k j+1 > k and e k X = D(l 1 , k 1 ) · · · D(l j−1 , k j−1 )D(l j , k)D(l j+1 , k j+1 ) · · · D(l d , k d ). In particular, we haveẽ k X ∈ B. (iii) Ifẽ r X = 0, then there exists j (1 ≤ j ≤ d) such that either following (a) or (b) holds: (a) k j = r − 1, k j+1 > r − 1 and e r X = D(l 1 , k 1 ) · · · D(l j−1 , k j−1 )D(l j − 1, r)D(l j+1 , k j+1 ) · · · D(l d , k d ). (b) k j = r, k j+1 > r − 1 and e r X = D(l 1 , k 1 ) · · · D(l j−1 , k j−1 )D(l j , r − 1)D(l j+1 , k j+1 ) · · · D(l d , k d ). Lemma 8.15. For X := D(l 1 , k 1 )D(l 2 , k 2 ) · · · D(l d , k d ) ∈ B and s ∈ Z (0 ≤ s ≤ r−m+1), we suppose that 1 ≤ k 1 < · · · < k d−s ≤ r−1 and k d−s+1 , · · · , k d−1 , k d ∈ {r, · · · , 1} ∪ {r}, that is, s := D(l 1 , k 1 )D(l 2 , k 2 ) · · · D(l d−s , k d−s ) = D(m − 1, k ′ 1 )D(m − 2, k ′ 2 ) · · · D(γ, k ′ m−γ ) Y γ,r−1 Y γ,r where {k ′ 1 , k ′ 2 , · · · , k ′ m−γ } := {1, 2, · · · , r − 1} \ {k 1 , k 2 , · · · , k d−s } and k ′ 1 < k ′ 2 < · · · < k ′ m−γ . Thus, X = D(m − 1, k ′ 1 ) · · · D(γ, k ′ m−γ )D(l d−s+1 , k d−s+1 ) · · · D(l d , k d ) Y γ,r−1 Y γ,r . Putting κ := k ′ m−γ , we suppose that k j < κ ≤ k j+1 for some j (d − s ≤ j ≤ d), where k d+1 := 1. The remaining part of the proof is the same as in Lemma 8.11, that is, acting the Kashiwara operators {f i } on X properly, we can send the factor D(l d−s+1 , k d−s+1 ) to D(l d−s+1 + 1, r − 1). Similar to the proof of Lemma 8.11, using induction on s and by its assumption, we see that the monomial X can be written as in (8.27). [Proof of Theorem 6.5 for type D r ] The case i k = d < r − 1: Similar to the cases of type C r and type B r , we can verify µ(B − (Λ d ) u ≤k ) = B by using Lemma 8.14 and 8.15. The case i k = d = r − 1 or r: Next, let us prove Theorem 6.5 for type D r of the cases i k = r −1 and i k = r. Since we can prove both cases in almost the same way, we shall prove the case i k = r only. We use the notation in (7.1) and (7.2). Set the subset B (ii) For X := D(m − 1, k 1 ) · · · D(m − s, k s )D(m − s, r + 1) ∈ B (+) sp and k ∈ [1, r − 1], ifẽ k X = 0, then there exists j (1 ≤ j ≤ s) such that k j = k + 1, k j+1 < k and e k X = D(m − 1, k 1 ) · · · D(m − j, k) · · · D(m − s, k s )D(m − s, r + 1). In particular, we haveẽ k X ∈ B sp . Similar to the cases of type B r , C r and D r (i k < r − 1), we can prove µ(B − (Λ r ) u ≤k ) ⊂ B Using Lemma 8.16 repeatedly, we obtain (ẽ ks+1 · · ·ẽ r−2ẽr−1 )(ẽ ks−1+1 · · ·ẽ r−2ẽr ) · · · (ẽ k2+1 · · ·ẽ r−2ẽr )(ẽ k1+1 · · ·ẽ r−2ẽr )D(m, r+1) = D(m − 1, k 1 )D(m − 2, k 2 ) · · · D(m − s, k s )D(m − s, r + 1) = X. Recall that D(m, r+1) = Y m,r = µ(v Λr ), where v Λr is the lowest weight vector of the crystal base B(Λ r ). Therefore, by Theorem 5.7, we get X ∈ µ(B − (Λ r ) u ≤k ). Hence, we obtain µ(B − (Λ r ) u ≤k ) ⊃ B (+) sp . Due to the above arguments, we have completely shown Theorem 6.5. A counter example In this final subsection, we shall see an example such that ∆(k; i)(Y) can not be described as the total sum of monomials in any Demazure crystals. Example 8.17. Let G be the simple algebraic group of type C 3 , and we set u := (s 1 s 2 s 3 ) 3 ∈ W , i := (1, 2, 3, 1, 2, 3, 1, 2, 3) and k = 3. In the notation of Theorem 6.5, we have m = 3, d = i k = 3, and i k belongs to (m − 2)th cycle. Hence, this setting does not satisfy the condition in Theorem 6.5 such that i k belongs to (m − 1)th cycle. In this case, following [10], we obtain ∆ L (3; i)(Y) = 1 Y 2,3 Y 3,3 + Y 1,3 Y 2 2,2 Y 3,3 + Y 1,3 Y 2,3 Y 2 2,2 Y 2 3,2 + 2Y 1,2 Y 2,1 Y 2,2 Y 3,3 + Y 2 1,2 Y 1,3 Y 2 2,1 Y 3,3 + · · · + Y 2 1,1 Y 1,3 Y 2 2,1 Y 2,3 + 2 Y 1,2 Y 2,1 Y 3,1 Y 3,2 + Y 1,1 Y 3,1 Y 3,2 + Y 1,1 Y 2,1 Y 2,2 Y 3,2 + Y 1,1 Y 1,2 Y 1,3 Y 3,2 + Y 1,1 Y 1,2 Y 2,2 Y 1,3 Y 2,3 Y 3,1 . The set of the monomials in the part ( Y1,3Y2,3Y3,1 ) coincides with a monomial realization of certain subset of the crystal B(2Λ 2 ), but it does not coincide with any Demazure crystals B(2Λ 2 ) w (w ∈ W ). ( 3 . 4 ) 34Theorem 3.2.[3] For u ∈ W and its reduced word i, the map x G i is a biregular isomorphism from H × (C × ) l(u) to a Zariski open subset of G u,e . Theorem 3. 3 . [ 2 ] 32For u ∈ W and its reduced word i, the map x L i is a biregular isomorphism from (C × ) l(u) to a Zariski open subset of L u,e . Definition 4. 2 . 2LetB(i) be an integer matrix with rows labelled by all the indices in [−1, −r] ∪ [1, l(u)] and columns labelled by all the indices in e(i). For k ∈ [−1, −r] ∪ [1, l(u)] and l ∈ e(i), an entry b kl ofB(i) is determined as follows: 7. 4 , 4The properties of paths in X d (m, m − 1) Lemma 7.9. Let p ∈ X d (m, m − 1) be a path in the form · · · , a (m−1) d ) → vt(0; a (m) 1 , · · · , a (m) d ). (7.15) =. K i . We can verify (7.23) and the path p satisfies the conditions of X d (m, m − 1) in Definition 7.4. Hence, by Proposition 7.Let us define the directed graph (V sp , E sp ) as follows: We define the set V sp = V sp (m) of vertices as V sp (m) := {vt(s; k (s) Remark 8. 5 . 5(i) We use different notation from the one in[10]. It was defined C(l, k) in Lemma 8.3 follows from (i) and (ii). subset B ⊂ Y as B := {B(l 1 , k 1 )B(l 2 , k 2 ) · · · B(l d , k d )\|{k i } satisfy (8.19), (8.20) and (8.21)}. Similar to Lemma 8.7, we can verify the following lemma. Lemma 8.10. (i) For k ∈ [1, r − 1], we haveẽ k B(l, k) = A l−1,k · B(l, k) = B(l − 1, k + 1) andẽ k B(l, k + 1) = A l,k · B(l, k + 1) = B(l, k) in Y. Furthermore,ẽ r B(l, r) = A l−1,r · B(l, r) = B(l − 1, 0) andẽ r B(l, 0) = A l,r · B(l, 0) = B(l, r). in (8.25), if s = 0, the summand of ∆ L (k; i) is B(m, r + 1). Set the subset B sp ⊂ Y as B sp := {B(m − 1, k 1 ) · · · B(m − s, k s )B(m − s, r + 1)\|{k i } and s satisfy (8.25)}. Lemma 8.13. (i) We haveẽ r B(l, r + 1) = B(l − 1, r)B(l − 1, r + 1). e r X = B(m−1, k 1 ) · · · B(m−s, k s )B(m−s−1, r)B(m−s−1, r+1). (8.26) B (+) sp := {D(m − 1, k 1 ) · · · D(m − s, k s )D(m − s, r + 1)\|{k i } and s satisfy (7.7)}. Similar to Lemma 8.13, we can verify the following lemma: Lemma 8.16. (i) We haveẽ r D(l, r + 1) = D(l − 1, r − 1)D(l − 2, r)D(l − 2, r + 1). . Ifẽ r X = 0 then k s ≤ r − 2 ande r X = D(m−1, k 1 ) · · · D(m−s, k s )·D(m−s−1, r − 1)D(m−s−2, r)D(m−s−2, r+1).Now, let us complete the proof of Theorem 6.5. All we have to show isµ(B − (Λ r ) u ≤k ) = B (+) by usingLemma 8.16 (ii).Let us prove µ(B − (Λ r ) u ≤k ) ⊃ B (+) sp . For an arbitrary monomial X = D(m − 1, k 1 ) · · · D(m − s, k s )D(m − s, r + 1) ∈ B (+)sp , it follows from Lemma 8.16 that e k1+1 · · ·ẽ r−3ẽr−2ẽr D(m, r + 1) = D(m − 1, k 1 )D(m − 2, r)D(m − 2, r + 1), and (ẽ k2+1 · · ·ẽ r−3ẽr−2ẽr−1 )(ẽ k1+1 · · ·ẽ r−3ẽr−2ẽr )D(m, r+1) = D(m−1, k 1 )D(m−2, k 2 )D(m−2, r+1). be a monomial. By Theorem 5.7, we need to show that there exist non-negative integers {a(i, j)} such that which has been already obtained in Lemma 8.8. Thus, we have µ(B − (Λ d ) u ≤k ) ⊃ B.X =ẽ a(1,1) 1 · · ·ẽ a(1,r) rẽ a(2,1) 1 · · ·ẽ a(2,r) r · · ·ẽ a(m−1,1) 1 · · ·ẽ a(m−1,d) d 1 Y m,d , (8.17) 1}, that is, s := #{i ∈ [1, d]\|k i ∈ {0, r, · · · , 1}}. Then there exist nonnegative integers {a(i, j)} 1≤i≤r−d+s, 1≤j≤r such that Proof. Let us prove Lemma 8.11 by the induction on s. In the case s = 0, we can prove (8.22) by the same argument in the proof of Lemma 8.8. So, we assume s > 0. Similar to(8.13), we see thatX =ẽ a(r−d+s,1) 1 · · ·ẽ a(r−d+s,r) r · · ·ẽ a(2,1) 1 · · ·ẽ a(2,r) rẽ a(1,1) 1 · · ·ẽ a(1,d) d 1 Y m,d . (8.22) #{i ∈ [1, d]\|k i ∈ {r, · · · , 1} ∪ {r}}. Then there exist non-negative integers {a(i, j)} 1≤i≤r−d+s, 1≤j≤r such that Proof. We set γ := m − r + d − s + 1. 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[ "Improved underwater image enhancement algorithms based on partial differential equations (PDEs)", "Improved underwater image enhancement algorithms based on partial differential equations (PDEs)" ]	[ "U A Nnolim \nDepartment of Electronic Engineering\nUniversity of Nigeria Nsukka\nEnuguNigeria\n" ]	[ "Department of Electronic Engineering\nUniversity of Nigeria Nsukka\nEnuguNigeria" ]	[]	The experimental results of improved underwater image enhancement algorithms based on partial differential equations (PDEs) are presented in this report. This second work extends the study of previous work and incorporating several improvements into the revised algorithm. Experiments show the evidence of the improvements when compared to previously proposed approaches and other conventional algorithms found in the literature.	null	[ "https://arxiv.org/pdf/1705.04272v1.pdf" ]	3,903,981	1705.04272	373144ae89dae769f0084ead6ee9be69403c7d75	Improved underwater image enhancement algorithms based on partial differential equations (PDEs) U A Nnolim Department of Electronic Engineering University of Nigeria Nsukka EnuguNigeria Improved underwater image enhancement algorithms based on partial differential equations (PDEs) 1 The experimental results of improved underwater image enhancement algorithms based on partial differential equations (PDEs) are presented in this report. This second work extends the study of previous work and incorporating several improvements into the revised algorithm. Experiments show the evidence of the improvements when compared to previously proposed approaches and other conventional algorithms found in the literature. 1. Introduction Numerous works involving underwater image enhancement have been published in the literature [ [27] with newer techniques adding to the continuously expanding body of work. These algorithms span a wide array of methods and approaches. However, it is difficult to observe extensive testing of several well-known images using several of the developed algorithms. Additionally, it is a challenge to obtain all the implementations of the published works from the authors for evaluation and testing with additional image data. Previous work [28] involved the development of several algorithms based on partial differential equations for underwater image enhancement based on modifications and combinations of natural image enhancement algorithms [29] [30] such as the contrast limited adaptive histogram equalization (CLAHE) [31]. The results showed several advantages and improvements of the proposed algorithms over previous works from the literature [29] [30]. Additionally, the PDE-based framework enabled greater control over the various combined processes [32]. This was impressive considering that single image-based enhancement methods do not operate with any image formation or acquisition data [1]. Thus, it would not be unexpected if such algorithms were to fail or be wholly inadequate for processing underwater images, which suffer from unique problems due to the aquatic environment [1]. However, the proposed algorithms were unable to process adequately a few images, which had unique problems that could not be resolved completely with the earlier proposed approaches. This led to the evaluation of the key algorithms that were consistently effective and combining them with additional colour correction methods suited to adequately processing the affected images [29] [30]. The results of these improvements are presented for comparison with previous approaches and other algorithms from established works found in the literature. Motivation for improving on this work was partially due to the encouraging results and confirmation from other authors who have mentioned the usage of conventional algorithms in underwater image enhancement [5] [10] [17] [22]. On the other hand, improving the results of the previous approaches presents an opportunity for further study. 2. Background on PDE formulation for image enhancement The initial framework for the proposed algorithms is rooted in earlier works by Shapiro, et and the basic formulation [33] [32] was given as; We modified this approach in previous work [29] [30] to yield the modified and updated models; In (1), (2) and ( We continue to build on this approach by cascading certain various algorithms to ensure that local and global image features are adequately enhanced in the process. This further boosts the strengths of the individual algorithms, minimizing their disadvantages while amplifying their strengths. For example, the CLAHE method is a powerful localized operator when applied correctly. However, colour distortion is unavoidable with such algorithms. Thus, globalized operators help to remedy this problem and the sequence in which each of these operators are applied depends on the image as will be seen in experiments [29] [30] [34]. Improved PDE-based algorithm Based on experiments, mathematical derivation and proof, we select the PWL algorithm since it is a generalization of the various contrast stretching approaches. Thus, based on findings from this work [34] and previous study [29] [30], we combine the local and global operations in cascaded form such as; { { ( , , )}} = { ( , , )} = ( ({ ( , , )}) ) and introduce additional control parameters to further regularize and regulate the various processes within the new PDE [34]. The improved PDE-based approaches are discussed in this section and utilize additional components for processing specific images. The results of the new approach [34] and the previous work [29] [30] were subsequently compared. Comparison of selected PDE-based approaches The results of using the various selected contrast enhancement and colour correction algorithms are presented in this subsection. These include the key selected algorithms from previous work incorporated into a PDE-based framework [29] [30]. We present the images used in the various experiments in Fig. 1. In previous work, we performed numerous experiments with these images using a number of conventional image enhancement algorithms to obtain suitable candidates for augmentation within a PDE-based framework [29] [30]. In this work, we now further explore the modified approaches. It was observed that some images yielded better results using a global-local process than a local-global one [34]. Thus we show results using alternative sequences of global and local contrast operators and based on results, certain images are favoured by a particular sequence of operations over the other [34]. Based on the results in Fig. 2, we obtain fairly consistent performance indicating that these algorithms are effective for most images. We focus on trying to improve results for the few images for which the approaches perform poorly. Further improvements and additions to the proposed approaches Not all images are adequately processed by the various substitutions of the selected contrast enhancement and colour correction operators. For example, in Fig. 2, the images used in works by Bianco, et al [6] (based on the work by Reinhard, et al [35]) and Gouinaud, et al [16] show poor colour correction results. Based on the image histogram analysis in previous work [29] and shown in Fig. 3, it is observed that the red, green and blue channels are not in alignment with each other. Thus, it would be difficult to realign the colours with RGB-based colour correction schemes. Thus, we explore the incorporation of the methods used in Gouinaud and Bianco's work into the proposed algorithms. The proposed modifications are shown in Fig. 4. For easy notation, we denote the additional approaches as PA-1 and PA-2 as shown in Fig. 4. Based on investigations, the main component for the colour correction for such images is the RGB2XYZ operation, which can be performed prior or after enhancement using the proposed PDE-based PWL-CLAHE scheme. The greyed out boxes in Fig. 4(b) are for the Fuzzy Homomorphic Enhancement [36] [37] and Piecewise linear transform-based (PWL) [38] enhancement components, which are optional. They are normally used when the output image is dark or faded to restore contrast to the processed image. Based on Fig. 4, PA-1 is relatively more complicated than PA-2 and though both algorithms clearly eliminate the colour cast, each is better suited to a particular image than the other. Utilizing perceptual colour spaces such as HSI or HSV fails in this case to yield pleasing results for these particular images. For example, using Iqbal's scheme in this case yielded no effect on the colour cast effects. This is easily understood from the histogram analysis as discussed in previous work [29]. Experiments and results Further experiments are performed to verify the efficacy of incorporating these algorithms into the proposed framework. The results are shown in Fig. 5 and 6. Based on the visual results in Fig. 5, we can see that the hybrid approach yields much better colour correction and contrast enhancement results. The image results from Bianco still appear faded and hazy compared to the results using the proposed hybrid approach. However, for the results by Gouinaud, et al [16], the results are darker using the proposed modifications. Thus, in order to refine the process, we devise a new scheme to process these images in order to avoid the complexity of the Colour Logarithmic Image Processing (CoLIP) approach used in work by Gouinaud et al [16]. The results obtained using the second scheme (PA-2) are shown in Fig. 6 and compared with the results using Gouinaud, et al [16]. It is clearly seen that there is still a considerable amount of blue haze in the results obtained using the CoLIP approach. In comparison, the modified approach yields images with a larger portion of the dominant blue hue removed. However, there is colour distortion in addition to contrast enhancement compared with the CoLIP method. Also, some colour distortion is observed with the images used by Bianco et al [6] though there is a great deal of colour cast removal (first row of Fig. 6). Nevertheless, the additional proposed colour correction scheme of PA-2 is much simpler than the CoLIP. Ultimately, we have devised solutions to adequately process the outlier images in addition to the initial scheme that works well for most images. Thus, the proposed approaches show much improved results, though there are still other issues to be resolved for these type of underwater images. Additional future work may involve the possibility of devising a scheme to classify underwater images based on certain unique features. However, it will be not an easy task to numerically quantify such a subjective attribute. Conclusion and future work We have presented an improved PDE-based scheme for single underwater image enhancement in addition to colour correction pre-and processing steps based on colour space conversions. The proposed schemes are stable and effective for most images in addition to yielding better results than most algorithms from the literature. Additional improvements are also devised and proposed for handling the rare cases of failure of the proposed approaches. The key idea behind the schemes is the use of modified algorithms in the absence of image information to enhance underwater images. Thus, the approach is quite effective and some evidence in the literature validates the fundamental ideas used in developing the proposed techniques. Future work would involve automated recognition and classification of such outlier images and improving results for the specific cases in terms of colour enhancement, where possible. ( , , ) = λ (‖ ( , , )‖)div ( ( , , ) ‖ ( , , )‖ ) + [ { ( , , )} − ( , , )] + { ( , , )} + [ ( , , )] 3), ( , , ) is the continuous image, with coordinates in (horizontal and vertical) spatial and temporal coordinates, x, y and t respectively. The smoothing term, λ (‖ ( , , )‖)div (( , , ) ‖ ( , , )‖ ) is the anisotropic diffusion term with control parameter, λ > 0, is the gradient operator, ‖ ‖ as the norm and div as the divergent operator. The local and global contrast operators, , in addition to the colour correction term, ( , , ) are defined by contrast enhancement and colour correction functions. The functions used are given as; could be used. For (2), we use the term ( , , ) = ( ( , , )− ) to ensure a gradual evolution using mode, m, in the absence of a fidelity term while for (3) we use ( , , ) = ( ( , , )− ) with mean, , to speed up the process since the presence of the fidelity term in the contrast enhancement in the latter ensures stability and convergence. Fig. 2(a) Images processed with various PDE-based configurations (b) key to figures Fig. 3 3RGB colour histograms of original Ocean jar and Ocean floor images[6], Divers and Diver images[16] Fig. 4(a) PA-1 and (b) PA-2 for processing the images inFig. 3 Fig. 5 Fig. 6 56Results of Bianco, et al (first row) and PA-1 (second and third row) Results of PA-2 (first row) and CoLIP method by Gouinaud, et al (second row) Underwater Image Processing: State of the Art of Smoothing and Image Enhancement Methods. Raimondo Schettini, Silvia Corchs, EURASIP Journal on Advances in Signal Processing. 2010Raimondo Schettini and Silvia Corchs, "Underwater Image Processing: State of the Art of Smoothing and Image Enhancement Methods," EURASIP Journal on Advances in Signal Processing, vol. 2010, pp. 1-14, 2010. Application of underwater hyperspectral data for color correction purposes. J Ahlen, D Sundgren, E Bengtsson, Pattern Recognition and Image Analysis. 171J. Ahlen, D. Sundgren, and E. 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[ "Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm", "Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm" ]	[ "Salman Beigi \nSchool of Mathematics\nInstitute for Research in Fundamental Sciences (IPM)\nTehranIran\n\nDepartment of Information Engineering\nThe Chinese University of Hong Kong\nHong Kong\n", "Christopher King \nDepartment of Mathematics\nNortheastern University\n02115BostonMA\n" ]	[ "School of Mathematics\nInstitute for Research in Fundamental Sciences (IPM)\nTehranIran", "Department of Information Engineering\nThe Chinese University of Hong Kong\nHong Kong", "Department of Mathematics\nNortheastern University\n02115BostonMA" ]	[]	We develop the notions of hypercontractivity (HC) and the log-Sobolev (LS) inequality for completely bounded norms of one-parameter semigroups of super-operators acting on matrix algebras. We prove the equivalence of the completely bounded versions of HC and LS under suitable hypotheses. We also prove a version of the Gross Lemma which allows LS at general q to be deduced from LS at q = 2.	10.1063/1.4934729	[ "https://arxiv.org/pdf/1509.02610v3.pdf" ]	119,323,985	1509.02610	4bc63799008c63ffb8f2cb6c07822d3ed7513cee	Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm 6 Nov 2015 November 9, 2015 Salman Beigi School of Mathematics Institute for Research in Fundamental Sciences (IPM) TehranIran Department of Information Engineering The Chinese University of Hong Kong Hong Kong Christopher King Department of Mathematics Northeastern University 02115BostonMA Hypercontractivity and the logarithmic Sobolev inequality for the completely bounded norm 6 Nov 2015 November 9, 2015 We develop the notions of hypercontractivity (HC) and the log-Sobolev (LS) inequality for completely bounded norms of one-parameter semigroups of super-operators acting on matrix algebras. We prove the equivalence of the completely bounded versions of HC and LS under suitable hypotheses. We also prove a version of the Gross Lemma which allows LS at general q to be deduced from LS at q = 2. Introduction The notions of hypercontractivity (HC) and the logarithmic Sobolev (LS) inequalities were originally introduced in the context of quantum field theory [16,20,10]. The HC inequality can be formulated as follows: for 1 ≤ q ≤ p, and for a suitable operator A, e −tA q→p = sup f q ≤1 e −tA f p ≤ 1, if and only if t ≥ 1 2 log p − 1 q − 1 . The related concept of the logarithmic Sobolev inequality is an infinitesimal version of hypercontractivity, obtained by setting q = 2, p(t) = 1 + e 2t and taking the derivative at t = 0. In the original quantum field theory setting, A was the Hamiltonian for the free bosonic field in two spacetime dimensions, and f was a state in the bosonic Fock space. These results were later extended to the case of the free fermion field [9,4]. Recently, HC and LS inequalities have found applications in quantum information theory. For such applications, A = L is often the generator of a one-parameter semigroup of completely positive maps on an open quantum system, representing its dissipative evolution in the memoryless (Markovian) approximation, and f is an observable on the system. In this setting the norm · q is usually a Schatten norm on a matrix algebra. The theory of HC and LS inequalities in this setting has been developed by Olkiewicz and Zegarlinski [15], and more recently by Kastoryano and Temme [13], who also used these methods to derive mixing time bounds for a variety of quantum channel semigroups [14]. Many of the results derived for classical Markov chains using HC and LS inequalities can be extended to quantum channel semigroups. One notable exception is the 'tensoring up' property. This is the issue of finding bounds for products of independent copies of channels (super-operators), and is concerned with norms of the type e −t 1 L 1 ⊗e −t 2 L 2 q→p . For classical channels this operator norm is multiplicative, and thus the 'time to contraction' for a product of channels is the maximum of the times to contraction for each individual channel. For quantum channels this need not be true (although at this time there is no explicit example known of a channel semigroup which violates this classical 'additivity' result, it is widely believed that violations are generic). In order to handle this non-additivity, one approach is to use a different norm for which additivity is guaranteed, namely the completely bounded (CB) norm. The CB norm · CB,q→p was introduced by Pisier [19] and reviews can be found in [6,7]. This norm satisfies the property Φ 1 ⊗ Φ 2 CB,q→p = Φ 1 CB,q→p Φ 2 CB,q→p ,(1) for all p, q ≥ 1 and all completely positive maps Φ 1 and Φ 2 . One special case of the CB norm is the well-known diamond norm, which is the case q = p = 1. Applying the CB norm to a semigroup e −tL of completely positive maps, we can investigate the time to contraction and derive the corresponding LS inequality. In this case, by the above multiplicativity, the time to contraction (under CB norm) of a product of channels can be computed in terms of the time to contraction of individual ones, as in the classical setting. In this paper we introduce LS inequalities associated with CB norms. We show equivalence of the HC condition and the corresponding LS inequality for the CB norm of quantum channel semigroups. The proof requires some novel ingredients which are not required in the usual setting of the Schatten matrix norms. We also establish the CB version of the Gross Lemma, which allows LS for general q to be deduced from LS at q = 2. Furthermore, we show how this leads to an 'additivity' result for the LS constants of a product channel. The rest of this paper is organized as follows. We first establish notation, and review the definitions of the CB norm. We then state our main result which is a formulation of the LS inequality for the CB norm. This LS inequality displayed in (13) is strongly similar to the usual LS inequality, but with the partial trace appearing in some places. The following sections contain our analysis of the CB norm, which requires some careful characterization of the minimizers appearing in the definitions. The Appendix contains some technical results. Preliminaries We label systems and Hilbert spaces by uppercase letters such as H R , H S , and denote the tensor product H R ⊗ H S by H RS . All the spaces considered throughout this paper will be finite dimensional, and we will use the notation d R = dim H R < ∞. The space of linear operators acting on H R will be denoted by L(H R ), and we will often attach a label to an operator X R ∈ L(H R ) to indicate the underlying space. The adjoint of X R is denoted by X * R . For X R ∈ L(H R ) the normalized trace is defined by τ (X R ) = d −1 R Tr(X R ). We will work mostly with τ (·) rather than the unnormalized matrix trace, and this will enter the various definitions of norms that we will use. Thus the (normalized) p-Schatten norm of X R ∈ L(H R ) is defined by X R p = τ (\|X R \| p ) 1/p = d −1/p R Tr(\|X R \| p ) 1/p , p ≥ 1, where \|X R \| = (X * R X R ) 1/2 . For an operator X R ∈ L(H R ), we write X R ≥ 0 to indicate that X R is positive semidefinite, and X R > 0 to indicate that X R is positive definite. We denote by D + R ⊂ L(H R ) the set of positive definite matrices normalized with respect to τ , that is, D + R = {σ R ∈ L(H R ) : σ R > 0, τ (σ R ) = 1}.(2) For σ R ∈ D + R and ǫ > 0 we define B ǫ (σ R ) = ξ ∈ D + R : σ − ξ 1 ≤ ǫ = D + R ∩ X R ∈ L(H R ) : σ − X R 1 ≤ ǫ .(3) Non-commutative (q, p)-norm For operators Y RS acting on the product space H RS = H R ⊗ H S , we will use the noncommutative (q, p)-norms introduced by Pisier [19], which extend the classical l q (l p ) norms to bipartite matrices. See [7,6] for a review of this notion and its applications in quantum information theory. For 1 ≤ q ≤ p the (q, p)-norm is given by Y RS (q,p) = inf A,B,Z A R 2r B R 2r Z RS p : Y RS = (A R ⊗ I S )Z RS (B R ⊗ I S ) ,(4) where I S ∈ L(H S ) is the identity operator, and r is given by 1 r = 1 q − 1 p .(5) Note again that we are using the normalized Schatten norm, so our definition (4) of the (q, p)norm differs from the standard one in [19] by an overall multiplicative factor. As shown in [7], when Y is positive semidefinite, without loss of generality we may restrict the infimum in (4) to positive definite matrices A R = B R > 0, in which case we have Z RS = (A −1 R ⊗ I S )Y RS (A −1 R ⊗ I S ). Therefore, for positive semidefinite Y RS ≥ 0 we have Y RS (q,p) = inf A R >0 A R 2 2r · (A −1 R ⊗ I S )Y RS (A −1 R ⊗ I S ) p . Moreover, by rescaling we may assume that A R 2r = 1, which implies that A 2r R ∈ D + R as defined in (2). So for 1 ≤ q ≤ p and Y RS ≥ 0 we may write Y RS (q,p) = inf σ R ∈D + R (σ −1/2r R ⊗ I S )Y RS (σ −1/2r R ⊗ I S ) p .(6) If the ordering of q, p is reversed, so that 1 ≤ p ≤ q, the corresponding expression for the (q, p)-norm of a positive semidefinite operator Y RS is Y RS (q,p) = sup σ R ∈D + R (σ −1/2r R ⊗ I S )Y RS (σ −1/2r R ⊗ I S ) p ,(7) where again r is given by (5), and thus is negative in this case. When p = q we may use either of the definitions (6) or (7) to compute the (q, p)-norm. Indeed, for p = q these two definitions coincide and we have Y RS (q,q) = Y RS q . Completely bounded norm For a super-operator Φ : L(H S ) → L(H S ′ ), the completely bounded (CB) norm is defined by Φ CB,q→p = sup d R sup Y RS (I R ⊗ Φ)(Y RS ) (t,p) Y RS (t,q) ,(8) where I R is the identity super-operator acting on L(H R ), the first supremum is over the dimension d R = dim(H R ), and the second supremum is over Y RS ∈ L(H RS ). Moreover, t ≥ 1 is arbitrary [19]; the supremum is independent of the choice of t. We will generally use the value t = q. A super-operator Φ is positive if Φ(Y S ) ≥ 0 is positive semidefinite for any Y S ≥ 0. Moreover, Φ is called completely positive if I R ⊗ Φ is positive for all H R . We recall the following results which were proved in [7]. Φ CB,q→p = sup d R sup Y RS >0 (I R ⊗ Φ)(Y RS ) (t,p) Y RS (t,q) . Moreover, if p ≤ q then the supremum over d R is not required, i.e., Φ CB,q→p = Φ q→p = sup Y S >0 Φ(Y S ) p Y S q . It is also shown in [7] that the completely bounded norm is multiplicative for completely positive super-operators as in (1). Main results Let {Φ t : t ≥ 0} be a semi-group of completely positive super-operators on L(H S ) with generator L. That is, for any t ≥ 0, Φ t : L(H S ) → L(H S ) is completely positive, and we have Φ 0 = I S , and Φ s+t = Φ s Φ t . We let L = − d dt Φ t t=0 . The super-operator L is called the Lindblad generator of the semi-group. Thus, for every X S and t ≥ 0 we have d dt Φ t (X S ) = −L Φ t (X S ) = −Φ t (LX S ). Equivalently, for every t ≥ 0 and X S we have Φ t (X S ) = e −tL (X S ).(9) We assume that the semi-group {Φ t : t ≥ 0} implements the Markov approximation for a quantum dynamics on H S , in the Heisenberg representation. As a result, Φ t for every t is unital, meaning that Φ t (I S ) = I S . This in particular implies that L(I S ) = 0. The Schrödinger representation Φ * t is obtained by duality with respect to the Hilbert-Schmidt inner product, that is τ (Φ t (X S )) * ρ S = τ (X S ) * Φ * t (ρ S ) , for all observables X S ∈ L(H S ) and all states ρ S ∈ D + S . The generator of the semigroup {Φ * t : t ≥ 0} is L * , the adjoint of L. In the Schrödinger picture the quantum dynamics is trace-preserving, meaning that τ (Φ * t (ρ)) = τ (ρ) for all ρ. Indeed, Φ t is unital if and only if Φ * t is trace-preserving. In this paper we further assume that the semigroup is reversible, which means that Φ t = Φ * t for all t. By the above discussion a reversible semigroup is both unital and tracepreserving. Moreover, its generator is self-adjoint, that is L = L * .(10) A completely positive super-operator Φ t that is both unital and trace-preserving is a contraction under the Schatten q-norm, for every q ≥ 1, [17]. That is, for every q ≥ 1 we have Φ t CB,q→q = Φ t q→q ≤ 1,(11) where the equality Φ t CB,q→q = Φ t q→q holds by Theorem 1. Then the following question arises. For a given q, what is the largest p * = p * (t) such that Φ t CB,q→p * ≤ 1 , for all t ≥ 0? Observe that · p is a non-decreasing function of p, and so by our assumption (11), and the fact that {Φ t : t ≥ 0} forms a semigroup, it follows that p * (t) is a non-decreasing function of t with p * (0) = q. Definition 2 Consider q ≥ 1, and let p = p(t) ≥ q be defined for all t ≥ 0. We say that the semigroup {Φ t : t ≥ 0} is completely bounded-q-hypercontractive (CB-q-HC) for p(t) if for all t ≥ 0 we have Φ t CB,q→p(t) ≤ 1. Next we define our notion of completely bounded log-Sobolev inequality. Definition 3 We say that the semigroup Φ t with generator L satisfies the completely bounded (CB) log-Sobolev inequality at q with constant α > 0 if τ (Y q RS ln Y q RS ) − τ R τ S (Y q RS ) ln τ S (Y q RS ) ≤ αq 2 τ Y q−1 RS (I R ⊗ L)(Y RS ) ,(13) for all H R and all positive semidefinite Y RS ∈ L(H RS ). Here τ R and τ S denote the partial traces over H R and H S respectively. In the above definition, as usual, we extend the meaning of τ (Y ln Y ) to include positive semidefinite matrices, by restricting to the support of Y . The main result of this paper is the equivalence of the above two definitions in the following sense. Theorem 4 Let {Φ t : t ≥ 0} be a semigroup of completely positive super-operators that satisfy (11). Also consider q ≥ 1 and let p(t) ≥ q (defined for t ≥ 0) be a twice continuously differentiable increasing function with q = p(0). Then the following results hold: (i) If the semigroup is CB-q-HC for p(t), then it satisfies the CB log-Sobolev inequality at q with constant α = 1/p ′ (0). (ii) If the semigroup satisfies the CB log-Sobolev inequality at p(t) with constant α(t) = 1/p ′ (t) for all t ≥ 0, then it is CB-q-hypercontractive for p(t). We also prove a CB version of the 'Gross Lemma' [9] which relates the log-Sobolev inequalities at q = 2 and q > 2. This requires the additional assumption that the generator is self-adjoint. Theorem 5 Let {Φ t : t ≥ 0} be a semigroup of completely positive unital super-operators with self-adjoint generator L satisfying (10). Suppose Φ t satisfies the CB log-Sobolev inequality (13) at q = 2 with constant α > 0. Then Φ t also satisfies (13) for all q ≥ 2 with constant α(q − 1) −1 > 0. We note that unlike the usual log-Sobolev inequality, the CB log-Sobolev inequality in the non-commutative case satisfies the following tensorization property. Theorem 6 Suppose that for all i = 1, . . . , k the semigroup of completely positive superoperators Φ (i) t : L(H S i ) → L(H S i ) generated by L i satisfies the CB log-Sobolev inequality at q with constant α i . Then the semigroup Ψ t : L(H S 1 ...S k ) → L(H S 1 ...S k ) generated by L = iL i , whereL i is obtained from L i by tensoring with an appropriate identity super-operator, satisfies the CB log-Sobolev inequality at q with constant α = max{α 1 , . . . , α k }. In order to prove part (i) of Theorem 4 we will derive a formula for the derivative of the non-commutative (q, p)-norm at p = q. Since this result has independent interest we state it as a separate theorem. Theorem 7 Let p(t) ≥ 1 be a twice continuously differentiable increasing function with q = p(0) and p ′ (0) = 1/α > 0. Also let X RS (t) be a matrix-valued twice continuously differentiable function, where X RS (t) is positive definite in a neighborhood of 0. Let Y = X RS (0). Then d dt X RS (t) (q,p(t)) t=0 = 1 αq 2 Y q−1 q τ (Y q ln Y q ) − τ R τ S (Y q ) ln τ S (Y q ) + αq 2 τ (Y q−1 X ′ RS (0)) . (14) The main complication in proving the above theorems is that the definition of the (q, p)norm involves an infimum or supremum (depending on whether p ≥ q or q ≥ p) whose optimal point is not easy to compute. In the following section we derive some properties of the optimizer σ R in (6) and (7), and in subsequent sections we will use these properties to establish our results. We finish this section with a few remarks about applications of our results. Observe that our notion of CB log-Sobolev inequality is stronger than the usual log-Sobolev inequality. This can be verified by taking the Hilbert space H R in (13) to be trivial. As a result, the CB log-Sobolev constant is as large as the usual log-Sobolev constant. This fact can also be verified using the fact that the completely bounded norm is lower bounded by the usual operator norm. As a consequence of this observation, Theorems 4 and 6, and the results of [13], we find that the CB log-Sobolev constant at q = 2 is an upper bound for the mixing time of an arbitrarily large product of independent copies of a semigroup defined by a strongly regular generator. We emphasis that this statement (for a product of independent copies) has not been proven for the usual log-Sobolev constant, and is generally presumed to be false. Another application of our work is in computing the 'CB hypercontractivity ribbon' of [6] for certain bipartite density matrices. Analysis of the (q, p)-norm In this section we present some formulas for the derivative of the norm expression appearing in the definitions (6) and (7). We also partially characterize the optimizer in the definition of the (q, p)-norm. As in the statement of Theorem 7, let p(t) be an increasing twice continuously differentiable function with q = p(0) and p ′ (0) = 1/α > 0. Also let X RS (t) be a matrix-valued twice continuously differentiable function, where X RS (t) is positive definite for t ∈ [−η, η] for some η > 0. For simplicity we sometimes denote X RS (t) and p(t) by X RS and p, but keep in mind that they depend on t. Define s(t) = 1 q − 1 p(t)(15) and for σ R ∈ D + R define M(t, σ R ) = (σ −s(t)/2 R ⊗ I S )X RS (t)(σ −s(t)/2 R ⊗ I S ).(16) Thus M is positive definite for any t ∈ [−η, η]. Let F (t, σ R ) = M(t, σ R ) p(t) .(17) Note that p(0) = q so s(0) = 0, and F (0, σ R ) = X RS (0) q does not depend on σ R . Our first result establishes basic convexity and concavity properties of (17) as a function of σ R for fixed t (in the next section we will prove a more refined result for the case p ≥ q). Lemma 8 For fixed t in (−η, η), the function σ R → F (t, σ R ) p is convex for 1 ≤ q ≤ p(t) ≤ 2q and concave for 1 ≤ p(t) ≤ q. Proof: We are concerned with the function σ R → F (t, σ R ) p = (σ −s/2 R ⊗ I S )X RS (t)(σ −s/2 R ⊗ I S ) p p = X RS (t) 1/2 (σ R ⊗ I S ) −s X RS (t) 1/2 p p = τ X RS (t) 1/2 (σ R ⊗ I S ) −s X RS (t) 1/2 p .(18) Hiai [12, Theorem 1.1] has proven that the map ξ → τ W ξ −s W * p , on the set of positive definite matrices is convex if 0 ≤ s ≤ 1 and 1/2 ≤ p ≤ 1/s, and is concave if 0 ≤ −s ≤ 1 and 1/2 ≤ p ≤ −1/s. We apply this result to (18) with ξ = σ R ⊗ I S , and use the definition (15) to relate s, p, q. Hiai's conditions for convexity are satisfied when 1 ≤ q ≤ p(t) ≤ 2q, and the conditions for concavity are satisfied when 1 ≤ p(t) ≤ q. Our next result presents some smoothness properties of F , and also formulas for its derivative with respect to t. Lemma 9 (a) The function ∂ 2 F/∂t 2 is continuous on (−η, η) × D + R . (b) The function σ R → F (t, σ R ) is continuously differentiable for all σ R ∈ D + R and t ∈ (−η, η). (c) For all σ R ∈ D + R and t ∈ (−η, η) ∂ ∂t F (t, σ R ) = p ′ (t) F (t, σ R ) p 2 τ (M p ) − τ (M p ) ln τ (M p ) + τ (M p ln M p ) − τ R (τ S (M p ) ln σ R ) + p 2 p ′ (t) τ M p−1 (σ −s(t)/2 R ⊗ I S )X ′ RS (t)(σ −s(t)/2 R ⊗ I S ) . In particular, we have ∂ ∂t F (t, σ R ) t=0 = 1 αq 2 Y q−1 q − τ (Y q ) ln τ (Y q ) + τ (Y q ln Y q ) − τ R (τ S (Y q )(ln σ R )) + αq 2 τ Y q−1 X ′ RS (0) ,(19) where Y = X RS (0) and α = 1/p ′ (0). Our main tool in the proof of this lemma is the contour integral representation of M p ; the rest is a straightforward calculation, so we leave the proof for Appendix A. Next we will use these basic results about derivatives to provide estimates for F (t, σ R ) in a neighborhood of t = 0. Note first that since p(0) = q we have F (0, σ R ) = M(0, σ R ) q = X RS (0) q = Y q , where as before Y = Y RS = X RS (0). We define the normalized reduced density matrix of Y q RS by γ R = 1 τ (Y q ) τ S (Y q ),(20) Note that Y is positive definite and τ R (γ R ) = 1, so γ R ∈ D + R . Let us also define G(σ R ) to be the factor in braces on the right hand side of (19), that is, G(σ R ) = − τ (Y q ) ln τ (Y q ) + τ (Y q ln Y q ) − τ R (τ S (Y q )(ln σ R )) + αq 2 τ Y q−1 X ′ RS (0) .(21) Lemma 10 There is κ > 0 and K < ∞, such that for all t ∈ [−η/2, η/2] and σ R ∈ B κ (γ R ), F (t, σ R ) − Y q − t G(σ R ) αq 2 Y q−1 q ≤ K t 2 .(22) Proof: Let t ∈ [−η/2, η/2], and recall the definitions (3) and (20). Since D + R is open, there is κ > 0 such that B κ (γ R ) ⊂ D + R .(23) Since B κ (γ R ) is closed and bounded, it is a compact subset of D + R . Furthermore by Lemma 9, ∂ 2 F/∂t 2 is continuous on (−η, η) × D + R . Hence there is K < ∞ such that −2K ≤ ∂ 2 F ∂t 2 (t, σ) ≤ 2K,(24) for all t ∈ [−η/2, η/2], and all σ R ∈ B κ (γ R ). Therefore, for any t ∈ [−η/2, η/2] and σ R ∈ B κ (γ R ) we have F (t, σ R ) − F (0, σ R ) − t ∂F ∂u (u, σ R ) u=0 = t 0 (t − u) ∂ 2 ∂u 2 F (u, σ R )du ≤ Kt 2 . Noting that F (0, σ R ) = Y q and using the definition of G(σ R ) we find that F (t, σ R ) − Y q − t G(σ R ) αq 2 Y q−1 q ≤ K t 2 , for all t ∈ [−η/2, η/2], and σ R ∈ B κ (γ R ). Returning to the formula (21) and using the definition (20), we observe that for any σ R ∈ D + R , G(σ R ) =G(γ R ) + τ (Y q ) τ R (γ R ln γ R ) − τ (Y q ) τ R (γ R ln σ R ) =G(γ R ) + τ (Y q ) S d −1 R γ R d −1 R σ R ,(25) where S(· ·) is the relative entropy between density matrices γ R /d R and σ R /d R defined by S d −1 R γ R d −1 R σ R = Tr d −1 R γ R ln(d −1 R γ R ) − ln(d −1 R σ R )) = τ γ R (ln γ R − ln σ R ) .(26) Our final lemma in this section localizes the optimizer in the (q, p) norm for small t. Lemma 11 For any 0 < ǫ ≤ κ, where κ is the parameter described in Lemma 10, there is δ > 0 such that for all t ∈ [−δ, δ] there is σ R (t) ∈ D + R satisfying X RS (t) (q,p) = F (t, σ R (t)) and γ R − σ R (t) 1 ≤ ǫ. Proof: Given ǫ ≤ κ, where κ was defined in (23), we choose δ ′ > 0 to satisfy δ ′ < min η 2 , ǫ 2 τ (Y q ) 4K αq 2 Y q−1 q , where K is defined by (24). We have B ǫ (γ R ) ⊂ B κ (γ R ) ⊂ D + R , and so the boundary of B ǫ (γ R ) is contained in D + R . Suppose that σ R is on the boundary of B ǫ (γ R ), so that γ R − σ R 1 = ǫ.(27) Pinkser's inequality [18] implies that S d −1 R γ R d −1 r σ R ≥ 1 2 γ R − σ R 2 1 = ǫ 2 2 , where S is the relative entropy defined in (26). Thus from (25) we deduce G(σ R ) ≥ G(γ R ) + ǫ 2 τ (Y q ) 2 .(28) We consider first the case where t ≥ 0. From (22) we deduce that F (t, σ R ) ≥ Y q + t G(σ R ) αq 2 Y q−1 q − K t 2 ≥ Y q + t G(γ R ) αq 2 Y q−1 q + t ǫ 2 τ (Y q ) 2 αq 2 Y q−1 q − K t 2 . Our choice of δ ′ implies that for all 0 ≤ t ≤ δ ′ we have t ǫ 2 τ (Y q ) 2 αq 2 Y q−1 q − K t 2 > K t 2 , and thus F (t, σ R ) > Y q + t G(γ R ) αq 2 Y q−1 q + K t 2 .(29) Furthermore, from (22) we also deduce that F (t, γ R ) ≤ Y q + t G(γ R ) αq 2 Y q−1 q + K t 2 .(30) Combining (29) and (30) we find that F (t, γ R ) < F (t, σ R ). Since this inequality holds for all σ R on the boundary of B ǫ (γ R ), we conclude that for all 0 ≤ t ≤ δ ′ the function σ R → F (t, σ R ) has a local minimum σ R (t) in the interior of B ǫ (γ R ). We now choose 0 < δ + ≤ δ ′ so that q ≤ p(t) ≤ 2q for all 0 ≤ t ≤ δ + (the existence of δ + > 0 is guaranteed by our assumptions that p(0) = q ≥ 1 and that p(t) is increasing and differentiable). Applying Lemma 8 we conclude that the local minimum of the convex function σ R → F (t, σ R ) p(t) in the interior of B ǫ (γ R ) is in fact a global minimum for all 0 ≤ t ≤ δ + . Since F (t, σ R ) and F (t, σ R ) p share the same minimum σ R (t) ∈ B ǫ (γ R ), we conclude that X RS (t) (q,p) = F (t, σ R (t)) and γ R − σ R (t) 1 ≤ ǫ. Turning to the case t ≤ 0 we use (22) and (28) to deduce that F (t, σ R ) ≤ Y q + t G(σ R ) αq 2 Y q−1 q + K t 2 ≤ Y q + t G(γ R ) αq 2 Y q−1 q + t ǫ 2 τ (Y q ) 2 αq 2 Y q−1 q + K t 2 . Again using the definition of δ ′ and noting that t is negative, we have t ǫ 2 τ (Y q ) 2 αq 2 Y q−1 q + K t 2 < −K t 2 , and thus F (t, σ R ) < Y q + t G(γ R ) αq 2 Y q−1 q − K t 2 .(31) Combining this time with the lower bound for F (t, γ R ) obtained from (22) we deduce that F (t, γ R ) > F (t, σ R ), for all σ R on the boundary of B ǫ (γ R ). Thus we conclude that for all −δ ′ ≤ t ≤ 0 the function σ R → F (t, σ R ) has a local maximum in the interior of B ǫ (γ R ). We now choose 0 < δ − ≤ δ ′ so that 1 ≤ p(t) ≤ q for all −δ − ≤ t ≤ 0. Applying Lemma 8 we conclude that the local maximum of the concave function σ R → F (t, σ R ) p(t) in the interior of B ǫ (γ R ) is in fact a global maximum for all −δ − ≤ t ≤ 0. Finally we take δ = min{δ + , δ − } and deduce that for all t ∈ [−δ, δ] there is σ R (t) ∈ D + R satisfying X RS (t) (q,p) = F (t, σ R (t)) and γ R − σ R (t) 1 ≤ ǫ. Restriction to p > q We now restrict our attention to t > 0, in which case p > q. We will prove a refined characterization of the optimal σ R which holds for all p > q (and not just for small t > 0). Lemma 12 For a fixed t ∈ (0, η), for which p > q, the function σ R → F (t, σ R ),(32) is strictly convex, and there is a uniqueσ R ∈ D + R such that F (t,σ R ) = X RS (t) (q,p) .(33) Moreover, the optimizerσ R in (33) satisfieŝ σ R = 1 F (t,σ R ) p τ S (σ −s/2 R ⊗ I S )X RS (t)(σ −s/2 R ⊗ I S ) p . (34) Proof: We borrow ideas from the proof of Lemma 20 of [11] in order to prove this lemma. By the unitary invariance of the p-norm we can rewrite the function F as F (t, σ R ) = X(t) 1/2 (σ −s ⊗ I S )X(t) 1/2 p . Since p > q we have s ∈ (0, 1] (see (15)). Then the map σ → σ −s ,(35) is operator convex [1], and thus for any λ ∈ [0, 1] and σ R , ξ R ∈ D + R we have (λσ R + (1 − λ)ξ R ) −s ≤ λσ −s R + (1 − λ)ξ −s R . Next, the monotonicity of the map ζ → X 1/2 ζX 1/2 and of the p-norm imply F (t, λσ + (1 − λ)ξ) ≤ λX 1/2 (σ −s ⊗ I S )X 1/2 + (1 − λ)X 1/2 (ξ −s ⊗ I S )X 1/2 p . For all p ≥ 1 the Schatten p-norm is uniformly convex [2], and thus also strictly convex. Therefore λX 1/2 (σ −s ⊗ I S )X 1/2 + (1 − λ)X 1/2 (ξ −s ⊗ I S )X 1/2 p ≤ λ X 1/2 (σ −s ⊗ I R )X 1/2 p + (1 − λ) X 1/2 (ξ −s ⊗ I S )X 1/2 p = λF (t, σ) + (1 − λ)F (t, ξ), with equality if and only if X 1/2 (σ −s ⊗ I S )X 1/2 = c X 1/2 (ξ −s ⊗ I S )X 1/2 , for some c ∈ R. Since X is positive definite (and therefore invertible), the equality condition is equivalent to σ −s = c ξ −s which by the normalization τ (σ) = τ (ξ) = 1 gives σ = ξ. We conclude that F (t, λσ + (1 − λ)ξ) ≤ λF (t, σ) + (1 − λ)F (t, ξ), with equality if and only if σ = ξ. Therefore, the function F (t, σ R ) is strictly convex in σ R . Now we will show that the infimum in (33) is achieved. We argue by contradiction, so suppose that the infimum is not achieved in D + R . Then there must exist a non-convergent sequence {ξ n : n ≥ 1} ⊂ D + R such that lim n→∞ F (t, ξ n ) = X RS (t) (q,p) .(36) The closure of D + R is compact in L(H R ), and thus the sequence {ξ n : n ≥ 1} has a limit point in L(H R ). By assumption there is no limit point in D + R , thus the limit point belongs to the boundary ∂D + R . So there is a subsequence {ξ n j : j ≥ 1} which approaches ∂D + R as j → ∞. Since ∂D + R consists of singular matrices, and s > 0, we obtain lim j→∞ ξ −s n j p = ∞. Now since X RS (t) is positive definite (and thus invertible) we have F (t, ξ n j ) = (ξ −s/2 n j ⊗ I S )X RS (ξ −s/2 n j ⊗ I S ) p = X 1/2 RS (ξ −s n j ⊗ I S )X 1/2 RS p ≥ ξ −s n j ⊗ I S p X −1/2 RS −2 ∞ = ξ −s n j p X −1/2 RS −2 ∞ , which implies that F (t, ξ n j ) → ∞ as j → ∞. This contradicts our assumption (36). So we conclude that the infimum in (33) is achieved in D + R . Moreover, by the strict convexity proved above, the infimum is achieved at a unique point which we callσ R (t). Next we show thatσ R satisfies equation (34). For this purpose we recall Lemma 9(b), where we showed that ξ R → F (t, ξ R ) is a continuously differentiable function in D + R . Since the function has a minimum atσ R , its derivative must vanish at ξ R =σ R . To compute the derivative, let ̺ be a traceless hermitian matrix, and define ξ(x) =σ + x̺.(38) Then ξ(x) ∈ D + R for all sufficiently small \|x\|. Let B(x) = X 1/2 (ξ(x) −s/2 ⊗ I S ). Then we have F (t, ξ(x)) p = τ (B * B) p , and therefore, d dx F (t, ξ(x)) p = p τ (B * B) p−1 dB * dx B + B * dB dx . Define ψ(x) = ξ(x) s/2 d dx ξ(x) −s/2 .(39) Then we have dB dx = B(ψ ⊗ I S ). (40) Therefore, d dx F (t, ξ(x)) p = p τ (B * B) p−1 (ψ * ⊗ I S )B * B + B * B(ψ ⊗ I S ) = p τ (B * B) p (ψ * + ψ) ⊗ I S . Let N R = τ S (B(0) * B(0)) p = τ S (σ −s/2 R ⊗ I S )X RS (t)(σ −s/2 R ⊗ I S ) p . Then we have d dx F (t, ξ(x)) p x=0 = p τ R N R (ψ * (0) + ψ(0)) = p τ R σ −1/2 N Rσ −1/2 Γ(̺) ,(41) where Γ(̺) =σ s/2+1/2 d dx (ξ −s/2 )σ 1/2 x=0 +σ 1/2 d dx (ξ −s/2 )σ s/2+1/2 x=0 . We claim that ̺ → Γ(̺) maps the subspace of traceless Hermitian matrices into itself, and is onto. To see this, we first extend the definition of Γ to a linear operatorΓ on the space of all Hermitian matrices, by extending (38) to allow general Hermitian matrices ρ. We claim thatΓ is surjective. To see this, first note that the map ζ → ζ −s/2 is one-to-one on positive definite matrices, and hence its derivative ̺ → d dx (σ + x̺) −s/2 x=0 , is onto. The map ζ →σ s/2+1/2 ζσ 1/2 +σ 1/2 ζσ s/2+1/2 , is also onto. As a result their composition which isΓ is onto. Now we note that τ (Γ(̺)) = 2τ σ 1+s/2 d dx ξ −s/2 x=0 = −sτ d dx ξ x=0 = −sτ (̺). Therefore,Γ maps the subspace of traceless Hermitian matrices into itself, and is onto. Thus its restriction to the traceless Hermitian matrices, namely Γ, is also onto. Returning to (41), we conclude that for any traceless Hermitian matrix ζ we have τ σ −1/2 N Rσ −1/2 ζ = 0. Thereforeσ −1/2 N Rσ −1/2 is a multiple of the identity matrix. Thusσ is proportional to N R , and since τ (σ) = 1 we must have (34). According to Lemma 12, for any t > 0 there is a uniqueσ R (t) ∈ D + R such that X RS (t) (q,p) = F (t,σ R (t)). Moreover, from the results of Lemma 11 we can conclude that for sufficiently small t > 0,σ R (t) is close to γ R . We will use the following continuity result in the next section when we apply these lemmas to prove our main theorem. For t ≥ 0 we define ϕ(t) = X RS (t) (q,p(t)) = F (t,σ R (t)). (42) Lemma 13 ϕ(t) is continuous on [0, η). Proof: We first prove continuity at t = 0. Recalling Lemma 10, there is κ > 0 and K < ∞ such that for all σ R ∈ B κ (γ R ) and t ∈ [0, η/2) we have F (t, σ R ) − Y q ≤ t G(σ R ) αq 2 Y q−1 q + K t 2 .(43) On the other hand, for sufficiently small t > 0, Lemma 11 implies that the optimizerσ R (t) is in B κ (γ R ). Thus, noting that ϕ(0) = Y q , for sufficiently small t > 0, we deduce ϕ(t) − ϕ(0) ≤ t G(σ R ) αq 2 Y q−1 q + K t 2 .(44) Recalling (21) we note that the function σ R → G(σ R ) is continuous, and thus uniformly bounded on B κ (γ R ). Therefore the bound (44) implies continuity of ϕ(t) at t = 0. Now consider any t 0 ∈ (0, η). We will prove continuity of ϕ at t 0 . Let 0 < a < t 0 < b < η be arbitrary. For t ∈ [a, b], we have ϕ(t) = F (t,σ R (t)) = (σ R (t) −s(t)/2 ⊗ I S )X RS (t)(σ R (t) −s(t)/2 ⊗ I S ) p ≥ σ R (t) −s(t) p X RS (t) −1/2 −2 ∞ ≥ d −1 R λ min (σ R (t)) −s(t) X RS (t) −1/2 −2 ∞ , where λ min (σ R (t)) is the minimum eigenvalue ofσ R (t), and d R = dim H R . On the other hand, ϕ(t) = inf σ R F (t, σ R ) ≤ F (t, I R ) = X RS (t) p ≤ X RS (t) ∞ . Putting these together we conclude that λ min (σ R (t)) −s(a) ≤ λ min (σ R (t)) −s(t) ≤ d R X RS (t) −1/2 2 ∞ X RS (t) ∞ , where we use the fact that s(t) is increasing in t, and thatσ R (t) ∈ D + R which gives λ min (σ R (t)) ≤ 1. Now we note that X RS (t) is invertible and continuous. So there is C > 0 such that for all t ∈ [a, b] we have d R X RS (t) −1/2 2 ∞ X RS (t) ∞ ≤ C. Therefore, {σ R (t) : t ∈ [a, b]} ⊆ Λ where Λ = {σ R ∈ D + R : λ min (σ R ) ≥ C −1/s(a) }. The function F (t, σ R ) restricted to the compact set [a, b] × Λ is continuous, and therefore also uniformly continuous. Hence, for every ǫ > 0 there is δ > 0 such that for every t, t ′ ∈ [a, b] with \|t − t ′ \| < δ and σ R ∈ Λ we have \|F (t, σ R ) − F (t ′ , σ R )\| ≤ ǫ. This implies ϕ(t) = F (t, σ R (t)) ≤ F (t, σ R (t ′ )) ≤ F (t ′ , σ R (t ′ )) + ǫ = ϕ(t ′ ) + ǫ. We similarly have ϕ(t ′ ) ≤ ϕ(t) + ǫ. As a result, \|ϕ(t) − ϕ(t ′ )\| ≤ ǫ, for all \|t − t ′ \| < δ. Therefore, ϕ(t) is continuous in [a, b], and in particular at t = t 0 . Proof of Theorem 7 We now have all the tools required to prove Theorem 7. By assumptions α = p ′ (0) −1 is positive and finite. Using the definitions (20) and (21) we find G(γ R ) = τ (Y q ln Y q ) − τ R τ S (Y q ) ln τ S (Y q ) + αq 2 τ (Y q−1 X ′ RS (0)), which is the expression inside the braces on the right side of (14). We define ∆(t) = 1 t X RS (t) (q,p(t)) − X RS (0) (q,p(0)) − G(γ R ) α q 2 Y q−1 q . Thus our goal is to prove that ∆(t) → 0 as t → 0. Let 0 < ǫ be such that ǫ < min{κ, η, λ min (γ R ) 2d R },(45) where κ is the parameter described in Lemma 10, λ min (γ R ) is the minimum eigenvalue of γ R and as before d R = dim(H R ). According to Lemma 11, there is δ > 0 sufficiently small such that for every 0 < t < δ there is an optimizer σ R (t) such that σ R (t) − γ R 1 ≤ ǫ ≤ κ, and X RS (t) (q,p(t)) = F (t, σ R (t)). Then we have ∆(t) = 1 t F (t, σ R (t)) − Y q − G(γ R ) α q 2 Y q−1 q = 1 t F (t, σ R (t)) − Y q − t G( σ R (t)) α q 2 Y q−1 q + G( σ R (t)) − G(γ R ) α q 2 Y q−1 q .(46) Since σ R (t) ∈ B κ (γ R ), Lemma 10 implies that F (t, σ R (t)) − Y q − t G( σ R (t)) α q 2 Y q−1 q ≤ K t 2 .(47) Furthermore, from (25) and using Lemma 14 in Appendix B we obtain G( σ R (t)) − G(γ R ) = S d −1 R γ R d −1 R σ R (t) ≤ 2d R λ min (γ R ) γ R − σ R (t) 1 ≤ 2d R λ min (γ R ) ǫ.(48) Using (47) and (48) in (46) we obtain the bound \|∆(t)\| ≤ K t + 2d R λ min (γ R ) α q 2 Y q−1 q ǫ, for all ǫ satisfying (45), and all 0 < t < δ. Therefore lim sup t→0 \|∆(t)\| ≤ 2d R λ min (γ R ) α q 2 Y q−1 q ǫ, and since ǫ may be arbitrarily small, we deduce that Proof of Theorem 4 We prove parts (i) and (ii) of the theorem separately. Proof of (i) We need to show that (13) holds for any positive semidefinite Y RS . A continuity argument (using the Fannes inequality [8]) verifies that it suffices to prove (13) for positive definite Y RS . For this we apply Theorem 7 with X RS (t) = (I R ⊗ Φ t )(Y RS ). Since Y RS is positive definite, by Lemma 15, proved in Appendix C, we deduce that X RS (t) is also positive definite for all t ≥ 0. We note that X RS (0) = Y RS and d dt X RS (t) = −(I R ⊗ L)(X RS (t)), which gives X ′ RS (0) = −(I R ⊗ L)(Y RS ). Since by assumption Φ t CB,q→p(t) ≤ 1 we have X RS (t) (q,p) ≤ Y RS q , for all t in a neighborhood of 0. Since equality holds at t = 0, the derivative of X RS (t) (q,p) at t = 0 must be less than or equal to zero. Then from Theorem 7 we immediately conclude τ (Y q ln Y q ) − τ R τ S (Y q ) ln τ S (Y q ) − αq 2 τ (Y q−1 (I R ⊗ L)(Y RS )) ≤ 0, where as usual α = p ′ (0) −1 . Proof of (ii) Our goal is to show that for any Y RS > 0 and t ≥ 0 we have I R ⊗ Φ t (Y RS ) (q,p(t)) ≤ Y RS q . Without loss of generality we assume that Y RS q = 1, so that our goal becomes I R ⊗ Φ t (Y RS ) (q,p(t)) ≤ 1. We assume that the CB log-Sobolev inequality holds for all t ≥ 0, with constant α = p ′ (t) −1 . We will argue by contradiction, so let us suppose that I R ⊗ Φ t 0 (Y RS ) (q,p(t 0 )) > 1,(49) for some t 0 > 0. We will apply the results of Section 4 with X RS (t) = I R ⊗ Φ t (Y RS ). Note that by Lemma 15, X RS (t) is positive definite for all t ≥ 0, since by assumption Y RS is positive definite. Define ϕ(t) = X RS (t) (q,p(t)) − ǫt. Then by (49) for sufficiently small ǫ > 0 we have ϕ(t 0 ) > 1. Let U = {t ∈ [0, t 0 ] : ϕ(t) ≤ 1}. Since Φ 0 = I S and p(0) = q, we have ϕ(0) = 1 and thus U is non-empty. Let u = sup U. By Lemma 13 the function ϕ(t) is continuous, so u ∈ U and ϕ(u) ≤ 1. This means that u < t 0 . Moreover, for any t ∈ (u, t 0 ] we have ϕ(t) > 1 ≥ ϕ(u). For t > 0 letσ R (t) be the unique minimizer characterized in Lemma 12, and letσ R (0) = γ R where γ R is defined in (20). Define µ(t) = F (t,σ R (u)) − ǫt. Then for any t ≥ u we have µ(t) ≥ inf σ R F (t, σ R ) − ǫt = ϕ(t), and we have ϕ(u) = µ(u). The derivative of µ(t) = F (t,σ R (u)) − ǫt at t = u can be computed using the results of Lemma 9, and the characterization (34) ofσ R (u). The result is ∂ ∂t F (t,σ R (u)) t=u = p ′ (u)F p 2 τ (M p ) τ (M p ln M p ) − τ R τ S (M p ) ln τ S (M p ) − p 2 p ′ (u) τ M p−1 (I R ⊗ L)(M) ,(50) where M = (σ R (u) −s(u)/2 ⊗ I S )X RS (u)(σ R (u) −s(u)/2 ⊗ I S ). Then using the assumption that the semigroup satisfies the CB log-Sobolev inequality at p(u) with constant α(u) = 1/p ′ (u), we find that µ ′ (u) = ∂ ∂t F (t,σ R (u)) t=u − ǫ ≤ −ǫ. Therefore there exists δ > 0 such that u + δ ≤ t 0 and µ(u + δ) ≤ µ(u). We then have ϕ(u + δ) ≤ µ(u + δ) ≤ µ(u) = ϕ(u) ≤ 1. This contradicts the definition of u, therefore we conclude that the assumption (49) is false, and this establishes the Theorem. Proof of Theorem 5 We suppose that the CB log-Sobolev inequality holds at q = 2 with constant α, thus for any positive semidefinite Y = Y RS we have τ (Y 2 ln Y 2 ) − τ R τ S (Y 2 ) ln τ S (Y 2 ) ≤ 4 α τ (Y (I R ⊗ L)(Y )) .(51) We will prove that for any positive semidefinite V and q ≥ 2, 4 τ V q/2 (I R ⊗ L)(V q/2 ) ≤ q 2 q − 1 τ V q−1 (I R ⊗ L)(V ) .(52) Letting Y = V q/2 and combining the inequalities (51) and (52) we obtain the bound α q − 1 q 2 τ V q−1 (I R ⊗ L)(V ) ≥ τ (V q ln V q ) − τ R τ S (V q ) ln τ S (V q ) which is precisely the CB-log Sobolev inequality with constant α(q − 1) −1 . To prove (52) we will follow the method used in the recent paper [5], which is itself based on the Stroock-Varopoulos inequality [3,21]. The following inequality is proved in Appendix D: for all positive definite Z RS and all 2 ≤ r ≤ q, rr ′ τ Z 1/r (I R ⊗ L)(Z 1/r ′ ) ≤ qq ′ τ Z 1/q (I R ⊗ L)(Z 1/q ′ ) ,(53) where r ′ , q ′ are the usual conjugate values defined by 1 r ′ = 1 − 1 r , 1 q ′ = 1 − 1 q . The inequality (52) follows by taking r = 2, and Z = V q using the fact that L is self-adjoint. Proof of Theorem 6 The tensorization property of the CB log-Sobolev inequality can be proved using our main result Theorem 4, and the multiplicativity of the CB norm for completely positive maps (1). Here we present a direct proof. Let Y RS 1 ...S k be an arbitrary positive semidefinite matrix. By assumption for every i we have τ (Y q ln Y q ) − τ τ S i (Y q ) ln τ S i (Y q ) ≤ αq 2 τ Y q−1 (I R ⊗L i )(Y ) . Then the claim follows if we show that k i=1 τ (Y q ln Y q ) − τ τ S i (Y q ) ln τ S i (Y q ) ≥ τ (Y q ln Y q ) − τ τ S 1 ...S k (Y q ) ln τ S 1 ...S k (Y q ) .(54) Recall that the conditional entropy of ρ AB with Trρ AB = 1 is defined by H(A\|B) = −Tr(ρ AB ln ρ AB ) + Tr B Tr A (ρ AB ) ln Tr A (ρ AB ) = −d AB τ (ρ AB ln ρ AB ) + d AB τ B τ A (ρ AB ) ln τ A (ρ AB ) − ln d A , and satisfies the chain rule H(AC\|B) = H(A\|B) + H(C\|AB). Moreover, by the strong data processing inequality we have H(A\|B) ≥ H(A\|BC). Observe that in (54) with no loss of generality we may assume that Y q is normalized as TrY q = 1. Then this inequality can be rewritten as k i=1 H(S i \|RS ∼i ) ≤ H(S 1 . . . S k \|R),(55) where we use S ∼i = S 1 . . . S i−1 S i+1 . . . S k . Now using the chain rule we have H(S 1 . . . S k \|R) = k i=1 H(S i \|RS 1 . . . S i−1 ). On the other hand the strong data processing inequality gives H(S i \|RS ∼i ) ≤ H(S i \|RS 1 . . . S i−1 ). Using this inequality in the previous equation we arrive at (55). = − s ′ (t) 2 σ −s(t)/2 R ln σ R . To justify this equation we may assume without loss of generality that σ R is diagonal. Since s ′ (t) = p ′ (t)p −2 it follows that d dt M = − p ′ (t)p −2 2 (ln σ R ⊗ I S ) M − p ′ (t)p −2 2 M (ln σ R ⊗ I S ) + (σ −s(t)/2 R ⊗ I S )X ′ RS (t)(σ −s(t)/2 R ⊗ I S ). Thus we find τ (M p−1 M ′ ) = −p ′ (t)p −2 τ M p (ln σ R ⊗ I S ) + τ M p−1 (σ −s(t)/2 R ⊗ I S )X ′ RS (t)(σ −s(t)/2 R ⊗ I S ) . Combining these and using τ M p (ln σ R ⊗ I S ) = τ R τ S (M p ) ln σ R we get ∂ ∂t F (t, σ R ) = p ′ (t) F (t, σ R ) p 2 τ (M p ) − τ (M p ) ln τ (M p ) + τ (M p ln M p ) − τ R (τ S (M p ) ln σ R ) + p 2 p ′ (t) τ M p−1 (σ −s(t)/2 R ⊗ I S )X ′ RS (t)(σ −s(t)/2 R ⊗ I S ) . Also M(0) = X RS (0) = Y , and p(0) = q. Using these in the above equation gives (19). B Lipschitz constant of the relative entropy function Here we provide some estimates for the Lipschitz constant of the relative entropy function. As before, we will denote by λ min (σ) the smallest eigenvalue of σ ∈ D + R . Lemma 14 Let γ, σ, ξ ∈ D + R be such that γ − σ 1 < κ and γ − ξ 1 < κ where κ = 1 2d R λ min (γ). Then we have S(d −1 R γ d −1 R σ) − S(d −1 R γ d −1 R ξ) ≤ 4κ σ − ξ 1 .(59) Proof: Suppose that λ min (σ) ≥ λ min (γ), and that v, w are respectively the normalized eigenvectors of σ and γ for these eigenvalues. Then \|λ min (σ) − λ min (γ)\| = λ min (σ) − λ min (γ) = v * σv − w * γw ≤ w * (σ − γ)w ≤ σ − γ ∞ ≤ d R σ − γ 1 < d R κ. The same bound holds if λ min (σ) ≤ λ min (γ). We similarly have \|λ min (ξ) − λ min (γ)\| < d R κ. As a result we have λ min (σ), λ min (ξ) ≥ 1 2 λ min (γ). By definition S(d −1 R γ d −1 R σ) − S(d −1 R γ d −1 R ξ) = τ γ(ln ξ − ln σ) , and therefore S(d −1 R γ d −1 R σ) − S(d −1 R γ d −1 R ξ) ≤ ln σ − ln ξ ∞ .(61) Furthermore, ln σ − ln ξ = ∞ 0 1 t + ξ − 1 t + σ dt = ∞ 0 1 t + σ (σ − ξ) 1 t + ξ dt. Hence ln σ − ln θ ∞ ≤ ∞ 0 1 t + λ min (σ) σ − ξ ∞ 1 t + λ min (ξ) dt ≤ σ − ξ ∞ ∞ 0 t + 1 2 λ min (γ) −2 dt = σ − ξ ∞ 2 λ min (γ) ≤ σ − ξ 1 2d R λ min (γ) . where we used (60). Substituting this into (61) we get the desired bound. C Strict positivity of I R ⊗ Φ t Lemma 15 If Y RS is positive definite, then I R ⊗ Φ t (Y RS ) is positive definite for all t ≥ 0. Proof: By assumption I R ⊗ Φ t (Y RS ) is positive semidefinite. Then if it is not positive definite, it must be singular. That is, there is 0 = v ∈ H RS such that v * I R ⊗ Φ t (Y RS )v = 0. Let Z RS ≥ 0 be an arbitrary positive semidefinite matrix. Since Y RS is positive definite, Y RS − ǫZ RS ≥ 0 for sufficiently small ǫ > 0. Hence, I R ⊗ Φ t (Y RS ) ≥ ǫ I R ⊗ Φ t (Z RS ) ≥ 0, and then v * I R ⊗ Φ t (Y RS )v ≥ ǫ v * I R ⊗ Φ t (Z RS )v ≥ 0, which gives v * I R ⊗ Φ t (Z RS )v = 0 for all Z RS ≥ 0. Let Z RS = I R ⊗ e tL (I RS ). Then using (9) we find that v * v = 0 which is a contradiction since v = 0. Therefore, I R ⊗ Φ t (Y RS ) is positive definite for all t ≥ 0. D The quantum Gross Lemma Let Z RS be positive definite with spectral decomposition Z RS = i λ i w i w * i . Then for any a, b ∈ R we have τ Z a (I R ⊗ L)(Z b ) = i,j λ a i λ b j L ij , where L ij = τ w i w * i (I R ⊗ L)(w j w * j ) . Since the semigroup is reversible and L * = L, we have L ij = L ji for all i, j. Moreover, since the semigroup is unital, we have L(I S ) = 0 and j L ij = 0 for all i. Using these properties we can write τ Z a (I R ⊗ L)(Z b ) = − 1 2 i,j (λ a i − λ a j ) (λ b i − λ b j ) L ij .(62) On the other hand Φ t = e −tL is completely positive for t ≥ 0, so in particular τ w i w * i (I R ⊗ Φ t )(w j w * j ) = −t L ij + O(t 2 ) ≥ 0, for all i = j. Thus L ij ≤ 0 for all i = j. We apply the representation (62) to (53) on the left side with a = 1/r and b = 1/r ′ , and on the right side with a = 1/q and b = 1/q ′ . It is sufficient to prove the inequality for each index pair i = j: rr ′ (λ 1/r i − λ 1/r j ) (λ 1/r ′ i − λ 1/r ′ j ) ≤ qq ′ (λ 1/q i − λ 1/q j ) (λ 1/q ′ i − λ 1/q ′ j ).(63) We assume without loss of generality that λ i > λ j and let c = λ i /λ j > 1. Define f (u) = c u − 1 u . Then the left side of (63) is λ j f (1/r)f (1 − 1/r). Since 1/q < 1/r and λ j > 0, the bound will follow if we can show that for all 0 < u < v ≤ 1/2 we have f (u)f (1 − u) ≥ f (v)f (1 − v). Equivalently, we can show that log f (u) + log f (1 − u) ≥ log f (v) + log f (1 − v). The function g(u) = log f (u) + log f (1 − u) is symmetric around u = 1/2, so the above inequality follows if we show that it is convex. For this it is sufficient to show that log f (u) is convex, and this follows from a straightforward calculation of its second derivative. Theorem 1 [ 7 ] 17Let Φ : L(H S ) → L(H S ′ ) be completely positive. Then in (8) we may restrict the second supremum to include only positive definite Y RS , i.e., Thus we need to show that M ′ and M ′′ are continuous. For this we need to show that σ and X RS (t) are each twice continuously differentiable. X RS (t) is twice continuously differentiable by assumption. For σ −s(t)/2Finally we compute the derivative of M. First we note that−s(t)/2 R d dt σ −s(t)/2 R AcknowledgementsThis work was initiated at the BIRS workshop 15w5098, "Hypercontractivity and Log Sobolev Inequalities in Quantum Information Theory". We thank BIRS and the Banff Centre for their hospitality. SB was supported in part by Institute of Network Coding of CUHK and by GRF grants 2150829 and 2150785.AppendixA Proof of Lemma 9 (a) Since F (t, σ R ) = (τ (M p )) 1/p , it is sufficient to prove that t → τ (M p ) is twice continuously differentiable. We note thatwhere Γ is a closed contour which encloses the spectrum of M, which can be assumed to be in the open right half plane since M is positive definite. Moreover, the function z p = e p ln z is defined with a cut along the negative real axis, and is analytic for Re(z) > 0. So we are reduced to proving that τ ((z − M) −1 ) is twice continuously differentiable for all z outside the spectrum of M. Explicit calculation yieldswe again use the representationwhere Γ ′ is some contour which encloses the spectrum of σ R and is in the open right half plane. The proof finishes observing that the function z −s(t)/2 = e −s(t) ln z/2 is analytic in s, and then twice continuously differentiable in t.R for \|u\| sufficiently small. We define h(u) = F (t, ξ(u)). Following the reasoning from the proof of part (a), it is sufficient to prove differentiability of τ (z − M) −1 , which boils down to differentiability of ξ −s/2 . 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[ "Two-point functions of Neumann-Dirichlet open-string sector moduli ", "Two-point functions of Neumann-Dirichlet open-string sector moduli ", "Two-point functions of Neumann-Dirichlet open-string sector moduli ", "Two-point functions of Neumann-Dirichlet open-string sector moduli " ]	[ "Thibaut Coudarchet [email protected] \nCPHT\nCNRS\nEcole Polytechnique\nF-91128Paris, PalaiseauIPFrance\n", "Hervé Partouche [email protected] \nCPHT\nCNRS\nEcole Polytechnique\nF-91128Paris, PalaiseauIPFrance\n", "Thibaut Coudarchet [email protected] \nCPHT\nCNRS\nEcole Polytechnique\nF-91128Paris, PalaiseauIPFrance\n", "Hervé Partouche [email protected] \nCPHT\nCNRS\nEcole Polytechnique\nF-91128Paris, PalaiseauIPFrance\n" ]	[ "CPHT\nCNRS\nEcole Polytechnique\nF-91128Paris, PalaiseauIPFrance", "CPHT\nCNRS\nEcole Polytechnique\nF-91128Paris, PalaiseauIPFrance", "CPHT\nCNRS\nEcole Polytechnique\nF-91128Paris, PalaiseauIPFrance", "CPHT\nCNRS\nEcole Polytechnique\nF-91128Paris, PalaiseauIPFrance" ]	[ "9th International Conference on New Frontiers in Physics", "9th International Conference on New Frontiers in Physics" ]	We compute at one loop the two-point functions of massless scalars in the Neumann-Dirichlet open-string sector of the type IIB orientifold compactified on T 2 × T 4 /Z 2 , when N = 2 supersymmetry is spontaneously broken. This is done by evaluating correlation functions of "boundary-changing vertex operators" which are analogous to correlators of twist fields for closed strings. We use our results to compute the mass developed at one loop by the moduli fields arising in the Neumann-Dirichlet sector.	10.1142/s0217751x21410086	[ "https://arxiv.org/pdf/2012.14442v1.pdf" ]	229,924,340	2012.14442	3cd9956c2b79d575ddaa1633cc2b38e60b5eb582	Two-point functions of Neumann-Dirichlet open-string sector moduli * September 2020 Thibaut Coudarchet [email protected] CPHT CNRS Ecole Polytechnique F-91128Paris, PalaiseauIPFrance Hervé Partouche [email protected] CPHT CNRS Ecole Polytechnique F-91128Paris, PalaiseauIPFrance Two-point functions of Neumann-Dirichlet open-string sector moduli * 9th International Conference on New Frontiers in Physics Kolympari, GreeceSeptember 2020 We compute at one loop the two-point functions of massless scalars in the Neumann-Dirichlet open-string sector of the type IIB orientifold compactified on T 2 × T 4 /Z 2 , when N = 2 supersymmetry is spontaneously broken. This is done by evaluating correlation functions of "boundary-changing vertex operators" which are analogous to correlators of twist fields for closed strings. We use our results to compute the mass developed at one loop by the moduli fields arising in the Neumann-Dirichlet sector. Introduction Vertex operators of open-string states in Neumann-Dirichlet (ND) sectors change the nature of the boundary conditions at the insertion points. They involve "boundary-changing fields" whose operator-product expansions (OPE's) are analogous to those of twist fields [1] which create states in closed-string twisted sectors of orbifold theories [2]. Hence, computational techniques for twist fields [3] can be exploited, along with the method of images which relates correlation functions on open-string worldsheets to those for closed strings [4,5], to calculate string amplitudes involving states in ND sectors. In this work, we are interested in the evaluation at one loop of the two-point functions of the massless scalars sitting in the ND sector of a type IIB orientifold model compactified on T 2 × T 4 /Z 2 , [6][7][8] where N = 2 supersymmetry in Minkowski spacetime is spontaneously broken to N = 0 via a string version [9,10] of the Scherk-Schwarz mechanism [11]. The computation of this amplitude originates from the need to evaluate the masses generated at the quantum level by all moduli fields present in the model, in order to perform a stability analysis at one loop [12][13][14]. Since the dependance of the quantum effective potential on the vacuum expectation values of the moduli fields arising from the ND sector is unknown, the derivation of the quantum mass of these scalars relies on a direct computation of a two-point function. In Sect. 2, we briefly review the relevant content and specificities of the N = 2 → N = 0 model we are working in. We also define the vertex operators of the massless scalars in the ND sector and derive the expression of their two-point functions in terms of various correlators. Sect. 3 paves the road to the evaluation of the correlators of boundary-changing field and Sect. 4 makes use of these results to evaluate the two-point function we are interested in. In Sect. 5, we consider the regime of small supersymmetry breaking scale to obtain a more practical expression of the squared mass developed by the massless scalars of the ND sector. Further details beyond the results presented here can be found in Ref. [13]. Two-point function of massless states in the ND sector 2.1 The open-string model Our starting point is the type IIB orientifold model initially constructed in six dimensions by Bianchi and Sagnotti, and by Gimon and Polchinski [6][7][8]. Compactifying down to four dimensions, the full N = 2 supersymmetric background is R 1,3 × T 2 × T 4 Z 2 . (2. g : (X 6 , X 7 , X 8 , X 9 ) −→ (−X 6 , −X 7 , −X 8 , −X 9 ) . (2.2) Due to the orientifold projection, one O9-plane fills the entire space which requires the existence of 32 D9-branes for the model to be consistent. The orbifold action also implies the presence of one O5-plane at each fixed point of T 4 /Z 2 , along with 32 D5-branes orthogonal to the T 4 directions. Eventually, N = 2 supersymmetry is spontaneously broken to N = 0 thanks to the Scherk-Schwarz mechanism [9,10] implemented along the coordinate X 5 of the two-torus. For simplicity throughout this work, we will consider the tori T 2 and T 4 to be products of circles whose radii are denoted R I and R I . In that case, the scale of supersymmetry breaking, which is given by the gravitini masses, takes the form M 3/2 = 1 2R 5 . (2.3) Indeed, this expression is identical to that encountered in field theory [11] open-string sectors, as well as from the untwisted and twisted closed-string sectors [12]. In the present work, we focus on those arising from the ND sector by considering all classically massless scalars realized as strings stretched between D9-branes and D5-branes [13]. We will compute their squared masses in backgrounds corresponding to extrema of the one-loop effective potential with respect to all moduli fields except M 3/2 in general. 1 To characterize these backgrounds, it is convenient to consider two T-dual descriptions in which the moduli fields in the NN and DD sectors can be interpreted geometrically. T-dualizing T 2 , the internal space takes the formT 2 × T 4 /Z 2 modded by the involution (X I , X I ) → (−X I , −X I ) which has 2 6 = 64 fixed points, whereT 2 is the T-dual torus of coordinatesX I [17]. In this picture, there is one O3-plane at each fixed point and Vertex operators and amplitudes The amplitudes we consider, which are depicted in the left panel of Fig. 1, are evaluated on worldsheets with topologies of the annulus or Möbius strip. After conformal transformations, the surfaces can be realized as double-cover tori with Teichmüller parameters τ dc = i τ 2 2 for the annulus and τ dc = 1 2 + i τ 2 2 for the Möbius strip , τ 2 > 0 , (2.4) modded by the involution z → 1 −z, and with vertex operators inserted at z 1 , z 2 (see the right panel of Fig. 1). The external legs bring a Chan-Paton index α 0 referring to one of the 2n i 0 i 0 D9-branes (in green) and a Chan-Paton index β 0 which refers to one of the 2d j 0 i 0 D5branes (in orange). For the annulus, the two legs must be attached to the same boundary, while the second boundary carries a Chan-Paton index γ that refers either to one of the 32 D9-branes or one of the 32 D5-branes. There are therefore two annulus amplitudes to consider. Along the boundaries where the vertices are located, the fact that the boundary condi-tion changes from Neumann to Dirichlet and from Dirichlet to Neumann implies that the vertex operators involve boundary-changing fields. In ghost-picture −1, the vertex operators actually take the form α 0 β 0 α 0 β 0 γ 0 1 i τ 2 2 γ α 0 α 0 β 0 z 1 z 2 α 0 β 0 α 0 β 0 γ 0 1 i τ 2 2 γ α 0 α 0 β 0 z 1 z 2 α 0 β 0 α 0 β 0 0 1 1 2 + i τ 2 2 α 0 α 0 α 0 β 0 z 1 z 2V α 0 β 0 −1 (z 1 , k) = λ α 0 β 0 e −φ e ik·X e i 2 (H 3 −H 4 ) σ 3 σ 4 (z 1 ) , V β 0 α 0 −1 (z 2 , −k) = λ T β 0 α 0 e −φ e −ik·X e − i 2 (H 3 −H 4 ) σ 3 σ 4 (z 2 ) , (2.5) where our definitions are as follows: • k µ is the external momentum satisfying on-shell the condition k µ k µ = 0. • φ(z) is the ghost field encountered in the bosonization of the superconformal ghosts. • λ is the matrix that labels the states transforming in the bifundamental representation of U (n i 0 i 0 ) × U (d j 0 i 0 ) [7] , while λ T stand for its transpose. • Denoting ψ µ (z), ψ I (z), ψ I (z) the Grassmann fields superpartners of the bosonic coor- dinates X µ (z), X I (z), X I (z), we define complex degrees of freedom for u ∈ {0, . . . , 4}, Z u ≡ X 2u + iX 2u+1 √ 2 , Z u ≡ X 2u − iX 2u+1 √ 2 , Ψ u ≡ ψ 2u + iψ 2u+1 √ 2 ≡ e iHu , Ψ u ≡ ψ 2u − iψ 2u+1 √ 2 ≡ e −iHu ,(2.6) where H u are scalars introduced to bosonize the fermionic fields. • The operators e ±i(H 3 −H 4 ) are spin fields which enforce the fact that the ND-and DNsector states we consider are spinors of the T 4 /Z 2 space. • σ u , u ∈ {3, 4}, are the boundary-changing fields associated with the complex direc- tions Z u . The computation of the amplitudes is more easily done with vertex operators in ghostpicture 0. In that case, the operators naturally split into sums of two contributions referred to as "external" and "internal". The former, V α 0 β 0 0,ext (z 1 , k), V β 0 α 0 0,ext (z 2 , −k), involve the spacetime complexified momenta K u = k 2u + ik 2u+1 √ 2 , u ∈ {0, 1} ,(2.7) while the internal pieces, V α 0 β 0 0,int (z 1 , k), V β 0 α 0 0,int (z 2 , −k) , require the introduction of "excited boundary-changing fields" τ u and τ u , u ∈ {3, 4}, which appear in the OPE's ∂Z u (z)σ u (w) ∼ z→w (z − w) − 1 2 τ u (w) + finite , ∂Z u (z)σ u (w) ∼ z→w (z − w) − 1 2 τ u (w) + finite . (2.8) We thus define "external" and "internal" amplitudes to be evaluated on the annulus (Σ = A) and Möbius strip (Σ = M). In terms of various correlators, their expressions are given by A α 0 β 0 extΣ ≡ V α 0 β 0 0,ext (z 1 , k)V β 0 α 0 0,ext (z 2 , −k) Σ = α λ α 0 β 0 λ T β 0 α 0 e ik·X (z 1 )e −ik·X (z 2 ) e i 2 H 3 (z 1 )e − i 2 H 3 (z 2 ) e − i 2 H 4 (z 1 )e i 2 H 4 (z 2 ) × σ 3 (z 1 )σ 3 (z 2 ) σ 4 (z 1 )σ 4 (z 2 ) 1 u=0 \|K u \| 2 e iHu (z 1 )e −iHu (z 2 ) + e −iHu (z 1 )e iHu (z 2 ) ,(2.9) and A α 0 β 0 intΣ ≡ V α 0 β 0 0,int (z 1 , k)V β 0 α 0 0,int (z 2 , −k) Σ = 1 α λ α 0 β 0 λ T β 0 α 0 e ik·X (z 1 )e −ik·X (z 2 ) × e − i 2 H 3 (z 1 )e i 2 H 3 (z 2 ) e − i 2 H 4 (z 1 )e i 2 H 4 (z 2 ) τ 3 (z 1 )τ 3 (z 2 ) σ 4 (z 1 )σ 4 (z 2 ) + e i 2 H 3 (z 1 )e − i 2 H 3 (z 2 ) e i 2 H 4 (z 1 )e − i 2 H 4 (z 2 ) σ 3 (z 1 )σ 3 (z 2 ) τ 4 (z 1 )τ 4 (z 2 ) . (2.10) In these formulas, sums over fermionic spin structures are implicit, while for Σ = A a sum over the second boundary γ is also understood. Notice that it is the internal part of the amplitude, A α 0 β 0 intΣ , that captures the quantum mass we are looking for, while the external part, A α 0 β 0 extΣ , can be used to extract the one-loop correction to the Kähler potential of the scalars in the ND+DN sector. Twist-field correlators at genus-1 The main difficulty in the evaluation of the open-string amplitudes (2.9) and (2.10) resides in the computation of the correlators of boundary-changing fields, which are related to their counterparts for closed strings we now focus on. Ground-state twist fields The OPE's of ∂Z u (z) and ∂Z u (z) with σ u (z), u ∈ {3, 4}, are identical to those with Z 2 -twist fields σ u (z,z) which create the ground state of the closed-string twisted sector of a T 4 /Z 2 orbifold [1]. As a consequence, techniques to evaluate correlators of twist fields [2,3] can be adapted to the case of boundary-changing fields. The correlator of two Z 2 -twist fields on a genus-1 Riemann surface Σ can be written as a sum over instanton contributions. Decomposing the complex coordinates into a classical background and a quantum fluctuation, Z u (z,z) = Z u cl (z,z) + Z u qu (z,z) ,(3.1) we have σ u (z 1 ,z 1 )σ u (z 2 ,z 2 ) = Z u cl e −S Σu cl σ u (z 1 ,z 1 )σ u (z 2 ,z 2 ) qu ,(3.2) where S Σu cl is the classical action of Z u , S Σu cl = i 2πα Σ dz ∧ dz ∂Z u cl∂ Z u cl + ∂Z u cl∂ Z u cl ,(3.3) and σ u σ u qu is the "quantum correlator" arising from quantum fluctuations. The latter have been computed in Ref. [3] for an arbitrary number of twist fields by using the stress-tensor method that we summarize briefly. The starting point is the "Green's function in presence of twist fields" g(z, w) ≡ −∂Z u qu (z) ∂Z u qu (w) A σ u (z A ,z A ) qu α A σ u (z A ,z A ) qu . (3.4) Upon subtracting the double-pole singularity as z → w, this expression can be used to derive the quantity T u (z) ≡ T u (z) A σ u (z A ,z A ) qu A σ u (z A ,z A ) qu ,(3.5) thanks to the OPE of ∂Z qu (z)∂Z qu (w) which involves the stress tensor T u (w). Then, exploiting the OPE of T u (z)σ u (z A ,z A ) as z approaches z A , one finds differential equations ∂ z B ln A σ u (z A ,z A ) qu = lim z→z B (z − z B ) T u (z) − h (z − z B ) ,(3.6) where h = 1/8 is the conformal weight of the twist fields. Solving these equations, one obtains the desired correlators. The key point is that g(z, w), which is needed for applying this program, can be determined by writing the most general function satisfying double periodicity on the torus Σ and with correct behavior dictated by the OPE's as z or w approach some z A . This can be done since functions with these properties form a vectorial space that can be expanded against a basis of so-called "cut differentials," which are holomorphic one-forms on Σ. The coefficients of this expansion can be determined uniquely by imposing that Z u qu (z,z) is doubly periodic. In our case of interest, one obtains [3] σ u (z 1 ,z 1 )σ u (z 2 ,z 2 ) qu = f (τ dc ) (det W ) −1 ϑ 1 (z 1 − z 2 ) − 1 4 ϑ 1 (z 1 − z 2 ) − 1 4 ,(3.7) where f (τ dc ) is an "integration constant" that depends on the Teichmüller parameter τ dc of Σ and ϑ ν , ν ∈ {1, 2, 3, 4}, are the Jacobi modular forms. Moreover, W is a 2 × 2 matrix whose entries are the periods of the cut differential ω(z) = ϑ 1 (z − z 1 ) − 1 2 ϑ 1 (z − z 2 ) − 1 2 ϑ 1 z − z 1 + z 2 2 (3.8) along the cycles γ 1 : z → z + 1 and γ 2 : z → z + τ dc , W = W 1 W 1 W 2 W 2 , W a = γa dz ω , a ∈ {1, 2} . (3.9) Excited twist fields To compute correlators of excited twist fields, we use the OPE's (2.8) to write [18] τ u (z 1 ,z 1 )τ u (z 2 ,z 2 ) qu = lim z→z 1 w→z 2 (z − z 1 ) 1 2 (w − z 2 ) 1 2 ∂Z u (z)∂Z u (w)σ u (z 1 ,z 1 )σ u (z 2 ,z 2 ) qu . (3.10) Using the decomposition (3.1), this expression splits naturally into two pieces τ u (z 1 ,z 1 )τ u (z 2 ,z 2 ) (1) qu = σ u (z 1 ,z 1 )σ u (z 2 ,z 2 ) qu lim z→z 1 w→z 2 (z − z 1 ) 1 2 (w − z 2 ) 1 2 ∂Z u cl (z)∂Z u cl (w) , τ u (z 1 ,z 1 )τ u (z 2 ,z 2 ) (2) qu = lim z→z 1 w→z 2 (z − z 1 ) 1 2 (w − z 2 ) 1 2 ∂Z u qu (z)∂Z u qu (w)σ u (z 1 ,z 1 )σ u (z 2 ,z 2 ) qu . (3.11) To compute part (1) of the correlator, we determine ∂Z u cl (z) and∂Z u cl (z) by using the fact that they must be linear sums of cut differentials whose coefficients imply that Z u cl (z,z) are instanton solutions in the target space S 1 (R 2u )×S 1 (R 2u+1 ). Denoting winding and wrapping numbers as n I , l I ∈ Z, this conditions takes the form γa dz ∂Z u cl (z) + γa dz∂Z u cl (z) = v u a , a ∈ {1, 2} , (3.12) where we have defined v u 1 = 2πR 2u n 2u + 2iπR 2u+1 n 2u+1 √ 2 , v u 2 = 2πR 2u l 2u + 2iπR 2u+1 l 2u+1 √ 2 . (3.13) Using these notations, we obtain that τ u (z 1 ,z 1 )τ u (z 2 ,z 2 ) (1) qu = i (W −1 ) 1 a v u a (W −1 ) 2 bv u b ϑ 1 ( z 1 −z 2 2 ) 2 ϑ 1 (0) ϑ 1 (z 1 − z 2 ) σ u (z 1 ,z 1 )σ u (z 2 ,z 2 ) qu . (3.14) Notice that the determination of ∂Z u cl (z) and∂Z u cl (z) also yield an explicit expression of the action (3.3). Part (2) of the correlator can be expressed in terms of the Green's function (3.4). Using the explicit form of g(z, w) given in Ref. [3], one obtains τ u (z 1 ,z 1 )τ u (z 2 ,z 2 ) (2) qu = −i α σ u (z 1 ,z 1 )σ u (z 2 ,z 2 ) qu ×   C ϑ 1 ( z 1 −z 2 2 ) 2 ϑ 1 (0) ϑ 1 (z 1 − z 2 ) + ϑ 1 (0) F 1 (z 1 , z 2 ) 2 ϑ 1 (z 1 − z 2 )   , (3.15) where we have defined C ϑ 1 ( z 1 −z 2 2 ) 2 ϑ 1 (0) ϑ 1 (z 1 − z 2 ) = − 1 2 ϑ 1 (0) ϑ 1 z 1 − z 2 2 (W −1 ) 1 a γa dz F 1 (z, z 2 ) ϑ 1 (z − z 1 ) 1 2 ϑ 1 (z − z 2 ) 3 2 . (3.16) In these expressions, the function F 1 (z, w) is given by F 1 (z, w) = ϑ 1 (z − w) ϑ 1 (U 1 ) ϑ 1 z − w + z 2 −z 1 2 − U 1 ϑ 1 z 2 −z 1 2 − U 1 , (3.17) where U 1 is such that ∂ z F 1 (z, w) z=w = 0. Full open string amplitudes The instanton action and correlators for a genus-1 worldsheet can be used S Σ cl = π[(R 4 l 4 ) 2 + (R 5 l 5 ) 2 ] α τ 2 + \|W 1 \| 2 4πα Im(W 1 W 2 ) ×                    4 u=3 \|v u 2 \| 2 for NN and N , where we have defined for u ∈ {3, 4} v u 2 = 2πR 2u l 2u + 2iπR 2u+1 l 2u+1 √ 2 ,ṽ u 2 = 2π α R 2ul 2u + 2iπ α R 2u+1l 2u+1 √ 2 . (4.2) Second, the method of images can be applied during the course of the derivation of the ground-state twist-field correlators to obtain those of "ground-state boundary-changing fields." The result amounts to take the "square root" of the closed-string result [18] σ u (z 1 )σ u (z 2 ) qu = f op (τ dc ) (det W ) − 1 2 ϑ 1 (z 1 − z 2 ) − 1 4 ,(4.3) where f op (τ dc ) depends on the Teichmüller parameter of the double-cover torus. The twist fields τ u (z,z), τ u (z,z) being excited only on their holomorphic sides, their correlators are identical to those of "excited boundary-changing fields", up to an extra division by 2 of part (1) in Eq. (3.14) which follows from the same rescaling of the instanton action. Gathering all results and denoting z 12 ≡ z 1 − z 2 , we find the following expression for the external part of the amplitudes A α 0 β 0 extΣ = α k 2 λ α 0 β 0 λ T β 0 α 0 ϑ 1 (z 12 ) ϑ 1 (0) e − 2π τ 2 [Im(z 12 )] 2 −2α k 2 1 det W ϑ 1 (z 12 ) 2 × νext =1 ϑ 1 (0) ϑ νext (z 12 ) ϑ νext (0) ν int (−1) δ ν int ,1 ϑ ν int z 12 2 2 l e − π α τ 2 I (R I l I ) 2 ×   l e − \|W 1 \| 2 (\|v 3 2 \| 2 +\|v 4 2 \| 2 ) 4πα Im (W 1 W 2 ) C Σ l l ν int + l e − \|W 1 \| 2 (\|ṽ 3 2 \| 2 +\|ṽ 4 2 \| 2 ) 4πα Im (W 1 W 2 )C Σ l l ν int   ,(4.4) while for the internal parts we obtain A α 0 β 0 intΣ = − s i α λ α 0 β 0 λ T β 0 α 0 ϑ 1 (z 12 ) ϑ 1 (0) e − 2π τ 2 [Im(z 12 )] 2 −2α k 2 ϑ 1 ( z 12 2 ) 2 det W ϑ 1 (z 12 ) 2 ϑ 1 (0) × 4 ν int ϑ ν int z 12 2 2 l e − π α τ 2 I (R I l I ) 2 ×    l e − \|W 1 \| 2 (\|v 3 2 \| 2 +\|v 4 2 \| 2 ) 4πα Im (W 1 W 2 ) C Σ l l ν int   W 2 1 (\|v 3 2 \| 2 + \|v 4 2 \| 2 ) 8[Im (W 1 W 2 )] 2 + 2α (C +Ĉ)   + l e − \|W 1 \| 2 (\|ṽ 3 2 \| 2 +\|ṽ 4 2 \| 2 ) 4πα Im (W 1 W 2 )C Σ l l ν int   W 2 1 (\|ṽ 3 2 \| 2 + \|ṽ 4 2 \| 2 ) 8[Im (W 1 W 2 )] 2 + 2α (C +Ĉ)      , (4.5) whereĈ ≡ ϑ 1 (0) 2 2 ϑ 1 ( z 12 2 ) 2 F 1 (z 1 , z 2 ) . (4.6) In these formulas, ν ext , ν int ∈ {1, 2, 3, 4} label the spin structures of the worldsheet fermions Ψ 0 , Ψ 1 , Ψ 2 and Ψ 3 , Ψ 4 , respectively. Moreover, we have introduced normalization functions C Σ l l ν int andC Σ l l ν int , which are the τ dc -dependent "integration constants" arising in the stresstensor method applied for the correlators of the boundary-changing field and spin-fields. They can be identified by taking the limit z 12 → 0 in the external parts of the amplitudes which then reduce to sectors of the known one-loop partition function. In the annulus case, the results are C A l l 1 = C τ 2 2 η 3 f A l l α 0 D ,C A l l 1 = C τ 2 2 η 3 f A l l β 0 N , C A l l 2 = C τ 2 2 η 3 ϑ 2 3 ϑ 2 4 f A l l α 0 D − C ϑ 2 2 τ 4 2 η 9 f A l l α 0 N e 2iπ l · a S ,C A l l 2 = C τ 2 2 η 3 ϑ 2 3 ϑ 2 4 f A l l β 0 N − C ϑ 2 2 τ 4 2 η 9 f A l l β 0 D e 2iπ l · a S , C A l l 3 = C ϑ 2 3 τ 4 2 η 9 f A l l α 0 N − C τ 2 2 η 3 ϑ 2 2 ϑ 2 4 f A l l α 0 D e 2iπ l · a S ,C A l l 3 = C ϑ 2 3 τ 4 2 η 9 f A l l β 0 D − C τ 2 2 η 3 ϑ 2 2 ϑ 2 4 f A l l β 0 N e 2iπ l · a S ,C A l l 4 = − C ϑ 2 4 τ 4 2 η 9 f A l l α 0 N ,C A l l 4 = − C ϑ 2 4 τ 4 2 η 9 f A l l β 0 D ,(4.7) where C is a numerical factor, a S is the two-vector whose components are (a 4 S , a 5 S ) = (0, 1 2 ) and f A l l α 0 N = v 2 v 4 α 3 i,i 2n ii e 2iπ l·( a i 0 − a i ) e 2iπ l ·( a i 0 − a i ) , f A l l β 0 D = v 2 α 2 α v 4 i,i 2d ii e 2iπ l ·( a j 0 − a i ) e 2iπ l ·( a i 0 − a i ) , f A l l α 0 D = δ l, 0 v 2 α i,i 2d ii e 2iπ l ·( a i 0 − a i ) , f A l l β 0 N = δ l , 0 v 2 α i,i 2n ii e 2iπ l ·( a i 0 − a i ) . (4.8) In the above definitions, v 2 ≡ R 4 R 5 and v 4 ≡ R 6 R 7 R 8 R 9 , while a i , a i are respectively two-and four-vectors parametrizing the positions of the fixed points i and i in the T-dual pictures: 2πR I a I i ∈ {0, πR I }, 2πR I a I i ∈ {0, πR I }. For the Möbius strip, the normalization functions are C M l l 1 = 0 ,C M l l 1 = 0 , C M l l 2 = C ϑ 2 2 τ 4 2 η 9 v 2 v 4 α 3 e 2iπ l · a S ,C M l l 2 = C ϑ 2 2 τ 4 2 η 9 v 2 α 2 α v 4 e 2iπ l · a S , C M l l 3 = − C ϑ 2 3 τ 4 2 η 9 v 2 v 4 α 3 ,C M l l 3 = − C ϑ 2 3 τ 4 2 η 9 v 2 α 2 α v 4 , C M l l 4 = C ϑ 2 4 τ 4 2 η 9 v 2 v 4 α 3 ,C M l l 4 = C ϑ 2 4 τ 4 2 η 9 v 2 α 2 α v 4 . (4.9) Squared masses in the limit of small M 3/2 In principle, the squared masses of the moduli arising from the ND+DN sector can be extracted from the internal parts of the amplitudes. However, all expressions being so far complicated, it is illuminating to derive the masses in the regime where the scale of supersymmetry breaking is lower than all other mass scales of the classical spectrum. This can be done by considering a field theory limit α → 0 which implies all stringoscillator states to be supermassive. To this end, the imaginary parts of the Teichmüller parameters of the double-covering tori are rescaled, Im τ dc ≡ τ 2 2 ≡ t 2πα 1 , where t ∈ (0, +∞) . For all compactification scales to be greater than 1/R 5 , we also rescale all radii and T-dual radii other than R 5 as follows, R 4 = r 4 √ α 1 , R I = r I √ α 1 , α R I = √ α r I 1 , r 4 , r I finite . (5.3) In this regime, the light states of the theory are the Kaluza-Klein modes propagating in the Scherk-Schwarz direction X 5 . Before taking the small α limit in Eq. (4.5), it is convenient to apply a Poisson summation over the indices l 4 , l I andl I . The reason is that the dominant contributions arise from zero modes of the new lattices, while all other terms are exponentially suppressed. In the brackets appearing in the expression of A α 0 β 0 intΣ , the terms involving v u 2 ,ṽ u 2 and those proportional to C +Ĉ arise respectively from part (1) and part (2) the 32 D5-branes become 32 D3-branes transverse toT 2 × T 4 /Z 2 . Their positions in this internal space parametrize the expectation values of all the moduli fields arising from the DD sector of the initial theory. Similarly, denotingT 4 the four-torus of coordinatesX I Tdual to T 4 , one obtains by T-dualizing all six internal directions a description with internal spaceT 2 ×T 4 /Z 2 modded by (X I ,X I ) → (−X I , −X I ). There is one O3-plane at each of the 64 fixed points and 32 D3-branes T-dual to the initial D9-branes. Their positions iñ T 2 ×T 4 /Z 2 parametrize all NN-sector moduli expectation values of the original description. The backgrounds we are interested in correspond to distributing all D3-branes T-dual to the D5-branes or D9-branes on the fixed points of their respective internal spaces. In that case, the gauge symmetries supported by the stacks of D3-branes are unitary. Labelling the 64 fixed points of each T-dual picture by two indices ii , where i ∈ {1, . . . , 16} for those of T 4 /Z 2 orT 4 /Z 2 and i ∈ {1, 2, 3, 4} for those associated with theT 2 directions, the brane configurations are characterized by the numbers 2n ii and 2d ii of D3-branes T-dual to D9and D5-branes located at the fixed points ii . In the following, we compute the two-point functions of the massless scalars in the ND+DN sector, which are in the bifundamental representation of some U (n i 0 i 0 ) × U (d j 0 i 0 ) for some given i 0 , j 0 and i 0 . 2 Figure 1 : 1Open-string diagrams with two external legs in the ND and DN sectors (left panel). On the double-cover tori (right panel), the external legs are mapped to boundary-changing vertex operators at positions z 1 , z 2 . all, open-string instantons exist for worldsheets with NN, DD, ND or DN boundary conditions in the annulus case, and N or D boundary conditions for the Möbius strip. In the NN and N case, all winding numbers n I , n I must vanish, while for DD or D boundary conditions, denoting with tildes the T-dual winding and wrapping numbers,ñ I andñ I must also be zero. For ND and DN boundary conditions, instantons can only wrap T 2 . Dividing by 2 the double-cover worldsheet classical action (3.3) for u ∈ {2, 3, 4}, one obtains for Σ ∈ {A, M} the full open-string instanton actions the imaginary parts of the insertion points can be redefined asIm z A = u A Im τ dc = t u A 2πα 1 , u A ∈ (0, 1) , A ∈ {1, 2} . (5.2) of the correlator of excited boundarychanging fields. They both yield contributions of the same order of magnitude in the limit of small α . Integrating over the proper times t of the diagrams, as well as over the position of one insertion point along one boundary of the annulus and the single boundary of the Möbius strip, one obtains the squared mass [13] λλ T )\|2l 5 + 1\| 3 M 2 3/2 (n i 0 i 0 − n i 0î 0 − 1 + d j 0 i 0 − d j 0î 0 − 1) + O(α M 4 3/2 ) . (5.4)In this formula,î 0 denotes the fixed point in the T-dual pictures whose coordinates satisfy 1. As expected, the final answer in the regime where M 3/2 is lower than all other mass scales of the spectrum is dominated by the contributions of the infinite towers of Kaluza-Klein states in 4 + 1 dimensions running in the loop. 1 ) 1The spacetime coordinates are labeled by Greek indices µ ∈ {0, . . . , 3}, while the T 2 directions are denoted by primed Latin indices I ∈ {4, 5}. 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[ "Automated Screening for Distress: A Perspective for the Future", "Automated Screening for Distress: A Perspective for the Future" ]	[ "Rajib Rana \nUniversity of Southern Queensland\nAustralia\n", "Siddique Latif \nUniversity of Southern Queensland\nAustralia\n", "Raj Gururajan \nUniversity of Southern Queensland\nAustralia\n", "Anthony Gray \nUniversity of Southern Queensland\nAustralia\n", "Geraldine Mackenzie \nUniversity of Southern Queensland\nAustralia\n", "Gerald Humphris \nUniversity of St Andrews\nUnited Kingdom\n", "Jeff Dunn \nUniversity of Southern Queensland\nAustralia\n\nGriffith University\nAustralia\n\nUniversity of Technology Sydney\nAustralia\n" ]	[ "University of Southern Queensland\nAustralia", "University of Southern Queensland\nAustralia", "University of Southern Queensland\nAustralia", "University of Southern Queensland\nAustralia", "University of Southern Queensland\nAustralia", "University of St Andrews\nUnited Kingdom", "University of Southern Queensland\nAustralia", "Griffith University\nAustralia", "University of Technology Sydney\nAustralia" ]	[]	Distress is a complex condition which affects a significant percentage of cancer patients and may lead to depression, anxiety, sadness, suicide and other forms of psychological morbidity. Compelling evidence supports screening for distress as a means of facilitating early intervention and subsequent improvements in psychological well-being and overall quality of life. Nevertheless, despite the existence of evidence based and easily administered screening tools, for example the Distress Thermometer, routine screening for distress is yet to achieve widespread implementation. Efforts are intensifying to utilise innovative, cost effective methods now available through emerging technologies in the informatics and computational arenas.	10.1111/ecc.13033	[ "https://arxiv.org/pdf/1902.09944v1.pdf" ]	67,855,992	1902.09944	f222862c3c89edb1d1172d6de0f82d299d9b41cd	Automated Screening for Distress: A Perspective for the Future 22 Feb 2019 Rajib Rana University of Southern Queensland Australia Siddique Latif University of Southern Queensland Australia Raj Gururajan University of Southern Queensland Australia Anthony Gray University of Southern Queensland Australia Geraldine Mackenzie University of Southern Queensland Australia Gerald Humphris University of St Andrews United Kingdom Jeff Dunn University of Southern Queensland Australia Griffith University Australia University of Technology Sydney Australia Automated Screening for Distress: A Perspective for the Future 22 Feb 20191 Distress is a complex condition which affects a significant percentage of cancer patients and may lead to depression, anxiety, sadness, suicide and other forms of psychological morbidity. Compelling evidence supports screening for distress as a means of facilitating early intervention and subsequent improvements in psychological well-being and overall quality of life. Nevertheless, despite the existence of evidence based and easily administered screening tools, for example the Distress Thermometer, routine screening for distress is yet to achieve widespread implementation. Efforts are intensifying to utilise innovative, cost effective methods now available through emerging technologies in the informatics and computational arenas. I. INTRODUCTION Distress is described as emotional suffering with high global prevalence, which can result in disabling conditions and impairment for patients. It is highly prevalent in cancer patients affecting 25 to 60% of patients [1] and can cause severe harm to this cohort with diminished Quality of Life (QoL) is one of the key adverse effects of distress [1]. Other implications include shortened survival time [2] and negative outcomes for physical health through impaired immune functioning [3]. Distress may also negatively influence treatment adherence in cancer patients and non-adherence substantially increases healthcare costs through increased likelihood of recurrence and other disease complications [4]. Overall, distress is estimated to increase the cost of cancer care by as much as 20% [5], [6]. Early assessment and screening for distress will enable timely management of distress leading to (1) improved adherence to treatment, (2) more effective communication between patient and clinician, (3) fewer visits to the hospital, and (4) early intervention for and prevention of severe anxiety or depression [7]. This is why International organisations have endorsed the quality care standard of whole-patient care that is achieved through routine comprehensive distress screening [8], however, routine screening for distress has yet to be widely adopted [9]. Reasons for limited system wide update of screening for distress are reported to include time constraints, limitations in the skills of health care providers, cost, and attitudes of health care providers to the use of standardised tools [1], [5], [8], [9]. Automating screening for distress with the assistance of emerging technologies may serve to alleviate many of the challenges associated with its routine applications. However, there are also a number of challenges need to be addressed to develop a robust automated distress detection system. In this paper, we aim to provide critiques on distress detection research in terms of assessment tools, available datasets and existing methods for automatic distress screening. A number of studies [10]- [12] have reviewed the research on automatic depression detection but none of them has highlighted the research gap for distress detection. This study attempts to highlight the core differences of distress from other emotions and mental disorders, discuss existing methodologies for inferring distress and provide future directions for developing an automated distress system. II. WHAT IS DISTRESS Distress is considered as a continuum of psychological symptoms with varying severity [13]. In the forensic sciences the term "distress" is specifically used for the affective states that arise in violent situations. The National Comprehensive Cancer Network provides the widely accepted definition of distress [7]: Distress is a multifactorial unpleasant emotional experience of a psychological (cognitive, behavioural, emotional), social, and/or spiritual nature. Distress encompass a range of common feelings of vulnerability, sadness, and fears that can cause depression, anxiety, panic, social isolation, and existential and spiritual crisis. Distress may not always be caused by some unexpected external events, but can also be caused by internal states such as feelings, thoughts, and habitual behaviours [14], [15]. It is an uncomfortable feeling and can impact individuals' capacity of working, social life, bodily part and the mind. It is a subjective experience, and different people manifest it differently with varied range of symptoms. However, the most common symptoms [16] of distress are: sleep disturbances, memory problems, anger management issues, obsessive thoughts, fatigue, sadness, weight gain, hallucinations, delusions, etc. A. How Distress is different from Emotions Emotion is an essential component of human life and plays important role for their survival [17]. As a human being, we feel a whole range of emotions that may be comfortable or uncomfortable [18], [19]. Emotional discomfort is a universal human experience. In fact, negative emotions including sadness, anger and fear are important and useful in various situations. For instance, fear is helpful when there is real threat to our safety (for e.g., gun pointed at us or wild ferocious animal in the vicinity) and helps humans to effectively withstand such threatening situations. Similarly, sadness inadvertently helps in spotlighting the things that we care about in our life and it is important reinforce that negative emotions are not necessarily distress [20], for example: disgust [21]. In our daily life, emotions are transient [22] and they fluctuate like waves as they plateau, subside and eventually pass. In other words, emotions are transient, continually moving and changing. In contrast, distress is a prevailing situation that, if not addressed, escalates until emotional combust [20]. Emotions such as fear or anger are aroused to prevent, solve, cope with, or get away from specific situation. Distress is different, it can be felt strongly, it compromises a person's ability to cope and if left untreated may escalate to more serious conditions [23]. B. How Distress is different from Stress Often stress and distress are used interchangeably, which blurs and confuses the distinctions between these concepts. It is however important to distinguish these two terms. Stress is an important element of life, as it has both positive and negative effect. As pointed by Spielberger [24], "Stress is an integral part of the natural fabric of life, and coping with stress is an everyday requirement for normal human growth and development". The body uses behavioural or physiological mechanisms to counter the perturbation caused by stress and come back to normality. People usually adapt stress but when this adaptive process is compromised stress may develop into distress. Stress may present as either chronic and acute [25] and any transition of stress to distress depends on various factors including duration, intensity, and controllability. C. How Distress is different from Depression Other dimensions of psychopathology such as depression and anxiety are also closely related to distress. In particular, most assessment tools and the consequent treatment of distress is based on the depression symptoms [26]. Patients with depression need to meet at least five of the DSM-5 (Diagnostic and Statistical Manual of Mental Disorder, Fifth Edition) criterion for major depressive disorder nearly every day during a two-week period. However, distress has different symptoms such as poor self-management, feeling angry and scared, and feeling of unsupported by family and friends [27], which are not included in DSM-5. This suggests the need to formulate an alternative screening for distressed people, who are not clinically depressed. III. ASSESSMENT SCALES Distress remains undetected in most patients [9], however, surprisingly, there are many scales available to gauge distress. In this section we present the most popular scales used to screen distress. We also present (see Table I) the number of questions/items in each scale and time to conduct the screening to indicate the complexity of each scale. The Disability Distress Assessment Tool (DisDAT) [36] was designed by a palliative care team to assess distress and is not a scoring tool, rather it documents a wide range of behaviours and signs related to distress. A distress scale based on ten symptoms was designed by Mccorkle et al. [37]. This scale was tested on 53 patients, where distress score was ranged from 10-41. The Distress Thermometer [38], is another scale which enables patients to rate their distress level on visual scale ranging from 0 (no distress) to 10 (extreme distress). The SCL-90 (Symptom Checklist-90) and BSI (Brief Symptom Inventory) have been widely used for screening of psychological distress in medical patients and demonstrated high levels of specificity and sensitivity [13], [39]. The 12item General Health Questionnaire (GHQ-12) is designed to study of psychological disorders in general clinical setting and has been used in various studies [40], [41]. A recently proposed scale for distress assessment is the K10 [42]. It is a a 10-item scale specifically designed to assess distress in population surveys. This scale evaluates the individuals on anxio-depressive symptoms over the last 30 days and provides a total score as an index of distress. The Functional Assessment of Chronic Illness Therapy (FACIT) Measurement System [30] is used for the management of chronic illness using questionnaires related to health-related quality of life. Its generic version known as the Functional Assessment of Cancer Therapy-General (FACT-G) is compiled to use in four primary quality of life domains including physical well-being, social/family well-being, emotional well-being, and functional well-being. A six-item sub-scale of Somatic and Psychological Health Report (SPHERE-12) measures the aspects of distress and related conditions [43]. This scale is based on GHQ [29] and each item is scored on a three-point scale between 0 and 2, which gives a maximum score of 12. A number of scales for depression are also used to scale distress. Hospital Anxiety and Depression Scale (HADS) is a screening instrument that is used to assess anxiety and depression of physically ill patients [34]. It includes 14 items for anxiety and depression with 4 alternative answers, which are used to measure total distress score. Self-report scales including Beck's Depression Inventory (BDI) [33], and Patient Health Questionnaire -Anxiety and Depression Scale (PHQ-ADS) [44] have also been shown to have some relevance with distress for particular patient groups. IV. AUTOMATIC DISTRESS ASSESSMENT Distress is highly prevalent in patients with chronic disease. Despite the fact that it can cause serious harm, clinicians are reluctant to use the existing distress screening for various reasons, most importantly for cost and time requirements [45]. Emerging information technologies are playing promising role to automate the screening of different health issues [46] and they also have great potential to be exploited for the automated screening of distress that may greatly alleviate these problems and facilitate widespread update. The potential benefit [28] Psychological distress 18/53 3-7 GHQ-12 [29] Psychological disorders 12 3-7 FACT-G [30] Quality of life 21 < 15 Distress Thermometer [31] Distress 37 -SCL-90 [32] Psychiatric disorders 90 >20 BDI [33] Depression 21 5-10 HADS [34] Depression/Anxiety 14 2-5 PHQ-9 [35] Depression 9 <5 of automated screening for distress has encouraged research efforts and we discuss progress in this section. Besides the application in health, automated distress detection has also been studied in two other areas and these are Aged Care and Forensics. In homes for elderly people distress calls arise if there is fall or a fire or other such events [47]. In the forensic scenario, automated distress assists the Police prioritise the crime response based on the intensity of distress of the caller [48]. Also, automated distress detection can assist the forensic phoneticians by providing them an objective measure of distress of victims in recorded attacks. In this section, we discuss the methodologies used in these three sectors. A. Health For automated distress detection in health, most of the studies focused on distress related conditions such as depression, anxiety, PTSD, and suicidal behaviour; very few studies [49], [50] have reported their results on distress detection. For instance, an automated distress management system [51] is piloted in outpatient medical oncology practice using tablet or computer for tailored psychosocial coping recommendations or referrals to individuals after immediate analysis. The authors used Distress Thermometer and problem list proposed by National Comprehensive Cancer Network as a screening tool. Their system matches patients identified concerns with the problem list and proposes evidence-based treatment suggestions and referrals. Verona coding definitions of emotional sequences (VR-CoDES) was developed for the detection and categorisation of patients' emotions and their corresponding healthcare physicians [52]. Different studies have exploited VR-CoDES [53]- [55], however, the need for training of researchers on its usage and skilled labour necessary for labelling consultation recording are its major practical limitations. In this regard, Birkett et al. [56] developed computer-based tools to assist VR-CoDES in the labelling of patients-physicians' recordings. The authors tried different representations of patients' utterances and evaluated wellknown classifiers including naïve Bayes, logistic regressions, support vector machines, and boosted ensemble decision trees for the labelling of recordings as an explicit concern, an emotional cue, or neither. Researchers are predominantly attempting to infer distress based on the after effect such as depression, anxiety, PTSD, and suicidality, have developed various techniques. In [57], authors analysed 33 individuals from a clinical trial of depres-sion [58] and investigated the relationship between nonverbal behaviour and severity of depression using video recording over the course of treatment. Scherer at al. [59] evaluated different visual features for psychological disorder analysis. They found that depressed individuals tend to gaze downwards more, give less intense and shorter duration of smile, and show longer self-touches and fidgeting. The inclusion of gender information with the visual is found to be helpful in detecting of distress related situations [60]. In addition to the visual indicators, Space-Time Interest Points (STIP) features are also exploited to detect depression with significantly improved results [61], [62]. These features include gestures related to head, face, shoulder, hands movements. Recent studies have shown the promise of using speech as an effective marker for diagnosis and monitoring of depression. Speech can provide a wide range of prosodic and spectral features that can be effectively being used for human emotion [63], [64] and depression detection. Many researchers have used speech as an objective indicator for the detection of depression [65]- [67]. An interactive voice response (IVR) system was used to collect speech samples for automated HAM-D measures of depression severity [68]- [71]. Acoustic features such as spectral, prosodic, cepstral, glottal, and features obtained from Teager energy operators (TEO) were investigated for clinical depression detection in adolescents [72]. TEO based features were produced more promising results compared to all other features and their combinations. Other studies [73]- [76] also investigated different acoustic features and identified more relevant identifier for depression. Ozdas et al. [77] studied excitation related speech parameters including glottal flow spectrum and vocal jitter for identification of major depressed, high-risk near-term suicidal, and non-suicidal patients. Vocal jitter was found a significant discriminator clue suicidal and non-depressed control, where glottal flow spectrum related parameters provided discrimination of all three groups with significantly improved results. Scherer et al. [67] used prosody and voice quality related speech parameters for identification of suicidal and non-suicidal adolescents. They found that suicidal adolescents tend to have more breathy voice qualities compared to nonsuicidal. A comparative study performed in [78] using acoustic and prosodic features to detect depression in spontaneous speech. Authors found that voice features such as intensity, root mean square, and loudness performed best to detect depression in the dataset. Other studies (for example [79]- [83]) also exploited different machine learning techniques and suggested that the speech can be effectively utilised to detect distress and related conditions. B. Aged Care Life expectancy is increasing globally, leading us to a higher number of older people in our society [84]. This increasing share of the elderly population is in part responsible for a shift in the cause of death from infectious and parasitic illnesses to chronic non-communicable diseases [85], [86]. Ageing can lead to physical limitations that need to be compensated by technical assistance and the help of aged care services. In aged care residential communities, feeling of isolation, fear, and a sense of helplessness, such as an inability to perform routine tasks, may lead to distress [87], [88]. Distress in elderly people often goes unrecognised for a range of reasons including confusing or unknown symptoms of distress [89], avoidance from checkups [90], and lack of systematic method or tool for distress detection [91]. The early detection and treatment of distress among elderly people is important because it can enhance recovery from illness and improve overall quality of life [89]. There exist different innovative products and solutions which promote independence and better quality of life among seniors with physical or cognitive diseases, for instance, the CIRDO project [92] aims to automatically detect the situations of falls and distress in residential care to promote autonomy for elderly people. This system involves video and audio analysis to detect the risky situation and make necessary emergency call using e-lio system 1 For distress detection, CIRDO evaluated the proposed system using Automatic Speech Recognition (ASR) to detect distress sentences in AS80 [47] corpus and achieved promising results. The SweetHome project [93] used home equipped noise robust multisource automatic speech recognition (ASR) to detect vocal command or distress sentences in the realistic noisy environment of a smart home. Twenty three subjects or "speakers", participated in this experiment where the closest distance between speakers and microphone was two meters. The authors performed voice order recognition of speech command belonging to three classes: distress calls, neutral sentences, and home automation orders. Alternatively, a sound based surveillance system [94] to detect alarming sounds in home situation has been described. This system performed real-time audio analysis for the detection of distress situation without compromising patients' privacy. Distress detection in elders using ASR system is a very challenging task due age-related degeneration of vocal cords, problems of laryngeal cartilages, and changes in larynx muscles [95], [96]. Some studies have empirically shown that ASR models performed poorly on elderly voice when they are trained on young or middle-aged adult speech [97], [98]. For such situations, speaker adaptation techniques or training ASR model on elderly voice can help improvement in recognition rate [99]. To explore the performance of ASR in distress situation, Aman et al. [47] presented word error rate in aged voice compared to non-aged speech. They showed that ASR 1 http://www.technosens.fr/. system gives higher word error rate equal to equal to 43.5% for the aged group and 9% on young speakers. C. Forensic Distress detection has an increasing presence in forensics, particularly in informing opinions about the authenticity of distress in criminal investigations. In forensic investigation, the lie can occur from distortion, denial, evasion, concealment, and outright fabrication by people to appear non-accountable for their exertions [100] and distress surveillance systems are used to identify the presence of reliable emotional clues to detect malingering or deception. Forensic examinations are performed by psychologists using different techniques including interviews, observations, home or institutional visits, psychological tests and instruments, as well as other methods recognised by the Forensic Council [101]. Automatic distress detection can play a crucial role in the assisting the forensic examination practice with an objective measure that can assist the judicial authorities. A comprehensive study was performed by Lisa [102] to investigate distress in speech using acoustic and perceptual cues and empirically compare the results for real-life victims and actors in life-threatening situations. Based on the results of the acoustic analysis, it is concluded that acoustic parameters can be utilised to detect distress situations for actors and victims. In another study, Lisa [103] reported that two acoustic parameters intensity and formant bandwidth are helpful in differentiating between acted and genuine victims' speech. Similarly, Fundamental Frequency (F0) mean, range and vowel formant can be used to distinguish between baseline and distress conditions for both victims and actors. V. DISTRESS DATASETS Development of automated systems require historical data that contains the correlation of physical properties such as speech, facial expressions with distress labels. In this section we discuss various datasets that have been used previously for the purpose of distress identification are identified and discussed. As depression is a possible after effect of distress most of these datasets are built to diagnose depression. This section concludes with a summary of the sector wise studies (health, age care, and forensic) with the datasets used within, in Table II. A. Distress Assessment Interview Corpus (DAIC) The Distress Analysis Interview Corpus (DAIC) [168] includes semi-structured clinical interviews of participants to enable the diagnosis of psychological distress conditions such as depression, anxiety, and post-traumatic stress disorder. The interviews of participants were conducted by humans, human controlled agents and autonomous agents. Overall data consists of audio, video, and questionnaire responses of participants and each interview is labelled with a depression score using PHQ-9. A portion of this dataset was released in Audio/Visual Emotion Recognition Depression Sub-challenge (AVEC) [169] 2016, which also contains transcription of the interviews. [113], [116] Forensic Self Generated by Author Distressed/non-distressed speech Speech Acoustic Analysis Distressed/non-distressed speech detection [102], [103] PCL-C, PHQ-9 Depression and PTSD [67], [117], [ B. Aged and Non-Aged Corpus (AS80) This corpus was recorded for adaptation of standard Automatic Speech Recognition (ASR) system to aged voice [47]. This corpus contains recording of from 95 speakers who were asked to read distress and casual sentences. These sentences contain a list of home automation orders and of distress calls that could be uttered by an elderly person in distress or fall situations. C. AVEC Corpus 2013 This dataset contains 340 video recordings of subjects performing a Human-Computer Interaction tasks [148]. There were total 292 speakers and the length of each recorded video clip is between 20 to 50 minutes. The level of depression for recordings was labelled using Beck Depression Inventory (BDI-II) [170]. The AVEC corpus 2014 [171] is a portion of this dataset which contains 300 video with the duration from 6 seconds to 4 minutes. D. SDC (suicidal, depressed, and control subjects) This database is the collection of different dataset. Suicidal corpus was collected from the existing datasets [172] that was recorded from phone conversations, treatment sessions, and suicide notes. Depression related samples were obtained from Vanderbilt II and depression dataset used by Hollon et al [173]. DSM-IV and, ICD-9-CM (International Classification of Diseases, ninth edition, Clinical Modification) criteria were used for depressed patients. For the control group sample, Vanderbilt II dataset was used. E. Pitt Depression Dataset This is a clinically validated depression dataset collected during the treatment of depressed patients at University of Pittsburgh (Pitt) [174]. All participants from a clinical trial were met with DSM-IV criteria for major depression. Total 57 patients were accessed using the HRSD clinical interview for depression severity. Interviews were recorded in audio-video format and depression was evaluated by the clinicians. F. Black Dog dataset This audio-visual dataset was recorded by the Black Dog Institute Australia [175]. Over 40 depressed individuals (both male and female) were interviewed and asked to read sentences. Audio-video recordings of subjects include selfdirected speech, related facial expressions, and body language. G. Cincinnati Children's Interview Corpus (CCIC) This dataset [67] includes the interview of 60 children patients (average age 15.47 years) at the Emergency Department of Cincinnati Children's Hospital Medical Center. These children came to the hospital due to suicidal ideation, gestures, and attempts. Data was collected by a professional social worker. Due to lengthy interviews of suicidal and non-suicidal patients, only 60 seconds of speech for each participant is utilised for the analysis [176]. VI. DISCUSSIONS A search of the literature for research focused on the application of automation in screening for distress found most of the papers in the health area utilizing post-distress conditions including depression, anxiety, and Post Traumatic Stress Disorder (PTSD) as a proxy to determine the presence of distress, rather than screen for distress itself. More broadly, it is evident from Table II where a summary of 75 studies relating to automated approaches to screening for distress in the aged care, forensic and health care settings is presented, that the focus is mostly on depression and PTSD. These studies have statistically analysed depression, anxiety, and PTSD against distress and reported that distress is highly correlated with these measuring dimensions [177]- [179]. Only two studies were found, refer Table II, [59], [122] targeted distress specifically using automated methods. However, even these two studies statistically correlated depression, anxiety, and PTSD to categorise (high, low, unclear) distress. In cancer care, this approach, where other conditions such as depression, are used as a proxy or signal for distress has limitations as as patients with distress may not have depression when measured with the existing scales [180] and as such may not be detected. Future research needs to focus on automated approaches to screening for distress in cancer patients which are independent from other related conditions (such as anxiety and depression) and are designed specifically to identify symptoms associated with distress. Guidance can be found in promising work from the Forensic [103] and Health [52] areas where it has been shown that speech independently carries latent properties for inferring distress. Existing tools to screen for distress vary in relation to complexity, have been criticised as being costly, in terms of time and resources, and have failed to attract widespread or routine implementation, refer Table I. Evidence based automated approaches which efficiently and effectively screen for distress without adding to patient or staff burden may well be the future of screening for distress. Such an approach would triage high distress individuals for the attention of professional staff for further assessment or referral, consistent with a tiered model of care approach [181]. Currently, a number of distress screening tools are available, but as we report in Table I, most of these tools have many questions and require a considerable amount of time to complete. More importantly, it has been found that screening patients with multiple scales can appreciably improve the accuracy of results compared to single scale [182]. However, such multiscale approach will further increase the screening time. Due to busy practices, oncologists are already reluctant to use distress screening tools, so a further increase in screen time will not be welcome by the oncology practices. Moreover, for aged care, and forensic scenarios, real-time distress inference is sought, so the time taking screening techniques will not be very useful. An automated distress detection/screening is therefore inevitable. Datasets will play a vital role to develop an independent and automated distress detection system. From Section V, we found that most of the available datasets are recorded for depression, anxiety, and assessment of suicidal behaviour. Very few datasets such as AD80 is designed for distress in the elderly population. 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[ "Heuristics for the Variable Sized Bin Packing Problem Using a Hybrid P System and CUDA Architecture", "Heuristics for the Variable Sized Bin Packing Problem Using a Hybrid P System and CUDA Architecture" ]	[ "Qadha'a Alenezi \nComputer Science Department\nKuwait University\nKuwait\n", "Hosam Aboelfotoh \nComputer Science Department\nKuwait University\nKuwait\n\nComputer Science Department\nKuwait University\nKuwait\n", "Bader Albdaiwi [email protected] \nComputer Science Department\nKuwait University\nKuwait\n", "Mohammad Ali Almulla [email protected] \nComputer Science Department\nKuwait University\nKuwait\n\nComputer Science Department\nKuwait University\nKuwait\n" ]	[ "Computer Science Department\nKuwait University\nKuwait", "Computer Science Department\nKuwait University\nKuwait", "Computer Science Department\nKuwait University\nKuwait", "Computer Science Department\nKuwait University\nKuwait", "Computer Science Department\nKuwait University\nKuwait", "Computer Science Department\nKuwait University\nKuwait" ]	[]	The Variable Sized Bin Packing Problem has a wide range of application areas including packing, scheduling, and manufacturing. Given a list of items and variable sized bin types, the objective is to minimize the total size of the used bins. This problem is known to be NP-hard.In this article, we present two new heuristics for solving the problem using a new variation of P systems with active membranes, which we call a hybrid P system, implemented in CUDA. Our hybrid P-system model allows using the polarity and labels of membranes to represent object properties which results in reducing the complexity of implementing the P-system. We examine the performance of the two heuristics, and compare the results with those of other known algorithms. The numerical results0 F 1 show that good solutions for large instances (10000 items) of this problem could be obtained in a very short time (seconds) using our CUDA simulator.	null	[ "https://arxiv.org/pdf/1602.08735v2.pdf" ]	16,510,795	1602.08735	4418834098be5548f15ca510b919b4f27241e54b	Heuristics for the Variable Sized Bin Packing Problem Using a Hybrid P System and CUDA Architecture Qadha'a Alenezi Computer Science Department Kuwait University Kuwait Hosam Aboelfotoh Computer Science Department Kuwait University Kuwait Computer Science Department Kuwait University Kuwait Bader Albdaiwi [email protected] Computer Science Department Kuwait University Kuwait Mohammad Ali Almulla [email protected] Computer Science Department Kuwait University Kuwait Computer Science Department Kuwait University Kuwait Heuristics for the Variable Sized Bin Packing Problem Using a Hybrid P System and CUDA Architecture 1Variable Sized Bin Packing ProblemNatural ComputingHybrid P-systemP- systemGPUGPGPUCUDA The Variable Sized Bin Packing Problem has a wide range of application areas including packing, scheduling, and manufacturing. Given a list of items and variable sized bin types, the objective is to minimize the total size of the used bins. This problem is known to be NP-hard.In this article, we present two new heuristics for solving the problem using a new variation of P systems with active membranes, which we call a hybrid P system, implemented in CUDA. Our hybrid P-system model allows using the polarity and labels of membranes to represent object properties which results in reducing the complexity of implementing the P-system. We examine the performance of the two heuristics, and compare the results with those of other known algorithms. The numerical results0 F 1 show that good solutions for large instances (10000 items) of this problem could be obtained in a very short time (seconds) using our CUDA simulator. INTRODUCTION The Variable Sized Bin Packing Problem (VSBPP) is an NP hard problem. It contains the classical Bin Packing Problem (BPP), as the BPP has only one type of bins. There are many applications in practical life for the VSBPP. For example, in the truck loading problem, we use trucks with different sizes to load items. We can use as many trucks of each size as we want. The goal is to minimize the total capacity (or cost) of selected trucks. Also, the machine-scheduling problem can be considered as a VSBPP. This problem appears when different classes of processors are used to parallelize the processing of a set of processes with a known processing time. The goal is to minimize the total cost of the associated processors (Coreia, I., Goveia, L. and da Gamma, F., 2008). Transmitting requests through the network is another application of the VSBPP. It is assumed that a set of unit-time tasks is to be transmitted through a network, each task requiring a specific network capacity. The network In this paper, we present two new heuristics for the VSBPP based on a P system and implemented in CUDA. We use a new implementation of P systems which we call a hybrid P system. Section 2 is an introduction to P systems and related concepts. Section 3 gives more details on P systems with Active Membranes. In Section 4, we give an overview of General-Purpose Computing on Graphics Processing Units (GPGPU) and Nvidia Compute Unified Device Architecture (CUDA). Section 5 includes previous work related to VSBPP and cites the work done on simulating P systems using GPGPU and CUDA. Section 6 presents our two new heuristics. Section 7 contains the numerical results. Section 8 is conclusion and future work. MEMBRANE COMPUTING -P SYSTEM Membrane computing was first introduced by Gheorghe Păun on 1998 (Paun, P Systems with Active Membranes: Attacking NP Complete Problems, May 1999). Membrane computing is a branch of Natural computing which refers to any computational model that is based on an inspiration from nature (Castro, 2006). This includes Cellular Automata, Neural operations we can get 2 n copies of that membrane. This helps creating an exponential working space for NP-complete problems (Paun, Introduction of Memebrane Computing, 2006). P system with active membranes can be defined formally as: π = (O, H, e, µ, m 1 , m 2 , … , m p , R) , where m ≥ 1: is the initial degree of the P system, O: the alphabets that represents the objects, H: finite set of membrane labels, e: finite set of signs representing the polarity of the membrane {-, +, 0}, µ: membrane structure consists of a number (p) of membranes that are initially with neutral polarity 0, each membrane is labeled by an element of H. For example [ [ [ ] 2 ] 1 [ ] 3 ] 0 represents an outer membrane labeled 0 (usually called the skin) that contains two internal membranes labeled 1 and 3. Membrane 1 contains an internal membrane labeled 2. The simplest membrane structure is: [ ] ℎ , where, h is the label of the membrane and e is the membrane polarity. m 1 , m 2 , … , m p : strings of alphabets over O, representing the multisets of objects in the membrane structure. R: Finite set of development rules associated with the regions of the P system. There are five types of the development rules (we give an example of each type): A. Object Evolution Rules: This type of rules is evolving only the objects -not the membranes-and it depends on the polarity and labels of the membranes. [ → ] ℎ , where, h ∈ H, e ∈ {+, −,0}, a ∈ O, v ∈ O* The element a will be evolved to v if the polarity of the membrane h is e. B. In-Communication Rules: In this type of rules an object is injected into the membrane and it could be evolved during this process for another object. The polarity of the membrane could be changed but not the label of the membrane. [ ] ℎ 1 → [ ] ℎ 2 , where, h ∈ H, e1, e2 ∈ {+, −,0}, a, b ∈ O The object a is injected into the membrane h, during the transformation process the object a evolved to b. The polarity of the membrane h changed from e 1 to e 2 . C. Out-Communication Rules: In this type of rules an object is sent out the membrane and it could be evolved during this process for another object. The polarity of the membrane could be changed but not the label of the membrane. [ ] ℎ 1 → [ ] ℎ 2 , where, h ∈ H, e1, e2 ∈ {+, −,0}, a, b ∈ O The object a is sent outside the membrane h, during the transformation process the object a evolved to b. The polarity of the membrane h changed from e 1 to e 2 . Generally the label of the membrane could be changed in the (in and out) communication rules. D. Dissolving Rules: As a result of interaction of the objects the membrane could be dissolved and the objects modified. [ ] ℎ → , where, h ∈ H, e ∈ {+, −,0}, a, b ∈ O The membrane h is dissolved and the object a evolved to b and dumped to the parent membrane. E. Division Rules: In this rule as a result of object interaction the membrane will be divided into two membranes. The new membranes could be with different polarities and different labels. Some objects will evolve for new objects; the lasting components -objects or inner membranes-will be duplicated in all the new membranes. [ ] ℎ1 1 → [ ] ℎ2 2 [ ] ℎ3 3 , where h1, h2, h3 ∈ H, e1, e2, e3∈ {+, −,0}, a, b, c ∈ O The membrane h 1 will be divided into two membranes one of label h 2 and one of label h 3 . The new membranes could be with different or the same polarity of the membrane h 1 . The object a will evolve to new objects b and c. Using the P system as a computational model requires testing for the applicability of each rule and choosing randomly between those eligible for application. In parallel execution of rules, care must be taken when more than one rule are applicable to the same object. The P system model is a parallel model that can exploit the new technology of GPGPU to allow for developing new heuristics for NP-hard problems. GPGPU and CUDA GPGPU Graphics Processing Units (GPUs) are massively parallel processors. They are accelerating the graphics computation by taking the job from the CPUs. Today's GPU is extremely fast because of its high parallelism. GPGPU is the use of GPUs to solve general computational problems (other than Graphics). In the recent years, a large interest in the GPGPU approach has appeared aiming at obtaining near-optimal or approximations to NP-Complete problems in a relatively short time. The start of GPGPU was in 2002 and its evolution was slow until 2007 when NVIDIA released CUDA (McClanahan, History and Evolution of GPU Architecture, 2011) (Owens, J., Luebke,D., Govindaraju N., Harris, M., Krüger,J., Lefohn, A., and Purcell, T., August 2005), (NVIDIA CUDA, Programming Guide, 2007). When CUDA was released, the approach of GPGPU grew rapidly. Still the GPGPU is growing with the improvement of NVIDIA's products (Zibula, 2010). CUDA CUDA is a hardware and software architecture that was developed by NVIDIA in 2007 to manage the GPGPU and support heterogeneous computation. With CUDA, the serial part will be executed by the CPU and parallel parts by the GPU. CUDA allowed the developer to use C language, Fortran, OpenCL, Java ... etc. This feature makes GPU computation easier even for the novice programmer. CUDA succeeded in qualifying the GPU for the non-graphical operations by overcoming the difficulties that were facing this approach. The GPGPU computation has made a significant progress after the emergence of CUDA. • CUDA Programming Model: With CUDA, the GPU operates as a coprocessor for the CPU. Programming through CUDA makes the CPU perform as a host and the GPU performs as a compute device with the capability of executing a large number of threads in parallel. The portion of the code that needs to be executed in parallel with independent data will be executed on the GPU device on several different threads. This piece of code is called kernel, it is compiled to the device instruction set then loaded on the device with all required data. The kernel call has the following format: Thread_code_name<<<number_of_blocks,number_of_threads >>> (parameter_list). There are independent DRAMs for both host and device; however the data can be copied back-and-forth between host memory and device memory. A thread is a set of instructions that are to be processed by GPU (device). Each thread has two indices (block index and thread index). The set of threads executing the same kernel are organized in a grid. The grid is composed of a number of thread blocks. Thread block is a batch of threads to be executed in parallel that can cooperate together by sharing data through a fast shared memory. A grid of thread blocks is a batch of thread blocks of the same dimensions, sizes, and they are executing the same kernel. Threads can synchronize their execution by suspending the threads of a block until all of them reach the synchronization point. Threads that are in different blocks of the same grid cannot communicate or synchronize with each other. This hierarchy allows the kernel to be compiled only once for each device. • CUDA Memory Model: There are various types of memory spaces available on a GPU device, these memory spaces are classified as: The memory space that can be accessed by a thread is thread's registers (32-bit) and thread's local memory (parallel data cache). The memory space that can be accessed by all threads of a thread block is the block's shared memory. The memory space that can be accessed by all thread blocks of a grid are grid's global memory, grid's constant memory (read only), and grid's texture memory (read only). • CUDA Hardware Implementation: The GPU is built of a set of multithreaded, multicore, streaming multiprocessors. The multiprocessors consist of Single Instruction Multiple Thread architecture (SIMT). SIMT means that each processor in the multiprocessor executes the same instruction with different threads (and subsequently different data). Each multiprocessor in the GPU has a (read and write) shared memory accessed by its processors only. A read-only texture cache and constant cache are shared by all processors of the multiprocessor. A set of 32-bit registers is assigned to each processor of the multiprocessor. • CUDA Execution Model: A CUDA program is executed on the CPU (the host), as soon as it reaches a call for a kernel on the GPU, it will invoke that kernel on the GPU. The kernel will be executed by a grid of thread blocks. Each multiprocessor will handle one or two blocks using time slicing. Each block will be divided into SIMT sets of threads (Warps). Each warp has the same number of the threads. The warp is executed by a multiprocessor. A thread scheduler will swap from one warp to another to maximize the performance of the multiprocessor. A block can be executed only on one multiprocessor. More than one block can be executed on the same multiprocessor. The order of warp execution within a block is undefined but it can be synchronized to arrange the accessing of the shared and global memory. The order of the thread blocks execution is undefined and it cannot be synchronized, because of that threads from different blocks cannot communicate to keep the shared and global memory safe from wrong updates (NVIDIA CUDA, Programming Guide, 2007). PREVIOUS WORK On 2003, Kang and Park introduced two greedy algorithms (Kang, J. and Park, S., 2003). On 2002, Guochuan Zhang proposed a number of approximation algorithms to solve online VSBPP (ZHANG, 2002). Haouari and Serairi proposed six optimization-based heuristics for the VSBPP (Haouari, M., and Serairi, M., 2009). Coreia, Goveia, and da Gamma studied the use of a discretized function for solving the VSBPP (Coreia, I., Goveia, L. and da Gamma, F., 2008). Crainic, perboli, Rei, and Tadie proposed a series of lower bounds then they computed the means to measure the quality of their solution. They used these bound to build their heuristics for the problem (Crainic, T., perboli, G., Rei, W., and Tadie, R., 2010). Later on 2011 the authors introduced Lower bounds for the Generalized BPP. They presented two formulations for the problem, an aggregate knapsack lower bound, and column generationbased lower bound (Baldi, M., Crainic, T., Perpoli, G. and Tadei, R., 2011). On 2012 they proposed two methods to solve the problem, an exact method using the branch-and-price search and a heuristic using the beam search (Baldi, M., Crainic, T., Perpoli, G. and Tadei, R., 2012). On 2014 they applied a worst case analysis to the Generalized BPP of FFD, and BFD heuristics. Also they proposed two semi-online algorithms to tackle the problem (Baldi, M., Crainic, T., Perpoli, G., and Tadei R., 2014). As for the P system, Zandron , Ferretti and Mauri published a paper in 2001. They showed that the P systems with active membrane which use one type of membrane division: division of elementary membrane can be used to solve SAT and Hamiltonian Problems in linear time with respect to the input length (Zandron, C., Ferretti, C. and Mauri, G., 2001). On 2004 Perez-Jimenez and Romero-Campero published a paper that provides an effective solution of the standard BPP. They used a family of recognizer P system with active membranes to solve an instance of the BPP (Pérez-Jiménez, M., and Romero-Campero F., 2004). Linqiang Pan, and Carlos Martín-Vide solved the multidimensional 0-1 Knapsack Problem by an algorithm based on the recognizer P system with input and active membrane (Pan, L. and Martin-Vide, C., December 2005). Alhazov, and Perez-Jimenez tried to form a uniform family of P system to solve the Satisfiability of a Quantified Boolean formula (QSAT), which is a PSPACEcomplete Problem in their paper, which was published in 2007 (Alhazov A. and Pérezjiménez, M., 2007). In 2010, Chun Lu, and Xingyi Zhang provided a solution for the Vertex Cover Problem using the Tissue-like P system with cell separation (Lu, C. and Zhang, X., 2010). Aman, and Ciobanu published a paper in 2011 where they provide semi-uniform polynomial solutions for some weak NP-complete problems such as Knapsack, Subset Sum, and Partition Problems (Aman, B. and Ciobanu, G., December 2011). The research on simulating P systems with GPGPUs has recently attracted many researchers. (Cecilia & etal., 2010) , (Martinez-del-Amor & etal., 2013), (Martinez-del-Amor & etal, Simulating P systems on GPU devices: a Survey, 2015) and (Maroosi & etal, 2014). To our knowledge, this paper is the first to propose a solution (heuristics) to the VSBPP based on P systems and CUDA. A NEW HYBRID P SYSTEM FOR SOLVING THE VSBPP USING CUDA In this paper we propose two P system-based heuristics to find a solution for the VSBPP using CUDA. However, we define a new type of P systems which we call a hybrid P-system. A hybrid P-system differs from the P-system in two ways. First, it extends the set of polarity symbols and labels to represent dynamic properties of the objects as well as the membranes. This feature allows for using the labels to represent the item weight instead of representing the object weight using the multiplicity of the object as defined in the original definition of Psystems. For example, in P systems an item s k of weight three is represented in the initial configuration as the multiset { s k , s k, s k } (Pérez-Jiménez, M., and Romero-Campero F., 2004). Second, the model allows for selecting an object within a membrane based on a selection criterion (FF, WF, and BF). The selection criterion is represented as a component of an item object label within the membrane. Similar to the recognizer P system model as defined in (Pérez-Jiménez, M., and Romero-Campero F., 2004) two distinguished objects {Yes, No} are used to signal a termination condition in the computation. Problem Description Given a list of indivisible items of different weights list = (w 1 , …, w m ), where w i is the weight of the ith item, 1 ≤ i < m, and m is the number of the items in the list, and n types of bins, B i , 1 ≤ i ≤ n, where B i denotes the capacity of each bin of type i, find a packing of the items such that the total capacity of used bins is minimum. We assume that for each type of bin, there is an infinite number (unlimited supply) of bins. We also assume that B 1 > B 2 > ... > B n > 0, and the maximum weight of an item is less than or equal B 1 . Both heuristics perform an initialization step on the host (CPU) then call a kernel to perform the item packing. Let nb and nt be the number of CUDA blocks and threads respectively. The choice of nb and nt will be illustrated in Section 7. Initialization Step: -Rule 1: w i [ … ] → [ … w i ] +1 , k < s . This (In-Communication) rule injects item objects into membranes S1 , …. Sl . Polarity is incremented. First Heuristic 1. Distribute the items randomly among the l subsets; until each subset contains s items (Rule 1). This step is executed by the host code (or an initialization kernel). 2. Copy the P system parameters from host memory to device memory. 3. Call the kernel Pack_1 <<< nb, nt >>> ( P system parameters). 4. //Let bx be the block index, tx be the thread index. Pack_1 has the following steps: (executed by each thread (bx, tx) ) − Compute subset index I = bx * nt + tx. //Thread(bx, tx) will pack subset SI. -Check applicability. -Select randomly a rule for execution. -Check termination condition. allowing for Rule 4 to pack remaining items in the membrane. / 5. All threads within a block will synchronize (Syncthreads() ) before return. 6. The host executes a call to (cudaDeviceSynchronize() ) to make sure that the kernel has terminated then the results can be copied from device memory to host. Remarks on correctness: 1. No synchronization is required in rule execution since: 1-items list is partitioned between threads (in the first heuristic) or blocks (in the second heuristic), 2-each thread is executing one of the applicable rules at a time. (The parallel execution is implemented by CUDA blocks and threads.) 2. An item can be packed in a bin only if there is a sufficient space (Rule 4). 3. A new bin is generated for each type only if all bins of the same type are used to at least half of their capacity (Rule 5). Therefore, the rule guarantees that only a finite number of bins of each type will be generated in the system. Second Heuristic The second heuristic differs from the first heuristics in that: • Distribute the items into subsets of size s, such that s! ≤ Maximum block size. • Find all the permutations of each subset in parallel and assign each permutation to a thread within a block. • Each thread packs (Pack_2) the items in its subset according to their order in the permutation using rules 2-6. • After all thread items are packed, thread 0 of each block computes the minimum capacity used by any thread in its block, and the index of the corresponding thread. • The best results are accumulated by the host. NUMERICAL RESULTS In this experiment we have used Intel(R) Core (TM) i5-2410M CPU @ 2.30 GHz, 6GB RAM, GeForce GTX 560M, Driver 285.86, CUDA Cores 192, and CUDA Toolkit CUDA 4.1.28. We use the benchmarks that were used by (Coreia, I., Goveia, L. and da Gamma, F., 2008). They generate two classes of benchmarks. In this experiment, we used the first class. For some experiments we need large number of items which are not available in the tested benchmarks; for that we created new instances such that: 1. Items weights are selected randomly from the set {1, 2, …, 20} 2. Number of items (m) ∈ {3, 5, 10, 15, 30, 5000, 10000, 50000, 100000} We grouped the whole instances which are used in our experiments into the following groups: (Coreia, I., Goveia, L. and da Gamma, F., 2008) which are mentioned at the beginning of this section. It is important to know that the computation capability of the used machine is limited. For that, we had to divide some problem into parts (sub-kernels). A separate kernel will execute each part. At the end, we combine the results of the kernels to get the final result. For example for 100000 items and heuristic 2, we use 625 kernels each with 32 blocks, and each Block size will not exceed 120 threads. This size is very small and affects the performance of the heuristics, but it is suitable for our machine and guarantees that the system will work without any unexpected memory over flow or device timeout. Two types of kernels are used. The first kernel is for packing the items and the second kernel is for computing the total capacity of the used bins. The notation 1 st K stands for the execution time of the first kernel. Where, the notation 2 nd K stands for the execution time of the second kernel. In some cases there is only one block of threads, which mean the total capacity of the used bins is computed in the first kernel, and the execution time of 2 nd K will be equal to 0. All the times that were reported in our experiments are GPU time, except in serial implementation it is the CPU time. Comparison between the Two P System Heuristics We use Group 3 of instances. In Table 1 we compare the execution structure of our two heuristics. Grid size and the number of the kernels depend on the total number of the items m and the used heuristic. It is clear in Table 1 that the first heuristic needs less number of blocks and kernels which implies a faster performance as shown in Table 2. Table 3 shows the best solution that was reached by the two heuristics. We notice that the first heuristic is able to bring better solution than the second heuristic because it deals with less number of blocks and the number of items that was assigned to a thread is larger than the second heuristic. These results show the efficiency of the first heuristic, and its speed comparing to the second heuristic. Also it shows the ability of the first heuristic to bring same or better results comparing to the second heuristic with less time and space for when using small instances. Comparison with a Serial and Parallel Implementations To know the effectiveness of the parallelism of our P system, we compare its CUDA implementation with a serial and parallel implementation. For this purpose, we implement a serial application that computes exact solution by computing all the permutation of a list of up to 10 items. We then pack the each permutation inside bins with different sizes. Finally, we find the permutation that yield the minimum capacity of the used bins. Also, we implemented another CUDA application to find the exact solution of the VSBPP by packing the permutations in parallel using BF, WF, and FF. Then find the permutation that uses the least capacity of bins. In this implementation, we assigned each permutation to a thread. Then within a block find the permutation that used the least capacity of bins. Then, within a grid find the permutation that used the least capacity of bins which will be the final results. We used the instances in Group 1. Table 4 shows the execution time and the number of the permutations that have been processed by our second heuristic and the serial implementation. Table 5 lists the processing time of the second heuristic and the parallel implementation And Table 6 shows the execution structure of our second heuristic and the parallel implementation. It is clear from these results the second heuristic is faster than the Serial and the Parallel Implementation. Also, the increase of the processing time of the second heuristic is steady and slow unlike the Serial and the Parallel Implementation. Also, we can notice that the Parallel Implementation can find a solution for VSBPP in a reasonable time for a list with at most 10 items. However, the second heuristic is the best in achieving the minimum processing time for large instances. The Packing Utilization of The second heuristic To measure the efficiency of a heuristic for the VSBPP, two criteria are required, the execution time and the packing utilization. The packing utilization is computed using the formula: = Total weight of items best Solution Reached If the achieved utilization of a packing is large that means that the waste space in the used bin is minimized, which is an indication of the robustness of the tested heuristic (Ortmann, F., Ntene, N., and Van Vuuren, J., 2010). Table 7 shows the best results that achieved when packing the instances of Group 2 using the second heuristic. The packing utilization is about 89% of the capacity of the used bins. This is an indication that our heuristic is efficient and robust. Comparison with Discretized Formulations To ensure the robustness and the efficiency of our P system heuristics we need to use the instances of Group 4 which are the instances of (Coreia, I., Goveia, L. and da Gamma, F., 2008). In their paper, they introduced results of number of Discretized Formulations. They used CPLEX solver to implement their formulations. In Table 8, we compare the execution time of 3 of their Discretized Formulations and our second heuristic, in case of packing 1000 items using (3, 6, 12) types of bins. It's clear that the processing time of the second heuristic is much less than the processing time of the three discretized formulations. Also we notice that in our implementation there is a small difference between the execution time in case of using 3, 6, or 12 bins types. This is because in our CUDA implementation we rely on the local and shared variables and simple arrays structures. Meanwhile in the three discretized formulations there was a big difference between the execution time in case of using 3, 6, or 12 different bins types, which mean another advantage added to our P system heuristic. In Table 9 we are listing the results of the instances in Group 4. Dr. Isabel Coreia provided us with (commendable) their detailed results. According to their paper D is a constraint representing "the minimum number of bins of each size that must be considered in order to have a total available capacity greater than or equal than the total requirement" (Coreia, I., Goveia, L. and da Gamma, F., 2008). In our experiment it is not possible to put this constraint because of the nature of our P system and our CUDA implementation. In our P system we are dealing with sublists of the main list which could use number of bins less than the number of bins types. For that we list in table 9 the best reached solution, such that it is using the maximum number of bins types. Table 9. The results of the second heuristic compared to the optimum results of Coreia Paper in case of using 3, 6, and 12 types of Bins CONCLUSION AND FUTURE WORK Finding minimum space or cost to pack a certain list of items is a critical problem to many applications; the VSBPP is a variant of this problem. The VSBPP is one of the classical combinatorial NP-Hard Problems. In the literature of the VSBPP, there are few parallel heuristics that were presented to find a solution for some instances of this Problem. In this paper, we propose two parallel heuristics that use the approach of the membrane computing to find a solution for the VSBPP using NVIDIA's CUDA. Membrane computational model is parallel, distributed, scalable and nondeterministic. These features makes it very suitable model to use in GPGPU and CUDA. In this paper we propose a new variation of membrane computing that we call a Hybrid P system. Our model allows for using objects and membrane label and polarity to represent properties instead of using object multiplicity as in the classical model. This new model improves significantly the time and space complexity of implementing the classical P-system. Based on the numerical results of the CUDA simulation of our two heuristics, the first heuristic is faster than the second one in all cases. But the second heuristic succeeds in achieving better results in case of using benchmarks with large number of items. The results also show that the performance of the second heuristic is surpassing the performance of the serial implementation and the performance of all permutations parallel implantation. The average utilization achieved by the second heuristic is about 89% of the capacity of used bins, which indicates that it is an efficient and robust heuristic. The execution time required by the second heuristic is much less than the time required by the discretized formulations (Coreia, I., Goveia, L. and da Gamma, F., 2008). Also the total capacity of the used bins achieved by the heuristics is less than the best results obtained by these discretized formulations. The membrane computational model is a good framework for parallel heuristics since it is making the non-determinism and parallelization process easy to implement. The exploitation of the power of the NVIDIA's GPU helped us in getting good results and strengthened the success of the membrane computing. The merge of Membrane Computing model and NVIDIA's CUDA is expected to lead to good results for most of the NP-Complete problems. In future, we are planning to enhance our Recognizer P system with Active Membrane of the VSBPP in order to increase the packing utilization. We will use the features of the NVIDIA's Kepler to improve the CUDA implementation. Also, we will find a way to make a general reusable P system so that it can be used to solve different kinds of problems. This step is common to both heuristics and performed by the host (main): -Create the initial configuration of the P system ( Π ) as follows: Π = (O, H, e, µ, m 0 , m 1 , … , m p , R) Objects: O = { { w 1 , …, w m } ∪ { FF, BF, WF } ∪ {Yes, No} } Membrane labels: H = { S 1 , … , S l , (b,x,B 1 ), … , (b,x,B n ) }, x > 0. The first set of labels S denote l sublists of items, and (b,x,B y ) denotes a bin number x with capacity (size) B y . l = nb × nt . The maximum sublist size s = m/l. Polarity: e= { 0 , … , n} ∪ { 0 … s } . Membrane structure: µ = [ skin membrane contains l membranes (for subsets of items) and one membrane for each type of bin. / -Initialize skin membrane objects: m 0 = { w 1 , …, w m }./* Skin membrane contains all items. / -Initialize internal membranes: m 1 , … , m p = Φ . − Rule 2 :−− 2[ Create a new membrane [ ] 1 enclosing SI and all bin membranes: [ [ … ] [… ] ,1, , , ∈ {1 .For Rules 3 -6: repeat the following until Termination condition is true: − Rule 3: [… ] , → w i,c [… ] −1 , k ≥ 1 , c ∈{FF, BF, WF}. / This out-communication rule sends out (of the subset membrane) an item. The item evolves to indicate the selection criterion used in its future packing. The three criterions are used with equal probabilities. / − Rule 4: w i,c [… ] , , , → [… ] , , , + , Bi > k ≥ 0 , c ∈{FF, BF, WF}. / This in-communication rule sends out an item. Bi is determined using the selection criterion c. division rule creates a new bin of the same type. The change of polarity prevents the rule to be executed more than once for the same bin. / − Rule 6: [… ] 0 → . // All items are sent out of SI membrane. / This rule allows for detecting the end of thread computation (Termination condition). A simple atomic counter can be checked to make sure that all items are packed. Otherwise, the execution loop continues Group 1 : 1consists of nine instances where: m ∈ {3, 5, 10, 15, 30, 100, 200, 500, 1000}, b = 3, types of bins = {100, 200, 300} Group 2: consists of five instances, where the weight of the items is selected manually: m = 1000, b = 4, types of bins = {10, 20, 30, 40}, the weights of the items and the number of the existence of the item in each instance are: Group 3: consists of eight instances: m ∈ {100, 200, 500, 1000, 5000, 10000, 50000, 100000}, b = 3, types of bins = {100, 200, 300} Where, the instances with number of m ∈ {100, 200, 500, 1000} are taken from (Coreia, I., Goveia, L. and da Gamma, F., 2008) the lists of items number: (100_1, 200_1, 500_1, 1000_1). Meanwhile, the instances with number of m ∈ {5000, 10000, 50000, 100000} are generated randomly. Group 4: consists of twelve instances: These instances are the instances of Table 1 .Table 2 . 12The execution structure of the two heuristics The execution time of the two heuristics using instances of Group 3 Table 3 . 3The results of the two heuristics in case of using instances of Group 3M First Heuristic Second Heuristic Grid size Items/ thread Number of Kernels Grid size Items/ block Number of Kernels Block Thread block thread 100 1 10 10 1 20 120 5 1 200 1 20 10 1 40 120 5 2 500 1 50 10 1 100 120 5 4 1000 1 100 10 1 200 120 5 7 5000 1 500 10 1 1000 120 5 32 10000 1 1000 10 1 2000 120 5 63 50000 5 1000 10 1 10000 120 5 313 100000 10 1000 10 1 20000 120 5 625 M First Heuristic Second Heuristic 100 0.017623 1 st K: 0.019020 2 nd K: 0.08876 200 0.017679 1 st K: 0.023359 2 nd K: 0.09188 500 0.017626 1 st K: 0.040346 2 nd K: 0.01241 1000 0.018638 1 st K: 0.094445 2 nd K: 0.01151 5000 0.019600 1 st K: 0.376696 2 nd K: 0.012366 10000 0.019165 1 st K: 0.765959 2 nd K: 0.024270 50000 1 st K: 0.013015 2 nd K: 0.013174 1 st K: 3.575214 2 nd K: 0.111391 100000 1 st K: 0.014671 2 nd K: 0.011624 1 st K: 5.782224 2 nd K: 0.202711 m Total weight of items First Heuristic Second Heuristic Best Solution Reached Best Solution Reached 100 1119 1,600 2,300 200 2171 3,800 4,300 500 5511 9,900 10,600 1000 10459 21,400 21,200 5000 53252 117,100 112,600 10000 105516 226,600 200,800 50000 526,327 1,153,900 1,009,200 100000 1,051,854 2,308,800 2,016,200 Table 4 . 4The execution time required by the Second heuristic and the Serial Implementation Table 5 . 5The execution time of the second heuristic and the parallel implementationM Second Heuristic Serial Implementation Permutations Time (sec.) Permutations Time (sec.) 3 6 1 st K: 0.011561 6 0.009386 5 120 1 st K: 0.013344 120 0.282215 10 240 1 st K: 0.016758 2 nd K: 0.01098 3628800 592.81012 15 360 1 st K: 0.017598 2 nd K: 0.01091 NA NA 30 720 1 st K: 0.018640 2 nd K: 0.01112 NA NA 100 2400 1 st K: 0.019020 2 nd K: 0.08876 NA NA 200 4800 1 st K: 0.023359 2 nd K: 0.09188 NA NA 500 12000 1 st K: 0.040346 2 nd K: 0.01241 NA NA 1000 24000 1 st K: 0.094445 2 nd K: 0.01151 NA NA M Second Heuristic All permutations parallel Implementation Time of All Kernels (sec.) Time of All Kernels (sec.) 3 1 st K: 0.011561 2 nd K: 0.0 0.011293 5 1 st K: 0.013344 2 nd K: 0.0 0.016814 Table 6 . 6The execution structure of the second heuristic and the Parallel Implementation Table 7 . 7The packing utilization of the second heuristic using instances of Group 210 1 st K: 0.016758 2 nd K: 0.01098 1.022951 15 1 st K: 0.017598 2 nd K: 0.01091 15.677255 100 kernels only 30 1 st K: 0.018640 2 nd K: 0.01112 NA 100 1 st K: 0.019020 2 nd K: 0.08876 NA 200 1 st K: 0.023359 2 nd K: 0.09188 NA 500 1 st K: 0.040346 2 nd K: 0.01241 NA 1000 1 st K: 0.094445 2 nd K: 0.01151 NA M Second Heuristic All Permutations Parallel Implementation Grid size Items/ block Number of Kernels Grid size Items/Th read Number of Kernels block thread block thread 3 1 6 3 1 1 6 3 1 5 1 120 5 1 1 120 5 1 10 2 120 5 1 5184 70 10 10 15 3 120 5 1 5184 70 15 3603600 30 6 120 5 1 NA NA NA NA 100 20 120 5 1 NA NA NA NA 200 40 120 5 2 NA NA NA NA 500 100 120 5 4 NA NA NA NA 1000 200 120 5 7 NA NA NA NA Table 8 . 8The execution time of the Second heuristic and three discretized formulations in case of using 3, 6 and 12 types of Bins Models 3 Bins Types (sec.) 6 Bins Types (sec.) 12 Bins Types (sec.)Second Heuristic 1 st K: 0.094445 2 nd K: 0.011513 1 st K: 0.098116 2 nd K: 0.010878 1 st K: 0.104158 2 nd K: 0.011004 P1+(6)+(7) 576 1611 2885 P2+(16)+(17)+(19)+(20) 1614 3323 5419 P2+(13)+(14)+(16)+(17)+(19)+(20) - - - This work is extracted from M.Sc. 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[ "A CONVERGENT LAGRANGIAN DISCRETIZATION FOR A NONLINEAR FOURTH ORDER EQUATION", "A CONVERGENT LAGRANGIAN DISCRETIZATION FOR A NONLINEAR FOURTH ORDER EQUATION" ]	[ "Daniel Matthes ", "Horst Osberger " ]	[]	[]	A fully discrete Lagrangian scheme for numerical solution of the nonlinear fourth order DLSS equation in one space dimension is analyzed. The discretization is based on the equation's gradient flow structure in the L 2 -Wasserstein metric. We prove that the discrete solutions are strictly positive and mass conserving. Further, they dissipate both the Fisher information and the logarithmic entropy. Numerical experiments illustrate the practicability of the scheme.Our main result is a proof of convergence of fully discrete to weak solutions in the limit of vanishing mesh size. Convergence is obtained for arbitrary non-negative initial data with finite entropy, without any CFL type condition. The key ingredient in the proof is a discretized version of the classical entropy dissipation estimate.	10.1007/s10208-015-9284-6	[ "https://arxiv.org/pdf/1410.1728v1.pdf" ]	28,830,177	1410.1728	89101ba2cb293601566d28c540e95b17429298ed	A CONVERGENT LAGRANGIAN DISCRETIZATION FOR A NONLINEAR FOURTH ORDER EQUATION Daniel Matthes Horst Osberger A CONVERGENT LAGRANGIAN DISCRETIZATION FOR A NONLINEAR FOURTH ORDER EQUATION A fully discrete Lagrangian scheme for numerical solution of the nonlinear fourth order DLSS equation in one space dimension is analyzed. The discretization is based on the equation's gradient flow structure in the L 2 -Wasserstein metric. We prove that the discrete solutions are strictly positive and mass conserving. Further, they dissipate both the Fisher information and the logarithmic entropy. Numerical experiments illustrate the practicability of the scheme.Our main result is a proof of convergence of fully discrete to weak solutions in the limit of vanishing mesh size. Convergence is obtained for arbitrary non-negative initial data with finite entropy, without any CFL type condition. The key ingredient in the proof is a discretized version of the classical entropy dissipation estimate. Introduction 1.1. The equation and its properties. In this paper, we study a full discretization of the following initial boundary value problem on the one-dimensional interval Ω = [a, b]: ∂ t u + ∂ x u ∂ x ∂ xx √ u √ u = 0 for t > 0 and x ∈ Ω,(1)∂ x u = 0, u ∂ x ∂ xx √ u √ u = 0 for t > 0 and x ∈ ∂Ω,(2) u = u 0 at t = 0. Equation (1) is known as the DLSS equation, where the acronym refers to Derrida, Lebowitz, Speer and Spohn, who introduced (1) in [18,19] for studying interface fluctations in the anchored Toom model. In the context of semi-conductor physics, (1) appears as a simplified quantum drift diffusion equation [16,30]. The analytical treatment of (1) is far from trivial: see e.g. [4,22,28,24,32,33] for results on existence and uniqueness of solutions in various different settings, and [8,13,9,24,32,35,39] for qualitative and quantitative descriptions of the long-time behavior. The main difficulty in the development of the time-global well-posedness theory has been that the nonlinear operator in (1) is defined only for positive functions u, but there is no maximum principle available which would provide an a priori positive lower bound on u. Ironically, solutions are known to be C ∞ -smooth as long as they remain strictly positive [4], but the question if strict positivity of the initial datum u 0 is sufficient for that remains open, despite much effort and some recent progress in that direction, see [23]. In order to deal with the general case -allowing arbitrary non-negative initial data u 0 of finite entropy -a theory for non-negative weak solutions has been developed [24,32] on grounds of the a priori regularity estimate √ u ∈ L 2 loc R + ; H 2 (Ω) , which gives a meaning to (1) in the formally equivalent representation ∂ t u + ∂ xxxx u − ∂ xx ∂ x √ u 2 = 0.(5) On the other hand, the problem (1)-(3) has several remarkable structural properties, and these eventually paved the way to a rigorous analysis. We list some of those properties: • The evolution is mass preserving, b a u(t; x) dx = M := b a u 0 (x) dx for all t > 0. • There are infinitely many (formal) Lyapunov functionals [4,8,31]. The two most important ones are the (logarithmic) entropy, H(u) = b a u ln u dx − H 0 with H 0 = M ln M b − a ,(6) and the Fisher information, F(u) = b a ∂ x √ u 2 dx.(7) • The Fisher information is more than just a Lyapunov functional: in [24], it has been shown that (1)&(2) is a gradient flow in the potential landscape of F with respect to the L 2 -Wasserstein metric W. That is, formally one can write (1)&(2) as ∂ t u = − grad W F(u).(8) • Also H is not "an arbitrary" Lyapunov functional: the L 2 -Wasserstein gradient flow of H is the heat equation [29], ∂ s v = − grad W H(v) = ∂ xx v,(9) and the Fisher information F equals the dissipation of H along its own gradient flow, F(v(s)) = − d ds H(v(s)).(10) In view of (8), this relation makes the DLSS equation the "big brother" of the heat equation, see [17,39] for structural consequences. 1.2. Fully discrete approximation. For the numerical approximation of solutions to (1)-(3), it is natural to ask for structure preserving discretizations that inherit at least some of the nice properties listed above. At the very least, the scheme should produce non-negative (preferably positive) discrete solutions, but there is no reason to expect that behavior from a standard discretization approach. Several (semi-)discretizations for (1)-(3) that guarantee positivity have been proposed in the literature [6,10,34,36]. In all of them, positivity actually appears as a consequence of another, more fundamental feature: each of these schemes also inherits a Lyapunov functional, either a logarithmic/power-type entropy [6,10,34], or a variant of the Fisher information [6,20,36]. An exception is the discretization from [20], which preserves the Lagrangian representation of (1), see below, and thus enforces positivity by construction. Apparently, at least some structure preservation seems necessary to obtain an acceptable numerical scheme. Here we follow further the ansatz from [20], which lead to a discretization with a very rich structure: the scheme is positivity and mass preserving, it dissipates the Fisher information, it has the same Lagrangian structure as (1), and it even inherits (in a certain sense) the gradient flow structure (8). By a small change of that discretization, we obtain a new scheme which still has all of these properties, but in addition also dissipates the logarithmic entropy. Thus, we have two discrete Lyapunov functionals at our disposal, and the interplay between these allows us to give a proof of convergence in the limit of vanishing mesh size. We emphasize that our scheme is the first one to preserve more than one Lyapunov functional for (1), and it is the only fully discrete scheme for which a rigorous convergence analysis is available. Below, we give the "pragmatic" definition of our full discretization, which is actually very simple. In Section 2, we show how this scheme arises from a structure preserving discretization of the gradient flow structure. The starting point is the Lagrangian representation of (1)& (2). Since each u(t; ·) is of mass M , there is a Lagrangian map X(t; ·) : [0, M ] → Ω -the so-called pseudo-inverse distribution function of u(t; ·) -such that ξ = X(t;ξ) 0 u(t; x) dx, for each ξ ∈ [0, M ]. (11) Written in terms of X, the Wasserstein gradient flow (8) for F turns into an L 2 -gradient flow for F(X) = M 0 ∂ ξ 1 ∂ ξ X 2 dξ, that is, ∂ t X = ∂ ξ Z 2 ∂ ξξ Z , where Z(t; ξ) := 1 ∂ ξ X(t; ξ) = u t; X(t; ξ) . At this point, a standard discretization of (12) with parameter ∆ = (τ ; δ) is performed: we use the implicit Euler method for time discretization with fixed time step τ > 0, and central finite differences for equidistant discretization on the mass space [0, M ] with mesh width δ > 0. More explicitly: denote by x ∆ = (x n k ) a fully discrete solution on the ∆-mesh, so that x n k approximates X(nτ ; kδ), then the x n k satisfy where the values z n − 1 2 = δ/(x n − x n −1 ) are associated to the mid-points of the spatial grid. At each time step n ∈ N, x n ∆ = (x n 1 , . . . , x n K−1 ) approximates a Lagrangian map, so we assume that x n ∆ is monotone, i.e., x n k > x n k−1 , and in accordance with (11), we associate to x n ∆ a piecewise constant functionū n ∆ : Ω → R + with u n ∆ (x) = δ x n k − x n k−1 for x n k−1 < x < x n k . As replacements for the entropy H and the Fisher information F, we introduce H δ ( x n ) = δ K k=1 log z n k− 1 2 , F δ ( x n ) = K−1 k=1 z n k+ 1 2 − z n k− 1 2 δ 2 . These choices are made such that H δ is the restriction of H, i.e., H δ ( x n ∆ ) = H(ū n ∆ ), and such that F δ is related to H δ in the same way (10) as F is related to H; see Section 2 for datails. 1.3. Results. Our first result is concerned with qualitative properties of the discrete solutions. For the moment, fix a discretization parameter ∆ = (τ ; δ). Theorem 1. From any monotone discrete initial datum x 0 ∆ , a sequence of monotone x n ∆ satisfying (13) can be constructed by inductively defining x n ∆ as a global minimizer of x → δ 2τ k x k − x n−1 k ) 2 + F δ ( x). This sequence of vectors x n ∆ and the associated densitiesū n ∆ have the following properties: • Positivity:ū n ∆ is a strictly positive function. • Mass conservation:ū n ∆ has mass equal to M . • Dissipation: Both the entropy and the discrete Fisher information are dissipated, H δ ( x n ∆ ) ≤ H δ ( x n−1 ∆ ) and F δ ( x n ∆ ) ≤ F δ ( x n−1 ∆ ). • Equilibration: There is a constant r > 0 only depending on b − a such that H δ ( x n ∆ ) ≤ H δ ( x 0 ∆ )e −rnτ .(14) Some of these properties follow immediately from the construction, while others (like the equilibration) are difficult to prove. Note that even well-posedness (which involves existence of a monotone minimizer for the functional) is a non-trivial claim. To state our main result about convergence, we need to introduce the time-interpolation {ū ∆ } τ : R + × Ω → R + , which is given by {ū ∆ } τ (t; x) =ū n ∆ (x) for (n − 1)τ < t ≤ nτ . Further, ∆ symbolizes a whole sequence of mesh parameters from now on, and we write ∆ → 0 to indicate that τ → 0 and δ → 0 simultaneously. Theorem 2. Let a non-negative initial condition u 0 with H(u 0 ) < ∞ be given. Choose initial conditions x 0 ∆ such thatū 0 ∆ converges to u 0 weakly as ∆ → 0, and H := sup ∆ H δ ( x 0 ∆ ) < ∞ and lim ∆→0 (τ + δ)F δ ( x 0 ∆ ) = 0.(15) For each ∆, construct a discrete approximation x ∆ according to the procedure described in Theorem 1 above. Then, there are a subsequence with ∆ → 0 and a limit function u * ∈ C(R + × Ω) such that: • {ū ∆ } τ converges to u * locally uniformly on R + × Ω, • √ u * ∈ L 2 loc (R ≥0 ; H 1 (Ω)), • there are non-increasing functions f, h : R + → R such that F(u * (t)) = f (t) and H(u * (t)) = h(t) for a.e. t > 0, and additionally h(t) ≤ He −rt with the constant r > 0 from (14), • u * satisfies the following weak formulation of (1)&(2), see (5): ∞ 0 Ω ∂ t ϕ u * + ∂ xxx ϕ ∂ x u * + 4∂ xx ϕ ∂ x √ u * 2 dx dt + Ω ϕ(0, x)u 0 (x) dx = 0 (16) for every test function ϕ ∈ C ∞ c (R ≥0 × Ω) satisfying ∂ x ϕ(t; a) = ∂ x ϕ(t; b) = 0. Remark 3. (1) Quality of convergence: Since {ū ∆ } τ is piecewise constant in space and time, uniform convergence is obviously the best kind of convergence that can be achieved. (2) Rate of convergence: The scheme (13) is formally consistent of order τ + δ 2 , see Proposition 27, and this is also the observed rate of convergence in numerical experiments with smooth initial data u 0 , see Section 6.2.4. (3) Initial condition: We emphasize that our only hypothesis on u 0 is H(u 0 ) < ∞, which allows the same general initial conditions as in [24,32]. If F(u 0 ) happens to be finite, and also sup ∆ F δ ( x 0 ∆ ) < ∞, then the uniform convergence of {ū ∆ } τ holds up to t = 0. (4) Long time behavior: By means of the Csiszar-Kullback inequality, the exponential decay of H(u * (t)) to zero implies exponential convergence of u * to the constant function u ∞ ≡ M/(b − a) in L 1 (Ω). (5) No uniqueness: Since our notion of solution is too weak to apply the uniqueness result from [22], we cannot exclude that different subsequences of {u ∆ } τ converge to different limits. The idea to derive numerical discretizations for solution of Wasserstein gradient flows from the Lagrangian representation is not new in the literature, see e.g. [37] for a general treatise. Several practical schemes have been developed on grounds of the Lagrangian representation for this class of evolution problems, mainly for second-order diffusion equations [5,7,38,41], but also for chemotaxis systems [3], for non-local aggregation equations [11,14], and for variants of the Boltzmann equation [27]. For certain nonlinear fourth order equations, Lagrangian numerical schemes have been developed as well, e.g., for the Hele-Shaw flow [15] and for a class of thin film equations [27]. On the other hand, a rigorous analysis of stability and convergence of the fully discrete schemes is rare and apparently limited to the case of nonlinear diffusion in one space dimension, see [26,40]. There are, however, results available for semi-discrete Lagrangian approximations, see e.g. [2,21]. The primary challenge in our convergence analysis is to carry out all estimates under no additional assumptions on the regularity of the limit solution u * . In particular, we do not exclude a priori the formation of zeros -and the induced loss of regularity -in the limit u * , since this cannot be excluded by the existing theory. Also, we allow extremely general initial conditions u 0 . Without sufficient a priori smoothness, we cannot simply use Taylor approximations and the like to estimate the difference between {ū ∆ } τ and u * . Instead, we are forced to derive new a priori estimates directly from the scheme, using our two Lyapunov functionals. On the technical level, the main difficulty is that our scheme is fully discrete, which means that we are working with spatial difference quotients instead of derivatives. Lacking a discrete chain rule, the derivation of the relevant estimates turns out to be much harder than for the original problem (1)- (3). For instance, we are able to prove a compactness estimate forū ∆ , but not for its inverse distribution function, although both estimates would be equivalent in a smooth setting. This forces us to switch back and forth between the original (1) and the Lagrangian (12) formulation of the DLSS equation. We further remark that the convergence of a family of gradient flows to a limiting gradient flow has been thoroughly investigated on a very abstract level, see e.g. in [1,42], using methods of Γconvergence. Unfortunately, these appealing abstract results would not help to simplify our proof significantly, since the verification of their main hypothesis (Γ-convergence of the subdifferentials) is essentially equivalent to the derivation of the a priori estimates, which is the main part of our work. Therefore, we decided to give a "hands-on proof", which requires only very few elements from the general theory of metric gradient flows. 1.4. Structure of the paper. We start with a description of our Lagrangian discretization in Section 2; the fully discrete scheme is defined in Subsection 2.5. In Section 3, we derive various a priori estimates on the fully discrete solutions. This leads to the main convergence results in Propositions 19 and (20), showing the existence of a limit function u * for ∆ → 0. In Section 4, it is verified that u * is indeed a weak solution to (1)-(3). The formal conclusion of the proofs for Theorems 1 and 2 is contained in the short Section 5. Finally, Section 6 provides a consistency analysis and results from numerical simulations of (13). Discretization in space and time 2.1. Inverse distribution functions. Before defining the discrete quantities, let us recall some basic facts from the continuous context. We denote by P(Ω) = u : Ω → R + : Ω u(x) dx = M the space of densities of total mass M on Ω, and we endow P(Ω) with the L 2 -Wasserstein metric W. We refer to [43] for a comprehensive introduction to the topic. For our purposes here, it suffices to know that convergence with respect to W is equivalent to weak-convergence in L 1 (Ω), and that the L 2 -Wasserstein distance on P(Ω) is isometrically equivalent to the usual L 2 -distance on the space X = {X : [0, M ] → Ω : X continuous and strictly increasing, with X(0) = a, X(M ) = b} of inverse distribution functions X. The isometry is given as follows. Lemma 4. Given u 0 , u 1 ∈ P(Ω), introduce their Lagrangian maps X 0 , X 1 ∈ X such that ξ = X j (ξ) 0 u j (x) dx for all ξ ∈ [0, M ]. Then W(u 0 , u 1 ) = X 0 − X 1 L 2 ([0,M ]) . Above, the name Lagrangian map is underlined by the following change of variables formula, Ω ϕ(x)u(x) dx = M 0 ϕ X(ξ) dξ,(17) that holds for every bounded and continuous test function ϕ ∈ C 0 ([a, b]). 2.2. Ansatz space. Fix a discretization parameter K ∈ N, which is the number of degress of freedom plus one. We will need both the integers and the half-integers between 0 and K, that is For discretization of Ω = [a, b], we consider (non-equidistant) grids from I + K = {1,x δ = x = (x 1 , . . . , x K−1 ) a < x 1 < . . . < x K−1 < b ⊆ (a, b) K−1 . By definition, x ∈ x δ is a vector with K −1 components, but we shall frequently use the convention that x 0 = a and x K = b. In the convex set X of inverse distribution functions, we single out the (K − 1)-dimensional open and convex subset X δ = X ∈ X X is affine on each [ξ k−1 , ξ k ] . Functions X ∈ X δ are called Lagrangian maps, since they map the (fixed reference) mesh (ξ 0 , ξ 1 , . . . , ξ K ) to a (variable) mesh x ∈ x δ . There is a one-to-one correspondence between grid vectors x ∈ x δ and inverse distribution function X ∈ X δ , explicitly given by X = X δ [ x] = k∈I 0 K x k θ k ,(18) where the θ k : [0, M ] → R are the usual affine hat functions, with θ k (ξ ) = δ k, . Further, the density function u δ [ x] ∈ P(Ω) associated to X δ [ x] is u δ [ x](x) = κ∈I 1/2 K z κ 1 (x κ− 1 2 ,x κ+ 1 2 ] (x),(19) where the vector z = z δ [ x] = (z 1/2 , . . . , z K−1/2 ) of weights z κ = δ x κ+ 1 2 − x κ− 1 2(20) is such that each interval (x κ− 1 2 , x κ+ 1 2 ] contains the same amount δ of total mass. The following convention reflects the no-flux boundary conditions: z − 1 2 = z 1 2 , z K+ 1 2 = z K− 1 2 .(21) We finally introduce the associated (K − 1)-dimensional submanifold P δ (Ω) := u δ [x δ ] ⊂ P(Ω) as the image of the injective map u δ : x δ → P δ (Ω). 2.3. A metric on the ansatz space. Below, we define a "Wasserstein-like" metric W δ on the ansatz space P δ (Ω). For motivation of that definition, observe that P δ (Ω) is a geodesic submanifold of P(Ω), hence the restriction W δ of the genuine L 2 -Wasserstein distance W to P δ (Ω) appears as a natural candidate for W δ . Thanks to the flatness of W in one space dimension, see Lemma 4, the pull-back metric of W on x δ induced by u δ is a homogeneous quadratic form. More precisely, W u δ [ x 0 ], u δ [ x 1 ] 2 = K−1 k=1 ( x 1 k − x 0 k ) W k ( x 1 − x 0 ) for all x 0 , x 1 ∈ x δ ,(22) where the positive matrix W ∈ R (K−1)×(K−1) is tridiagonal. This approach has been followed in our previous work [40]. Here, we take a modified approach and use (22) to define a metric W δ on P δ (Ω), but with the simpler matrix δ1 K−1 in place of W above. In other words: up to a factor δ 1/2 , the pull-back metric of W δ via u δ is the usual Euclidean distance on x δ . Remark 5. Our proof of convergence heavily relies on several explicit estimates of quantities with respect to the metric W δ . With the rescaled scalar product ·, · δ and norm · δ defined for v, w ∈ R K−1 by v, w δ = δ K−1 k=1 v k w k , v δ = δ K−1 k=1 v 2 k 1/2 , the distance W δ is conveniently written as W δ (u δ [ x 0 ], u δ [ x 1 ]) = x 1 − x 0 δ . In [40,Lemma 3.2], we have shown the following. Lemma 6. W δ is equivalent to the Wasserstein metric restricted to P δ (Ω), uniformly in K: 1 6 W δ (u 0 , u 1 ) 2 ≤ W(u 0 , u 1 ) 2 ≤ W δ (u 0 , u 1 ) 2 for all u 0 , u 1 ∈ P δ (Ω).(23) Note that, as a direct consequence of (23), we obtain that X δ [ x 0 ] − X δ [ x 1 ] L 2 ([0,M ]) ≤ x 0 − x 1 δ . We shall not elaborate further on the point in which sense the thereby defined metric W δ is a good approximation of the L 2 -Wasserstein distance on P δ (Ω). However, Theorem 2 validates our choice a posteriori. For results concerning the Γ-convergence of discretized transport metrics to the Wasserstein distance see [25]. 2.4. Functions on P δ (Ω). When discussing functions on P δ (Ω) in the following, we always assume that these are given in the form f : x δ → R. We denote the first and second derivatives of f by ∂ x f : x δ → R K−1 and by ∂ 2 x f : x δ → R (K−1)×(K−1) , respectively, with components [∂ x f ( x)] k = ∂ x k f ( x) and [∂ 2 x f ( x)] k,l = ∂ x k ∂ x l f ( x).(24)Example 7. Each component z κ of z = z δ [ x] is a function on x δ , and ∂ x z κ = −z 2 κ e κ+ 1 2 − e κ− 1 2 δ ,(25) where e k ∈ R K−1 is the kth canonical unit vector, with the convention e 0 = e K = 0. We introduce further the gradient ∇ δ f ( x) = δ −1 ∂ x f ( x), where the scaling by δ −1 is chosen such that, for arbitrary vectors v ∈ R K−1 , v, ∇ δ f ( x) δ = K−1 k=1 v k ∂ x k f ( x). The gradient flow of a function f on P δ (Ω) with respect to W δ is then defined as the solution x : [0; ∞) → x δ for the system of ordinary differential equationṡ x = −∇ δ f ( x), or, more explicitly,ẋ k = −δ −1 ∂ x k f ( x), for each k ∈ I + K .(26) 2.4.1. The discretized Boltzmann entropy. The Boltzmann entropy H as defined in (6) is a nonnegative functional on P(Ω), which vanishes precisely on the constant function u ≡ M/(b − a). In analogy to [40], we introduce a discretization H δ : x δ → R of the Boltzmann entropy H by restriction to P δ (Ω): H δ ( x) := H(u δ [ x]) = Ω u δ [ x] ln u δ [ x] dx − H 0 = δ κ∈I 1/2 K ln z κ − H 0 , where H 0 was defined in (6), and z = z δ [ x]. Naturally, H δ inherits non-negativity, and vanishes only for x with x k = a + (b − a)k/K. For the derivatives, we obtain -using the rule (25) - ∂ x H δ ( x) = −δ κ∈I 1/2 K z κ e κ− 1 2 − e κ+ 1 2 δ = δ k∈I + K z k+ 1 2 − z k− 1 2 δ e k ,(27)∂ 2 x H δ ( x) = δ κ∈I 1/2 K z 2 κ e κ− 1 2 − e κ+ 1 2 δ e κ− 1 2 − e κ+ 1 2 δ T .(28) It is obvious that ∂ 2 x H δ is positive semi-definite, i.e., that H δ is convex. 2.4.2. The discretized Fisher information. The discrete Fisher information F δ : x δ → R is not defined by restriction of F from (7). Instead, we mimick (10) and define accordingly F δ ( x) = 1 2 ∇ δ H δ ( x) 2 δ = δ 2 k∈I + K z k+ 1 2 − z k− 1 2 δ 2 , using (27). Thanks to this simple structure, the gradient flow equation for F δ has an explicit and compact representation. Using the rule (25), the representation (28) and the convention (21), we obtain with z = z δ [ x]: ∇ δ F δ ( x) = δ −2 ∂ 2 x H δ ( x)∂ x H δ ( x) = κ∈I 1/2 K , k∈I + K z 2 κ z k+ 1 2 − z k− 1 2 δ e κ+ 1 2 − e κ− 1 2 δ e κ+ 1 2 − e κ− 1 2 δ T e k (29) = κ∈I 1/2 K z 2 κ z κ+1 − 2z κ + z κ−1 δ 2 e κ+ 1 2 − e κ− 1 2 δ .(30) This should be understood as a discretization of the differential operator (Z 2 Z ξξ ) ξ appearing on the right hand side of (12). Remark 8. Without calculating the second derivative ∂ 2 x F δ explicitly, we remark that it is unbounded from below on X δ , hence F δ is not λ-convex for any λ ∈ R. This is in agreement with the fact that already the original Fisher information F is not geodesically λ-convex in the Wasserstein metric, see [12]. Time stepping. For the definition of the fully discrete scheme for solution of (8), we discretize the spatially discrete gradient flow equatioṅ x = −∇ δ F δ ( x)(31) also in time, using minimizing movements. To this end, fix a time step with τ > 0; we combine the spatial and temporal mesh widths in a single discretization parameter ∆ = (τ ; δ). For each y ∈ x δ , introduce the Yosida-regularized Fisher information F ∆ (·; y) : (31) is now defined inductively from a given initial datum x 0 ∆ by choosing each x n ∆ as a global minimizer of F ∆ (·; x n−1 ∆ ). Below, we prove that such a minimizer always exists, see Lemma 10. In practice, one wishes to define x n ∆ as -preferably unique -solution of the Euler-Lagrange equations associated to F ∆ (·; x n−1 ∆ ), which leads to the implicit Euler time stepping: x δ → R by F ∆ ( x; y) = 1 2τ x − y 2 δ + F δ ( x). A fully discrete approximation ( x n ∆ ) ∞ n=0 ofx − x n−1 ∆ τ = −∇ δ F δ ( x).(32) Using the explicit representation (30) of ∂ x F δ , it is immediately seen that (32) is indeed the same as (13). Equivalence of (32) and the minimization problem is guaranteed at least for sufficiently small τ > 0. Proposition 9. For each discretization ∆ and every initial condition x 0 ∈ x δ , the sequence of equations (32) can be solved inductively. Moreover, if τ > 0 is sufficiently small with respect to δ and F δ ( x 0 ), then each equation (32) possesses a unique solution with F δ ( x) ≤ F δ ( x 0 ), and that solution is the unique global minimizer of F ∆ (·; x n−1 ∆ ). The proof of this proposition is a consequence of the following rather technical lemma. Lemma 10. Fix a spatial discretization parameter δ and a bound C > 0. Then for every y ∈ x δ with F δ ( y) ≤ C, the following are true: • for each τ > 0, the function F ∆ (·; y) possesses at least one global minimizer x * ∈ x δ ; • there exists a τ C > 0 independent of y such that for each τ ∈ (0, τ C ), the global minimizer x * ∈ x δ is strict and unique, and it is the only critical point of F ∆ (·; y) with F δ ( x) ≤ C. Proof. First, observe that the sublevel A C := F −1 δ ([0, C + 1]) ⊂ x δ is a compact subset of R K−1 . Indeed, A C is a relatively closed subset of x δ by continuity of F δ . Moreover, thanks to (103), every x ∈ A C satisfies x κ+ 1 2 − x κ− 1 2 ≥ x for all κ ∈ I 1/2 K with a positive constant x that depends on C only. Thus A C does not touch the boundary (in the ambient R K−1 ) of x δ . Consequently, A C is closed and bounded in R K−1 . Let y ∈ x δ with F δ ( y) ≤ C be given. The restriction of the continuous function F ∆ (·; y) to the compact and nonempty (since it contains y) set A C possesses a minimizer x * ∈ A C . We clearly have F δ ( x * ) ≤ F δ ( y) ≤ C, and so x * lies in the interior of A C and therefore is a global minimizer of F ∆ (·; y). This proves the first claim. Since F δ : x δ → R is smooth, its restriction to A C is λ C -convex with some λ C ≤ 0, i.e., ∂ 2 x F δ ( x) ≥ λ C 1 K−1 for all x ∈ A C . Independently of y, we have that ∂ 2 x F ∆ ( x; y) = ∂ 2 x F δ ( x) + δ τ 1 K−1 , which means that x → F ∆ ( x; y) is strictly convex on A C if 0 < τ < τ C := δ (−λ C ) . Consequently, each such F ∆ (·; y) has at most one critical point x * in the interior of A C , and this x * is necessarily a strict global minimizer. 2.6. Spatial interpolations. Consider a fully discrete solution ( x n ∆ ) ∞ n=0 . For notational simplification, we write the entries of the vectors x n ∆ and z n ∆ = z δ [ x n ∆ ] as x k and z κ , respectively, whenever there is no ambiguity in the choice of ∆ and the time step n. Recall that u n ∆ = u δ [ x n ∆ ] ∈ P δ (Ω) defines a sequence of densitites on [a, b] which are piecewise constant with respect to the (non-uniform) grid (a, x 1 , . . . , x K−1 , b). To facilitate the study of convergence of weak derivatives, we introduce also piecewise affine interpolations z n ∆ : [0, M ] → R + and u n ∆ : [a, b] → R + . In addition to ξ k = kδ for k ∈ I 0 K , introduce the intermediate points ξ κ = κδ for κ ∈ I 1/2 K . Accordingly, introduce the intermediate values for the vectors x n ∆ and z n ∆ : x κ = 1 2 x κ+ 1 2 + x κ− 1 2 ) for κ ∈ I 1/2 K , z k = 1 2 z k+ 1 2 + z k− 1 2 for k ∈ I + K . Now define • z n ∆ : [0, M ] → R as the piecewise affine interpolation of the values (z 1 2 , z 3 2 , . . . , z K− 1 2 ) with respect to the equidistant grid (ξ 1 2 , ξ 3 2 , . . . , ξ K− 1 2 ), and • u n ∆ : [a, b] → R as the piecewise affine function with u n ∆ • X n ∆ = z n ∆ .(33) Our convention is that z n ∆ (ξ) = z 1 2 for 0 ≤ ξ ≤ δ/2 and z n ∆ (ξ) = z K− 1 2 for M − δ/2 ≤ ξ ≤ M , and accordingly u n ∆ (x) = z 1 2 for x ∈ [a, x 1 2 ] and u n ∆ (x) = z K− 1 2 for x ∈ [x K− 1 2 , b]. The definitions have been made such that x k = X n ∆ (ξ k ), z k = z(ξ k ) = u(x k ) for all k ∈ I 0 K ∪ I 1/2 K .(34) Notice that u n ∆ is piecewise affine with respect to the "double grid" (x 0 , x 1 2 , x 1 , . . . , x K− 1 2 , x K ), but in general not with respect to the subgrid (x 0 , x 1 , . . . , x K ). By direct calculation, we obtain for each k ∈ I + K that ∂ x u (x k− 1 2 ,x k ) = z k − z k− 1 2 x k − x k− 1 2 = z k+ 1 2 − z k− 1 2 x k − x k−1 = z k− 1 2 z k+ 1 2 − z k− 1 2 δ , ∂ x u (x k ,x k+ 1 2 ) = z k+ 1 2 − z k x k+ 1 2 − x k = z k+ 1 2 − z k− 1 2 x k+1 − x k = z k+ 1 2 z k+ 1 2 − z k− 1 2 δ .(35) Trivially, we also have that ∂ x u vanishes identically on the intervals (a, x 1 2 ) and (x K− 1 2 , b). A priori estimates and compactness Throughout this section, we consider a sequence ∆ = (τ ; δ) of discretization parameters such that δ → 0 and τ → 0 in the limit, formally denoted by ∆ → 0. We assume that a fully discrete solution ( x n ∆ ) ∞ n=0 is given for each ∆-mesh, defined by inductive minimization of the respective F ∆ . The sequencesū ∆ , u ∆ , z ∆ and X ∆ of spatial interpolations are defined from the respective x ∆ accordingly. For the sequence of initial conditions x 0 ∆ , we assume that u 0 ∆ → u 0 weakly in L 1 (Ω), that there is some finite H with H δ ( x 0 ∆ ) ≤ H for all ∆,(36) and that (τ + δ)F δ ( x 0 ∆ ) → 0 as ∆ → 0.(37) Further, we use {q} τ to denote the constant in time interpolations of sequences (q n ) ∞ n=0 with step size τ > 0, that is {q} τ (t) := q n for t ∈ ((n − 1)τ ], nτ ], {q} τ (0) := q 0 . 3.1. Energy inequality. The following basic energy estimates are classical for gradient flows. Lemma 11. One has that F δ is monotone, i.e., F δ ( x n ∆ ) ≤ F δ ( x n−1 ∆ ), and further: F δ ( x n ∆ ) ≤ F δ ( x 0 ∆ ) for all n ≥ 0,(38)x n ∆ − x n ∆ 2 δ ≤ 2F δ ( x 0 ∆ ) (n − n)τ for all n ≥ n ≥ 0,(39)τ ∞ n=1 x n ∆ − x n−1 ∆ τ 2 δ = τ ∞ n=1 ∇ δ F δ ( x n ∆ ) 2 δ ≤ 2F δ ( x 0 ∆ ).(40) Proof. The monotonicity (38) follows (by induction on n) from the definition of x n ∆ as minimizer of F ∆ (·; x n−1 ∆ ): F δ ( x n ∆ ) ≤ 1 2τ x n ∆ − x n−1 ∆ 2 δ + F δ ( x n ∆ ) = F ∆ ( x n ∆ ; x n−1 ∆ ) ≤ F ∆ ( x n−1 ∆ ; x n−1 ∆ ) = F δ ( x n−1 ∆ ) . Moreover, summation of these inequalities from n = n + 1 to n = n yields τ 2 n n=n+1 x n ∆ − x n−1 ∆ δ τ 2 ≤ F δ ( x n ∆ ) − F δ ( x n ∆ ) ≤ F δ ( x 0 ∆ ). For n = 0 and n → ∞, we obtain the first part of (40). The second part follows by (32). If instead we combine the estimate with Jensen's inequality, we obtain x n ∆ − x n ∆ δ ≤ τ n n=n+1 x n ∆ − x n−1 ∆ δ τ ≤ τ n n=n+1 x n ∆ − x n−1 ∆ δ τ 2 1/2 τ (n − n) 1/2 , which leads to (39). Entropy dissipation. The key to our convergence analysis is a refined a priori estimate, which follows from the dissipation of the entropy H δ along the fully discrete solution. Lemma 12. One has that H δ is monotone, i.e., H δ ( x n ∆ ) ≤ H δ ( x n−1 ∆ ), and further: τ ∞ n=1 δ κ∈I 1/2 K z 2 κ z κ+1 − 2z κ + z κ−1 δ 2 2 ≤ H δ ( x 0 ∆ ).(41) Proof. By convexity of H δ and the discrete evolution (32), we have H δ ( x n−1 ∆ ) − H δ ( x n ∆ ) ≥ ∇ δ H δ ( x n ∆ ), x n−1 ∆ − x n ∆ δ = τ ∇ δ H δ ( x n ∆ ), ∇ δ F δ ( x n ∆ ) δ for each n = 1, 2, . . . Evaluate the (telescopic) sum with respect to n and use that H δ ≥ 0 to obtain τ ∞ n=1 ∇ δ H δ ( x n ∆ ), ∇ δ F δ ( x n ∆ ) δ ≤ H δ ( x 0 ∆ ). It remains to make the scalar product explicit, using (27) and (30): ∇ δ H δ , ∇ δ F δ δ = δ κ∈I 1/2 K , k∈I + K z 2 κ z κ+1 − 2z κ + z κ−1 δ 2 z k+ 1 2 − z k− 1 2 δ e κ+ 1 2 − e κ− 1 2 δ T e k = δ κ∈I 1/2 K z 2 κ z κ+1 − 2z κ + z κ−1 δ 2 2 , using that z − 1 2 = z 1 2 and z K+ 1 2 = z K− 1 2 , according to our convention (21). We draw several conclusions from (41). The first is an a priori estimate on the the ξ-derivative of the affine functions z n ∆ . Lemma 13. One has that τ ∞ n=1 ∂ ξ z n ∆ 4 L 4 ([0,M ]) = τ ∞ n=1 δ k∈I + K z n k+ 1 2 − z n k− 1 2 δ 4 ≤ 9H.(42)Remark 14. Morally, a bound on ∂ ξ z in L 4 ([0, M ]) corresponds to a bound on ∂ x 4 √ u in L 4 (Ω). Proof. Fix n ∈ N. Invoking our convention (21), one obtains ∂ ξ z n ∆ 4 L 4 (Ω) = k∈I 0 K (z k+ 1 2 − z k− 1 2 ) z k+ 1 2 − z k− 1 2 δ 3 = − k∈I 1/2 K z κ z κ+1 − z κ δ 3 − z κ − z κ−1 δ 3 = (A) Using the elementary identity (p 3 −q 3 ) = (p−q)(p 2 +q 2 +pq) and Young's inequality, one obtains further (A) = −δ κ∈I 1/2 K z κ z κ+1 − 2z κ + z κ−1 δ 2 × × z κ+1 − z κ δ 2 + z κ − z κ−1 δ 2 + z κ+1 − z κ δ z κ − z κ−1 δ ≤ 3δ 2 κ∈I 1/2 K z κ z κ+1 − 2z κ + z κ−1 δ 2 z κ+1 − z κ δ 2 + z κ − z κ−1 δ 2 ≤ 3 2   δ κ∈I 1/2 K z 2 κ z κ+1 − 2z κ + z κ−1 δ 2 2    1/2   4δ k∈I 0 K z k+ 1 2 − z k− 1 2 δ 4   1/2 . Note that the last sum above is again the L 4 -norm of ∂ ξ z n . Taking the square on both sides, dividing by the L 4 -norm, summing over n = 1, 2, . . ., and finally applying the entropy dissipation estimate (41), one arrives at (42). The a priori estimate (42) is the basis for almost all of the further estimates. For instance, the following control on the oscillation of the z-values at neighboring grid points is a consequence of (42). Lemma 15. One has τ ∞ n=1 δ k∈I + K z n k+ 1 2 z n k− 1 2 − 1 4 + z n k− 1 2 z n k+ 1 2 − 1 4 ≤ 18(b − a) 4 H δ ( x 0 ∆ ).(43) Moreover, given T > 0, then for each N ∈ N with N τ ≤ T , one has τ N n=1 δ k∈I + K z n k+ 1 2 z n k− 1 2 − 1 2 + z n k− 1 2 z n k+ 1 2 − 1 2 ≤ 6(b − a) 2 T 1/2 H δ ( x 0 ∆ ) 1/2 δ 1/2 .(44) Proof. Recall that z κ ≥ δ/(b − a) for all κ, see (102). Consider the first term in the inner summation in (43): δ k∈I + K z n k+ 1 2 z n k− 1 2 − 1 4 = δ k∈I + K δ z n k− 1 2 4 z n k+ 1 2 − z n k− 1 2 δ 4 ≤ (b − a) 4 z n ∆ 4 L 4 (Ω) . The same estimate holds for the second term. The claim (43) is now directly deduced from (42) above. The proof of the second claim (44) is similar, using the Cauchy-Schwarz inequality instead of the modulus estimate: δ k∈I + K z n k+ 1 2 z n k− 1 2 − 1 2 = δ k∈I + K δ z n k− 1 2 2 z n k+ 1 2 − z n k− 1 2 δ 2 ≤   δ k∈I + K δ z n k− 1 2 4   1/2 z n ∆ 2 L 4 (Ω) . Use estimate (101), sum over n = 1, . . . , N , and apply the Cauchy-Schwarz inequality to this second summation. This yields τ N n=1 δ k∈I + K z n k+ 1 2 z n k− 1 2 − 1 2 ≤ δ 1/2 (b − a) 2 τ N n=1 1 1/2 τ ∞ n=1 z n ∆ 4 L 4 (Ω) 1/2 . Invoking again (42), and recalling that N τ ≤ T , we arrive at (44). We are now going to prove the main consequence from the entropy dissipation (38), namely a control on the total variation of u n ∆ . This estimate is the key ingredient for obtaining strong compactness in Proposition 20. Recall that several equivalent definitions of the total variation of f ∈ L 1 (Ω) exist. Most generally, TV [f ] = sup Ω f (x)∂ x φ(x) dx ; φ ∈ C 0,1 (Ω), sup x \|φ(x)\| ≤ 1 .(45) Since we are dealing with functions f : Ω → R that are piecewise smooth on intervals and only have jump discontinuities, the following definition is most appropriate: TV [f ] = sup    J−1 j=1 \|f (r j+1 ) − f (r j )\| : J ∈ N, a < r 1 < r 2 < · · · < r N < b    .(46) Further recall the notation f x = lim x↓x f (x) − lim x↑x f (x). for the height of the jump in f (x)'s value at x =x. Lemma 16. One has τ ∞ n=1 TV ∂ x u n ∆ 2 ≤ 10(b − a)H.(47) Proof. Fix n. Observe that u n ∆ is smooth on Ω except for the points x 1 2 , x 1 , . . . , x K− 1 2 , with derivatives given by ∂ x u n ∆ = 1 2 u n ∆ ∂ x u n ∆ , ∂ xx u n ∆ = − 1 4 u n ∆ 3 ∂ x u n ∆ 2 ≤ 0. Therefore, ∂ x ũ n ∆ is monotonically decreasing in between the (potential) jump discontinuities at the points x 1 2 , x 1 , . . . , x K− 1 2 . Further, recall that ∂ x u n ∆ (x) = 0 for all x ∈ (a, a + δ/2) and all x ∈ (b − δ/2, b), It follows that the supremum in (46) can be realized (in the limit ε ↓ 0) for a sequence of just J = 2(2K − 1) many points r ε j , chosen as follows: r ε 2i−1 = x i/2 − ε and r ε 2i = x i/2 + ε, for i = 1, . . . , 2K − 1. On the one hand, lim ε↓0 ∂ x u ∆ (r ε 2i−1 ) − ∂ x u ∆ (r ε 2i ) = ∂ x u n ∆ x i/2 .(49) On the other hand, since ∂ x u n ∆ is monotone decreasing in between r ε 2i and r ε 2i+1 , and vanishes near the boundary by (48), we have that lim ε↓0 2K−2 i=1 ∂ x u n ∆ (r ε 2i ) − ∂ x u n ∆ (r ε 2i+1 ) = lim ε↓0 2K−1 i=1 ∂ x u n ∆ (r ε 2i ) − ∂ x u n ∆ (r ε 2i−1 ) ≤ k∈I + K ∂ x u n ∆ x k + κ∈I 1/2 K ∂ x u n ∆ xκ .(50) Summarizing (49) and (50), we obtain the estimate TV ∂ x u n ∆ ≤ 2 k∈I + K ∂ x u n ∆ x k + 2 κ∈I 1/2 K ∂ x u n ∆ xκ ,(51) In view of (35), we have that ∂ x u n ∆ x k = 1 2 √ z k (z k− 1 2 − z k+ 1 2 ) 2 δ for k ∈ I + K , ∂ x u n ∆ xκ = 1 2 √ z κ z κ+1 − 2z κ + z κ−1 δ for κ ∈ I 1/2 K . Accordingly, using that 1/z k ≤ (1/z k+ 1 2 +1/z k− 1 2 )/2 by the arithmetic-harmonic mean inequality, k∈I + K ∂ x u n ∆ x k = δ 2 k∈I + K (z k− 1 2 − z k+ 1 2 ) 2 δ 2 · 1 √ z k ≤ 1 2   δ k∈I + K z k− 1 2 − z k+ 1 2 δ 4   1/2 k∈I + K δ z k 1/2 = 1 2 ∂ ξ z n ∆ 2 L 4 (Ω) (b − a) 1/2 ,(52) and also κ∈I 1/2 K ∂ x u n ∆ xκ = δ 2 κ∈I 1/2 K z κ z κ+1 − 2z κ + z κ−1 δ 2 · 1 √ z κ ≤ 1 2   δ κ∈I 1/2 K z 2 κ z κ+1 − 2z κ + z κ−1 δ 2 2    1/2 (b − a) 1/2 .(53) Combine (52) with the L 4 bound from (42), and (53) with the entropy dissipation inequality (41). Inserting this into (51) to obtain the claim (47). Convergence of time interpolants. Recall that we require the a priori bound (36) on the initial entropy, but only (37) on the initial Fisher information. This estimate improves over time. Lemma 17. One has, for every N ≥ 1, F δ ( x N ∆ ) ≤ 3 2 (M H) 1/2 (N τ ) −1/2 .(54) Consequently, {F δ ( x ∆ )} τ (t) is bounded for each t > 0, uniformly in ∆. Proof. Since F δ ( x n ∆ ) is monotonically decreasing in n (for fixed ∆), it follows that F δ ( x N ∆ ) ≤ 1 N N n=1 F δ ( x n ∆ ) = 1 2N N n=1 δ k∈I + K z n k+ 1 2 − z n k− 1 2 δ 2 ≤ 1 2N τ   τ N n=1 δ k∈I + K 1   1/2   τ ∞ n=1 δ k∈I + K z n k+ 1 2 − z n k− 1 2 δ 4   1/2 ≤ 1 2N τ (N τ M ) 1/2 (9H) 1/2 = 3 2 (M H) 1/2 (N τ ) −1/2 , as desired. In the following, we use the notation [t, t] R + to denote time intervals with 0 < t < t < ∞. Lemma 18. We have that, for each [t, t] R + , sup ∆ sup t∈[t,t] { u ∆ } τ (t) H 1 (Ω) < ∞,(55) and that, as ∆ → 0, sup t∈R+ { u ∆ } τ (t) − {ū ∆ } τ (t) L ∞ (Ω) → 0.(56) Proof. For each n ∈ N, ∂ x u n ∆ 2 L 2 (Ω) = k∈I + K (x n k+ 1 2 − x n k ) z n k+ 1 2 − z n k x n k+ 1 2 − x n k 2 + (x n k − x n k− 1 2 ) z n k − z n k− 1 2 x n k − x n k− 1 2 2 ≤ δ k∈I + K z n k+ 1 2 + z n k− 1 2 2 z n k+ 1 2 − z n k− 1 2 δ 2 ≤ F δ ( x n ∆ ) max κ∈I 1/2 K z n κ . Now combine this with the estimates (54) from above and (103) from the appendix to obtain (55). Estimate (56) follows directly from the elementary observation that sup x∈Ω \|ū n ∆ (x) − u n ∆ (x)\| 2 ≤ max k∈I + K z n k+ 1 2 − z n k− 1 2 2 ≤ δF δ ( x n ∆ ) ≤ δF δ ( x 0 ∆ ), and an application of (37). Proposition 19. There exists a function u * : R ≥0 × Ω → R ≥0 with u * ∈ C 1/2 loc (R + ; P(Ω)) ∩ L ∞ loc (R + ; H 1 (Ω)),(57) and there exists a subsequence of ∆ (still denoted by ∆), such that, for every [t, t] R + , the following are true: {u ∆ } τ (t) −→ u * (t) in P(Ω), uniformly with respect to t ∈ [t, t],(58){u ∆ } τ , { u ∆ } τ −→ u * uniformly on [t, t] × Ω,(59){X ∆ } τ (t) −→ X * (t) in L 2 ([0, M ]), uniformly with respect to t ∈ [t, t],(60) where X * ∈ C 1/2 (R + ; L 2 ([0, M ])) is the Lagrangian map of u * . Proof. Fix t > 0. From the discrete energy inequality (39), the bound on the Fisher information in Lemma 17, and the equivalence (23) of W δ with the usual L 2 -Wasserstein metric W, it follows by elementary considerations that W {ū ∆ } τ (t), {ū ∆ } τ (s) 2 ≤ C(t) \|t − s\| + τ ,(61) for all t, s ≥ t. Moreover, since Ω = [a, b] is compact, also P(Ω) is compact. Hence the generalized version of the Arzela-Ascoli theorem from [1, Proposition 3.3.1] is applicable and yields the convergence of a subsequence of ({ū ∆ } τ ) to a limit u t in P(Ω), locally uniformly with respect to t ∈ [t, ∞). The Hölder-type estimate (61) implies u t ∈ C 1/2 ([t, ∞); P(Ω)). The claim (60) is a consequence of the equivalence between the Wasserstein metric on P(Ω) and the L 2 -metric on X, see Lemma 4. Clearly, the previous argument applies to every choice of t > 0. Using a diagonal argument, one constructs a limit u * defined on all R + , such that u t is the restriction of u * to [t, ∞). For the rest of the proof, let some [t, t] R + be fixed. For proving (59), it suffices to show that { u ∆ } τ → u * uniformly on [t, t] × Ω: indeed, (56) implies that if { u ∆ } τ converges uniformly to some limit, so does {ū ∆ } τ . As an intermediate step towards proving uniform convergence of { u ∆ } τ , we show that u ∆ (t) −→ u * (t) in L 2 (Ω), uniformly in t ∈ [t, t].(62) For t ∈ [t, t], we expand the L 2 -norm as follows: { u ∆ } τ (t) − u * (t) 2 L 2 (Ω) = Ω { u ∆ } τ − u * {ū ∆ } τ (t; x) dx + Ω { u ∆ } τ − u * { u ∆ } τ − {ū ∆ } τ (t; x) dx − Ω { u ∆ } τ − u * u * (t; x) dx. On the one hand, observe that sup t∈[t,t] Ω { u ∆ } τ − u * { u ∆ } τ − {ū ∆ } τ (t; x) dx ≤ sup t∈[t,t] { u ∆ } τ (t) L 1 (Ω) + u * (t) L 1 (Ω) { u ∆ } τ (t) − {ū ∆ } τ (t) L ∞ ≤ sup t∈[t,t] ((2M + (b − a) { u ∆ } τ (t) − {ū ∆ } τ (t) L ∞ ) { u ∆ } τ (t) − {ū ∆ } τ (t) L ∞ ) , which converges to zero as ∆ → 0, using both conclusions from Lemma 18. On the other hand, we can use property (17) to write Ω { u ∆ } τ − u * {ū ∆ } τ (t; x) dx − Ω { u ∆ } τ − u * u * (t; x) dx = M 0 { u ∆ } τ − u * t; {X ∆ } τ (t; x) dξ − M 0 { u ∆ } τ − u * t; X * (t; ξ) dξ. We regroup terms under the integrals and use the triangle inequality. For the first term, we obtain sup t∈[t,t] M 0 { u ∆ } τ t; {X ∆ } τ (t; ξ) − { u ∆ } τ t; X * (t; ξ) dξ ≤ sup t∈[t,t] M 0 {X∆} τ (t;ξ) X * (t,ξ) \|∂ x { u ∆ } τ \| (t; y) dy dξ ≤ sup t∈[t,t] M 0 { u ∆ } τ H 1 (Ω) \|X * − {X ∆ } τ \|(t, ξ) 1/2 dξ ≤ sup t∈[t,t] { u ∆ } τ (t) H 1 (Ω) X * (t) − {X ∆ } τ (t) 1/4 L 2 ([0,M ]) . A similar reasoning applies to the integral involving u * in place of { u ∆ } τ . Together, this proves (62), and it further proves that u * ∈ L ∞ (R + [t, t]; H 1 (Ω)), since the uniform bound on u ∆ from 55 is inherited by the limit. Now the Gagliardo-Nirenberg inequality (107) provides the estimate { u ∆ } τ (t) − u * (t) C 1/6 (Ω) ≤ C { u ∆ } τ (t) − u * (t) 2/3 H 1 (Ω) { u ∆ } τ (t) − u * (t) 1/3 L 2 (Ω) .(63) Combining the convergence in L 2 (Ω) by (62) with the boundedness in H 1 (Ω) from (55), it readily follows that u ∆ (t) → u * (t) in C 1/6 (Ω), uniformly in t ∈ [t, t]. This clearly implies that { u ∆ } τ → u * uniformly on [t, t] × Ω. Proposition 20. Under the hypotheses and with the notations of Proposition 19, we have that √ u * ∈ L 2 (R ≥0 ; H 1 (Ω)), and u ∆ τ → √ u * strongly in L 2 loc (R + ; H 1 (Ω))(64) as ∆ → 0. Notice that ∂ x √ u * ∈ L 2 ([0, t] × Ω) for each t > 0, but strong convergence takes place only on each [t, t] × Ω. Proof. Fix [t, t] R + . By definition (45) of the total variation, ∂ x f 2 L 2 (Ω) = Ω ∂ x f (x) 2 dx ≤ sup x∈Ω \|f (x)\| TV [∂ x f ] holds for every Lipschitz function f : Ω → R. The functions u n ∆ are obviously Lipschitz continuous. Moreover, thanks to weak lower semi-continuity of the total variation, it follows from (47) that ∞ 0 TV [∂ x √ u * ] 2 dt ≤ 10(b − a)H. In particular, the weak derivative x → ∂ x √ u * (t; x) is in L ∞ (Ω) -and thus x → √ u * (t; x) is Lipschitz -for almost every t > 0. Using that TV [f − g] ≤ TV [f ] + TV [g], we obtain that t t ∂ x u ∆ τ − √ u * 2 L 2 (Ω) dt ≤ (t − t) 1/2 sup [t,t]×Ω u ∆ τ − √ u * 2 ∞ 0 TV ∂ x { √ u ∆ } τ 2 + TV [∂ x √ u * ] 2 dt 1/2 . For ∆ → 0, the first term on the right-hand side converges to zero by (59), and the second term remains bounded by (47). This proves (64). To show square integrability of the limit, fix some T > 0. Below, N is always such that T < N τ < T + 1. A direct calculation yields that 4 Ω ∂ x u n ∆ 2 (x) dx = M 0 ∂ ξ z n ∆ (ξ) 2 z n ∆ (ξ)∂ ξ X n ∆ (ξ) dξ. From the properties of X n ∆ and z n ∆ as linear interpolations, one easily deduces that 1 z n ∆ (ξ)∂ ξ X n ∆ (ξ) ≤ z n k+ 1 2 z n k− 1 2 + z n k− 1 2 z n k+ 1 2 for all ξ ∈ (ξ k− 1 2 , ξ k+ 1 2 ). Therefore, 4 T 0 Ω u ∆ 2 τ (t; x) dx dt ≤ τ N n=1 δ k∈I + K z n k+ 1 2 − z n k− 1 2 δ 2 z n k+ 1 2 z n k− 1 2 + z n k− 1 2 z n k+ 1 2 ≤ 2   τ N n=1 δ k∈I + K z n k+ 1 2 − z n k− 1 2 δ 4   1/2   τ N n=1 δ k∈I + K   z n k+ 1 2 z n k− 1 2 2 + z n k− 1 2 z n k+ 1 2 2     1/2 . The two sums are ∆-uniformly bounded, thanks to the estimates (42) and (44). By lower semicontinuity of norms, √ u * obeys the same bound. Weak formulation of the limit equation To finish our discussion of convergence, we verify that the limit u * obtained in the previous section is indeed a weak solution to (1). From now on, ( x n ∆ ) ∞ n=0 with its derived functionsū ∆ , u ∆ , X ∆ is a (sub)sequence for which the convergence results stated in Proposition 19 and Proposition 20 holds. We continue to assume (36) and (37). The goal of this section is to prove the following. Proposition 21. For every ρ ∈ C ∞ (Ω) with ρ (a) = ρ (b) = 0, and for every ψ ∈ C ∞ c (R ≥0 ), ∞ 0 ψ (t) Ω ρ(x)u * (t; x) dx dt + ψ(0) Ω ρ(x)u 0 (x) dx + ∞ 0 ψ(t) Ω ρ (x)∂ x u * (t; x) + 4ρ (x)∂ x √ u * (t; x) 2 dx dt = 0.(65) For definiteness, fix a spatial test function ρ ∈ C ∞ (Ω) with ρ (a) = ρ (b) = 0, and a temporal test function ψ ∈ C ∞ c (R ≥0 ) with supp ψ ⊂ [0, T ) for a suitable T > 0. Let B > 0 be chosen such that ρ C 4 (Ω) ≤ B, ψ C 1 (R+) ≤ B.(66) For convenience, we assume δ < 1 and τ < 1. Further, we introduce the short-hand notation ρ ( x n ∆ ) = ρ (x n 1 ), . . . , ρ (x n K−1 ) ∈ R K−1 .(67) In the estimates that follow, the non-explicity constants possibly depend on (b − a), T , B, and H, but not ∆. The two main steps in the proof of Proposition 21 are to establish the following estimates, respectively: e 1,∆ := T 0 ψ (t) Ω ρ(x) {ū ∆ } τ (t; x) dx + ψ(t) { ρ ( x ∆ ), ∇ δ F δ ( x ∆ ) δ } τ (t) dt +ψ(0) Ω ρ(x)ū 0 ∆ (x) dx ≤ C (δF δ ( x 0 ∆ )) 1/2 + (τ F δ ( x 0 ∆ )) ,(68) and e 2,∆ : = T 0 ψ(t) Ω ρ (x)∂ x { u ∆ } τ (t; x) + 4ρ (x)∂ x u ∆ τ (t; x) 2 dx − { ρ ( x n ∆ ), ∇ δ F δ ( x n ∆ ) δ } τ (t) dt ≤ Cδ 1/4 .(69) We proceed by proving (68) Proof of (68). Choose N τ ∈ N such that N τ τ ∈ (T, T + 1). Then, using that ψ(N τ τ ) = 0, we obtain after "summation by parts": − T 0 ψ (t) Ω ρ(x) {ū ∆ } τ (t; x) dx dt = − Nτ m=1 mτ (m−1)τ ψ (t) dt Ω ρ(x)ū m ∆ (x) dx = −τ Nτ m=1 ψ(mτ ) − ψ((m − 1)τ ) τ M 0 ρ • X m ∆ (ξ) dξ = τ Nτ n=1 ψ((n − 1)τ ) M 0 ρ • X n ∆ − ρ • X n−1 ∆ τ (ξ) dξ + ψ(0) M 0 ρ • X 0 ∆ (ξ) dξ.(70) A Taylor expansion of the term in the inner integral yields ρ • X n ∆ − ρ • X n−1 ∆ τ = ρ • X n ∆ X n ∆ − X n−1 ∆ τ + τ 2 ρ • X X n ∆ − X n−1 ∆ τ 2 .(71) where X symbolizes suitable "intermediate values" in [0, M ]. We analyze the first term on the right-hand side of (71): using the representation (18) of X ∆ in terms of hat functions θ k , we can write its integral as follows, M 0 ρ • X n ∆ X n ∆ − X n−1 ∆ τ dξ = k∈I + K x n k − x n−1 k τ ξ k+1 ξ k−1 ρ • X n ∆ θ k dξ.(72) On the other hand, since ξ k+1 ξ k−1 θ k dξ = δ,(73) the discrete evolution equation (32) yields that − ρ ( x n ∆ ), ∇ δ F δ ( x n ∆ ) δ = ρ ( x n ∆ ), x n ∆ − x n−1 ∆ τ δ = k∈I + K x n k − x n−1 k τ ξ k+1 ξ k−1 ρ(x n k )θ k (ξ) dξ.(74) Finally, observing that \|X n ∆ (ξ) − x n k \| ≤ (x n k+1 − x n k−1 ) for each ξ ∈ (ξ k−1 , ξ k+1 ), we can estimate the difference of the terms in (72) and (74) with the help of the bound (66) on ρ as follows: M 0 ρ • X n ∆ (ξ) X n ∆ − X n−1 ∆ τ (ξ) dξ − ρ ( x n ∆ ), ∇ δ F δ ( x n ∆ ) δ ≤ k∈I + K x n k − x n−1 k τ ξ k+1 ξ k−1 ρ • X n ∆ (ξ) − ρ (x n k ) θ k (ξ) dξ ≤ Bδ k∈I + K x n k − x n−1 k τ (x n k+1 − x n k−1 ).(75) As a final preparation for the proof of (68), observe that (40). We are now ready to estimate e 1,∆ in (68): R := T 0 ψ(t) { ρ ( x ∆ ), ∇ δ F δ ( x ∆ ) δ } τ (t) dt − τ Nτ n=1 ψ((n − 1)τ ) ρ ( x n ∆ ), ∇ δ F δ ( x n ∆ ) δ ≤   τ Nτ n=1 1 τ nτ (n−1)τ ψ(t) dt − ψ((n − 1)τ ) 2   1/2 τ ∞ n=1 B 2 ∇ δ F δ ( x n ∆ ) 2 δ 1/2 ≤ (T + 1)B 2 τ 2 1/2 (2B 2 F δ ( x 0 ∆ )) 1/2 = C F δ ( x 0 ∆ ) 1/2 τ, using the energy estimatee 1,∆ (70) ≤ R + τ Nτ n=1 ψ((n − 1)τ ) M 0 ρ • X n ∆ − ρ • X n−1 ∆ τ (ξ) dξ − ρ ( x n ∆ ), ∇ δ F δ ( x n ∆ ) δ (71) ≤ R + Bτ Nτ n=1 M 0 ρ • X n ∆ (ξ) X n ∆ − X n−1 ∆ τ (ξ) dξ − ρ ( x n ∆ ), ∇ δ F δ ( x n ∆ ) δ + Bτ 2 M 0 X n ∆ − X n−1 ∆ τ 2 (ξ) dξ (75) ≤ R + B 2   τ ∞ n=1 δ k∈I + K x n k − x n−1 k τ 2   1/2   τ Nτ n=1 δ k∈I + K (x n k+1 − x n k−1 ) 2   1/2 + B 2 τ 2 τ ∞ n=1 X n ∆ − X n−1 ∆ τ 2 L 2 ([0,M ]) ≤ C (τ F δ ( x 0 ∆ )) + B 2 2(b − a) 2 T 1/2 (δF δ ( x 0 ∆ )) 1/2 + B 2 (τ F δ ( x 0 ∆ )) , where we have used the energy estimate (40) and the the bound (101). The proof of (69) requires more calculations, which are distributed in a series of lemmata below. The first step is to derive a fully discrete weak formulation from (32). Lemma 22. With (67), one has that − ρ ( x n ∆ ), ∇ δ F δ ( x n ∆ ) δ = A n 1 − A n 2 + A n 3 + A n 4 ,(76) where A n 1 = δ k∈I + K z n k+ 1 2 − z n k− 1 2 δ 2 z n k+ 1 2 + z n k− 1 2 2 ρ (x n k+1 ) − ρ (x n k−1 ) δ , A n 2 = δ k∈I + K z n k+ 1 2 − z n k− 1 2 δ 2 (z n k+ 1 2 ) 2 + (z n k− 1 2 ) 2 2z n k+ 1 2 z n k− 1 2 ρ (x n k ), A n 3 = δ k∈I + K z n k+ 1 2 − z n k− 1 2 δ (z n k+ 1 2 ) 2 + (z n k− 1 2 ) 2 2 ρ (x n k+1 ) − ρ (x n k ) − (x n k+1 − x n k )ρ (x n k ) δ 2 , A n 4 = δ k∈I + K z n k+ 1 2 − z n k− 1 2 δ (z n k+ 1 2 ) 2 + (z n k− 1 2 ) 2 2 ρ (x n k−1 ) − ρ (x n k ) − (x n k−1 − x n k )ρ (x n k ) δ 2 . Proof. Fix some time index n ∈ N (omitted in the calculations below). Recall the representation of ∇ δ F δ from (30). By definition of ρ ( x ∆ ), it follows via a "summation by parts" that − ∇ x F δ ( x ∆ ), ρ ( x ∆ ) δ = −δ κ∈I 1/2 K 1 δ z κ+1 − z κ δ − z κ − z κ−1 δ z 2 κ ρ (x κ+ 1 2 ) − ρ (x κ− 1 2 ) δ = δ k∈I + K z k+ 1 2 − z k− 1 2 δ 1 δ z k+ 1 2 ρ (x k+1 ) − ρ (x k ) δ − z k− 1 2 ρ (x k ) − ρ (x k−1 ) δ . Using the elementary identity (for arbitrary numbers α ± and β ± ) α + β + − α − β − = α + + α − 2 (β + − β − ) + (α + − α − ) β + + β − 2 , we obtain further: − ∇ x F δ ( x ∆ ), ρ ( x ∆ ) δ = δ k∈I + K z k+ 1 2 − z k− 1 2 δ z 2 k+ 1 2 − z 2 k− 1 2 2δ ρ (x k+1 ) − ρ (x k−1 ) δ (77) + δ k∈I + K z k+ 1 2 − z k− 1 2 δ z 2 k+ 1 2 + z 2 k− 1 2 2 ρ (x k+1 ) − 2ρ (x k ) + ρ (x k−1 ) δ 2 .(78) The sum in (77) equals to A n 1 . In order to see that the sum in (78) equals to −A n 2 + A n 3 + A n 4 , simply observe that the identity x k+1 − x k δ + x k−1 − x k δ = 1 z k+ 1 2 − 1 z k− 1 2 = − z k+ 1 2 − z k− 1 2 z k+ 1 2 z k− 1 2 , makes the coefficient of ρ (x n k ) vanish. Lemma 23. There is a constant C 1 > 0 such that for each N with N τ < T , one has R 1 := τ N n=1 A n 1 − 2 M 0 ∂ ξ z n ∆ (ξ) 2 ρ • X n ∆ (ξ) dξ ≤ C 1 δ 1/4 . Proof. First, observe that by definition of z, M 0 ∂ ξ z n ∆ (ξ) 2 ρ • X n ∆ (ξ) dξ = k∈I + K z n k+ 1 2 − z n k− 1 2 δ 2 ξ k+ 1 2 ξ k− 1 2 ρ • X n ∆ (ξ) dξ, and therefore, by Hölder's inequality, R 1 ≤ R 1/2 1a R 1/2 1b ,(79) with, recalling (42), R 1a = τ N n=1 δ k∈I + K z n k+ 1 2 − z n k− 1 2 δ 4 ≤ τ ∞ n=1 z n ∆ 4 L 4 (Ω) ≤ 9H,(80)R 1b = τ N n=1 δ k∈I + K z n k+ 1 2 + z n k− 1 2 2 ρ (x n k+1 ) − ρ (x n k−1 ) δ − 2 δ ξ k+ 1 2 ξ k− 1 2 ρ • X n ∆ dξ 2 .(81) To simplify R 1b , let us fix n (omitted in the following), and introducex + k ∈ (x k , x k+1 ) and x − k ∈ (x k−1 , x k ) such that ρ (x k+1 ) − ρ (x k−1 ) δ = ρ (x k+1 ) − ρ (x k ) δ + ρ (x k ) − ρ (x k−1 ) δ = ρ (x + k ) x k+1 − x k δ + ρ (x + k ) x k+1 − x k δ = ρ (x + k ) z k+ 1 2 + ρ (x − k ) z k− 1 2 . For each k ∈ I + K , we have that -recalling (73) - z k+ 1 2 + z k− 1 2 2 ρ (x + k ) z k+ 1 2 + ρ (x − k ) z k− 1 2 − 2 δ ξ k+ 1 2 ξ k− 1 2 ρ • X ∆ dξ = 1 2 z k− 1 2 z k+ 1 2 + 1 ρ (x + k ) + z k+ 1 2 z k− 1 2 + 1 ρ (x − k ) − 2 δ ξ k+ 1 2 ξ k− 1 2 ρ • X ∆ dξ = 1 2 z k− 1 2 z k+ 1 2 − 1 ρ (x + k ) + z k+ 1 2 z k− 1 2 − 1 ρ (x − k ) − 2 δ ξ k+ 1 2 ξ k ρ • X ∆ − ρ (x + k ) dξ − 2 δ ξ k ξ k− 1 2 ρ • X ∆ − ρ (x − k ) dξ. Since X ∆ (ξ) ∈ [x k , x k+ 1 2 ] for each ξ ∈ [ξ k , ξ k+ 1 2 ], andx + k ∈ [x k , x k+1 ], it follows that \|X ∆ (ξ) − x + k \| ≤ x k+1 − x k , and therefore 2 δ ξ k+ 1 2 ξ k ρ • X ∆ (ξ) − ρ (x + k ) dξ ≤ B(x k+1 − x k ).(82) A similar estimate holds for the other integral. Thus R 1b ≤ B 2 τ N n=1 δ k∈I + K z n k− 1 2 z n k+ 1 2 − 1 2 + z n k+ 1 2 z n k− 1 2 − 1 2 + 2(x n k+1 − x n k−1 ) 2 . Recalling the estimates (44) and (101), we further conclude that R 1b ≤ B 2 6(b − a) 2 (HT δ) 1/2 + 4T (b − a) 2 δ .(83) In combination with (79) and (80), this proves the claim. Lemma 24. There is a constant C 2 > 0 such that for each N with N τ < T , one has R 2 := τ N n=1 A n 2 − M 0 ∂ ξ z n ∆ (ξ) 2 ρ • X n ∆ (ξ) dξ ≤ C 2 δ 1/4 . Proof. The proof is almost identical to (and even easier than) the one for Lemma 23 above. Again, we have a decomposition of the form R 2 ≤ R 1/2 2a R 1/2 2b , where R 2a equals R 1a from (80), and R 2b = τ N n=1 δ k∈I + K z n k+ 1 2 + z n k− 1 2 2z n k+ 1 2 z n k− 1 2 ρ (x n k ) − 1 δ ξ k+ 1 2 ξ k− 1 2 ρ • X n ∆ dξ 2 . By writing (z n k+ 1 2 ) 2 + (z n k− 1 2 ) 2 2z n k+ 1 2 z n k− 1 2 = 1 2 z n k− 1 2 z n k+ 1 2 − 1 + 1 2 z n k+ 1 2 z n k− 1 2 − 1 + 1, and observing -in analogy to (82) -that 1 δ ξ k+ 1 2 ξ k− 1 2 ρ • X ∆ (ξ) − ρ (x k ) dξ ≤ B(x k+ 1 2 − x k− 1 2 ), we obtain the same bound on R 2b as the one on R 1b from (83). Lemma 25. There is a constant C 3 > 0 such that for each N with N τ ≤ T , one has R 3 := τ N n=1 A n 3 − 1 2 M 0 ∂ ξ z n ∆ (ξ)ρ • X n ∆ (ξ) dξ ≤ C 3 δ 1/4 . Proof. Arguing like in the previous proofs, we first deduce -now by means of Hölder's inequality instead of the Cauchy-Schwarz inequality -that R 3 ≤ R 1/4 3a R 3/4 3b , where R 3a = R 1a , and R 3b = τ N n=1 δ k∈I + K (z n k+ 1 2 ) 2 + (z n k− 1 2 ) 2 2 ρ (x n k+1 ) − ρ (x n k ) − (x n k+1 − x n k )ρ (x n k ) δ 2 − 1 2δ ξ k+ 1 2 ξ k− 1 2 ρ • X n δ dξ 4/3 . Introduce intermediate valuesx + k such that ρ (x n k+1 ) − ρ (x n k ) − (x n k+1 − x n k )ρ (x n k ) = 1 2 (x n k+1 − x n k ) 2 ρ (x + k ) = δ 2 2(z n k+ 1 2 ) 2 ρ (x + k ). Thus we have that (z n k+ 1 2 ) 2 + (z n k− 1 2 ) 2 2 ρ (x n k+1 ) − ρ (x n k ) − (x n k+1 − x n k )ρ (x n k ) δ 2 − 1 2δ ξ k+ 1 2 ξ k− 1 2 ρ • X n δ dξ = 1 4   z n k− 1 2 z n k+ 1 2 2 + 1   ρ (x + k ) − 1 2δ ξ k+ 1 2 ξ k− 1 2 ρ • X n δ dξ = 1 4 z n k− 1 2 z n k+ 1 2 + 1 z n k− 1 2 z n k+ 1 2 − 1 ρ (x + k ) − 1 2δ ξ k+ 1 2 ξ k− 1 2 ρ • X n ∆ − ρ (x + k ) dξ. By the analogue of (82), it follows further that R 3b ≤ 2B 4/3 τ N n=1 δ k∈I + K   z n k− 1 2 z n k+ 1 2 + 1 4/3 z n k− 1 2 z n k+ 1 2 − 1 4/3 + (x n k+1 − x n k−1 ) 4/3   ≤ 2B 4/3   τ N n=1 δ k∈I + K z n k− 1 2 z n k+ 1 2 + 1 4   1/3   τ N n=1 δ k∈I + K z n k− 1 2 z n k+ 1 2 − 1 2   2/3 + 2B 4/3 T (b − a) 4/3 δ, where we have used (101). At this point, the estimates (43) and (44) are used to control the first and the second sum, respectively. Along the same lines, one proves the analogous estimate for A 4 in place of A 3 . It remains to identify the integral expressions inside R 1 to R 3 with those in the weak formulation (65). Lemma 26. One has that M 0 ∂ ξ z n ∆ (ξ)ρ • X n ∆ (ξ) dξ = Ω ∂ x u n ∆ (x)ρ (x) dx,(84)R 5 := τ N n=1 M 0 ∂ ξ z n ∆ (ξ) 2 ρ • X n ∆ (ξ) dξ − 4 Ω ∂ x u n ∆ 2 (x)ρ (x) dx ≤ C 5 δ 1/4 ,(85) where (85) holds for each N with N τ ≤ T . Proof. The starting point is relation (33), that is z n ∆ (ξ) = u n ∆ • X n ∆ (ξ)(86) for all ξ ∈ [0, M ]. Both sides of this equation are Lipschitz continuous in ξ, and are differentiable except possibly at ξ 1 2 , ξ 1 , . . . , ξ K− 1 2 . At points ξ of differentiability, we have that ∂ ξ z n ∆ (ξ) = ∂ x u n ∆ • X n ∆ (ξ)∂ ξ X n ∆ (ξ) . Substitute this expression for ∂ ξ z n ∆ (ξ) into the left-hand side of (84), and perform a change of variables x = X n ∆ (ξ) to obtain the integral on the right. Next, take the square root in (86) before differentiation, then calculate the square and divide by ∂ ξ X n ∆ (ξ) afterwards: ∂ ξ z n ∆ (ξ) 2 4 z n ∆ (ξ)∂ ξ X n ∆ (ξ) = ∂ x u n ∆ 2 • X n ∆ (ξ)∂ ξ X n ∆ (ξ). Performing the same change of variables as before, this proves that M 0 ∂ ξ z n ∆ (ξ) 2 z n ∆ (ξ)∂ ξ X n ∆ (ξ) ρ • X n ∆ (ξ) dξ = 4 Ω ∂ x u n ∆ 2 (x)ρ (x) dx.(87) It remains to estimate the difference between the ξ-integrals in (85) and in (87), respectively. To this end, observe that for each ξ ∈ (ξ k , ξ k+ 1 2 ) with some k ∈ I + K , one has ∂ ξ X n ∆ (ξ) = 1/z n k+ 1 2 and z ∆ (ξ) ∈ [z k− 1 2 , z k+ 1 2 ]. Hence, for those ξ, interchanged. Consequently, using once again (42) and (44), 1 − 1 z n ∆ (ξ)∂ ξ X n ∆ (ξ) ≤ 1 − z n k+ 1 2 z n k− 1 2 . If instead ξ ∈ (ξ k− 1 2 , ξ k ),τ N n=1 M 0 ∂ ξ z n ∆ (ξ) 2 ρ • X n ∆ (ξ) dξ − M 0 ∂ ξ z n ∆ (ξ) 2 z n ∆ (ξ)∂ ξ X n ∆ (ξ) ρ • X n ∆ (ξ) dξ ≤ Bτ N n=1 M 0 ∂ ξ z n ∆ (ξ) 2 1 − 1 z n ∆ (ξ)∂ ξ X n ∆ (ξ) dξ ≤ B τ ∞ n=1 ∂ ξ z n ∆ 4 L 4 1/2   τ N n=1 δ k∈I + K   1 − z n k+ 1 2 z n k− 1 2 2 + 1 − z n k+ 1 2 z n k− 1 2 2     1/2 ≤ 3H 1/2 6(b − a) 2 T 1/2 H 1/2 δ 1/2 1/2 , since N τ ≤ T by hypothesis. This shows (85). Proof of (69). Again, let N τ ∈ N be such that N τ τ ∈ (T, T + 1). Combining the discrete weak formulation (76), the change of variables formulae (84)&(85), and the definitions of R 1 to R 5 , it follows that e 2,∆ ≤ BR 5 + Bτ Nτ n=1 M 0 ∂ ξ z n ∆ ρ • X n ∆ (ξ) + ∂ ξ z n ∆ (ξ) 2 ρ • X n ∆ (ξ) dξ − A n 1 − A n 2 + A n 3 + A n 4 ≤ B(R 1 + R 2 + R 3 + R 4 + R 5 ) ≤ B(C 1 + C 2 + C 3 + C 4 + C 5 )δ 1/4 . This implies the desired inequality (69). We are now going to finish the proof of this section's main result. Proof of Proposition 21. Thanks to (68)&(69), we know that T 0 ψ (t) Ω ρ(x) {ū ∆ } τ (t; x) dx dt + ψ(0) Ω ρ(x)ū 0 ∆ (x) dx + T 0 ψ(t) Ω ρ (x)∂ x { u ∆ } τ (t; x) + 4ρ (x)∂ x u ∆ τ (t; x) 2 dx dt ≤ e 1,∆ + e 2,∆ ≤ C (τ F δ ( x 0 ∆ )) + (δF δ ( x 0 ∆ )) 1/2 + δ 1/4 . By our assumption (37) on F δ ( x 0 ∆ ), the expression on the right hand side vanishes as ∆ → 0. To obtain (65) in the limit ∆ → 0, we still need to show the convergence of the integrals to their respective limits. A technical tool is the observation that, for each p ∈ [1,4], Q p := sup ∆ τ Nτ n=1 δ κ∈I 1/2 K (z n κ ) p < ∞, thanks to the estimates (102) and (42). For the first integral, we use that {ū ∆ } τ converges to u * w.r.t. W, locally uniformly with respect to t ∈ (0, T ). Thus clearly Ω ρ(x) {ū ∆ } τ (t; x) dx → Ω ρ(x)u * (t; x) dx for each t ∈ (0, T ). In order to pass to the limit with the time integral, we apply Vitali's theorem. To this end, observe that T 0 ψ (t) Ω ρ(x) {ū ∆ } τ (t; x) dx 2 dt ≤ B 2 (b − a)τ Nτ n=1 Ωū n ∆ (x) 2 dx = B 2 (b − a)τ Nτ n=1 δ κ∈I 1/2 K z n κ ≤ Q 1 B 2 (b − a). Next, using the strong convergence from (64), it follows that ∂ x { u ∆ } τ = 2 u ∆ τ ∂ x u ∆ → 2 √ u * ∂ x √ u * = ∂ x u * strongly in L 1 (Ω) , for almost every t ∈ (0, T ). Again, we apply Vitali's theorem to conclude convergence of the time integral, on grounds of the following estimate: T 0 ψ(t) Ω ρ (x)∂ x { u ∆ } τ dx 2 dt ≤ B 2 (b − a)τ Nτ n=1 Ω ∂ x u n ∆ (x) 2 dx = B 2 (b − a)τ Nτ n=1 δ k∈I + K z n k+ 1 2 − z n k− 1 2 δ 2 z n k+ 1 2 + z n k− 1 2 2 ≤ B 2 (b − a)   τ ∞ n=1 k∈I + K z n k+ 1 2 − z n k− 1 2 δ 4   1/2   τ Nτ n=1 δ κ∈I 1/2 K (z n κ ) 2    1/2 ≤ 3H 1/2 Q 1/2 2 B 2 (b − a), where we have used (42). Finally, the strong convergence implies (64) also implies that ∂ x { u ∆ } τ 2 → ∂ x √ u * 2 strongly in L 1 (Ω), for almost every t ∈ (0, T ). One more time, we invoke Vitali's theorem, using that T 0 ψ(t) Ω ρ (x)∂ x u ∆ 2 τ (t; x) dx 2 dt ≤ B 2 τ Nτ n=1 Ω ∂ x u n ∆ 4 (x) dx ≤ 1 2 B 2 τ Nτ n=1 δ k∈I + K z n k+ 1 2 − z n k− 1 2 δ 2   1 − z n k+ 1 2 z n k− 1 2 2 + 1 − z n k− 1 2 z n k+ 1 2 2   . ≤ B 2   τ ∞ n=1 δ k∈I + K z n k+ 1 2 − z n k− 1 2 δ 4   1/2   τ ∞ n=1 δ k∈I + K   1 − z n k+ 1 2 z n k− 1 2 4 + 1 − z n k− 1 2 z n k+ 1 2 4     1/2 . The two terms in the last line are uniformly controlled in view of (42) and (44), respectively. Proof of Theorems 1 and 2 Below, we collect the results derived up to here to formally conclude the proofs of our main theorems. Proof of Theorem 1. Well-posedness of the discrete scheme follows from Proposition 9. Positivity and mass conservation are immediate consequences of the construction: recall thatū ∆ = u δ [ x ∆ ], with u δ defined in (19). The monotonicity of H δ and F δ have been obtained in Lemma 12 and 11, respectively. It remains to show the exponential decay (14) of H δ . From (the proof of) Lemma 16, it follows for each n = 1, 2, . . . that H(ū n ∆ ) − H(ū n−1 ∆ ) = H δ ( x n ∆ ) − H δ ( x n−1 ∆ ) ≤ − τ 10(b − a) TV ∂ x u n ∆ 2 .(88) By the logarithmic Sobolev inequality on Ω and thanks to the fact that ∂ x u n ∆ (0) = 0, we further have that H( u n ∆ ) ≤ (b − a) 2 2π 2 Ω u n ∆ 2 (x) dx ≤ (b − a) 3 2π 2 TV ∂ x u n ∆ 2 .(89) Now combine (88) and (89) with the estimate (105) from the Appendix to conclude that 1 + π 2 τ 5(b − a) 4 H(ū n ∆ ) ≤ H(ū n−1 ∆ ). From here, the claim (14) is obtained by induction on n. Proof of Theorem 2. Local uniform convergence of the {ū ∆ } τ to a continuous limit function u * is part of the conclusions of Proposition 19, see (57) and (59). The regularity √ u * ∈ L 2 (R ≥0 ; H 1 (Ω)) has been observed in Proposition 20. The strong convergence stated in the same proposition implies that F(u * ) is "almost monotone": indeed, thanks to (64) we may assume -passing to a further subsequence with ∆ → 0 if necessary -that u ∆ τ (t) → u * (t) strongly in H 1 (Ω), for a.e. t > 0, and therefore also 2 Ω ∂ x u ∆ 2 τ (t; x) dx → F(u * (t)), for a.e. t > 0.(90) On the other hand, arguing just like in the proof of (85), it follows that ∞ 0 {F δ ( x ∆ )} τ − 2 Ω ∂ x u ∆ 2 τ (t; x) dx dt ≤ Cδ 1/4 .(91) Now combine (90) and (91) with the fact that {F δ ( x ∆ )} τ is decreasing in t, for each ∆, and is ∆-uniformly bounded above according to (54). By Helly's selection principle, there exists a monotone f : R + → R + such that F(u * (t)) = f (t) for a.e. t > 0. The proof of monotonicity for t → H(u * ) is similar, but easier: here it suffices to use the local uniform convergence from (59). Finally, the weak formulation (16) has been shown in Proposition 21. Simply observe that any ϕ ∈ C ∞ c (R ≥0 × Ω) can be approximated by linear combinations of products ψ(t)ρ(x) with functions ψ ∈ C ∞ (R ≥0 ) and ρ ∈ C ∞ (Ω). expansion must vanish, thus the approximation error on the right-hand side is actually of order O(δ 2 ) rather than O(δ). Further, using (94) and (93), the term of order δ 0 can be written as 2z * z * z * + z 2 * z * = ∂ ξ Z (nτ ; ξ k ) 2 ∂ 2 ξZ (nτ ; ξ k ) = ∂ ξ Z(nτ ; ξ k ) 2 ∂ 2 ξ Z(nτ ; ξ k ) + O(δ 2 ). On the left-hand side of (13), we obtain 6.2.1. Non-uniform meshes. In order to make our discretization more flexible, we are going to change our setting and allow non-equidistant mass grids. That is, the mass discretization of [0, M ] is determined by a vector δ = (ξ 0 , ξ 1 , ξ 2 , . . . , ξ K−1 , ξ K ), with x n k − x n−1 k τ = 1 τ (X(nτ ; ξ k ) − X((n − 1)τ ; ξ k )) = ∂ t X(nτ ; ξ k ) + O(τ ),(96)0 = ξ 0 < ξ 1 < · · · < ξ K−1 < ξ K = M, and we introduce accordingly the distances δ κ = ξ κ+ 1 2 − ξ κ− 1 2 , and δ k = 1 2 (δ k+ 1 2 + δ k− 1 2 ) = 1 2 (ξ k+1 − ξ k−1 ) for κ ∈ I 1/2 K and k ∈ I + K , respectively. The piecewise constant density functionū ∈ P δ (Ω) corresponding to a vector x ∈ R K−1 is now given bȳ u(x) = z κ for x κ− 1 2 < x < x κ+ 1 2 , with z κ = δ κ x κ+ 1 2 − x κ− 1 2 . The Wasserstein-like metric needs to be adapted as well: the scalar product ·, · δ is replaced by v, w δ = k∈I + K δ k v k w k . Hence the metric gradient ∇ δ f ( x) ∈ R K−1 of a function f : x δ → R at x ∈ x δ is given by ∇ δ f ( x) k = 1 δ k ∂ x k f ( x). Otherwise, we proceed as before: the entropy is discretized by restriction, and the discretized Fisher information is the self-dissipation of the discretized entropy. Explicitly, the resulting fully discrete gradient flow equation x n ∆ − x n−1 ∆ τ = −∇ δ F δ ( x n ∆ ) attains the form x n k − x n−1 k τ = 1 δ k (z n k+ 1 2 ) 2 δ k+ 1 2 z n k+ 3 2 − z n k+ 1 2 δ k+1 − z n k+ 1 2 − z n k− 1 2 δ k − (z n k− 1 2 ) 2 δ k− 1 2 z n k+ 1 2 − z n k− 1 2 δ k − z n k− 1 2 − z n k− 3 2 δ k−1 .(97) with = 10 −3 . The mass grid δ is chosen in such a way that u 0 ∆ is a piecewise constant approxiation of u 0 with respect to a spatially uniform grid. That is, we choose δ such that the initial condition x 0 ∆ for x attains the simple form x 0 k = a + b − a K k.(99) To construct δ, we first calculate the cummulative distribution function U 0 : [0, 1] → [0, M ] by numerical integration of u 0 , U 0 (x) = x a u 0 (y) dy, and then define ξ k := U 0 (x 0 k ), for k = 0, 1, . . . , K. Remark 28. An equidistant mass grid leads to a good spatial resolution of regions where the value of u 0 is large, but provides a very poor resolution in regions where u 0 is small. Since the evolution of the zones with low density are of particular interest in numerical studies of the DLSS equation, it is natural to use a non-uniform mass grid with an adapted spatial resolution, like the one defined above. 6.2.3. Implementation. From the initial condition x 0 ∆ , the fully discrete solution is calculated inductively by solving the implicit Euler scheme (97) for x n ∆ , given x n−1 ∆ . In each time step, a damped Newton iteration is performed, with the solution from the previous time step as initial guess. Slow convergence of the Newton iteration has been observed in situations where the densityū n−1 ∆ has steep gradients and/or intervals of very low values. Our reference solution is calculated with the scheme described in [20], which is fully variational as well, but uses different ansatz functions for the Lagrangian maps. Even without a rigorous result on uniqueness of weak solutions, it seems reasonable to expect that both schemes should approximate the same solution. A technical issue with the comparison of our solution to the reference solution is that both use a different way for the reconstruction of the density from the Lagrangian map. This difference camouflages the true approximation error in the plain L 2differences. For a fair comparison, we calculate the L 2 -difference of the linear interpolations of the values for the density with respect to the nodes of the Lagrangian maps. 6.2.4. Observed rate of convergence. Figure 1 provides a qualitative picture of the evolution with initial condition u 0 : the plot on the left shows the density functionū ∆ at several instances in time, the plot on the right visualizes the motion of the mesh points {x k } τ associated to the Lagrangian maps X ∆ in continuous time. It is clearly seen that the initial density has a very flat minimum (which is degenerate of order 16) at x = 1/2, which bifurcates into two sharper minima at later times, and eventually becomes one single minimum again. This behavior underlines that the comparison principle does obviously not hold for the DLSS equation. Both figures has been generated using K = 200 spatial grid points and the time step size τ = 10 −6 . For numerical analysis of the convergence rate, we have carried out two series of experiments. In the first series, we fix the time step size τ = 10 −8 and vary the number of spatial grid points, using K = 25, 50, 100, 200. Figure 2/Left shows the corresponding L 2 -error between the solution to our scheme and the reference solution, evaluated at time T = 10 −5 . It is clearly seen that the error decays with an almost perfect rate of δ 2 ∝ K −2 . For the second series of experiments, we keep the spatial discretization parameter K = 800 fixed and run our scheme with the time step sizes τ = 10 −5 , 5 · 10 −6 , 10 −6 , 5 · 10 −7 , 10 −7 , 5 · 10 −8 , respectively. The corresponding L 2 -error at T = 10 −5 is plotted in Figure 2/Right. It is proportional to τ . 6.2.5. Discontinuous initial data. One of the conclusions of Theorem 2 is that the discrete approximations u ∆ converge also for (a large class of) non-regular initial data u 0 . For illustration of this feature, we consider the discontinuous initial density function u 0 discont = 1 x ∈ [0, 1 3 ] ∪ [ 2 3 , 1], 10 −3 , x ∈ ( 1 3 , 2 3 )(100) instead of u 0 from (98). According to our hypothesis (15), we need to use a sufficiently high spatial and temporal resolution. In practice, this is done in an adaptive way: the K points of the initial grid x 0 ∆ are not placed equidistantly, but with a higher refinement around the points of discontinuity; the applied time step τ is extremely small (down to 10 −13 ) during the initial phase of the evolution, and is larger (up to 10 −9 ) at later times. Figure 3 provides a qualitative picture of the fully discrete evolution for K = 200 grid points: snapshots of the discrete density functionū ∆ are shown on the left, corresponding snapshots of the logarithmic density are shown on the right. Note that within a very short time, peaks of relatively high amplitudes are generated near the points where u 0 is discontinuous. The associated Lagrangian maps are visualized in Figure 4/Left. Notice the fast motion of the grid points near the discontinuities. To estimate the rate of convergence, we performed a series of experiments using K = 25, 50, 100 and 200 spatial grid points. For comparison, we calculated a highly refined solution of the following semi-implicit reference scheme, u n+1 ref − u n ref τ = −∆ 2 u n ref ∆ 2 ln(u n+1 ref ) , where ∆ 2 is the standard central difference operator ∆ 2 . The reference scheme is run with K = 800 spatial grid point. An adaptive choice of the time step τ needs to be made in order to avoid that the reference solution u ref breaks down due to loss of positivity. The L 2 -differences of the densities and of their logarithms have been evaluated at T = 10 −8 , see Figure 4/Right. As expected, the rate of convergence is no longer quadratic in δ ∝ K −1 ; instead, the error decays approximately linearly. 2, . . . , K − 1}, I 0 K = I + K ∪ {0, K}, and I For discretization of [0, M ], introduce the equidistant mass grid (ξ 0 , . . . , ξ K ) with ξ k = kδ for δ := M/K. and (69). At the end of this section, it is shown how the claim (65) follows from (68)&(69) on basis of the convergence for {ū ∆ } τ obtained previously. then this estimate holds with the roles of z Figure 1 . 1Left: snapshots of the densitiesū ∆ for the initial condition (98) at times t = 0 and t = 10 i , i = −6, . . . , −3, using K = 200 grid points and the time step size τ = 10 −6 . Right: associated particle trajectories. Figure 2 . 2Numerical error analysis for u 0 from (98). Left: fixed time step size τ = 10 −8 and K = 25, 50, 100, 200 spatial grid points. The L 2 -errors are evaluated at T = 5·10 −6 . Right: fixed K = 800 using τ = 10 −5 , 5·10 −6 , 10 −6 , 5· 10 −7 , 10 −7 , 5 · 10 −8 . The error is evaluated at T = 10 −5 . Figure 3 . 3Snapshots of the densitiesū ∆ for the initial condition (100) at times t = 0 and t = 10 i , i = −13, −11, . . . , −5, −3, using K = 200 grid points with linear (left) and logarithmic (right) scaling. Figure 4 . 4Left: associated particle trajectories ofū ∆ using the initial condition (100). Right: Numerical error analysis for u 0 discont from (100) with fixed τ and K = 25, 50, 100, 200 spatial grid points. The L 2 -errors are evaluated at T = 10 −8 . thanks to the smoothness of X in time. Comining (95)&(96) with the continuous equation (92), we arrive at (13), with an error of O(τ ) + O(δ 2 ).6.2. Numerical experiments. 6.2.2. Initial condition.Our main experiments are carried out using the by now classical test case from[4], that is u 0 (x) = + cos 16 (πx), on Ω = [0, 1], This research was supported by the DFG Collaborative Research Center TRR 109, "Discretization in Geometry and Dynamics".Acknowledgement. The authors are indebted to Giuseppe Savaré for fruitful discussions on the subject, and especially for contributing the initial idea for the entropy preserving discretization scheme.Numerical results and order of consistencyThe proof of convergence for our discretization given above is purely qualitative. In this last section, we study quantitative aspects of the convergence. First, we calculate the order of consistency for approximation of smooth and strictly positive solutions. Second, we report on the observed order of convergence in several numerical experiments.6.1. Order of consistency. The following proposition shows that our scheme is (formally) of first order in time and of second order in space.which is further such that Z = 1/∂ ξ X is smooth and strictly positive. Let ∆ = (τ ; δ) be a family of discretization parameters. Then the corresponding restrictions ( x ∆ ) of X to the respective meshes, given by x n k := X(nτ ; ξ k ) for n ∈ N and k ∈ {1, . . . , K}, satisfy (13) with an errorwhich is a smooth and strictly positive function, thanks to the properties of X. It is immediately seen thatfor each m ∈ N and locally uniformly in (t; ξ) as ∆ → 0. Observe that, by definition of ( x ∆ ) as restriction of X to ∆, one hasFix indices n ∈ N and k ∈ {1, . . . , K − 1}. In the following, we abbreviateRelation (94) and a standard Taylor expansion ofZ around ξ = ξ k yield (z nThe same expansion -with (−δ) in place of δ -is obtained for z n , respectively. Therefore,Proof. The first equality is simply the definition (20) of z κ . Since trivially x κ+ 1 2 − x κ− 1 2 < b − a for each κ ∈ I 1/2 K , and since p − 1 > 0, it follows thatand consequently,Proof. The first estimate in(102)is an immediate consequence of the definition of z κ in(20).To prove the second estimate, let κ * ∈ I 1/2 K be such that z κ * = max z k . Observe that there exists a κ * ∈ I 1/2 K such thatWriting out z κ * − z κ * as a sum over differences of adjacent values of z k and applying the triangle and Cauchy Schwarz inequality, one obtainsNow combine this with (104).Lemma 31. With u andū being, respectively, the piecewise linear and the piecewise constant densities associated to a given vector x, thenProof. First observe thatwhich is an easy consequence of a Taylor expansion for the function s → (1+s) ln s around s = 1, substituting s = p/q. On the one hand, we have thatand on the other hand,where we have used (106). This clearly implies (105).Lemma 32 (Gargliardo-Nirenberg inequality). For each f ∈ H 1 (Ω), one has thatProof. Assume first that f ≥ 0. 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[ "Colossal dielectric constants in transition-metal oxides", "Colossal dielectric constants in transition-metal oxides" ]	[ "P Lunkenheimer \nExperimental Physics V\nCenter for Electronic Correlations and Magnetism\nUniversity of Augsburg\n86135AugsburgGermany\n", "S Krohns \nExperimental Physics V\nCenter for Electronic Correlations and Magnetism\nUniversity of Augsburg\n86135AugsburgGermany\n", "S Riegg \nSolid State Chemistry\nUniversity of Augsburg\n86135AugsburgGermany\n", "S G Ebbinghaus \nSolid State Chemistry\nMartin-Luther University Halle-Wittenberg\n06120HalleGermany\n", "A Reller \nSolid State Chemistry\nUniversity of Augsburg\n86135AugsburgGermany\n", "A Loidl \nExperimental Physics V\nCenter for Electronic Correlations and Magnetism\nUniversity of Augsburg\n86135AugsburgGermany\n" ]	[ "Experimental Physics V\nCenter for Electronic Correlations and Magnetism\nUniversity of Augsburg\n86135AugsburgGermany", "Experimental Physics V\nCenter for Electronic Correlations and Magnetism\nUniversity of Augsburg\n86135AugsburgGermany", "Solid State Chemistry\nUniversity of Augsburg\n86135AugsburgGermany", "Solid State Chemistry\nMartin-Luther University Halle-Wittenberg\n06120HalleGermany", "Solid State Chemistry\nUniversity of Augsburg\n86135AugsburgGermany", "Experimental Physics V\nCenter for Electronic Correlations and Magnetism\nUniversity of Augsburg\n86135AugsburgGermany" ]	[]	Many transition-metal oxides show very large ("colossal") magnitudes of the dielectric constant and thus have immense potential for applications in modern microelectronics and for the development of new capacitance-based energystorage devices. In the present work, we thoroughly discuss the mechanisms that can lead to colossal values of the dielectric constant, especially emphasising effects generated by external and internal interfaces, including electronic phase separation. In addition, we provide a detailed overview and discussion of the dielectric properties of CaCu3Ti4O12 and related systems, which is today's most investigated material with colossal dielectric constant. Also a variety of further transition-metal oxides with large dielectric constants are treated in detail, among them the system La2−xSrxNiO4 where electronic phase separation may play a role in the generation of a colossal dielectric constant. a	10.1140/epjst/e2010-01212-5	[ "https://arxiv.org/pdf/1003.4272v1.pdf" ]	54,896,199	1003.4272	6ab8c0f8321e6642318cda7794119388840b7361	Colossal dielectric constants in transition-metal oxides 22 Mar 2010 P Lunkenheimer Experimental Physics V Center for Electronic Correlations and Magnetism University of Augsburg 86135AugsburgGermany S Krohns Experimental Physics V Center for Electronic Correlations and Magnetism University of Augsburg 86135AugsburgGermany S Riegg Solid State Chemistry University of Augsburg 86135AugsburgGermany S G Ebbinghaus Solid State Chemistry Martin-Luther University Halle-Wittenberg 06120HalleGermany A Reller Solid State Chemistry University of Augsburg 86135AugsburgGermany A Loidl Experimental Physics V Center for Electronic Correlations and Magnetism University of Augsburg 86135AugsburgGermany Colossal dielectric constants in transition-metal oxides 22 Mar 2010arXiv:1003.4272v1 [cond-mat.str-el] EPJ manuscript No. (will be inserted by the editor) Many transition-metal oxides show very large ("colossal") magnitudes of the dielectric constant and thus have immense potential for applications in modern microelectronics and for the development of new capacitance-based energystorage devices. In the present work, we thoroughly discuss the mechanisms that can lead to colossal values of the dielectric constant, especially emphasising effects generated by external and internal interfaces, including electronic phase separation. In addition, we provide a detailed overview and discussion of the dielectric properties of CaCu3Ti4O12 and related systems, which is today's most investigated material with colossal dielectric constant. Also a variety of further transition-metal oxides with large dielectric constants are treated in detail, among them the system La2−xSrxNiO4 where electronic phase separation may play a role in the generation of a colossal dielectric constant. a Introduction Hot topics like high-T c superconductivity, colossal magnetoresistance and multiferroicity have led to a tremendous boost of solid state physics during the last 25 years. These and other interesting phenomena to a large extend first have been revealed and intensely investigated in transition-metal oxides. The complexity of the ground states of these materials arises from strong electronic correlations, enhanced by the interplay of spin, orbital, charge and lattice degrees of freedom. These phenomena are a challenge for basic research and also bear enormous potentials for future applications as the related ground states are often accompanied by so-called "colossal" effects, which are possible building blocks for tomorrows correlated electronics. The measurement of the response of transition-metal oxides to ac electric fields is one of the most powerful techniques to provide detailed insight into the underlying physics that may comprise very different phenomena, e.g., charge order, molecular or polaronic relaxations, magnetocapacitance, hopping charge transport, ferroelectricity or density-wave formation. The present work concentrates on materials showing so-called colossal dielectric constants (CDC), i.e. values of the real part of the permittivity ε ′ exceeding 1000. Since long, materials with high dielectric constants are in the focus of interest, not only for purely academic reasons but also because new high-ε ′ materials are urgently sought after for the further development of modern electronics. In general, for the miniaturisation of capacitive electronic elements materials with high-ε ′ are prerequisite. This is true not only for the common silicon-based integrated-circuit technique but also for stand-alone capacitors. For example, the latter, if constructed using materials with CDCs, can reach capacitances high enough to enable their use for energy storage, without the disadvantage of escalating volume and weight. Such capacitors can be used, e.g., to replace batteries in hybrid vehicles. Most of the currently available capacitor materials with CDCs are based on ferroelectrics, which reach very high values of the dielectric constant often exceeding 10 4 . However, ferroelectric materials exhibit a strong temperature dependence of ε ′ , limiting their straightforward application in electronic devices. Currently, the most prominent non-ferroelectric material showing colossal values of ε ′ is CaCu 3 Ti 4 O 12 (CCTO). Initiated by the first reports of extremely high dielectric constants in CCTO in 2000 [1] and further boosted by the article of Homes et al. [2] appearing in "Science" in the following year, until now more than 380 papers have been published on this and related materials. The fact that among them there are more than ten so-called "highly-cited" papers [3] (e.g., [1,2,4,5,6,7,8,9,10,11,12,13,14]) demonstrates the tremendous interest in new high-ε ′ materials. The big advantage of CCTO compared to ferroelectric-based dielectrics is its nearly temperature independent CDC around room temperature. Only below about 200 K (depending on frequency) it shows a marked and strongly frequency-dependent decrease of ε ′ (T ) from values up to 10 5 to magnitudes of the order of 100. Many speculations about the origin of the CDCs in CCTO have been put forward. Already in the original work by Subramanian et al. [1] a barrier layer mechanism [10,15,16,17,18] was suggested, arising from twin boundaries. However, in some other early works intrinsic bulk mechanisms were proposed for the explanation of the CDCs in CCTO. This includes relaxational excitations of highly polarizable entities of unspecified origin [4], geometrically frustrated ferroelectric order caused by symmetrical off-centre displacements of Ti ions [2] and a special kind of defects in the perovskite structures relaxing between different equivalent configurations [19]. After the thorough investigation of this material in numerous works during the last ten years, nowadays it is quite commonly accepted that a barrier mechanism is the correct explanation (see, e.g., [5,8,9,10,11,12,20,21,22,23]). However, the nature of these barriers leading to the CDCs in CCTO still is an open question. There are various reports on experimental results that seem to support internal [5,8,9,12,20,21,22,23] or surface barrier layer capacitors [11,24,25,26,27] (IBLCs or SBLCs, respectively) giving rise to the detected CDCs. The first could stem from grain boundaries (in ceramic samples) and/or from boundaries between twins (or other planar defects) within single crystals or within the crystallites of ceramic samples. SBLCs may arise from the depletion layers of Schottky or metal-insulator-semiconductor (MIS) diodes at the interfaces between the metallic electrodes and the bulk sample. Despite huge efforts, the properties of CCTO seem to be not ideal for straightforward application if considering the amplitude of its dielectric loss, the downscaling of its properties to miniaturised devices and its applicability at frequencies in the increasingly important frequency range around GHz [11,26]. Thus there is still an ongoing search for new, better materials. There is a vast number of materials isostructural to CCTO, which may also show CDCs but have only partly been characterised so far [1,6,28,29,30,31]. But also completely different transition-metal oxides have been reported to exhibit CDCs and it seems that this phenomenon is quite common in this class of materials (e.g., [10,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52]). In addition other mechanisms for the generation of CDCs are considered. For example, the formation of CDCs by internal interfaces spontaneously arising via electronic phase separation (e.g., stripe-ordering) seems one of the most feasible ways to accomplish materials that are suitable for application. In the following we will give an overview of the different ways of generating CDCs, laying special emphasis on interface-related mechanisms. Then we will provide and discuss various results on CCTO, CCTO-related materials and other transition-metal oxides with CDCs. Mechanisms giving rise to colossal dielectric constants The most prominent mechanisms that can give rise to enhanced values of the dielectric constant are ferroelectricity, charge-density wave formation, hopping charge transport, the metalinsulator transition and various kinds of interface effects. In a first step we would like to get a rough guess about the maximum dielectric constant that can be reached in ionic solids without permanent dipole moments and without invoking any of these mechanisms. The static dielectric constant ε s in an ionic solid is given by ε s = ε ∞ + (Ω p /ω T ) 2 . Here ε ∞ is the electronic dielectric constant determined at frequencies beyond the phonon modes, Ω p is the effective ionic plasma frequency and ω T the eigenfrequency of a transverse optical vibration, which can be directly observed in infrared experiments. The ionic plasma frequency is given by Ω 2 p = ε ∞ N (Ze) 2 /(V ε 0 µ), where N is the number of ionic pairs of effective charge Ze per volume V , ε 0 is the vacuum dielectric permittivity and µ the reduced mass of the ion pair involved in the eigenmode. Large dielectric constants demand large ionic plasma frequencies and small eigenfrequencies of the infrared active mode. Large plasma frequencies are reached via large effective charges and low masses. It is clear that in the oscillator model small masses and small eigenfrequencies are related via ω ∝ 1/ √ µ and can hardly be reached at the same time if soft-mode scenarios are not taken into consideration. To get some estimate of the largest possible value of the dielectric constant we make the following assumptions: ε ∞ = 10, Z = 3, N = 1 per unit cell volume V = 64Å 3 and µ = 12 in units of the atomic mass constant. These numbers can be reached assuming an ion pair with Ti 4+ vibrating against O 2− and yield an effective ionic plasma frequency Ω p /2π ≈ 72 THz. The resulting dielectric constant now depends significantly on the eigenfrequency of the mode under consideration. Assuming ω T /2π = 2.3 THz yields ε s ≈ 1000; taking an eigenfrequency which better correlates with the low reduced masses, namely ω T /2π = 7 THz, results in a dielectric constant ε s ≈ 120, which comes close to the maximum value of experimentally observed dielectric constants in purely ionic solids. Thus for the generation of CDCs, other mechanisms have to be considered as discussed in the following. Ferroelectricity It is known since long that in ferroelectrics very high values of the dielectric constant can arise [53,54]. Approaching the ferroelectric phase transition at T c from high temperatures, ε ′ (T ) strongly increases, usually following a Curie-Weiss behaviour and starts to decrease again below T c . In addition, ferroelectrics have pronounced non-linear dielectric properties, e.g., showing characteristic hysteresis loops of the electric-field dependent polarisation [54]. Both phenomena represent problems for application in electronic devices. This partly can be overcome by doping and special processing thereby adjusting microstructure and internal interfaces. The well-known ceramic Barrier Layer Capacitors use a combination of interface polarisation effects and ferroelectric materials like BaTiO 3 to achieve high capacitance values with temperature and voltage dependences that are tolerable at least for some applications [55,56,57]. Ferroelectric transitions often are classified as displacive or order-disorder type. The latter case corresponds to the ordering of dipolar degrees of freedom already present at T > T c . In these systems the hopping of the dipoles can lead to strong frequency dependence of ε ′ at technically relevant frequencies (Hz-GHz) making them less suited for application [54,58,59]. In contrast, ferroelectrics of displacive type usually show no frequency dependence up to infrared frequencies, where the well-known soft-phonon modes appear [54]. A special variant of ferroelectrics are the so-called relaxor ferroelectrics [60,61]. Their static dielectric constant shows a strong increase with decreasing temperature just as for canonical ferroelectrics. However, this is superimposed by a marked relaxation mode that leads to peaks in ε ′ (T ) at temperatures that are strongly dependent on frequency. Different explanations have been proposed for this behaviour, e.g., in terms of polar cluster dynamics, but no consensus has been achieved so far [60,61,62,63,64]. While in conventional and relaxor ferroelectrics ions or dipoles are the relevant entities achieving ferroelectric order, also the ordering of electronic degrees of freedom has been considered [42,65,66,67]. For example, the occurrence of CDCs of magnitude > 4000 detected in the mixed-valent transition-metal oxide LuFe 2 O 4 was ascribed to an electronic polarisation mechanism involving charge ordering of Fe 2+ and Fe 3+ ions [42]. Also in certain charge-transfer salts a ferroelectric transition of electronic origin recently was discussed [58,68]. Finally, it should be mentioned that in some materials, the so-called incipient ferroelectrics, long-range ferroelectric order is prevented by quantum fluctuations at low temperature, setting in before complete order is achieved at T c . At temperatures sufficiently above T c , ε ′ (T ) of these materials follows the Curie-Weiss law and thus CDCs are observed. The most prominent incipient ferroelectric is SrTiO 3 [69], which shows a tendency of ε ′ (T ) to saturate at low temperatures, setting in below about 30 K due to the mentioned quantum effects [70]. Besides BaTiO 3 , also SrTiO 3 is often employed as dielectric material in commercial ceramic Barrier Layer Capacitors. Charge-Density Waves In highly anisotropic low-dimensional materials a metal-insulator transition can arise with lowering of the temperature, which is accompanied by the formation of a charge-density wave (CDW). Here the electronic charge density is a periodic function of position and its period can be incommensurate with the crystal lattice. A very well-known example of a CDW system is also found in the group of transition-metal oxides, namely the blue bronze, K 0.30 MoO 3 [71]. The dielectric behaviour of CDW systems shows two characteristic features: A harmonic oscillator mode at GHz frequencies caused by the CDW being pinned at defects and a huge relaxation mode at kHz-MHz involving colossal values of the dielectric constant [72,73,74]. In CDW systems the highest intrinsic dielectric constants of any materials are observed, reaching magnitudes of up to 10 8 . Littlewood [75] has proposed screening effects of the pinned CDW by the normal electrons, not participating in the CDW, to explain the occurrence of the lowfrequency relaxation mode and CDCs in this class of materials. Due to the strong frequency dependence of ε ′ and the high dielectric losses associated with the relaxational modes, CDWs are not applied in capacitive devices. Hopping charge transport Hopping conductivity is the most common charge-transport process in condensed matter. As it is intimately related with the occurrence of a power law with negative exponent in the frequencydependent dielectric constant ε ′ (ν), it will always lead to a divergence of ε ′ for low frequencies [76,77,78,79]. Hopping conduction is the typical charge-transport process of localised charge carriers. In electronic conductors, electrons (or holes) can localise due to disorder. Disorder may arise from an amorphous structure, from doping (substitutional disorder) or occur even in nominally pure crystals due to slight deviations from stoichiometry or lattice imperfections. Hopping conduction leads to a characteristic signature in the frequency dependence of the complex conductivity, namely a power-law increase σ ′ = σ 0 ν s with the exponent s < 1 [76,77,80]. This power law was shown by Jonscher [78,79] to be a quite universal phenomenon in all types of disordered matter and termed "Universal Dielectric Response" (UDR). This behaviour can be understood in the framework of various models on the charge transport of localised charge carriers, including the often-employed variable-range hopping (VRH) model [76,77,80]. These models originally were developed for amorphous and highly doped semiconductors like doped silicon but also, e.g., for thin scandium-oxide films [76,77,80,81,82]. The typical signature of hopping transport in measurements of the ac conductivity was also found in numerous transition-metal oxides (e.g., [11,26,44,45,46,48,49,50,51,82,83,84,85]). Via the Kramers-Kronig relation, the ν s power law also leads to a corresponding power law in the imaginary part of the ac conductivity, namely σ ′′ = tan(sπ/2)σ 0 ν s [79]. As the dielectric constant is directly related to σ ′′ via ε ′ = σ ′′ /(2πνε 0 ) (with ε 0 the permittivity of vacuum) hopping conduction is expected to lead to a power law ε ′ ∝ ν s−1 . Thus, as s < 1, the dielectric constant can easily reach colossal magnitudes for low frequencies. However, as the factor tan(sπ/2) usually is of the order of one, the dielectric loss ε ′′ = σ ′ /(2πνε 0 ) is relatively high (which of course is reasonable for a conducting material) rendering this effect unsuited for application. Metal-insulator transition It is known since about 100 years that the Clausius-Mosotti relation will lead to a polarisation catastrophe, i.e. a divergence of the dielectric constant when approaching the metal-insulator (MI) transition from the insulating side [86]. It is naively clear that the reduction of the restoring forces experienced by electrons localised at atomic sites, which should occur when the material approaches the metallic state with its itinerant electron states, will lead to an increase of the electronic polarisability and thus the dielectric constant. Indeed, such a divergence has been observed in some cases [86,87,88,89,90], the most prominent one being measurements of the dielectric properties of doped silicon for increasing doping level [88]. Several theoretical approaches have appeared treating this topic and going beyond the simple arguments based on the Clausius-Mosotti equation [91,92,93,94]. Prototypical MI transitions are regularly found in transition-metal oxides, the most well-known ones being those in magnetite, Fe 3 O 4 , and vanadium oxide, V 2 O 3 . One may speculate if some of the observations of CDCs in transitionmetal oxides [10,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,90] could be due to the fact that these materials often are at the verge of a MI transition. However, in most cases interfacial effects as treated in the following section seem the more likely explanation. The expected divergence of ε ′ at the MI transition of course is also accompanied by an increase of the conductivity, preventing any technical application of this effect for the construction of capacitive components. Interface effects Interfaces of any kind can generate very high apparent values of the dielectric constant because they act as parallel-plate capacitors with very small plate distances, thus having high capacitances [10]. This corresponds to the long-known Maxwell-Wagner (MW) polarisation effects. Originally, Maxwell and Wagner developed their models for three different types of heterogeneous samples, namely for a dielectric sample consisting of layers with different dielectric properties (i.e. ε ′ and σ ′ ) arranged perpendicular to the electric field, for an ideal dielectric sample covered with a bad conductor of varying thickness and for a homogeneous dielectric medium in which spherical particles with different dielectric properties are suspended [15,16]. For all these cases, strong dispersion of the dielectric constant and of the loss was deduced, which can be completely understood from the heterogeneity of the investigated samples without invoking any frequency-dependent microscopic processes within the dielectric materials. Fig. 1. Equivalent circuit representing the simplest model for a heterogeneous sample, composed, e.g., of an interfacial and bulk region. bulk C b C i G b G i interface A straightforward approach to understand the dielectric behaviour of such a heterogeneous system is an equivalent circuit analysis [18,79]. Let us assume the simplest case of the sample being composed of two regions with different dielectric properties. Each region, of course, has some conductivity and dielectric constant, i.e. represents a lossy capacitor. It shall be modeled by a simple parallel RC circuit, i.e. we assume that there is no intrinsic frequency dependence of conductivity or dielectric constant. For the cases treated by Maxwell and Wagner [15,16] and for any cases where the second dielectric region is of interfacial type as, e.g., a grain boundary or a surface depletion layer, it seems reasonable to assume a serial connection of the two RC circuits as depicted in Fig. 1. Then the total admittance of this circuit is given by Y t = G ′ t + iG ′′ t = (G i + iωC i )(G b + iωC b )/[(G i + iωC i ) + (G b + iωC b )] (ω = 2πν is the circular frequency). When resolved into real and imaginary part and calculating the capacitances via C ′ t = G ′′ /ω and C ′′ t = G ′ /ω, we obtain: C ′′ t = G i G b (G i + G b ) + ω 2 (G i C 2 b + G b C 2 i ) ω(G i + G b ) 2 + ω 3 (C i + C b ) 2 (1) C ′ t = (G 2 i C b + G 2 b C i ) + ω 2 C i C b (C i + C b ) (G i + G b ) 2 + ω 2 (C i + C b ) 2(2) Here G b and G i are the conductances and C b and C i are the capacitances of the different sample regions with indices b and i standing for bulk and interface, respectively. If neglecting the geometry factors connecting C and ε ′ , these relations lead to exactly the same frequency dependence as the Debye relaxation laws, ε ′ = ε ∞ + ε s − ε ∞ 1 + ω 2 τ 2(3) and ε ′′ = (ε s − ε ∞ )ωτ 1 + ω 2 τ 2 + σ dc ωε 0 ,(4) describing the relaxational response of an ideal dipolar system ( Fig. 2) [95]. Here ε s and ε ∞ are the low and high frequency limits of ε ′ (ω), respectively, and τ is the relaxation time, describing the dipolar dynamics. The last term in Eq. (4) accounts for the contribution of dc charge transport to the loss, with σ dc the dc conductivity. The quantities of Eqs. 1 and 2 corresponding to ε s , ε ∞ and τ are: C ′ s = G 2 i C b + G 2 b C i (G i + G b ) 2 (5) C ′ ∞ = C i C b C i + C b (6) τ = C i + C b G i + G b(7) Nowadays any relaxational response generated by heterogeneities in the sample is termed MW relaxation. Let us now assume that one of the two dielectric regions of the sample indeed is an insulating interface like the depletion layer of a Schottky diode at the electrodes or like barrier layers between the grains of a polycrystalline sample. In the latter case we make the simplifying assumption that all grain boundaries act like a single capacitor, which is justified in most cases (there are more detailed treatments like the brick layer model [18,96,97] but they lead to essentially the same results). Then we can assume that C i ≫ C b (the interface is much thinner than the bulk) and that G i ≪ G b (the interface is nearly insulating). In this limit we get C ′ s = C i , C ′ ∞ = C b and τ = C i /G b . Thus at low frequencies, the dielectric constant is completely governed by the large capacitance of the thin insulating interface layer and only at high frequencies the intrinsic bulk response is detected (by the way, the same goes for the con- ductance, i.e. G ′ t (ν → 0) = G i and G ′ t (ν → ∞) = G b ) . This can be rationalised by the bridging of the high contact resistance by the contact capacitance, acting like a short at high frequencies [10]. When evaluating dielectric measurements, the dielectric constant usually is calculated from the measured capacitance data via ε ′ = C ′ /C 0 . C 0 is the geometrical capacitance, for a parallel-plate capacitor given by C 0 = ε 0 A/t with A the area and t the plate distance of the capacitor. At low frequencies, in the interface dominated regime, of course C 0 determined from the bulk geometry is the wrong quantity and can be many decades smaller than the geometrical capacitance of the thin insulating layer(s). Thus, despite the true dielectric constant of the interface region (and of course that of the bulk) have "normal" magnitudes, say of the order of ten, very large apparent values of ε ′ can arise [10]. It is difficult to distinguish these effects from true bulk contributions as the resulting relaxation modes have all the characteristics of intrinsic relaxations, where dipolar entities are assumed to reorient in accord with the ac field at low frequencies but to be unable to follow its quick variations at high frequency. Also the typical frequency-dependent shifts of the relaxation features, mirroring in the intrinsic case the freezing of dipolar dynamics at low temperatures [95], are observed: The bulk conductance G b of the considered materials usually has semiconducting temperature characteristics, i.e. it increases exponentially with temperature. Thus, via the relation τ = C i /G b the relaxation time will exhibit the typical strong temperature variation of dipolar systems leading to pronounced shifts of the step in ε ′ (ν) and the peak in ε ′′ (ν) to low frequencies when temperature is reduced, just as for an intrinsic dipolar relaxation. Real-life samples exhibiting interfacial polarisation usually do not exactly follow the behaviour suggested by Eqs. 1 and 2 and depicted in Fig. 2. For example, the step in ε ′ (ν) and the peak in ε ′′ (ν) sometimes are smeared out and significantly broader than expected. Just as for systems showing intrinsic dipolar relaxations [98,99], this can be ascribed to a distribution of relaxation times. For heterogeneity-generated MW relaxations it can be assumed to arise, e.g., from the roughness of the surface in case of a Schottky-diode mechanism or from the distribution of grain (and thus also grain-boundary) sizes in ceramic samples. In addition, the ubiquitous hopping conductivity usually leads to the typical power-law frequency dependences of the UDR [79] , i.e. σ ′ ∝ ν s , σ ′′ ∝ ν s , ε ′ ∝ ν s−1 and ε ′′ ∝ ν s−1 with s < 1 (see Sec. 2.3), which determines the spectra in the high-frequency regime where the intrinsic bulk behaviour governs the dielectric response. The resulting curves are schematically shown in Fig. 3 (solid lines) [10]. The dashed lines represent the pure intrinsic behaviour of the bulk sample, which would be observed in the absence of any interface contributions. Finally, in heterogeneity dominated samples often an additional relaxation leading to even higher values of ε ′ at the lowest frequencies is found as schematically illustrated by the dotted lines in Fig. 3. This implies the presence of two different interfacial regions in the sample, which can be rationalised, e.g., by the simultaneous presence of a surface depletion layer and insulating grain or twinning boundaries [11,26,27]. Thus, the most general equivalent circuit is provided by two parallel RC circuits, connected in series to the bulk element, the latter consisting of a capacitance, dc resistor and a frequency-dependent complex conductance element accounting for the hopping-generated UDR (inset of Fig. 3(c)). As mentioned above, insulating interfaces can occur at the surface of the sample, e.g., due to the formation of a Schottky diode at the electrode/sample contact or due to a thin insulating surface layer with slightly different stoichiometry (e.g., oxygen content). Also internal interfaces can arise, e.g., from grain boundaries in ceramic samples or planar crystal defects (e.g., twin boundaries). In general, surface-related effects (SBLCs) represent a problem for technical application as they do not scale down in accord with sample size and especially in the common thin-film techniques employed in integrated electronics, much smaller CDCs, if colossal at all, may arise. One may run into similar problems for grain-boundary or defect-generated IBLCs, at least if the film thickness approaches the spatial dimensions of the regions separated by these types of interfaces (grains or crystallites). Nevertheless, both kinds of IBLCs are used in some types of stand-alone high-capacitance condensers as the mentioned ceramic barrierlayer capacitors and so-called double-layer capacitors (also termed supercapacitors) [100]. The latter can reach capacitances up to several kF making them suitable for energy storage instead of batteries. They use the well-known blocking-electrode effect [18] observed at low frequencies in ionically conducting materials. In contrast to electrons or holes in electronic conductors, ionic charge carriers cannot penetrate into the metallic electrodes used to contact the samples and instead accumulate in thin layers immediately below the sample surface. These insulating and very thin layers represent large capacitors leading, by means of the equivalent circuit discussed above, to an enhanced capacitance of the sample. For both supercapacitors and ceramic barrier layer capacitors, special preparation techniques are necessary for optimisation of the microstructure. For example in supercapacitors, electrode materials with large inner surfaces like activated charcoal are commonly used. Of high interest for application may also be any spontaneously arising interfaces caused, e.g., by electronic phase separation or charge order. These phenomena were found in many transitionmetal oxides as the colossal magnetoresistance manganites [101,102,103,104] or the cuprate superconductors [105,106,107,108]. Also the system La 2−x Sr x NiO 4 attracted much interest due to the formation of stripe-like electronic phase separation in large parts of its phase diagram [109,110,111,112,113,114,115,116,117]. In the charge-ordered or phase-separated phases of many materials, a large or even colossal magnitude of the dielectric constant has been detected (e.g., [38,41,43,51,52,118,119,120,121,122]). The true reason for the found high values of ε ′ often is not completely clarified but it seems well possible that at least in some cases internal interfaces between the different phase-separated regions may play a role [51,119]. An enhanced dielectric constant due to interfaces between these spontaneously forming heterogeneous regions within the sample still represents a kind of MW effect and no intrinsically large ε ′ in any parts of the sample has to be assumed to explain the CDC. However, within this scenario the MW relaxation leading to the apparently colossal ε ′ would arise from an effect that is inherent to the material and does not depend on sample preparation and thus the observed CDCs could be regarded as quasi-intrinsic. In addition, in this case the heterogeneity arises on a much finer scale than, e.g., for contact or grain boundary effects and can be expected to persist also for the scaled down dimensions prevailing in integrated electronics. Distinguishing CDCs from interface and intrinsic bulk effects Distinguishing MW relaxations from intrinsic bulk ones is not a trivial task. As mentioned above, both phenomena can lead to absolutely identical frequency and temperature characteristics of the dielectric quantities. In general, if a strong relaxation mode is observed, with absolute values of ε ′ s exceeding 100 and without the typical temperature characteristics of ferroelectrics or relaxor ferroelectrics, a non-intrinsic mechanism is very likely. In such case no exotic mechanisms should be invoked to explain the experimental observations before some additional checks have been performed. Checking for SBLC effects is a relatively simple task. If the formation of Schottky diodes at the electrode-sample interface are suspected, two measurements of the same sample with contacts formed by a conducting paste or paint (e.g., silver paint) and by sputtering or evaporating a metal layer should be performed. Schottky diodes only form if metal and semiconductor are in direct contact. Conducting pastes or paints typically contain metallic grains of µm dimension. It is easy to imagine that these grains "touch" the sample surface only at distinct points and the "wetting" of the sample surface by the metal is poor. Quite in contrast, sputtered or evaporated contacts provide a much better wetting and thus a more effective formation of the thin depletion layer of the Schottky diodes. It should be noted that in case of silver paint at areas, where no direct contact between metal and sample is achieved, the depletion layer of the diode is replaced by air gaps. These gaps of course also act as capacitors, which, however, are of much smaller magnitude due to the larger gap width compared with the thickness of the diode depletion layers and the low ε ′ ≈ 1 of air. (If no Schottky diodes are formed, these air-gap capacitors obviously have too small capacitances to provide any contribution to the measured ε ′ as, e.g., in BaTiO 3 silver paint measurements provide the correct values [11].) Overall, the values of ε s can be expected to be strongly reduced when using metallic paints or pastes as electrodes. Such a finding clearly points to diode formation at the surface causing the observed CDCs. The thickness of Schottky-diode generated depletion layers depends on the difference of the work functions of metal and semiconductor. Thus at first glance it may seem feasible to simply use two different metals without any further variation of the type of contact. However, the work functions of typical electrode metals do not strongly differ (typically by the order of 10% only) and the effect is barely visible, having in mind that ε ′ (T, ν) often varies by several decades. In contrast, the effect of varying the contact type usually is much bigger and often of the order of one decade or more [11,26,49,51]. Another origin of SBLCs could be deviations from bulk stoichiometry at the surface, e.g., of the oxygen content that may be different to that of the bulk due to the contact with ambient atmosphere. This can lead to the direct generation of a contact capacitance by this layer. Also the formation of a MIS diode with a corresponding thin depletion layer seems possible, with the insulator being the mentioned surface layer of different stoichiometry [25,27,123]. In the latter case, again a variation of contact type should lead to a marked variation of ε s . In the former case, the same should be achieved by polishing the sample to remove the insulating surface layer and applying the same type of contacts. When doing this, some care should be taken to ensure the same surface roughness before and after polishing as it can be essential for the formation of diodes (see above). Finally, the presence of surface-related effects can also be simply checked for by varying the thickness of the sample [11]. For any kind of SBLC, the sample thickness should directly scale with the absolute value of the CDCs because the surface capacitance remains the same but the geometrical capacitance C 0 used to calculate ε ′ varies with thickness (see Sec. 2.5). Thus thicker samples should have higher CDC values. Again, when polishing down the sample, the same surface roughness as before must be ensured. Checking for IBLCs often is a more difficult task. The simplest way to exclude grain boundaries is measuring single-crystalline samples. Alternatively, different grinding of the powders before pressing and sintering and different sintering conditions should be applied to achieve different grain sizes. Insulating grain boundaries may arise from deviations of stoichiometry at the grain surfaces from the bulk one [12] and varying the conditions during preparation may lead to different conductivities, thicknesses etc. of these surface layers. The most commonly applied way to check for grain-boundary effects is the variation of the sintering time applied to the final pressed sample tablets (e.g., [12,20,22]). Usually longer sintering times lead to larger grain sizes. The concomitant reduction of the thickness of the internal barriers, if averaged over the whole sample, leads to an increase of the overall capacitance of the IBLCs. However, one should be aware that longer sintering also should have an effect on the surface of the sample, reducing, e.g., its roughness due to grain growth at the surface. This may also influence possible IBLC formation and makes the interpretation of such experiments somewhat ambiguous. In case of single-crystalline samples, the presence of possible IBLCs caused by twin or other defect-generated boundaries may also be checked by varying the preparation conditions. Sometimes, instead of presenting the response of CDC materials to ac electric fields via the dielectric constant, loss or conductivity, the complex impedance is provided (e.g., [5,29]). This usually is done by showing complex impedance-plane plots, i.e. Z ′′ (ν) vs. Z ′ (ν). In such plots, any parallel RC circuit being part of the equivalent circuit describing the sample (see, e.g., Figs. 1 and 3(c)) will lead to a separate semicircle (at least if their time constants τ = RC are well separated). As this kind of plots allows for a simple graphical evaluation (e.g., extrapolations of the semicircles enable the determination of the involved resistors of the circuit) they were often employed in earlier times when performing least-square fits of complex functions were a difficult task. However, one has to be aware that the frequency information is lost in this kind of plots and nowadays it is state of the art to perform fits of the dielectric spectra with the complete formula of the equivalent circuit (e.g., Eqs. 1 and 2), simultaneously applied to complemental quantities as, e.g., dielectric constant and conductivity or loss. Sometimes the observation of a second semicircle in the impedance-plane plot, found in addition to the one arising from the bulk, is taken as evidence for grain boundary or other IBLC mechanisms. However, just in the same way as for an IBLC, also an SBLC will lead to such a semicircle and no information concerning the nature of the barrier layers is provided from such type of plots. Interfacial barrier effects usually lead to relaxation steps in ε ′ (T, ν) with temperatureindependent values of ε s . The static dielectric constant in case of MW relaxations is determined by the interface capacitance C i (see Sec. 2.5). The thickness of grain boundaries or of depletion layers of diodes usually are not or only weakly temperature dependent, thus explaining the constant ε s (T ). However, if the interface conductance G i should be of similar magnitude as the bulk value G b , i.e. if the condition G i ≪ G b mentioned in Sec. 2.5 is no longer fulfilled, the relation C ′ s = C i becomes invalid. In this case the low-frequency dielectric constant ε s ∝ C ′ s may become temperature dependent. Assuming C i ≫ C b and G i ≈ G b we obtain from Eq. (5): C ′ s = G 2 b C i (G i + G b ) 2(8) The conductivity of bulk and of the barrier layer can be assumed to exhibit strong semiconducting temperature dependence but both quantities usually should not exhibit identical temperature variation and in some temperature region the condition G i ≪ G b may apply and in others G i ≈ G b might be valid. Thus, a strong temperature dependence of the static dielectric constant can result due to a transition from C ′ s = C i to C ′ s as given by Eq. 8, which, e.g., for G i = G b would lead to C ′ s = C i /4. Under certain conditions, this can lead to temperature characteristics of ε s , mimicking that of relaxor ferroelectrics. Colossal dielectric constants: Experimental results In the following, we provide an overview of results from broadband dielectric spectroscopy on various materials with CDCs as collected in the Augsburg dielectric group during recent years, also including new, so far unpublished data. For details on the dielectric experiments and sample preparation, the reader is referred to our earlier publications, e.g., [11,27,28,29,51,124,125]. As mentioned above, CaCu 3 Ti 4 O 12 is the most prominent material showing CDCs. The three-dimensional representation of Fig. 4 provides a convenient overview of the temperature and frequency dependence of its dielectric constant. Most investigations on CCTO reported in literature were performed on ceramic samples and there were also various efforts to prepare high-quality thin films as a first step to application [126,127,128]. However, the measurements of Fig. 4 were obtained on single-crystalline CCTO, which is only rarely investigated so far [2,26,27]. As pointed out in the original publications [1,2,4], in contrast to ferroelectric materials the CDC of CCTO remains unchanged in a relatively broad temperature range. In Fig. 4 this region corresponds to the plateau formed by the yellow, orange and red data points. Indeed at the lowest frequencies of the order of several Hz, this plateau extends over the complete temperature range. However, at higher frequencies, this range becomes successively restricted and at frequencies above some 100 MHz no CDCs are observed at all, even at room temperature [11,26]. The reason for this behaviour is the strong relaxational mode typically observed in CCTO. It leads to a step-like decrease of ε ′ (T, ν) with decreasing temperature or with increasing frequency. Such characteristics is commonly found also in materials with intrinsic relaxations due to dipolar degrees of freedom [95]. For example, it closely resembles the findings in glass forming liquids with dipolar molecules as, e.g., glycerol, the only difference being the very large values of the static dielectric constant reached at low frequencies and/or high temperatures [129]. In Fig. 5(a) data on the same single crystal, but now using sputtered contacts are provided in a conventional graph showing the temperature dependence of ε ′ for measurements at different frequencies [27]. The step-like decrease of ε ′ (T ) with low values at low temperatures arises from a combination of the Debye frequency-dependence shown in Fig. 2 and the semiconductor characteristics of the bulk conductance, which, via τ = C i /G b (see Sec. 2.5), leads to a strong increase of the relaxation time with decreasing temperature. Thus at low temperatures, the system no longer can follow the ac field and low ε ′ values are observed. In Fig. 5(b) the corresponding conductivity curves are shown. It should be noted that the conductivity is di-rectly related to the dielectric loss via σ ′ = ωε ′′ ε 0 . Thus σ ′ (T ) curves are identical to those of ε ′′ (T ), except for their absolute values. However, the additional frequency factor usually leads to a better distinguishability of the different curves and thus to a better readability of the graphs. In addition, in cases of typical interface-generated relaxational response the conductivity representation provides further insight, e.g., concerning the behaviour of the intrinsic bulk conductivity. In general, relaxation steps in ε ′ should lead to corresponding peaks in ε ′′ or σ ′ (for ε ′′ , this is valid for both the temperature and the frequency dependence, cf. Fig. 2). In Fig. 5(b) shoulders are observed instead of peaks because of the additional contribution from the conductance of the interface barrier, which is discussed in more detail below. In Fig. 6, the frequency dependence of dielectric constant and conductivity are shown for selected temperatures. This figure contains the results from two different measurements using silver paint or sputtered gold contacts applied to the same sample. While at high frequencies the results from both measurements agree, there are marked deviations at lower frequencies. Especially, the static dielectric constant (Fig. 6(a)) reaches values that are about one order of magnitude larger for the sputtered contacts and the low-frequency plateau of σ ′ (ν) (Fig. 6(b)) even differs by about two decades. These curves are qualitatively similar to the solid lines shown in the schematic plot of Fig. 3 and can be explained in the same way as discussed in detail in Sec. 2.5 in terms of interfacial effects. Within this framework, at low frequencies the response is dominated by the non-intrinsic barrier contribution. Obviously, for sputtered contacts a much higher interfacial capacitance arises. As discussed in detail in Sec. 2.6, this is just what is expected for an SBLC effect caused by the formation of Schottky or MIS diodes at the electrode-sample interfaces. Also the higher contact conductance of the sputtered contacts, corresponding to the plateau value at low frequencies in σ ′ (ν), can be understood in this way. It should be noted that even in the sputtered case the contact resistance, while lower than for silver paint, still is considerable and clearly no ohmic contacts are formed. The reason for the different contact conductances obtained for the two contact materials is not the better quality of sputtered contacts. Instead it is due to the variation in the formation of the diodes at the electrodes, which is caused by the different "wetting" as discussed in Sec. 2.6. Obviously, the much higher resistance of the air gaps, replacing parts of the diode depletion layers in the silver-paint case, must play a role here. According to the equivalent-circuit framework developed in Secs. 2.5 and 2.6, at high frequencies the bulk response should be detected, which is nicely corroborated by the agreement of the curves from both measurements shown in Fig. 6. At 40 K and 80 K, in Fig. 6(b) the bulk conductivity increases with frequency, approaching a power law for the highest frequencies. As discussed in Sec. 2.3, this is the typical UDR arising from hopping of localised charge carriers [76,77,79,80]. The corresponding contribution in ε ′ (ν) is the slight increase with decreasing frequency, best seen in the 40 K curve between about 100 Hz and 100 kHz (cf. dashed line in Fig. 3(a)). Coming now back to Fig. 5(b), the merging of the conductivity curves for different frequencies at low and high temperatures can be understood as follows: The lower merging curve corresponds to the low-frequency plateau seen in the frequency dependence ( Fig. 6(b)), i.e. it mirrors the temperature dependence of the contact conductance. The approach of this curve with increasing temperature prevents the observation of well-defined peaks in σ ′ (T ), which would become visible only for much smaller contact conductance (see, e.g., Fig. 1 in Ref. [27]). The upper merging curve in Fig. 5(b) corresponds to the intrinsic conductivity of CCTO. At T < 75 K deviations arise for the highest frequencies, which signifies just the same UDR contributions as revealed in Fig. 6(b). The intrinsic dc conductivity of CCTO does not follow conventional thermally activated behaviour. Instead it can be quite well described by the prediction of the VRH model [80], i.e. σ dc ∝ T 1/4 [27,130,131]. This is demonstrated in the inset of Fig. 5 where the x-axis was chosen to linearise the T 1/4 law of the VRH model. The curves for 1 MHz and 1.3 kHz, shown in the inset, correspond to the intrinsic dc conductivity for high and low temperatures, respectively. As demonstrated by the solid line, indeed VRH provides a good fit of the data. This is fully consistent with the detection of the typical UDR power law in the frequency dependence, which is characteristic for hopping transport. The strong dependence of the CDC of CCTO on contact preparation is also found in polycrystals as demonstrated in Fig. 7. Again, strong deviations show up at low frequencies while at high frequencies the bulk response governs the spectra for both types of contact. Taking together all these results, it seems clear that the CDCs in CCTO to a large extent are determined by surface-related effects. However, it should be noted that experiments with varying contacts were reported in literature that partly do not allow for such definite conclusions [14,132,133]. Surface variations introduced by annealing, polishing and other surface treatments obviously also can play an important role, in addition to the effect of the contact material, making the interpretation of experimental results difficult. Finally, it should be noted that for the ceramic sample with sputtered contacts, CDC values of the order of 10 5 are observed ( Fig. 7(a)). Usually such high magnitudes of ε ′ are found for CCTO single crystals only and for polycrystalline samples often much lower values are revealed [1,4,11]. However, annealing the sample for 48h to achieve large grains (see below) and using sputtered contacts to reach good contact wetting obviously can lift the dielectric constant of ceramic samples into similar regions as for single crystals. As mentioned in Sec. 2.6, sintering of ceramic samples to achieve different grain sizes is a quite commonly employed way of checking for grain-boundary related IBLC effects (e.g., [12,20,22]). Figure 8 shows the frequency-dependent dielectric constant of ceramic CCTO that has been subjected to tempering in air at 1000 • C for 3, 24 and 48 hours [27]. With increasing tempering time, i.e. increasing grain sizes, the magnitude of the CDC increases continuously, while the intrinsic value of ε ′ , read off at high frequencies, remains nearly unaltered. At first glance, this finding seems to strongly point to grain boundaries as the origin of the CDCs as the overall thickness of the interfaces between grains becomes less in relation to the sample thickness and thus the corresponding capacitance increases (see Sec. 2.6). However, as mentioned in Sec. 2.6 also the surface smoothness may change with sintering and therefore an SBLC effect cannot be fully excluded [27]. The effect of sintering on the sample surface is seen in the inset of Fig. 8 providing surface topographies detected by Environmental Scanning Electron Microscopy (ESEM). Indeed for the sample tempered for 48 h, the larger grains also lead to a much smoother surface. In any case, there are several sophisticated experiments reported in literature that seem to suggest that grain boundary or other SBLC mechanisms at least play some role for the generation of the CDCs [9,14,20,23,133]. A solution of the partly contradictory results in literature could be a second relaxation, which was reported in several publications to occur in ceramic samples [11,21,24,26,27] (cf. dotted line in Fig. 3). Thus, one may assume that one relaxation is due to SBLCs and the second one is caused by IBLCs. However, in Ref. [26] it was shown that the second relaxation also shows up in single crystals, excluding a grain boundary mechanism. In fact, this relaxation is also weakly seen in Fig. 6(a) as an additional step-like increase below about 10 kHz revealed by the curve measured at 300 K with silver-paint contacts. It also may well be possible that the domination of the dielectric response by IBLC or SBLC effects is strongly sample dependent and no clear-cut statement for CCTO in general can be made. For example, in Ref. [133] the prevailance of SBLC effects for single crystals and of grain boundary effects for ceramic samples was proposed. Differences may also arise for fine and coarse-grained ceramics or for varying surface resistivities [132]. As revealed by Figs. 5 -8, the intrinsic dielectric constant of CCTO, which can be read off at high frequencies and/or low temperatures is of the order of 100 and thus relatively high. HfO 2 , the most considered high-ε ′ material for applications in integrated circuits, e.g., has ε ≈ 22 only. For heterogeneity-dominated systems, high bulk ε ′ -values may enhance the CDCs. For example in case of diode-generated SBLCs the capacitance of the depletion layer, which determines the magnitude of the CDC of course depends on the dielectric constant in this region. The depletion layer of course still is CCTO, only depleted from any mobile charge carriers. Its ε ′ , determined by the ionic and electronic polarisability thus should be of similar magnitude at the intrinsic ε ′ of CCTO. Therefore the CDC should be directly proportional to the bulk value. To understand the generation of the high bulk ε ′ in CCTO, Fig. 9 shows combined spectra of dielectric, THz and infrared measurements [125]. As revealed by Fig. 9(a), in the narrow frequency region between about 3 and 20 THz, ε ′ (ν) decreases by about one order of magnitude. In this region a number of sharp resonances show up, caused by phononic excitations. Thus, one can conclude that the high intrinsic dielectric constant of CCTO mainly arises from the ionic polarisability. This notion is corroborated by a quantitative evaluation of the phonon modes performed by fitting the spectra using ten oscillators [125]. The resulting sum of the dielectric strengths of all phonon modes is about 77 at room temperature and about 96 at 5 K; the dielectric constant due to electronic polarisability is 6.5. Thus we obtain for the static dielectric constant of the infrared experiment at room temperature ε s,o ≈ 84, which agrees well with the intrinsic dielectric constant of the dielectric experiments, ε ∞ ≈ 83. With ε ′ ≈ 85 also the result from THz spectroscopy matches these values. In the combined plot of the conductivity of Fig. 9(b), the sublinear UDR power law arising from hopping conductivity is nicely seen (cf. lower dash-dotted line) but also additional contributions are detected. When also taking into account the THz and infrared results, another steeper power law σ ′ ∝ ν n with exponent n ≈ 1.4 shows up (cf. upper dash-dotted line). Indeed, fits assuming the sum of dc conductivity, a sublinear and a superlinear power law (dashed lines) are able to describe the spectra beyond the relaxation mode. It should be noted that via the Kramers-Kronig relation, a superlinear power law in σ ′ (ν) of course also leads to a contribution in ε ′ (ν), namely an additional decrease proportional to -ν n−1 . This decrease is too small to actually show up within the resolution of the experimental data. However, it is observed in the fits shown by the dashed lines in Fig. 9(a) as a slight decrease between 1 GHz and 1 THz. A superlinear power law, showing up at frequencies beyond the validity of the sublinear UDR power Fig. 9. Dielectric constant (a) and dynamic conductivity (b) of CCTO over 15 decades in frequency for various temperatures [125]. The symbols at ν < 10 12 Hz and the lines at ν > 10 12 Hz show experimental data. Dashed lines are fits of the data beyond the relaxation, taking into account dc and ac conductivities, the latter described by sub-and superlinear power laws, ν s and ν n . The inset shows the dynamic conductivity of CCTO at 295 K in the optical regime. law, was previously observed in various transition-metal oxides and other materials [48,134]. It is clear that additional contributions beyond the UDR must prevail in the region bridging the gap between dielectric spectroscopy (ν 1 GHz) and infrared experiments (ν 1 THz) because in most materials a simple extrapolation of the UDR does not match the absolute values of σ ′ at the lowest frequencies of the infrared experiments. In Ref. [134] the superlinear ν n power law was proposed to be a universal property of disordered matter but its microscopic origin still is unclarified. Interestingly, for high frequencies a transition from phonon-assisted to photon-assisted hopping with σ ′ ∝ ν 2 is predicted for a Fermi glass [80,135]. However, this transition should appear at hν > k B T , which is not fulfilled here. The inset of Fig. 9 shows a magnified view of the results in the infrared region. At low frequencies, ν < 20 THz, ten phonon modes are clearly revealed. As discussed in detail in Ref. [125], the low-lying modes show a decrease of their eigenfrequencies with decreasing temperature, i.e. their temperature dependence resembles the soft-phonon behaviour that is typical for ferroelectric materials [54]. Thus, while CCTO clearly is not ferroelectric, at least some ferroelectric correlations seem possible, which also is consistent with the increase of the bulk dielectric constant from 85 at room temperature to about 100 at 5 K revealed by both the dielectric and the infrared experiments [125]. Beyond 20 THz, the further increase of σ ′ (ν) is due to electronic excitations. They qualitatively agree with the predictions from LSDA band structure calculations [7] assuming transitions from filled hybridised O 2p and Cu 3d bands to empty O 2p/Cu 3d and Ti 3d states [125]. To make use of the CDCs of CCTO for technical applications but avoiding the disadvantages of this materials as, e.g., its relatively high dielectric losses, one can proceed along different routes. Aside of an optimisation of the pure material, for example by adjusting its microstructure, doping of CCTO or the complete substitution with different elements may seem a promising approach. In Fig. 10 we compare the complex permittivity at 121 Hz of pure and three doped CCTO samples [28]. It was previously reported that doping CCTO with Fe or Mn leads to a marked variation of its dielectric constant and a reduction of its dc conductivity [136,137,138,139,140,141]. In Fig. 10(a) the typical MW relaxation step is seen for all investigated materials with CDC values varying between 2000 and 17000. However, for the iron-and manganese-doped samples, the relaxation step is shifted to much higher temperatures than for the undoped and nickel-doped samples. Quite in general, shifts of relaxational features to higher temperatures correspond to shifts to lower frequencies in frequency-dependent plots and thus (because τ ∝ 1/ν p with ν p the loss-peak frequency) imply an increase of the relaxation times. Obviously, for the Mn-and Fe-doped samples, already shortly below 400 K the relaxation time becomes too large for the systems to follow the excitation frequency of 121 Hz. In contrast, for the pure and Ni-doped sample the relaxation time is much smaller and the materials can follow the frequency for temperatures as low as 100 K before ε ′ (ν) finally relaxes to the intrinsic bulk value of the order of 100. CaCu In Fig. 10(b) the loss peaks corresponding to the MW relaxation are nicely revealed for the undoped and Ni-doped sample. For the Fe-and Mn-doped sample this is not the case and the MW relaxation peak shows up as shoulder at 300-400 K only. There the situation is similar as, e.g., seen in Fig. 5(b) for pure CCTO, which is discussed in detail in Sec. 3.1. As noted above, the marked shift of the MW-relaxation features implies a strong variation of the relaxation time. As τ = C i /G b (see Sec. 2.5), and a variation of C i is unlikely, the conductivity must be strongly reduced for Fe and Mn doping. This is well corroborated by Fig. 10(b) where the left flanks of the MW-relaxation peaks (indicated by the solid lines) are directly proportional to the dc conductivity (see Sec. 2.5). Obviously, already for relatively small doping levels the dc conductivity becomes reduced by many orders of magnitude when doping CCTO with iron or manganese. Most likely, charge transport in pure CCTO mainly arises from slight oxygen deficiencies, which are compensated by doping with manganese or iron, which should be substituting on the titanium site [28]. In contrast, Ni doping, most likely of isoelectronic type substituted on the copper-place [28], leads to a much smaller reduction of conductivity and thus a smaller shift of the MW relaxation only. These findings demonstrate that reducing the intrinsic bulk conductivity of a material to minimise its dielectric losses is not a suitable approach for its optimisation for technical application in case of a MW-generated mechanism. Reducing the bulk conductance will lead to an increase of the relaxation time and, thus, to a strong restriction of the frequency range available for application. For example, for 1% Mn doping, even at a frequency as low as 121 Hz no CDCs are found anymore at room temperature ( Fig. 10(a)). Instead, reducing the conductance of the barrier layers seems to be the more feasible way of optimising MW-dominated CDC materials. This should reduce the losses without affecting the relaxation time. However, a simple increase of the layer thickness will not help as this would lead to a simultaneous reduction of the CDC values. For the iron-and manganese-doped samples, in Fig. 10(b) two, respectively one additional relaxation peak is revealed at temperatures below the dominating MW loss-peak. Interestingly, the peak located at about 180 K is found for both materials and an evaluation of the frequencydependent loss reveals that the relaxation time associated with this feature shows identical temperature dependence for both materials [28]. The inset of Fig. 10 demonstrates for the Mn-doped sample that the intrinsic relaxation also shows up in the real part as a small step superimposed to the general decreasing trend of ε ′ (ν). A closer look at the curve for the Ni-doped sample in Fig. 10(b) may also reveal a similar feature for this material. Thus one can suspect that this relaxation is an intrinsic property of CCTO and not related to the doping. It should be noted that for pure CCTO, the intrinsic dielectric properties in this temperature region are inaccessible due to the dominating non-intrinsic barrier contribution (cf. Fig. 5), except for very high frequencies when the MW relaxation is shifted towards high temperatures. However, in this case, also the intrinsic relaxation should be shifted and still may remain undetected. Only the low bulk conductivity of the doped compounds and the subsequent shift of the relaxation feature to higher frequencies could reveal this intrinsic relaxation. Its microscopic origin is unclear at present. The second relaxation in the Fe-doped sample does not show up for Mn doping and thus seems to be related to the Fe defects [137]. As revealed by the inset of Fig. 10, aside of the small relaxation step, the bulk ε ′ (T ) shows a clear increase with decreasing temperature. This agrees with the findings from measurements at GHz in pure CCTO where the frequency was sufficiently high to shift the MW-relaxation step to temperatures beyond room temperature and also with the results from an analysis of the phonon modes measured by infrared spectroscopy [125]. As discussed above, this finding may indicate ferroelectric-like correlations in CCTO and there are even speculations about incipient ferroelectricity in doped CCTO [140], similar to SrTiO 3 [70]. Another interesting finding is the peak of ε ′ (T ) occurring just at the transition into an antiferromagnetic state below about 25 K. This small but significant magnetocapacitive effect is discussed in detail in Ref. [28]. Finally, we want to mention the additional increase of ε ′ (T ) observed at high temperatures, T 200 K, in the pure and Ni-doped sample. It signifies the presence of a second relaxation, in addition to the main MW relaxation, which also is of non-intrinsic origin, as briefly discussed above. For a detailed treatment of the second relaxation, see Ref. [27]. Already in the pioneering work on CCTO by Subramanian et al. and a follow-up paper [1,6], a number of compounds, isostructural to CCTO were introduced. Until now, only few of them are satisfactorily characterised by dielectric spectroscopy and it is not clear why CCTO should be unique within this large group of materials. Indeed, dielectric constants of colossal magnitude were reported for some of these compounds [6,29,30,31,142,143,144,145]. Figure 11 shows the dielectric constant (a) and the loss tangent (= ε ′′ /ε ′ ) at 1.16 kHz (b) for a number of ceramic Ln 2/3 Cu 3 Ti 4 O 12 samples where Ca was replaced by various lanthanides [29]. In all cases, the typical interface-barrier induced MW relaxation is observed with a high-temperature plateau in ε ′ (T ) of large magnitude, ε ′ s reaching colossal values exceeding 1000 for Ln = La, Ce, Pr and Dy. Among the latter, the cerium compound has the highest loss tangent and seems less suited for application. The relaxation features of the dysprosium compound arise at clearly higher temperatures than for Ln = La and Pr. Thus its relaxation time is higher and the CDC can be expected to occur in a smaller frequency range only (cf. the above discussion of the shift of the relaxations in Fig. 10). La 2/3 Cu 3 Ti 4 O 12 and Pr 2/3 Cu 3 Ti 4 O 12 , which behave nearly identical in Fig. 11, seem the most promising materials and in Fig. 12 we provide a more detailed plot of the dielectric properties of the lanthanum compound. For an in-depth treatment of the dielectric properties of Pr 2/3 Cu 3 Ti 4 O 12 , see Ref. [29]. Figure 12 shows the temperature dependence of the dielectric constant and loss tangent of La 2/3 Cu 3 Ti 4 O 12 for a number of frequencies. Dielectric properties of this material were also reported in Refs. [30,31,144]. The overall behaviour of ε ′ (T ) (Fig. 12(a)) closely resembles that of pure CCTO as shown, e.g., in Fig. 5. In contrast to the measurements of La 2/3 Cu 3 Ti 4 O 12 , included in Fig. 11, a different sample was used, which was treated to increase the grain size. In this sample, indeed a truly colossal magnitude of ε ′ of about 10000 is achieved. Interestingly, it seems to arise from an IBLC effect as this value remained nearly unaffected by the type of electrodes used in the measurements (just as for Pr 2/3 Cu 3 Ti 4 O 12 ; see Ref. [29] for details). The intrinsic ε ′ , read off at low temperatures and high frequencies is about 100, similar as in CCTO. The loss tangent exhibits peaks, which, just as for ε ′′ (T ), are characteristic features of relaxations (see Sec. 2.5). The additional increase at high temperatures arises from the conductance of the barriers, just as the lower merging curve revealed in the conductivity plot of pure CCTO (Fig. 5(b); see also discussion of this figure in Sec. 3.1). Overall, here we have a material that has properties at least as good as CCTO. A similar statement can be made for Pr 2/3 Cu 3 Ti 4 O 12 [29]. Therefore it seems that there is nothing peculiar in CCTO and there may be many more isostructural materials with comparable or even better dielectric properties. Another promising member of the family of CCTO-related materials is Cu 2 Ta 4 O 12 . Its crystal structure is derived from that of CCTO by leaving the Ca sites unoccupied and replacing 1/3 of the copper sites by vacancies. Its dielectric behaviour was reported to be similar to that of CCTO, reaching CDCs of about 75000 [49]. For details the reader is referred to Ref. [49]. Fig. 13. Temperature-dependent dielectric constant (a) and loss tangent (b) of a LSNO single crystal with sputtered gold contacts for various frequencies [51]. As mentioned in Sec. 2.5, spontaneously arising interfaces caused, e.g., by electronic phase separation or charge order are of high interest for possible applications of CDC materials for capacitive circuit elements. Interestingly there are some reports on CDCs in the system La 2−x Sr x NiO 4 [38,41,43], which is known to exhibit electronic phase separation, namely a stripe-like ordering of holes, in large portions of its phase diagram [109,110,111,112,113,114,115,116,117]. In Fig. 13 the temperature dependences of the dielectric constant [51] and the loss tangent of single-crystalline La 15/8 Sr 1/8 NiO 4 (LSNO) are provided. Typical relaxation steps (a) and peaks (b), just as in CCTO, are revealed with extremely large values of ε s ≈ 600000. The loss tangent at high frequencies is comparable to or even lower than that of CCTO [2]. To elucidate the possible role of SBLCs for the generation of the observed CDCs, in Fig. 14 the frequency dependence of the dielectric constant (a) and the conductivity (b) are provided for both, contacts prepared by sputtering (as in Fig. 13) and by applying silver paint. Just as for CCTO single-and polycrystals (cf. Figs. 6 and 7), a marked variation of ε s is found with much higher values for the sputtered contacts. Thus, it is clear that surface generated MW effects indeed do contribute to the CDCs in LSNO. Again, at high frequencies the curves obtained with different contact types agree, showing the intrinsic response (for 300 K this regime is reached beyond the investigated frequency range only). The intrinsic σ ′ (ν) (Fig. 14(b)) is governed by a contribution from dc conductivity (e.g., the approximate plateau at about 2 × 10 −4 Ω −1 cm −1 in the 45 K curves) and a marked UDR power-law increase due to hopping charge transport (see Sec. 2.3). In ε ′ (ν) (Fig. 14(a)), in contrast to CCTO no clear saturation at high frequencies is seen. Obviously, up to the highest frequency the UDR contribution, ε ′ ∝ ν s−1 , is larger than ε ∞ arising from the ionic and electronic polarisabilites. This leads to relatively high values of the intrinsic bulk values of ε ′ at high frequencies, e.g., ε ′ (1 GHz) ≈ 300. The ε ′ spectra obtained with silver-paint contacts shown in Fig. 14 reveal clear indications for a second relaxation step at low frequencies, quite similar to CCTO [26]. Thus, a further mechanism enhancing ε ′ , in addition to the suspected SBLCs, must be active in this material. In Refs. [51,52] it was speculated that only the second, low-frequency relaxation may be due to an SBLC effect. In the spectra obtained for sputtered contacts, this second relaxation dominates the response and the first, high-frequency relaxation may become undetectable. Then a contribution from the electronic phase separation may well be possible, leading to a distinct separate relaxation step for the silver-paint sample only . Clearly further work is necessary to corroborate this scenario. In Fig. 15, a combined ε ′ -spectrum of LSNO is shown for room temperature, including dielectric and infrared data. For comparison also a corresponding spectrum for CCTO is included (same data as in Fig. 9 [125]). This plot reveals a marked difference of the dielectric response of both materials: While ε ′ (ν) of CCTO starts to strongly decrease for frequencies beyond 1 MHz and is far from being colossal at some 100 MHz and in the GHz region, this is not the case for LSNO. Due to the restrictions of the experimental technique employed to cover the MHz to GHz region when measuring very high capacitances, no measurements in LSNO at ν > 430 MHz were possible. However, the dashed line shows the most likely course of ε ′ (ν) beyond this frequency. It is based on an extrapolation of fit parameters from fits at lower temperatures, where the MW-relaxation step is well within the frequency window [51,52]. Overall, LSNO seems to be much better suited than CCTO for applications at frequencies in the MHz-GHz range, which is of high relevance, e.g., in modern telecommunications technology. Finally, we want to note that the infrared results in LSNO reveal a "static" dielectric constant (which would be denoted as ε ∞ from a dielectric-spectroscopy viewpoint) of approximately Fig. 15. Comparison of the frequency-dependent dielectric constant of LSNO and CCTO single crystals from dielectric and IR measurements [51,125]. The measurements were performed at room temperature. The upper dashed line was calculated from an extrapolation of the parameters obtained from fits of spectra at lower temperatures (cf. Fig. 2(a) of Ref. [51]). The lower dashed line is a guide to the eyes. 90, much lower than the bulk value of 300 observed at 1 GHz ( Fig. 14(a)). This corroborates the above-mentioned scenario of a dominating UDR contribution in ε ′ (ν) up to the highest frequencies covered in the dielectric measurements. The system La 2−x Sr x NiO 4 provides the possibility of tuning the Sr content over a wide range. Figure 16 shows first results demonstrating that, in addition to x = 1/8, CDCs are also found for other Sr contents, namely x = 0.2 and 0.25 (see also [38,41,43]). In all cases the typical MW relaxation is observed. The approximate agreement of the temperature location of the relaxation steps for the poly-and single-crystalline samples with x = 1/8 (cf. Figs. 13 and 16(a)) indicates that also for the ceramic sample the CDC can be expected to persist up to higher frequencies than in CCTO. The same can be said for the compound with x = 0.2 while for x = 0.25 the relaxation steps seem to be located at somewhat higher temperatures (i.e. the relaxation time is higher). In addition, they are smeared out or even composed of two separate relaxation steps and further investigations are necessary to clarify this issue. In any case it is clear that La 2−x Sr x NiO 4 is a promising system that deserves at least as much attention as CCTO and its relatives. Other transition-metal oxides with colossal dielectric constants As mentioned in the introductory section, there are many further transition-metal oxides showing CDCs. Two typical examples are provided in Fig. 17. It shows the frequency dependence of the dielectric constant of single crystalline La 2 CuO 4+δ , measured at various temperatures. Details have been reported in Refs. [84,146]. In this typical parent compound of high-T c superconductors, the carrier concentration is rather low (δ ∼ 0). Nevertheless, at the bulk-electrode interface a depletion layer is formed and the dielectric constant reaches values of almost 2000 in the audio frequency range even at low temperatures. As we were, at the time of this measurement, only interested in intrinsic properties of La 2 CuO 4+δ , we focused on low temperatures and high frequencies ν > 1 MHz. At elevated temperatures the contact contribution dominates the response far into the microwave regime. The lines in Fig. 17(a) are fits with the equivalent circuit shown in Fig. 3 but without the second interface-related RC element. The intrinsic dielectric constant reaches values of ε ∞ ≈ 35. Figure 17(b) shows results on single-crystalline Pr 0.65 Ca 0.28 Sr 0.07 MnO 3 [44], a colossal magnetoresistance material, which is very close to a metal-insulator phase boundary and reveals antiferromagnetism and charge order below about 200 K. Again the solid lines represent fits using the equivalent circuit. The dashed lines indicate the intrinsic response of the sample neglecting contact contributions. The UDR leads to a ω s−1 contribution, which smears out the contact-dominated step in ε ′ (ω) at high frequencies. In Pr(Ca : Sr)MnO 3 the intrinsic dielectric constant ε ∞ = 50. Finally, Fig. 18 provides the dielectric constant of La 1.2 Sr 2.7 IrO 7.33 [50]. This compound is formed by alternative stacking of hexagonal perovskite (A 2 IrO 6 ) and A ′ 2 O 1+δ layers (A = La/Sr; A ′ = Sr) [147]. In this material, both oxygen and peroxide ions are present, occupying large cavities formed by six AO 6 prisms. Within these cavities, the ions can occupy six different off-centre positions and also the A ′ ions can assume three different positions within the A ′ 2 O 1+δ layers. Ions in off-centre positions can generate ferroelectricity and also may lead to dipolar relaxation phenomena, which makes this material interesting from a dielectric-spectroscopy viewpoint. In addition, the strong substitutional disorder should give rise to charge-carrier localisation and the typical signatures of this phenomenon in the ac response. As seen in Fig. 18, for low frequencies ε ′ (ν) reaches the typical plateau of the MW relaxation with CDCs of the order of 1000. However, for further decreasing frequency an additional strong increase shows up. It exhibits no indication of a second plateau as would be expected for a second relaxation feature. Fits of these spectra, which were also simultaneously performed for the conductivity [50], were able to cover this extra increase without assuming a second relaxation (i.e. without a second interfacial RC element, as shown in the equivalent circuit of Fig. 3(c)). A detailed analysis revealed that this low-frequency increase of ε ′ (ν) is due to the strong intrinsic UDR contribution in this material (indicated by the dashed line for 450 K). It is sufficiently strong to also show up at frequencies below the CDC plateau of the MW relaxation and thus this material provides a nice example for CDCs generated by hopping charge transport (see Sec. 2.3). Measurements with different contact types in this material [50] indicate that the MW relaxation is due to an SLBC mechanism but even without this effect, very large values of ε ′ would be reached at low frequencies. For a detailed discussion of the dielectric properties of La 1.2 Sr 2.7 IrO 7.33 , also revealing an intrinsic dipolar relaxation, see Ref. [50]. Summary and Conclusions A large variety of physical mechanisms can give rise to a colossal magnitude of the dielectric constant. This includes ferroelectricity, with all its disadvantages for technical application, charge-density-wave formation and the approach of a metal-insulator transition, which mostly are of high academic interest only, and interfacial polarisation with its many subgroups as SBLC and IBLC generated effects, including, e.g., electronic phase separation. As revealed by the results provided in the present work, transition-metal oxides in general seem to be prone to the occurrence of colossal magnitudes of the dielectric constant. It is clear that in many cases interfacial barriers within or at the surface of the samples play an important role in the generation of the observed CDCs. Especially in the extensively investigated CCTO, despite some sophisticated attempts of invoking intrinsic mechanisms, a non-intrinsic MW process seems the most likely explanation of its CDCs. However, despite epic efforts of innumerous groups during the past ten years, no consensus on the nature of the involved interfaces has been achieved. In light of the results presented in this work it is clear that in CCTO and also in most other materials, thin insulating layers at the sample surface, most likely induced by diode formation between the metallic electrodes and the semiconducting sample, must play a role. However, also indications for IBLCs of various origins were found. It seems that the question "What causes the CDCs in CCTO?" cannot be unequivocally answered and following the line proposed in Ref. [133] the answer may depend on the specific sample investigated and also combinations of different effects seem likely. Another outcome of the present work is the finding that CCTO is only one member of a large group of isostructural materials with similar properties, examples being La 2/3 Cu 3 Ti 4 O 12 and Pr 2/3 Cu 3 Ti 4 O 12 . Those material show promising dielectric properties and one can expect to discover other, even better suited ones within this vast group of compounds, that indeed once may become new standard materials for high-capacitance applications. Irrespective of any application viewpoints one should not forget that CCTO and its relatives are highly interesting materials also for purely academic reasons. For example, its relatively high intrinsic dielectric constant, further increasing when lowering the temperature, the softening of several of its phonon modes and the occurrence of isosbestic points in the infrared reflectivity spectra [125] deserves further investigation. It seems likely that ferroelectric correlations are present in CCTO and based on its crystal structure, the Ti 4+ cation, rattling within the TiO 6 octaeder and tending to go off-centre at low temperatures [1] seems a likely mechanism. Even incipient ferroelectricity may be possible [140]. Among the many other transition-metal oxides with CDCs, the system La 2−x Sr x NiO 4 stands out by showing electronic phase separation that may well play an important role in the generation of the observed CDCs in this material. Of special significance is our finding that at room temperature ε ′ (ν) of this material remains colossal well up to the GHz frequency range, quite in contrast to CCTO. Fig. 2 . 2Dielectric response for a Debye relaxation arising from the polarisation of dipolar degrees of freedom with an additional contribution from dc charge transport. In the loss a peak arises in ωp = 1/τ . Identical spectra are also produced by the equivalent circuit shown inFig. 1without invoking any intrinsic frequency dependence of the circuit elements. Fig. 3 . 3Frequency-dependent dielectric response for the equivalent circuit shown in (c). Dashed lines: Intrinsic bulk response. Solid and dotted lines: Overall response with one, respectively two additional RC-circuits. The circuit parameters have been chosen to reveal the prototypical behaviour of doped semiconductors with Schottky-barrier-type contacts[10]. Fig. 4 . 4Temperature-and frequency-dependent dielectric constant of single-crystalline CCTO with silver-paint contacts. Fig. 5 . 5Temperature-dependent dielectric constant (a) and conductivity (b) of single-crystalline CCTO with sputtered gold contacts at various frequencies[27]. The inset shows the conductivity for 1.3 kHz and 1 MHz in VRH representation. The solid line demonstrates VRH behaviour of the dc conductivity. Fig. 6 . 6Frequency-dependent dielectric constant (a) and conductivity (b) of single-crystalline CCTO with silver-paint (open symbols) and sputtered gold contacts (closed symbols) at selected temperatures[27]. Fig. 7 . 7Frequency-dependent dielectric constant (a) and conductivity (b) of polycrystalline CCTO (48 h tempered) with silver-paint (open symbols) and sputtered gold contacts (closed symbols) at selected temperatures. Fig. 8 . 8Frequency-dependent dielectric constant of ceramic CCTO samples tempered for 3 h (open symbols), 24 h (symbols) and 48 h (closed symbols) with silver-paint contacts [27]. The insets show the surface topographies obtained by ESEM of the 3 h and 48 h tempered samples. Fig. 10 . 10Temperature-dependent dielectric constant (a) and dielectric loss (b) of undoped and doped (1% Mn, 0.5% Fe and 0.5% Ni) polycrystalline CCTO at 121 Hz[28]. The lines in (b) indicate the temperature-dependent development of the dc conductivity contribution, detected at the low-frequency flank of the MW relaxation peaks. The inset shows a magnified view of ε ′ (ν) for 1% Mn doping below room temperature. Fig. 11 . 11Temperature-dependent dielectric constant (a)[29] and loss tangent (b) of the investigated LnCTO-compounds at 1.16 kHz (silver-paint contacts). Fig. 12 . 12Temperature-dependent dielectric constant (a) and loss tangent (b) of ceramic La 2/3 Cu3Ti4O12 with silver-paint contacts for various frequencies. Fig. 14 . 14Frequency-dependent dielectric constant (a) and conductivity (b) of single-crystalline LSNO with silver-paint (open symbols) and sputtered gold contacts (closed symbols) at selected temperatures[51]. 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[ "Empirical Error Modeling Improves Robustness of Noisy Neural Sequence Labeling", "Empirical Error Modeling Improves Robustness of Noisy Neural Sequence Labeling" ]	[ "Marcin Namysl [email protected] \nFraunhofer IAIS\nSankt AugustinGermany\n\nAutonomous Intelligent Systems\nUniversity of Bonn\nGermany\n", "Sven Behnke [email protected] \nFraunhofer IAIS\nSankt AugustinGermany\n\nAutonomous Intelligent Systems\nUniversity of Bonn\nGermany\n", "Joachim Köhler [email protected] \nFraunhofer IAIS\nSankt AugustinGermany\n" ]	[ "Fraunhofer IAIS\nSankt AugustinGermany", "Autonomous Intelligent Systems\nUniversity of Bonn\nGermany", "Fraunhofer IAIS\nSankt AugustinGermany", "Autonomous Intelligent Systems\nUniversity of Bonn\nGermany", "Fraunhofer IAIS\nSankt AugustinGermany" ]	[]	Despite recent advances, standard sequence labeling systems often fail when processing noisy user-generated text or consuming the output of an Optical Character Recognition (OCR) process. In this paper, we improve the noise-aware training method by proposing an empirical error generation approach that employs a sequence-to-sequence model trained to perform translation from error-free to erroneous text. Using an OCR engine, we generated a large parallel text corpus for training and produced several real-world noisy sequence labeling benchmarks for evaluation. Moreover, to overcome the data sparsity problem that exacerbates in the case of imperfect textual input, we learned noisy language model-based embeddings. Our approach outperformed the baseline noise generation and error correction techniques on the erroneous sequence labeling data sets. To facilitate future research on robustness, we make our code, embeddings, and data conversion scripts publicly available.	10.18653/v1/2021.findings-acl.27	[ "https://arxiv.org/pdf/2105.11872v1.pdf" ]	235,187,329	2105.11872	008f036b32e9b118b78334553e9bc51176341543	Empirical Error Modeling Improves Robustness of Noisy Neural Sequence Labeling Marcin Namysl [email protected] Fraunhofer IAIS Sankt AugustinGermany Autonomous Intelligent Systems University of Bonn Germany Sven Behnke [email protected] Fraunhofer IAIS Sankt AugustinGermany Autonomous Intelligent Systems University of Bonn Germany Joachim Köhler [email protected] Fraunhofer IAIS Sankt AugustinGermany Empirical Error Modeling Improves Robustness of Noisy Neural Sequence Labeling Despite recent advances, standard sequence labeling systems often fail when processing noisy user-generated text or consuming the output of an Optical Character Recognition (OCR) process. In this paper, we improve the noise-aware training method by proposing an empirical error generation approach that employs a sequence-to-sequence model trained to perform translation from error-free to erroneous text. Using an OCR engine, we generated a large parallel text corpus for training and produced several real-world noisy sequence labeling benchmarks for evaluation. Moreover, to overcome the data sparsity problem that exacerbates in the case of imperfect textual input, we learned noisy language model-based embeddings. Our approach outperformed the baseline noise generation and error correction techniques on the erroneous sequence labeling data sets. To facilitate future research on robustness, we make our code, embeddings, and data conversion scripts publicly available. Introduction Deep learning models have already surpassed human-level performance in many Natural Language Processing (NLP) tasks 1 . Sequence labeling systems have also reached extremely high accuracy (Akbik et al., 2019;Heinzerling and Strube, 2019). Still, NLP models often fail in scenarios, where non-standard text is given as input (Heigold et al., 2018;Belinkov and Bisk, 2018). NLP algorithms are predominantly trained on error-free textual data but are also employed to process user-generated text (Baldwin et al., 2013;Derczynski et al., 2013) or consume the output of prior Optical Character Recognition (OCR) or Automatic Speech Recognition (ASR) processes (Miller et al., 2000). Errors that occur in any upstream Training Loss Sailing is a passion. Sailing 1s o passion. Seq2Seq Model Figure 1: Our modification of the NAT approach (green boxes). We propose a learnable seq2seq-based error generator and re-train FLAIR embeddings using noisy text to improve the accuracy of noisy neural sequence labeling. Γ is a process that induces noise to the input x producing erroneousx. E(x) is an embedding matrix. F(x) is a sequence labeling model. e(x) and e(x) are the embeddings of x andx, respectively. y(x) and y(x) are the outputs of the model for x andx, respectively. component of an NLP system deteriorate the accuracy of the target downstream task (Alex and Burns, 2014). In this paper, we focus on the problem of performing sequence labeling on the text produced by an OCR engine. Moreover, we study the transferability of the methods learned to model OCR noise to the distribution of the human-generated errors. Both misrecognized and mistyped text pose a challenge for the standard models trained using error-free data (Namysl et al., 2020). We make the following contributions ( Figure 1): • We propose a noise generation method for OCR that employs a sequence-to-sequence (seq2seq) model trained to translate from error-free to erroneous text ( §4.1). Our approach improves the accuracy of noisy neural sequence labeling compared to prior work ( §6.1). • We present an unsupervised parallel training data generation method that utilizes an OCR engine ( §4.2). Similarly, realistic noisy versions of popular sequence labeling data sets can be synthesized for evaluation ( §5.5). • We exploit erroneous text to perform Noisy Language Modeling (NLM; §4.5). Our NLM embeddings further improve the accuracy of noisy neural sequence labeling ( §6.3), also in the case of the human-generated errors ( §6.4). • To facilitate future research on robustness, we integrate our methods into the Noise-Aware Training (NAT) framework (Namysl et al., 2020) and make our code, embeddings, and data conversion scripts publicly available. 2 Related Work Errors of OCR, ASR, and other text generators always pose a challenge to the downstream NLP systems (Lopresti, 2009;Packer et al., 2010;Ruiz et al., 2017). Hence, methods for improving robustness are becoming increasingly popular. Data Augmentation A widely adopted method of providing robustness to non-standard input is to augment the training data with examples perturbed using a model that mimics the error distribution to be encountered at test time (Cubuk et al., 2019). Apparently, the exact modeling of noise might be impractical or even impossible-thus, methods that employ randomized error patterns for training recently gained increasing popularity (Heigold et al., 2018;Lakshmi Narayan et al., 2019). Although trained using synthetic errors, these methods are often able to achieve moderate improvements on data from natural sources of noise (Belinkov and Bisk, 2018;Karpukhin et al., 2019). Spelling-and OCR Post-correction The most widely used method of handling noisy text is to apply error correction on the input produced by human writers (spelling correction) or the output of an upstream OCR component (OCR post-correction). A popular approach applies monotone seq2seq modeling for the correction task (Schnober et al., 2016). For instance, Hämäläinen and Hengchen (2019) proposed Natas-an OCR post-correction method that uses character-level Neural Machine Translation (NMT). They extracted parallel training data using embeddings learned from the erroneous text and used it as input to their translation model. Grammatical Error Correction (GEC;Ng et al., 2013Ng et al., , 2014Bryant et al., 2019) aims to automatically correct ungrammatical text. GEC can be approached as a translation from an ungrammatical to a grammatical language, which enabled NMT seq2seq models to be applied to this task (Yuan and Briscoe, 2016). Due to the limited size of human-annotated GEC corpora, NMT models could not be trained effectively (Lichtarge et al., 2019), though. Grammatical Error Correction Several studies investigated generating realistic erroneous sentences from grammatically correct text to boost training data (Kasewa et al., 2018;Grundkiewicz et al., 2019;Choe et al., 2019;Qiu and Park, 2019). Inspired by back-translation (Sennrich et al., 2016;Edunov et al., 2018), Artificial Error Generation (AEG) approaches (Rei et al., 2017;Xie et al., 2018) train an intermediate model in reverse order-to translate correct sentences to erroneous ones. Following AEG, we generate a large corpus of clean and noisy sentences and train a seq2seq model to produce rich and diverse errors resembling the natural noise distribution ( §3.3, 4.2). Noise-Invariant Latent Representations Robustness can also be improved by encouraging the models to learn a similar latent representation for both the error-free and the erroneous input. Zheng et al. (2016) introduced stability training-a general method used to stabilize predictions against small input perturbations. Piktus et al. (2019) proposed Misspelling Oblivious Embeddings that embed the misspelled words close to their error-free counterparts. Jones et al. (2020) developed robust encodings that balance stability (consistent predictions across various perturbations) and fidelity (accuracy on unperturbed input) by mapping sentences to a smaller discrete space of encodings. Although their model improved robustness against small perturbations, it decreased accuracy on the error-free input. Recently, Namysl et al. (2020) proposed the Noise-Aware Training method that employs stability training and data augmentation objectives. They exploited both the error-free and the noisy samples for training and used a confusion matrixbased error model to imitate the errors. In contrast to their approach, we employ a more realistic empirical error distribution during training ( §3.3) and observe improved accuracy at test time ( §6.1). Namysl et al. (2020) pointed out that the standard NLP systems are generally trained using error-free textual input, which causes a discrepancy between the training and the test conditions. These systems are thus more susceptible to non-standard, corrupted, or adversarial input. Problem Definition Noisy Neural Sequence Labeling To model this phenomenon, they formulated the noisy neural sequence labeling problem, assuming that every input sentence might be subjected to some unknown token-level noising process Γ=P (x i \|x i ), where x i is the original i-th token, andx i is its distorted equivalent. As a solution, they proposed the NAT framework, which trains the sequence labeling model using auxiliary objectives that exploit both the original sentences and their copies corrupted using a noising process that imitates the naturally occurring errors (Figure 1). Confusion Matrix-Based Error Model Namysl et al. (2020) used a confusion matrix-based method to model insertions, deletions, and substitutions of characters. Given a corpus of paired noisy and manually corrected sentences P, they estimated the natural error distribution by calculating the alignments between the pairs (x, x) ∈ P of noisy and clean sentences using the Levenshtein distance metric (Levenshtein, 1966). Moreover, as P is usually laborious to obtain, they proposed a vanilla error model, which assumes that all types of edit operations are equally likely: c ∈ Σ\{ε} P ins (c\|ε) = P del (ε\|c) = c ∈ Σ\{c, ε} P subst (c\|c), where c andc are the original and the perturbed characters, respectively, Σ is an alphabet, and ε is a symbol introduced to model insertion and deletions. Realistic Empirical Error Modeling Namysl et al. (2020) compared the NAT models that used the vanilla-and the empirically-estimated confusion matrix-based error model and observed no advantages of exploiting the test-time error distribution during training. Would we make the same observation given a more realistic error model? Even though the methods that used randomized error patterns were often successful, we argue that leveraging the empirical noise distribution for training would be beneficial, providing additional accuracy improvements. The data produced by the naïve noise generation methods may not resemble naturally occurring errors, which could lead the downstream models to learn misleading patterns. Digitized text OCR-aware baseline model This work Vanilla baseline error model Figure 2: Distributions of the token error rates of sentences produced by the proposed and the baseline error models. For comparison, we plot the distribution of error rates in the text that contains naturally occurring errors. Each value n is the percentage of sentences with a token error rate in [n − 10, n). In Figure 2, we compare the distributions of error rates of sentences produced by the proposed and the prior noise models with the distribution of errors in the digitized text. We can observe that the distribution of naturally occurring errors follows Zipf's law, while the baseline noise models produce Bell-shaped curves. Interestingly, both the vanilla and the empirical models exhibit similar characteristics, which could explain the observations from the prior work. In practice, the error rate is not uniform throughout the text. Some passages are recognized perfectly, while others can barely be deciphered. Our objective is thus to develop a noise model that produces a smoother distribution, imitating the errors encountered at test time more precisely (cf. This work in Figure 2). Moreover, although the exact noise distribution in the test data cannot always be known beforehand, the noising process, e.g., an OCR engine, used to provide the input, can often be identified. We would thus take advantage of such prior knowledge to improve the efficiency of the downstream task. Data Sparsity of Natural Language Embeddings pre-trained on a large corpus of monolingual text are ubiquitous in NLP (Mikolov et al., 2013;Peters et al., 2018;Devlin et al., 2019). They capture syntactic and semantic textual features that can be exploited to solve higher-level NLP tasks. Embeddings are generally trained using corpora that contain error-free text. Due to the data sparsity problem that arises from the large vocabulary sizes and the exponential number of feasible contexts, the majority of possible word sequences do not appear in the input data. Even though increasing the size of the training corpora was shown to improve the performance of language processing tasks (Brown et al., 2020), most of the misrecognized or mistyped tokens would still be unobserved and therefore poorly modeled when using the errorfree text only. Would it be beneficial to pre-train the embeddings on data that includes realistic erroneous sentences? The Flaws of Error Correction Furthermore, we believe that the correction methods, although widely adopted, can only reliably manage moderately perturbed text (Flor et al., 2019). OCR post-correction has been reported to be challenging in the case of historical books that exhibit high OCR error rates (Rigaud et al., 2019). We note that correction methods have no information about the downstream task to be performed. Moreover, in the automatic correction setting, they only provide the best guess for each token. Comparing their performance with the NAT approach in the context of sequence labeling would be informative. Empirical Error Modeling Figure 1 presents our modifications of the NAT framework. Firstly, we propose to replace the confusion matrix-based noising process ( §3.2) with a noise induction method that generates a more realistic error distribution ( §4.1-4.4). Secondly, to overcome the data sparsity problem ( §3.4), we train language model-based embeddings using digitized text and use them as a substitution of the pre-trained model used in prior work ( §4.5). Sequence-to-Sequence Error Generator Motivated by the AEG approaches (Rei et al., 2017;Xie et al., 2018), we propose a learnable error generation method that employs a character-level seq2seq model to perform monotone string translation (Schnober et al., 2016). It directly models the conditional probability p(x\|x) of mapping errorfree text x into erroneous textx using an attentionbased encoder-decoder framework (Bahdanau et al., 2015). The encoder computes the representation h={h 1 , . . . , h n } of x, where n is the length of x. The decoder generatesx one token at a time: p(x\|x) = n i=1 p(x i \|x <i , x, c), where c=f attn ({h 1 , . . . , h n }) is a vector generated from h, and f attn is an attention function. Our models are trained to maximize the likelihood of the training data. At inference time, we randomly sample the subsequent tokens from the learned conditional language model. Error-Free Sentence Erroneous Sentence Sequence-to-Sequence Model Figure 3: Schematic visualization of the error generation (blue arrows) and the error correction (green arrows) methods. The parallel data can be utilized to train seq2seq models for both tasks. Note that our approach reverses the standard seq2seq error correction pipeline, which uses the erroneous text as input and trains the model to produce the corresponding error-free string ( Figure 3). By interchanging the input and the output data, we can also readily train sentence correction models. One difference is that at inference time we would prefer to perform beam search and select the best decoding result rather than sampling subsequent characters from the learned distribution. Unsupervised Parallel Data Generation To train our error generation model ( §4.1), we need a large parallel corpus P of error-free and erroneous sentences. AEG approaches use seed GEC corpora to learn the inverse models directly. Unfortunately, we are not aware of any comparably large resources for digitized text that could be used for this task. To address this issue, we propose an unsupervised sentence-level parallel data generation approach for OCR ( Figure 4). First, we collect a large seed corpus T that contains the error-free text. We then render each sentence and subsequently run text recognition on the rendered images using an OCR engine. Moreover, to increase the variation in training data, we sample different fonts for rendering. Furthermore, to simulate the distortions and degradation of the printed material, we induce pixel-level noise to the images before recognition. Note that our approach is universal and could be used to generate parallel data sets for other tasks, e.g., an ASR system could be trained on samples from a Text-to-Speech engine (Wang et al., 2018b). Sentence-and Word-Level Modeling We note that the sequence labeling problem is formulated at the word-level, i.e., each word has a class label assigned to it. To employ our method in Text Renderer OCR Rendered Text Images Noisy and Clean Sentence Pairs Error-free Sentences Figure 4: Our parallel data generation method for OCR. We render sentences extracted from a text corpus. Subsequently, an OCR engine recognizes the text depicted in the rendered images. Finally, the pairs of original and recognized sentences are gathered together to form a parallel corpus used to train translation models. this scenario, we develop (i) a sentence-level and (ii) a token-level variant of our error generator. Our sentence-level error generator uses a seq2seq model trained to translate from error-free to erroneous sentences. It can potentially utilize contextual information from surrounding words, which may improve the quality of the results. During the training of a NAT model, a learned seq2seq model translates the original input x to generatẽ x. Subsequently, we use an alignment algorithm ( §4.4) to transfer the word-level annotations from x tox. Our token-level error generator uses a seq2seq model trained to translate from error-free to erroneous words. It relies exclusively on the input and the output words. We use the alignment algorithm to build a training set for this task, i.e., extract word-level parallel data from the corpus of parallel sentences ( §4.2). During the training of a NAT model, a learned generator translates each word x i from x to produce the erroneous sentencex. Figure 5 illustrates the alignment procedure, which we developed to extract word-level parallel training data for our token-level generator and to transfer the labels to the erroneous sentences for the sentence-level generator in the sequence labeling scenario. Word-Level Sentence Alignment To this end, we align each pair of error-free and noisy sentences at the word-level using the Levenshtein Distance algorithm. Our alignment procedure produces pairs of aligned words. The annotations for words are transferred accordingly. Noisy Language Modeling Recently, Xie et al. (2017) drew a connection between input noising in neural network language models and smoothing in n-gram models. We believe that data noising could be an effective tech- Figure 5: Our sentence alignment procedure. We align the original and the recognized sentences (x andx, respectively) using the sequence of edit operations a, which include insertions "i", deletions "d", and substitutions "s" of characters. We use "¬" and "¦" as placeholders for the insertion and the deletion operation, respectively. Matched characters are marked with "-". The alignment procedure produces a list of paired error-free and possibly erroneous words with class labels (optional). nique for regularizing neural language models that could help to overcome the data sparsity problem of imperfect natural language text and enable learning meaningful representation of erroneous tokens. To this end, we propose to include the data from noisy sources in the corpora used to train LM-based embeddings. Specifically, in this work, we learn a noisy language model using the output of an OCR engine ( §4.2) that captures the characteristics of OCR errors. Any other noisy source could be readily used to model related domains, e.g., ASRtranscripts or ungrammatical text. Experimental Setup Sequence-to-Sequence Error Generator To learn our error generators ( §4.1), we utilize the OpenNMT 3 toolkit (Klein et al., 2017). 4 We encode the input sentence at the character-level before feeding it to the seq2seq model. Subsequently, the output produced by the seq2seq model is decoded back to the original form ( Figure 6). Sailing is a passion. S a i l i n g ¬ i s ¬ a ¬ p a s s i o n . S a i l i n g ¬ 1 s ¬ o ¬ p a s s i o n . Sailing 1s o passion. Seq2Seq Model Encoding Decoding Figure 6: Sentence encoding-decoding schema. The whitespace characters are first replaced with a placeholder symbol "¬". The sentences are tokenized at the character-level by adding whitespace between every pair of characters. Decoding reverses this process. Unsupervised Parallel Data Generation Following the approach from §4.2, we generated a large parallel corpus P to train our error generation and correction models. We sampled 10 million sentences 5 from the English part of the 1 Billion Word Language Model Benchmark 6 and used them as the source of error-free text, i.e., the seed corpus T . We rendered each sentence as an image using the Text Recognition Data Generator package 7 . We used 90 different fonts for rendering and applied random distortions to the rendered images. Subsequently, we performed OCR on each image of text using a Python wrapper 8 for Tesseract-OCR 9 (Smith, 2007). We present the distribution of error rates in our noisy corpus in Figure 2 (cf. the digitized text plot). Sequence Labeling Training Setup We employed the NAT framework 10 ( Figure 1) to study the robustness of sequence labeling systems. Following Akbik et al. (2018), we used a combination of FLAIR and GloVe embeddings in all experiments. 11 We employed the data augmentation (L AUGM ) and the stability training (L STAB ) objectives with default weights (α = 1.0), as proposed by Namysl et al. (2020). Consistent with prior work, erroneous sentencesx were generated dynamically in every epoch. Tasks We experimented with the Named Entity Recognition (NER) and Part-of-Speech Tagging (POST) tasks. NER aims to locate all named entity mentions in text and classify them into predefined classes, e.g., person names, locations, and organizations. POST is the process of tagging each word in the text with the corresponding part of speech. Evaluation Setup The evaluation pipeline is shown in Figure 7. Following Akbik et al. (2018), we report the entity-level micro-average F1 score for NER and the accuracy for POST. Baselines Error Generation We compared our error generator with the OCR-aware noise model from Namysl et al. (2020). We used the noisy part of the parallel corpus P to estimate the confusion matrix em- Error Correction To evaluate error correction, we trained the sequence labeling models using the standard objective (L 0 ) and employed the text correction method on the erroneous input before feeding it to the network (Figure 7). Figure 7: Evaluation pipeline. Γ is a noising process that transforms x intox. C(x) is an optional text cor- rection module that returnsx (x =x, if C(x) is absent). E(x) is an embedding matrix. F(x) is a sequence label- ing model. e(x ) We examined Natas 12 , the seq2seq OCR postcorrection method proposed by Hämäläinen and Hengchen (2019). We trained context-free error correction models compatible with Natas using our parallel corpus ( §5.2). Moreover, we also employed the widely adopted spell checker Hunspell 13 . Data Sets Original Benchmarks For NER, we employed the CoNLL 2003 data set (Tjong Kim Sang and De Meulder, 2003). To evaluate POST, we utilized the Universal Dependency Treebank (UD English EWT; Silveira et al., 2014). We present the detailed statistics of both data sets in Table 5 Noisy Benchmarks Unfortunately, we did not find any publicly available noisy sequence labeling data set that could be used to benchmark different methods for improving robustness. To this end, we generated several noisy versions of the original sequence labeling data sets (Table 1). We extracted the sentences from each original benchmark and applied the procedure described in §4.2. 14 We transferred the word-level annotations as described in §4.4. Finally, we produced the data in the CoNLL format (Table 7). Moreover, to evaluate the transferability of error generators, we followed Namysl et al. (2020) Empirical Noise Generation Approaches In this experiment, we compared the NAT models that employed either our seq2seq noise generators 14 We directly applied both Tesseract v3.04 and v4.0. We used different sets of distortions and image backgrounds than those employed to generate parallel training data. 15 We merged both sets of misspellings for evaluation. or the baseline error models (Table 2). In this evaluation scenario, we do not employ C(x) (Figure 7). Our error generators outperformed the OCRaware confusion matrix-based model on the noisy benchmarks generated using the Tesseract 4 engine. The advantage of our method was less emphasized in the case of the Tesseract 3 ♣ data sets. The tokenlevel translation method performed better than the sentence-level variant, while the latter was more efficient when the error rate of the input was lower (cf. the original data and the Tesseract 4 ♠ columns), although it often struggled with translating long sentences. Moreover, data augmentation generally outperformed stability training, which is consistent with the observation from Namysl et al. (2020). Furthermore, we observe a slight decrease in accuracy on the original UD English EWT with both auxiliary objectives. We believe that this was caused by the different proportions of the tokens that were perturbed during training by our seq2seq error generators (e.g., 18% and 19.5% in the case of our token-level model for CoNLL2003 and UD English EWT, respectively). The trade-off between accuracy for clean and noisy data has thus been shifted towards the latter. We also notice a greater advantage of the seq2seq method over the baseline on the noisy UD English EWT data sets. Additionally, in §A, we analyze the relationship between the size of the parallel corpus used for training and the F1 score of the NER task. Error Generation vs. Error Correction We compared the NAT approach with the baseline correction methods ( §5.4). Preliminary experiments revealed that these baselines underperformed due to the overcorrection problem. To make them more competitive, we extended their default dictionaries by adding all tokens from the corresponding test sets for evaluation. Although the vocabulary of a test set could rarely be entirely determined, this setting would simulate a scenario where accurate in-domain vocabularies could be exploited. Table 2 includes the results of this experiment. As expected, although more general, error correction techniques were outperformed by the NAT approach regardless of the noising method used. Surprisingly, Hunspell performed better than Natas on CoNLL 2003. We carried out a thorough inspection of the results of both methods and found out that Natas, although generally more accurate, had problems with recognizing tokens that were a part of entities. This behavior could be a flaw of data-driven error correction methods, as the entities are relatively rare in written text and are often out-of-vocabulary tokens (Alex and Burns, 2014). Noisy Language Modeling FLAIR (Akbik et al., 2018) learns a bidirectional LM to represent sequences of characters. We used the target side of our parallel data corpus ( §5.2) to re-train FLAIR embeddings on the noisy digitized text. 16 Subsequently, we compared the accuracy of the vanilla NAT models ( §3.2) that employed either the pre-trained or our NLM embeddings. Moreover, we do not use C(x) in this scenario (Figure 7). Note that the noise model and the embeddings are two distinct components of the NAT architecture (Γ and E(x) in Figure 1, respectively) and therefore they could be easily combined. However, in this work, we do not mix our NLM with empirically estimated error models to avoid the twofold empirical error modeling effect. We leave the evaluation of this combination to future work. Table 3 summarizes the results of this experiment. Our method significantly improved the accuracy across all training objectives, even when we employed exclusively the standard training objective for the sequence labeling task (L 0 ). Surprisingly, we also achieved evident improvements for the noisy data set generated using the Tesseract 3 engine, which confirms that NLM embeddings can model the features of erroneous tokens even in the out-of-domain scenarios. On the other hand, the NLM slightly decreased the accuracy on the original data for the standard training objective. We plan to investigate this effect in future work by eliminating possible differences in the pre-training procedure and comparing our NLM against a model trained on the original error-free text corpus instead of using the embeddings from Akbik et al. (2018). Human-Generated Errors In this experiment, we evaluated the utility of our seq2seq error generators learned to model OCR noise ( §6.1) and our NLM embeddings ( §6.3) in a scenario where the input contains human-generated 16 The hyper-parameters were consistent with prior work. Table 4: Transferability of the methods learned to model OCR noise to the distribution of the human-generated errors ( §6.4): (a) Comparison of the NAT approach with and without our NLM embeddings on the English CoNLL 2003 test set with human-generated errors. (b) Comparison of empirical error generation approaches on the English CoNLL 2003 and the UD English EWT test sets with human-generated errors. We report mean and standard deviation F1 scores (CoNLL 2003) and accuracies (UD English EWT) over five runs with different random initialization. L 0 , L AUGM , L STAB is the standard, the data augmentation, and the stability objective, respectively (Namysl et al., 2020). The NLM column indicates whether the model employed our NLM embeddings. Bold values indicate top results (within the models trained using the same objective) that are statistically inseparable (Welch's t-test; p < 0.05). errors. For evaluation, we used the noisy data sets with synthetically induced misspellings ( §5.5). We do not employ C(x) in this scenario (Figure 7). Table 4 summarizes the results of this experiment. The models with our NLM embeddings outperformed the baselines for all training objectives (Table 4a). The seq2seq error generation approach performed on par with the confusion matrix-based models on the CoNLL 2003 data set, while the latter achieved better accuracy on the UD English EWT data set (Table 4b). We believe that this difference was caused by the discrepancy between the data distributions. Note that although the data used in this experiment reflects the patterns of human-generated errors, the distribution of these errors does not necessarily follow the natural distribution of human-generated errors, as it was synthetically generated using a fixed replacement probability that was uniform across all candidates. 17 Nevertheless, our methods proved to be beneficial in this scenario, which would suggest that the errors made by human writers and by the text recognition engines have common characteristics that were exploited by our method. Conclusions In this work, we studied the task of performing sequence labeling on noisy digitized and humangenerated text. We extended the NAT approach and proposed the empirical error generator that per-17 For comparison, we visualized the error distributions of our noisy benchmarks in Figure 9. forms the translation from error-free to erroneous text ( §4.1). To train our generator, we developed an unsupervised parallel data synthesis method ( §4.2). Analogously, we produced several realistic noisy evaluation benchmarks ( §5.5). Moreover, we introduced the NLM embeddings ( §4.5) that overcome the data sparsity problem of natural language. Our approach outperformed the baseline noise induction and error correction methods, improving the accuracy of the noisy neural sequence labeling task ( §6.1-6.3). Furthermore, we demonstrated that our methods are transferable to the out-of-domain scenarios -human-generated errors ( §6.4) and the noise induced by a different OCR engine ( §6.1, 6.3). We incorporated our approach into the NAT framework and make the code, embeddings, and scripts from our experiments publicly available. Grundkiewicz and Junczys-Dowmunt (2019) showed that that unsupervised systems benefit from domain adaptation on authentic labeled data. For future work, we plan to fine-tune NAT models pretrained on synthetic samples using the labeled data generated directly by the noising process. Alan Akbik, Duncan Blythe, and Roland Vollgraf. 2018 A Relationship with the Corpus Size Empirical error generators are especially beneficial when we can approximate the noise distribution to be encountered at test time. In this experiment, we aimed to answer the question, how much parallel training data is required to train a solid seq2seq error generation model. Figure 8 shows that the NAT models that used our seq2seq error generator performed better than those employing the baseline vanilla error model proposed by Namysl et al. (2020) for all noisy benchmarks that were generated using the Tesseract 4 OCR engine. The improvements were observed even when we used as few as 1000 parallel training sentences. Our method also outperformed the baseline on the original CoNLL 2003 benchmark. On the contrary, the accuracy of models trained using our generator fell slightly behind the baseline on the Tesseract 3 ♣ and Typos data sets. B Sequence Labeling Data Sets Original Benchmarks Table 5 presents the detailed statistics of the original sequence labeling benchmarks used in our experiments. For NER, we employed CoNLL 2003 18 (Tjong Kim Sang and De Meulder, 2003). To evaluate POST, we utilized Universal Dependency Treebank (UD English EWT 19 ; Silveira et al., 2014). Table 6 presents the error rates and the correction accuracies of the Natas and Hunspell methods calculated on the test sets of the noisy sequence labeling benchmarks. Moreover, Table 7 shows an excerpt from a noisy sequence labeling data set generated for evaluation. Furthermore, Figure 9 presents the distribution of token error rates in relation to the percentage number of sentences in our noisy data sets. For comparison, we also included the distributions obtained by applying different noise generation methods -the vanilla-and the OCR-aware confusion We note that the error distribution of our noisy data sets is closer to the Zipf distribution in contrast to the results of prior methods that exhibit a Bell-Curve pattern. Note that the Typos data set was generated by randomly sampling possible lexical replacement candidates from the lookup tables, hence its distribution exhibits slightly different characteristics than the noisy data sets generated by directly applying the OCR engine to the rendered text images. Based on the above results, Number of sentences we believe that our noisy data sets are better suited for the evaluation of the robustness of sequence labeling models than the data generated by the prior approaches. Noisy Benchmarks Data Conversion Scripts Because of licensing and copyright reasons, we did not submit the noisy data sets directly. Our code includes the scripts for the conversion of the original benchmarks into their noisy variants. For reference, we added excerpts of the noisy UD English EWT data set in the supplementary materials. C Reproducibility In this section, we present additional information that could facilitate reproducibility. Hyper-parameters To train our seq2seq translation models, we generally used the default hyperparameters of the OpenNMT toolkit. We list all non-default values in Table 8. Moreover, we decayed the learning rate eight times during the training for all models. Furthermore, we utilized copy attention (See et al., 2017) for our error generation models and global attention (Luong et al., 2015) for the error correction model. Table 9 summarizes the validation accuracy of our seq2seq models for error generation. We trained the sentence-level models for 1.6×10 4 and the token-level models for Figure 9: Distributions of the token error rates of sentences in our noisy sequence labeling data sets (Tesseract 3 ♣ , Tesseract 4 ♦ , Tesseract 4 ♠ , and Typos). For comparison, we include error distributions obtained by applying our seq2seq token-level error generator and the baseline confusion matrix-based error models (Namysl et al., 2020) to the sentences extracted from the original benchmark. η CER is the character-level noising factor used by the vanilla error model. Each point is the percentage of sentences with a token error rate that falls into a specific token error range, i.e., the value of 50 corresponds to the sentences with a token error rate greater than 40 and lower than or equal to 50. . O Table 7: Example of a sentence from the noisy CoNLL 2003 data set. The first and the second column contains the noisy and the error-free tokens, respectively. The third column denotes the class label in BIO format. and achieved 96.9% accuracy on the validation set of 5000 sentences. Validation Accuracy Learnable Parameters The number of parameters in our sequence labeling models was constant among different models, as we used the same architecture in all experiments. The number of all model parameters was 60.3 million (including embeddings that were fixed during the training), and the number of all trainable parameters was 25.5 million. Moreover, all our seq2seq error generation and correction models had about 7.7 million parameters. Table 9: Validation accuracy of the seq2seq models for error generation. We trained both the token-level and the sentence-level variants. The first and the second column show the number of parallel sentences used for training and validation, respectively. Average Runtime The evaluation of the complete test set took 7 and 10 seconds on average in the case of UD English EWT and English CoNLL 2003, respectively. The runtime did not depend on the training method that was used. Nevertheless, when we employed the correction method, the runtime was significantly lengthened, e.g., it took almost 3 minutes to evaluate a model that employed the Natas correction method on English CoNLL 2003. Computing Architecture The evaluation was performed on a workstation equipped with an Intel Xeon CPU with 10 cores and an Nvidia Quadro RTX 6000 graphics card with 24GB of memory. and y(x ) are the embeddings and the output of the model forx , respectively. ployed by this baseline. Moreover, in the NLM experiment ( §6.3), we also evaluated the vanilla error model proposed by Namysl et al. (2020). and synthetically induced misspellings to the error-free data sets. To this end, we used the lookup tables of possible lexical replacements released by Belinkov and Bisk (2018) and Piktus et al. (2019). 15 6 Experimental Results 2003 contains the annotations for the following entity types: person names (PER), locations (LOC), organizations (ORG), and miscellaneous (MISC). For UD English EWT, the following universal POS tags were included: ADJ (adjective), ADP (adposition), ADV (adverb), AUX (auxiliary), CCONJ (coordinating conjunction), DET (determiner), INTJ (interjection), NOUN (noun), NUM (numeral), PART (particle), PRON (pronoun), PROPN (proper noun), PUNCT (punctuation), SCONJ (subordinating conjunction), SYM (symbol), VERB (verb), X (other). matrix-based models by Namysl et al. (2020), and our token-level seq2seq error generator. ) LAUGM (Tesseract 3 ♣ ). ) LSTAB (Typos). Figure 8 : 8F1 score in relation to the number of parallel sentences. The experiments were conducted on the original English CoNLL 2003 benchmark and its noisy variants: Tesseract 3 ♣ , Tesseract 4 ♦ , Tesseract 4 ♥ , Tesseract 4 ♠ , and Typos. We compare the accuracy of our token-level seq2seq approach with the vanilla error model(Namysl et al., 2020), and the standard objective (L 0 ). We present the results for both auxiliary objectives: the data augmentation (L ours AUGM , L base AUGM ) and the stability training (L ours STAB , L base STAB ). 4×10 5 iterations or at least one epoch of training. Moreover, the token-level error correction model employed by Natas was trained for one epoch (about 4×10 5 iterations) on one million parallel sentences baseline model ( CER = 10%) This work (d) Noise induction methods (UD English EWT). .Table 1: The noisy sequence labeling data sets that we generated either by applying OCR on rendered sentences from an original benchmark (first four rows) or by inducing misspellings (last row). We generated multiple variants of the former data sets by combining geometrical distortions and pixel-level noise induction. The last two columns present the token error rates (the column headers indicate the names of the original benchmarks).54±0.08 80.48±0.09 84.71±0.19 85.62±0.08 91.50±0.08 n/a Hunspell 92.54±0.08 82.17±0.11 85.80±0.11 86.70±0.07 91.73±0.11 n/a Natas 92.54±0.08 77.80±0.19 84.50±0.11 85.24±0.10 91.33±0.13 56±0.06 85.29±0.16 88.62±0.08 89.19±0.12 92.04±0.07 seq2seq (token-level, §4.3) -92.76±0.07 85.38±0.16 89.39±0.17 89.99±0.22 92.37±0.10 seq2seq (sentence-level, §4.3) -92.81±0.11 84.38±0.15 88.96±0.18 89.67±0.26 92.44±0.17Data Set Geom. Distort. Pixel-level Noise CoNLL 2003 UD English EWT Tesseract 3 ♣ 22.72% 23.31% Tesseract 4 ♦ 16.35% 22.12% Tesseract 4 ♥ 14.89% 20.38% Tesseract 4 ♠ 3.53% 5.83% Typos n/a n/a 15.53% 15.22% Training Loss Noise Model Correction Method Original Data Tesseract 3 ♣ Tesseract 4 ♦ Tesseract 4 ♥ Tesseract 4 ♠ L0 n/a - 92.LAUGM confusion matrix ( §3.2) - 92.LSTAB confusion matrix ( §3.2) - 92.23±0.12 84.49±0.10 87.58±0.13 88.40±0.20 91.65±0.14 seq2seq (token-level, §4.3) - 92.24±0.18 84.25±0.23 88.24±0.25 88.91±0.21 91.86±0.16 seq2seq (sentence-level, §4.3) - 92.45±0.12 83.89±0.30 88.14±0.23 88.88±0.11 91.99±0.11 (a) English CoNLL 2003 L0 n/a - 96.96±0.04 86.75±0.16 86.97±0.14 88.30±0.16 94.34±0.07 n/a Hunspell 96.96±0.04 87.53±0.14 86.74±0.14 88.12±0.16 94.49±0.08 n/a Natas 96.96±0.04 88.98±0.10 88.94±0.14 89.68±0.16 95.11±0.08 LAUGM confusion matrix ( §3.2) - 96.90±0.06 91.35±0.13 92.12±0.14 92.99±0.21 96.17±0.07 seq2seq (token-level, §4.3) - 96.76±0.04 91.44±0.11 93.65±0.13 94.19±0.10 96.26±0.07 seq2seq (sentence-level, §4.3) - 96.78±0.06 90.92±0.08 93.37±0.08 94.10±0.03 96.27±0.03 LSTAB confusion matrix ( §3.2) - 96.80±0.04 91.16±0.07 91.93±0.11 92.77±0.10 96.06±0.02 seq2seq (token-level, §4.3) - 96.65±0.07 91.36±0.12 93.34±0.09 93.97±0.05 96.14±0.07 seq2seq (sentence-level, §4.3) - 96.67±0.05 90.70±0.14 93.05±0.17 93.71±0.13 96.15±0.05 (b) UD English EWT Table 2 : 2Comparison of error generation ( §6.1) and error correction ( §6.2) approaches on the original and noisy English CoNLL 2003 and the UD English EWT test sets ( §5.5). We report mean and standard deviation F1 scores(CoNLL 2003) and accuracies (UD English EWT) over five runs with different random initialization. L 0 , L AUGM , L STAB is the standard, the data augmentation, and the stability objective, respectively(Namysl et al., 2020). Bold values indicate top results (within the models trained using the same objective) that are statistically inseparable (Welch's t-test; p < 0.05). Table 3 : 3Comparison of the NAT approach with and without our NLM embeddings ( §6.3) on the English CoNLL 2003 test set ( §5.5). We report mean and standard deviation F1 scores over five runs with different random initialization. L 0 , L AUGM , L STAB is the standard, the data augmentation, and the stability objective, respectively (Namysl et al., 2020). The NLM column indicates whether the model employed our NLM embeddings. Bold values indicate top results (within the models trained using the same objective) that are statistically inseparable (Welch's t-test; p < 0.05). Table 5 : 5Statistics of the English CoNLL 2003 and the UD English EWT data sets. We present statistics of the training (Train) development (Dev) and test (Test) sets, including the number of sentences and tokens. CoNLL Original 22.72 16.35 14.89 3.53 15.53 Natas 17.24 12.20 11.13 2.34 11.53 Hunspell 17.44 13.54 12.24 2.43 10.69 ACC Natas 24.13 25.40 25.24 33.70 25.75 Hunspell 23.26 17.19 17.76 31.20 31.17 ACC (entities) Natas 6.23 10.41 10.93 18.77 7.07 Hunspell 43.93 40.44 41.58 47.78 50.38 (a) English CoNLL 2003. Hunspell 19.14 19.74 18.09 4.75 11.22 ACC Natas 23.82 21.05 20.36 27.75 23.27 Hunspell 17.90 10.74 11.20 18.59 26.49 (b) UD English EWT.Measure Method Tesser- act 3 ♣ Tesser- act 4 ♦ Tesser- act 4 ♥ Tesser- act 4 ♠ Typos TER TER (entities) Original 29.66 16.70 15.00 3.61 8.20 Natas 27.81 14.97 13.36 2.93 7.62 Hunspell 16.63 9.95 8.76 1.89 4.07 Measure Method Tesser- act 3 ♣ Tesser- act 4 ♦ Tesser- act 4 ♥ Tesser- act 4 ♠ Typos TER Original 23.31 22.12 20.38 5.83 15.22 Natas 17.76 17.46 16.23 4.21 11.68 Table 6 : 6Token Error Rates (TER) and the correction accuracies (ACC) of Natas and Hunspell on the test sets of our noisy sequence labeling data sets. All values are percentages. Bold values represent the lowest TER and the highest ACC.Noisy Token Error-Free Token Class LabelNo No O nzw new O fixtuvzs fixtures O reported reported O from from O New New B-LOC Vork York I-LOC . Table 8 : 8The hyper-parameters of the OpenNMT toolkit used to train our seq2seq error generation models.Training set size Validation set size Validation accuracy token-level sentence-level 10 7 5000 98.3% 95.7% 10 6 5000 95.4% 94.9% 10 5 5000 95.1% 95.3% 10 4 1000 94.6% 90.1% 10 3 100 93.3% 91.6% GLUE benchmark(Wang et al., 2018a): https:// gluebenchmark.com/leaderboard https://github.com/mnamysl/nat-acl2021 https://github.com/OpenNMT/OpenNMT-py4 We list all non-default hyper-parameters inTable 8. Which accounts for about 253 million words. 6 https://www.statmt.org/lm-benchmark 7 https://pypi.org/project/trdg 8 https://github.com/sirfz/tesserocr 9 We used Tesseract v4.0 to generate the parallel data set. 10 https://github.com/mnamysl/nat-acl2020 11 Other hyper-parameters also followAkbik et al. (2018). https://github.com/mikahama/natas 13 https://hunspell.github.io https://www.clips.uantwerpen.be/ Pooled contextualized embeddings for named entity recognition. 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[ "ON THE MEAN VALUE OF A KIND OF ZETA FUNCTIONS", "ON THE MEAN VALUE OF A KIND OF ZETA FUNCTIONS" ]	[ "Kui Liu " ]	[]	[]	Let d α,β (n) = n=kl αl<k≤βl 1 be the number of ways of factoring n into two almost equal integers. For rational numbers 0 < α < β, we consider the following Zetafor ℜs > 1. It has an analytic continuation to ℜs > 1/3.We get an asymptotic formula for the mean square of ζ α,β (s) in the strip 1/2 < ℜs < 1.As an application, we improve an result on the distribution of primitive Pythagorean triangles.	10.4064/aa166-1-4	[ "https://arxiv.org/pdf/1212.6513v1.pdf" ]	119,332,141	1212.6513	98ca717781e1cf2b87a090d81d6ffbd5435c36f9	ON THE MEAN VALUE OF A KIND OF ZETA FUNCTIONS 28 Dec 2012 Kui Liu ON THE MEAN VALUE OF A KIND OF ZETA FUNCTIONS 28 Dec 2012 Let d α,β (n) = n=kl αl<k≤βl 1 be the number of ways of factoring n into two almost equal integers. For rational numbers 0 < α < β, we consider the following Zetafor ℜs > 1. It has an analytic continuation to ℜs > 1/3.We get an asymptotic formula for the mean square of ζ α,β (s) in the strip 1/2 < ℜs < 1.As an application, we improve an result on the distribution of primitive Pythagorean triangles. Introduction and main results All through this paper, we always suppose s = σ + it and x ≥ 2. Let Voronoi [13] improved Dirichlet's result to ∆(x) ≪ x 1 3 log x. It is conjectured that for any ε > 0, we have ∆(x) ≪ ε x 1 4 +ε . The best result to date is ∆(x) ≪ x 131 416 (log x) 26947 8320 , due to Huxley [5]. Let ζ (s) be the Riemann Zeta-function, then the generated function of d(n) is ζ 2 (s) = ∞ n=1 d (n) n s , for σ > 1. Hardy-Littlewood [6] considered the mean square of ζ 2 (s) I σ T, ζ 2 = 2T T \|ζ (σ + it)\| 4 dt, for 1/2 < σ < 1, and proved (2) I σ T, ζ 2 = ζ 4 (2σ) ζ (4σ) T + o (T ) . Note that their proof is based on the approximation (for example, see Section 3 of [7]) (3) ζ 2 (s) = n≤x d (n) n s + χ 2 (s) n≤y d (n) n 1−s + O x 1 2 −σ log t , for 1/2 < σ < 1, where x, y ≥ 2, 4π 2 xy = t 2 and χ (s) = (2π) s 2Γ (s) cos πs 2 is the Γ-factor in the functional equation (4) ζ (s) = χ (s) ζ (1 − s) . In this paper, we focus on the following type divisor function given by d α,β (n) = n=kl αl<k≤βl 1, where α, β are fixed rational numbers satisfying 0 < α < β. Define its generated Zeta function as ζ α,β (s) = ∞ n=1 d α,β (n) n s , for σ > 1. We prove that ζ α,β (s) has an analytic continuation to σ > 1/3 and get an asymptotic formula for the mean square of ζ α,β (s) in the strip 1/2 < σ < 1. Theorem 1. For any 1 2 < σ < 1 and rational numbers 0 < α < β, there exists a constant ε (σ) > 0 such that (5) 2T T \|ζ α,β (σ + it)\| 2 dt = T ∞ n=1 d 2 α,β (n) n 2σ + O α,β,σ T 1−ε(σ) . Theorem 1 can be used to study the distribution of primitive Pythagorean triangles (i.e. triples (a, b, c) with a, b, c ∈ N, a 2 + b 2 = c 2 , a < b and gcd (a, b, c) = 1). Let P (x) denote the number of primitive Pythagorean triangles with perimeter a + b + c ≤ x. D. H. Lehmer [9] proved P (x) = log 2 π 2 x + O x 1/2 log x . It is difficult to reduce the exponents 1/2 in the error term, which depends on the zero-free region of the Riemann zeta function. However, assuming the Riemann Hypothesis, it was showed in [11] that, for any ε > 0, we have (6) P (x) = log 2 π 2 x + O ε x 5805 15408 +ε . We improve this result by applying Theorem 1 and get Theorem 2. If the Riemann Hypothesis is true, then for any ε > 0, we have P (x) = log 2 π 2 x + O ε x 4 11 +ε . Note that 5805 15408 = 0.3767 · · · and 4 11 = 0.3636 · · · . 2. Main steps in the proof of Theorem 1 First, Let's recall a way to get the asymptotic formula (2). In Chapters 3 of [7], using the functional equation (4), Ivic derive the Voronoi formula for the error term ∆ (x) in (1). Then in Chapter 4 of [7], Ivic get the approximation (3) by the Voronoi formula, from which one can obtain (2) in a standard way. Now observing that ζ α,β (s) is similar to ζ 2 (s), we can realize 2T T \|ζ α,β (σ + it)\| 2 dt as an analogue of 2T T \|ζ(σ + it)\| 4 dt. Our main steps in the proof of Theorem 1 similar to the proof of (2). In Section 4, we study the asymptotic property of the summatory function (7) D α,β (x) = n≤x d α,β (n). In Section 5, we derive a Voronoi type formula for the error term ∆ α,β (x) = D α,β (x) − Main terms. In Section 6, using the asymptotic formula of D α,β (x) and the Voronoi type formula for ∆ α,β (x), we obtain the following approximation for ζ α,β (s), which is the key for the proof of Theorem 1. Proposition 1. The function ζ α,β (s) can be analytically extended to the half plane σ > 1 3 with simple poles at s = 1 2 , 1. Moreover, suppose T ≥ 2, s = σ + it and 4π 2 xy = t 2 , then for any 1 2 < σ < 1, T < t ≤ 2T and 0 < α < β, we have (8) ζ α,β (s) = n≤x d α,β (n) n s + χ 2 (s) n≤y d α,β (n) n s−1 + E α,β (s) , where χ (s) is given by (4) and E α,β (s) satisfies (9) 2T T \|E α,β (σ + it)\| 2 dt ≪ α,β,σ x −2σ T 2 + x 1−σ T 1 2 + x 1 2 −σ T + x −σ T 3 2 log 3 T. From (8), we can derive Theorem 1 in a standard way. Hence the main work of paper is to prove Proposition 1. Priliminary lemmas Denote the integral part of u by [u]. let ψ (u) = u − [u] − 1 2 and e(x) = e 2πix . It is well known that ψ (u) has a truncated Fourier expansion (for example, see [4] ψ (u) = − 1 2πi 1≤\|h\|≤H 1 h e (hu) + O (G (u, H)) , where (10) G (u, H) = min 1, 1 H \|\|u\|\| . We will use the first derivative test (for example, see Chapter 21 of [12]). Lemma 2. Let G (x) and F (x) be a real differentiable functions such that F ′ (x) G(x) is monotonic and F ′ (x) G(x) ≥ m > 0 or F ′ (x) G(x) ≤ −m < 0. Then we have b a G (x) e iF (x) dx ≤ 4m −1 . We will also use the following Van der Corput B-process (see [10], Lemma 2.2). Lemma 3. Let C i , i = 1, · · · , 7 be absolute positive constants. Suppose that g is a real-valued function which has four continuous derivatives on the interval [A, B]. Let L and W be real parameters not less than 1, such that C 1 L ≤ B − A ≤ C 2 L, g (j) (ω) ≤ −C j+2 W L 1−j , for ω ∈ [A, B], j = 1, 2, 3, 4, and g ′′ (ω) ≥ C 7 W L −1 or g ′′ (ω) ≤ −C 7 W L −1 , for ω ∈ [A, B]. Let φ denote the inverse function of g ′ . Define ǫ f = e πi 4 , if g ′′ (ω) > 0 for ω ∈ [A, B], e − πi 4 , if g ′′ (ω) < 0 for ω ∈ [A, B] and r (x) = 0, if g ′ (x) ∈ Z, min 1 \|\|g ′ (x)\|\| , L W , else, with \|\|·\|\| denoting the distance from the nearest integer. Then it follows that A<l≤B e (g (l)) = ǫ f ′′ min(g ′ (A),g ′ (B))≤k≤max(g ′ (A),g ′ (B)) e (g (φ (k)) − kφ (k)) \|g ′′ (φ (k))\| +O (r (A) + r (B) + log (2 + W )) , with the notation ′′ a≤m≤b Φ (n) = 1 2 (χ Z (a) Φ (a) + χ Z (b) Φ (b)) + a<m<b Φ (n) , where χ Z (·) is the indicator function of the integers and the O-constant depends on the constants C i , i = 1, · · · , 7. Asymptotic formula for the summatory function Proposition 2. Let α = p1 q1 and β = p2 q2 with p 1 , p 2 , q 1 , q 2 ∈ N, gcd (p 1 , q 1 ) = 1 and gcd (q 1 , q 2 ) = 1. We have D α,β (x) = c 1 x + c 2 √ x + ∆ α,β (x), where c 1 = c 1 (α, β) = log α − log β 2 , c 2 = c 2 (α, β) = 1 2 1 p 2 q 2 − 1 p 1 q 1 , and (11) ∆ α,β (x) = − √ x β <l≤ √ x α ψ x l + O α,β (1) . Proof. It is enough to consider d α (n) = n=kl k≤αl 1 and D α (x) = n≤x d α (n). Clearly, D α (x) = kl≤x k≤αl 1 = l≤x k≤min(x/l,αl) 1. Write D α (x) = 1 + 2 ,(12)with 1 = l≤ √ x α k≤αl 1 and 2 = √ x α <l≤x k≤x/l 1. It is easy to see that 1 = l≤ √ x α (αl − ψ (αl) − 1/2) = x 2 − √ αxψ x α − l≤ √ x α ψ (αl) − 1 2 x α + O α (1) .(13) Similarly, 2 = √ x α <l≤x (x/l − ψ(x/l) − 1/2) = x √ x α <l≤x 1/l − √ x α <l≤x ψ(x/l) − 1 2 x + 1 2 x α + O (1) .(14) By the Euler-Maclaurin summation, we have √ x α <l≤x 1/l = 1 2 log x + 1 2 log α + α x ψ x α + O a 1 x . (15) Combining (12)-(15), we get D α (x) = x 2 log x + log α 2 x − √ x α <l≤x ψ(x/l) − l≤ √ x α ψ (αl) + O α (1) . Note that − l≤ √ x α ψ (αl) = − l≤ q 1 x p 1 ψ p 1 l q 1 = 1 2 x p 1 q 1 + O α (1) . Hence D α (x) = x 2 log x + log α 2 x − √ x α <l≤x ψ(x/l) + 1 2 x p 1 q 1 + O α (1) . (16) Similarly, for d β (n) = n=kl k≤βl 1 and D β (x) = n≤x d β (n), we have (17) D β (x) = x 2 log x + log β 2 x − √ x β <l≤x ψ(x/l) + 1 2 x p 2 q 2 + O β (1) . Now Proposition 2 follows from (16), (17) and D α,β (x) = D β (x) − D α (x). Corollary 1. We have D α,β (x) = c 1 x + c 2 √ x + O α,β x 1 3 , where c 1 , c 2 are the same as Proposition 2. Proof. This can be proved easily (even with a better upper bound for the error term) by applying Lemma 1 and exponential pairs (see [3]) to Proposition 2. A Voronoi type formula Define d α,β (n, H) = 1≤h≤H ′′ hα≤k≤hβ n=hk 1, where the notation ′′ is the same as Lemma 3. Using the Van der Corput B-process and the same argument as Section 6.2 of [14], we can derive the following Voronoi type formula for ∆ α,β (x). Lemma 4. For any H ≥ 2 and rational numbers 0 < α < β, we have ∆ α,β (x) = M α,β (x, H) + E α,β (x, H) + F α,β (x, H) , where (18) M α,β (x, H) = x 1 4 π √ 2 n≤βH 2 d α,β (n, H) n 3 4 cos 4π √ nx − π 4 , (19) E α,β (x, H) ≪ √ x α <l≤ √ x β G x l , H and (20) F α,β (x, H) ≪ α,β log H. Proof. Applying Lemma 1 to (11), we get ∆ α,β (x) = 1 2πi 1≤\|h\|≤H 1 h √ x β <l≤ √ x α e hx l + E α,β (x, H) + O α,β (1) , with (21) E α,β (x, H) ≪ √ x α <l≤ √ x β G x l , H . Let (22) S α,β (x, H) = 1 2πi 1≤h≤H 1 h √ x β <l≤ √ x α e hx l , then we can write (23) ∆ α,β (x) = 1 2πi S α,β (x, H) − S α,β (x, H) + E α,β (x, H) + O α,β (1) . To treat the inner sum √ x β <l≤ √ x α e hx l for 1 ≤ h ≤ H in (22), we apply Lemma 3. Let A = x β , B = x α and g (l) = hx l , then we have g ′ (l) = − hx l 2 , g ′′ (l) = 2hx l 3 , g (3) (l) = − 6hx l 4 , g (4) (l) = 24hx l 5 , g ′ (B) = −hα, g ′ (A) = −hβ, 2α 3 2 h √ x < g ′′ (l) ≤ 2β 3 2 h √ x and \|g ′′′ (l)\| ≪ α,β h x . Hence we can take W = 1, L = √ x h , φ (k) = − hx k , g (φ (k)) − kφ (k) = 2 √ −hkx, and g ′′ (φ (k)) = 2 (−k) 3 hx . Noting α, β are rational numbers, we have (24) r (A) , r (B) ≪ α,β 1. Now for 1 ≤ h ≤ H, by Lemma 3, we get √ x α <l≤ √ x β e hx l = e πi 4 √ 2 ′′ −hβ≤k≤−hα h 1 4 x 1 4 (−k) 3 4 e 2 √ −hkx + O α,β (1) (25) = 1 √ 2 ′′ hα≤k≤hβ h 1 4 x 1 4 k 3 4 e 2 √ hkx + 1 8 + O α,β (1) . Inserting (25) to (22) gives S α,β (x, H) = 1 √ 2 1≤h≤H 1 h ′′ hα≤k≤hβ h 1 4 x 1 4 k 3 4 e 2 √ hkx + 1 8 + O α,β (log H) = x 1 4 √ 2 1≤h≤H ′′ hα≤k≤hβ 1 (hk) 3 4 e 2 √ hkx + 1 8 + O α,β (log H) = x 1 4 √ 2 n≤βH 2 d α,β (n, H) n 3 4 e 2 √ nx + 1 8 + O α,β (log H) . Thus 1 2πi S α,β (x, H) − S α,β (x, H) = x 1 4 π √ 2 n≤βH 2 d α,β (n, H) n 3 4 cos 4π √ nx − π 4 +O α,β (log H) . This combining with (23) and (21) yields Lemma 4. First, let's show that ζ α,β (s) can be analyticly extended to σ > 1 3 . For σ > 1 and any N ≥ 1, write ζ α,β (s) = n≤N d α,β (n) n s + n>N d α,β (n) n s = n≤N d α,β (n) n s + ∞ N + u −s dD α,β (u), where D α,β (u) is defined by (7). Applying Proposition 2, we get ζ α,β (s) = n≤N d α,β (n) n s + ∞ N + u −s d c 1 u + c 2 √ u + ∆ α,β (u) = n≤N d α,β (n) n s + c 1 ∞ N + u −s du + c 2 2 ∞ N + u −s−1/2 du + ∞ N + u −s d∆ α,β (u). By partial integration, we have (26) ζ α,β (s) = n≤N d α,β (n) n s − c 1 N 1−s 1 − s − c 2 N 1 2 −s 1 − 2s + s ∞ N + ∆ α,β (u)u −s−1 du + O N 1 3 −σ . From Corollary 1, we can see that the integral in (26) is absolutely convergent for σ > 1 3 , hence (26) gives an analytic continuation of ζ α,β (s) for σ > 1 3 . This proves the first assertion of Proposition 1. Now suppose σ > 1 3 and 2 ≤ T < t ≤ 2T . From now on, we take N = T A with A > 0 being a constant, sufficiently large. Break the sum in (26) into (27) n≤N d α,β (n) n s = n≤x d α,β (n) n s + x<n≤N d α,β (n) n s . For the second sum, applying Proposition 2 again, we have x<n≤N d α,β (n) n s = N x u −s dD α,β (u) = N x u −s d c 1 (α, β) u + c 2 (α, β) √ u + ∆ α,β (u) . By partial integration, we have x<n≤N d α,β (n) n s = c 1 (α, β) N 1−s 1 − s − x 1−s 1 − s + c 2 (α, β) N 1/2−s 1 − 2s − x 1/2−s 1 − 2s (28) + N −s ∆ α,β (N ) − x −s ∆ α,β (x) + s N x ∆ α,β (u)u −s−1 du. Combining In Section 7, we will show that the upper bound of E (s) is small, when H is large comparing to N and the mean square of F (s) has an acceptable estimate; see Lemma 6 and Lemma 7, respectively. In Section 8, we will pick out the second term in (8) from M(s); see Lemma 8, Combining (29) with Lemmas 6-8, we get Proposition 1. An upper bound and a mean square estimate To bound E (s) , we need the following mean value estimate for G (u, H) defined by (10). Proof. By (32) and trivial estimates, we get E (s) ≪ t N x √ u α <l≤ √ u β G u l , H u −σ−1 du ≪ tx −σ−1 √ x α <l≤ N β N x G u l , H du = tx −σ−1 √ x α <l≤ N β l N l x l G (u, H) du. This combining with Lemma 5 yields E (s) ≪ tx −σ−1 l≤ N β l N 0 G (u, H) du ≪ tx −σ−1 N 2 log H H . Now we consider the mean square of F (s). Lemma 7. For σ > 1/2, we have 2T T \|F (s)\| 2 dt ≪ α,β,σ x −2σ T 2 log 2 H log N. Proof. Noting F α,β (u) ≪ α,β log H and unfolding the square in the integral, we get 2T T \|F (s)\| 2 dt ≪ T 2 2T T N x F α,β (u) u −s−1 du 2 dt ≪ α,β T 2 log 2 H N x N x (u 1 u 2 ) −σ−1 2T T u 2 u 1 it dt du 1 du 2 Applying Lemma 2 to the above integral over t, we have 2T T \|F (s)\| 2 dt ≪ α,β T 2 log 2 H N x N x (u 1 u 2 ) −σ−1 min   T, 1 log u2 u1   du 1 du 2 ≪ α,β T 2 log 2 H N x N u1 (u 1 u 2 ) −σ−1 min T, 1 log u2 u1 du 1 du 2 . Write this as Let's deal with i , i = 1, 2, 3 respectively. For 1 , we have 1 ≪ T 3 log 2 H N x u −2σ−2 1 e 1 T u1 u1 du 2 du 1 ≪ T 3 log 2 H N x u −2σ−2 1 e 1 T u 1 − u 1 du 1 ≪ T 3 e 1 T − 1 log 2 H N x u −2σ−1 1 du 1 , which yields (35) 1 ≪ σ x −2σ T 2 log 2 H. For 2 , we have 2 = T 2 log 2 H N x u −σ−1 1 3 2 u1 e 1 T u1 u −σ−1 2 1 log u2 u1 du 2 du 1 = T 2 log 2 H N x u −σ−1 1 3 2 u1 e 1 T u1 u −σ−1 2 1 log 1 + u2−u1 u1 du 2 du 1 ≪ T 2 log 2 H N x u −2σ−1 1 3 2 u1 e 1 T u1 1 u 2 − u 1 du 2 du 1 ≪ T 2 log 2 H N x u −2σ−1 1 log u 1 du 1 , which yields (36) 2 ≪ σ x −2σ T 2 log 2 H log N. For 3 , we have (37) 3 ≪ T 2 log 2 H N x u −σ−1 du 2 ≪ σ x −2σ T 2 log 2 H. From (34)-(37), we get Lemma 7. Picking out the second term in Proposition 1 The second term in Proposition 1 is hidden in M (s). In this Section, we will pick it out and prove + O t − 1 2 x 1−σ log H + x 1/2−σ log H + x 1 2 −σ log t + x −σ t 1 2 log t .(38) The idea of the proof for Lemma 8 comes from Chapter 4 of [7]. By (31) and (18), we have M (s) = s π √ 2 N x u −s− 3 4 n≤βH 2 d α,β (n, H) n 3 4 cos 4π √ nu − π 4 du. Let η > 0 be a fixed, sufficiently small constant. Using cos z = e iz +e −iz 2 , we can write (39) M (s) = M 1 (s) + M 2 (s) + M 3 (s) + M 4 (s) with M 1 (s) = s 2π √ 2 N x u −s− 3 4 n≤(1+η)y d α,β (n, H) n 3 4 e 2 √ nu − 1 8 du, M 2 (s) = s 2π √ 2 N x u −s− 3 4 (1+η)y<n≤βH 2 d α,β (n, H) n 3 4 e 2 √ nu − 1 8 du, M 3 (s) = s 2π √ 2 N x u −s− 3 4 n≤y d α,β (n, H) n 3 4 e −2 √ nu + 1 8 du and M 4 (s) = s 2π √ 2 N x u −s− 3 4 y<n≤βH 2 d α,β (n, H) n 3 4 e −2 √ nu + 1 8 du. We will bound M 2 (s) , M 3 (s) and M 4 (s) in the following Lemmas 9-11 and pick out the first term on the right side of (38) in Lemma 12. From Lemmas 9-12 and (39). Lemma 9. For σ > 1/2, we have M 2 (s) ≪ t − 1 2 x 1−σ log H. Proof. Write M 2 (s) = s 2π √ 2 (1+η)y<n≤βH 2 d α,β (n; H) n 3/4 N x u −σ−3/4 e − t 2π log u + 2 √ nu − 1/8 du. In Lemma 2, taking G (u) = u −σ−3/4 and F t (u) = − t 2π log u + 2 √ nu − 1/8, we have F ′ t (u) = − t 2πu + n u and F ′ t (u) G (u) = − t 2π u σ−1/4 + √ nu σ+1/4 . Since n > (1 + η) y, u > x and 4π 2 xy = t 2 , then F ′ t (u) G (u) ′ = − (σ − 1/4) t 2π u σ−5/4 + (σ + 1/4) √ nu σ−3/4 > 0. Thus F ′ (u) G(u) is monotonic and F ′ t (u) G (u) = − t 2 4π 2 nu 1 2 √ nu σ+1/4 + √ nu σ+1/4 ≥ − t 2 4π 2 (1 + η)yx 1 2 √ nx σ+1/4 + √ nx σ+1/4 ≥ 1 − 1 √ 1 + η √ nx σ+1/4 ≫ √ nx σ+1/4 . Hence Lemma 2 gives N x u −σ−3/4 e − t 2π log u − √ nu + 1/8 ≪ x −σ−1/4 n − 1 2 , which yields M 2 (s) ≪ x −σ+3/4 (1+η)y<n≤βH 2 d α,β (n; H) n 5/4 ≪ t − 1 2 x 1−σ log H. Lemma 10. For σ > 1/2, we have M 3 (s) ≪ σ x 1 2 −σ + x −σ t 1 2 log t. Proof. Write M 3 (s) = s 2π √ 2 N x u −H (u) = 1, G (u) = u −σ−1/4 and F t (u) = − t 2π log u − 2 √ nu − 1/8, then we have F ′ t (u) = − t 2πu − n u and F ′ t (u) G (u) = − t 2π u σ− 3 4 − √ nu σ− 1 4 . Obviously, F ′ t (u) G (u) < − √ nu σ− 1 4 ≤ − √ nx σ− 1 4 . Noting F ′ t (u) G (u) ′ = − σ − 3 4 t 2π u σ− 7 4 − σ − 1 4 √ nu σ− 5 4 , let u 0 = (3/4−σ)t (σ−1/4)2π √ n be the root of F ′ t (u) G(u) ′ = 0. If u 0 ∈ [x, N ], then F ′ t (u) G(u) is monotonic in [x, u 0 ] and [u 0 , N ] respectively, otherwise F ′ t (u) G(u) is monotonic in [x, N ]. In either case, Lemma 2 is valid and gives N x u −σ−1/4 e − t 2π log u − 2 √ nu − 1/8 ≪ n − 1 2 x 1 4 −σ , which yields (43) M 33 (s) ≪ n≤y d (n) n 1/4 n − 1 2 x 1 4 −σ ≪ x 1 4 −σ y 1/4 log y ≪ x −σ t 1 2 log t. Then Lemma 10 follows from collecting (40)-(43). Lemma 11. For σ > 1/2, we have M 4 (s) ≪ x 1/2−σ log H. Proof. Write M 4 (s) = s 2π √ 2 y<n≤βH 2 d α,β (n; H) n 3/4 N x u −σ−3/4 e − t 2π log u − 2 √ nu + 1/8 du In Lemma 2, taking G (u) = u −σ−3/4 and F t (u) = − t 2π log u − 2 √ nu + 1/8, we have F ′ t (u) = − t 2πu − n u and F ′ t (u) G (u) = − t 2π u σ−1/4 − √ nu σ+1/4 . Thus F ′ (u) G(u) is monotonic and F ′ t (u) G (u) = − t 2π u σ−1/4 − √ nu σ+1/4 < − √ nx σ+1/4 Hence Lemma 2 gives N x u −σ−3/4 e − t 2π log u − √ nu + 1/8 ≪ x −σ−1/4 n − 1 2 , which yields M 4 (s) ≪ x −σ+3/4 y<n≤βH 2 d α,β (n; H) n 5/4 ≪ x 1/2−σ log H. Lemma 12. For σ > 1/2, we have M 1 (s) = χ 2 (s) n≤y d α,β (n) n s−1 + O x 1/2−σ log t . Proof. Similar to the the proof of Lemma 10, we rewrite M 1 (s) as Combining (44)-(46) gives Lemma 12. M 1 (s) = s 2π √ 2 N x u −with I n = N x u −σ− 1 4 e − t 2π log u + 2 √ nu − 1 8 du, 1     n s−1 + O x 1/2−σ log t (46) = χ 2 (s) n≤y     1≤h≤H hα≤k≤hβ n=hk 1     n s−1 + O   \|χ (s)\| 2 h≪ √ y h 2σ−2   +O x 1/2−σ log T = χ 2 (s) 9. Out line for the proof of Theorem 2 A primitive Pythagorean triangle is a triple (a, b, c) of natural numbers with a 2 + b 2 = c 2 , a < b and gcd(a, b, c) = 1. Let P (x) denote the number of primitive Pythagorean triangles with perimeter less than x. D. H. Lehmer [9] showed P (x) = log 2 π 2 x + O x 1/2 log x which was revisited by J. Lambek and L. Moser in [8]. The exponents 1/2 in the error term can not be reduced because the current technique depends on the best zero-free regions of the Riemann zeta function, which hard to be improved. In [11], the author showed if Riemann Hypothesis (RH) is true, then (6) holds. Let r (n) = We can prove that Z(s) has an analytic continuation to σ > 1/3 and has two simple poles at s = 1, 1 2 . The exponent 5805 15408 in (6) holds for any ε > 0. In [11], the author showed σ > 1064 1644 = 0.6472 · · · is admissible. Then by estimating the exponential sum (48) for M ≤ x 651 1926 , N ≤ x 3798 15408 , we get (6). In the review of [11], R. C. Baker mentioned that using the method in his paper [2], it is possible to prove σ > 3 5 = 0.6, which implies an improvement of (6). Now by Theorem 1, we have (49) holds for any σ > 1 2 , which forces us to deal with exponential sum (48) for M, N ≤ x 1 4 +ε . However, the estimate in this range has been investigated carefully by R. C. Baker in [1], which yields Theorem 2. Remark 1 . 1The bound (20) is important in the proof of Theorem 1. If α, β are not rational numbers, the author can't get the estimate (20). Because in this case the estimate (24) does not holds. 6. Proof of Proposition 1 F (26), (27) and (28), we get(29) ζ α,β (s) = n≤x d α,β (n) n s + s N x ∆ α,β (u)u −s−1 du + O α,β,σ x 1−σ t −1 + x 1/3−σholds for any σ > 1 3 . Our tool to prove Proposition 1 is the Voronoi formula for ∆ α,β (x). Using Lemma 4α,β (u, H) u −s−1 du. Lemma 5 . 5For any N ≥ 1 and H ≥ 2, we have N 0 G (u, H) du ≪ N log H H .Proof. Note that G (u, H) is a positive peridodic function with period 1, then we have Lemma 8 . 8For σ > 1/2, we have M (s) = χ 2 (s) n≤x d α,β (n) n s−1 M 3 M 3(s) = M 31 (s) + M 32 (s) + M 33 (s) + M 34 (32 (s) = x −s+1/4 2π √ 2 n≤y d α,β (n; H) n 3/4 e −2 √ nx + 1/8 , M 33 (s) = − i √ 2 n≤y d α,β (n; Using d α,β (n; H) ≤ d (n) and trivial estimates, it is easy to get (41) M 31 (s) , M 32 (s) Now we deal with M 33 (s). In Lemma 2, let i(t+ π 4 ) 1 + O t −1 ,for t ≥ 2. s , for ℜs > 1. a n ≪ 1 and c being a constant. Here the ranges of M, N are determined by the smallest σ such that(49) 2T T \|Z (σ + it)\| dt ≪ σ,ε T 1+ε ) . )Lemma 1. For any real number H > 2, we have (s) = M 11 (s) + M 12 (s) + M 13 (s) + M 14 (s) ,s− 3 4 n≤(1+η)y d α,β (n, H) n 3 4 e 2 √ nu − 1/8 du = − 1 2π √ 2 N x (−s + 1/4) n≤(1+η)y d α,β (n; H) n 3/4 e 2 √ nu − 1/8 u −s−3/4 du + 1 8π √ 2 N x n≤(1+η)y d α,β (n; H) n 3/4 e 2 √ nu − 1/8 u −s−3/4 du = − 1 2π √ 2 n≤(1+η)y d α,β (n; H) n 3/4 N x e 2 √ nu − 1/8 du −s+ 1 4 + 1 8π √ 2 n≤(1+η)y d α,β (n; H) n 3/4 N x e 2 √ nu − 1/8 u −s−3/4 du. By partial integration, we have (44) M 1 where M 11 (s) = − 1 i √ 2 n≤(1+η)y d α,β (n; H) n − 1 4 I n The square-free divisor problem II. R C Baker, Quart. J. Math. 472R. C. Baker, The square-free divisor problem II, Quart. J. Math. (Oxford)(2). 47 (1996), 133-146. Primitive lattice points in planar domains. R C Baker, Acta Arith. 142R. C. Baker, Primitive lattice points in planar domains, Acta Arith. 142 (2010), 267-302 Van der Corput's method of exponential sums. S W Graham, G Kolesnik, Lecture Note Series. 126London Mathematical SocietyS. W. Graham and G. Kolesnik, Van der Corput's method of exponential sums, (London Mathematical Society Lecture Note Series 126). The Piatetski-Shapiro prime theorem. D R Heath-Brown, J. of Number theory. 16D. R. Heath-Brown,The Piatetski-Shapiro prime theorem, J. of Number theory, Vol. 16(1983), 242-266. Exponential sums and Lattices points III. M N Huxley, Proc. London Math. Soc. 873M. N. Huxley, Exponential sums and Lattices points III, Proc. London Math. Soc. Vol. 87 (3) (2003), 591-609. The approximate functional equation in the theory of the zetafunction, with applications to the divisor-problems of Dirichlet and Piltz. G H Hardy, J E Littlewood, Proc. London Math. Soc. 2G. H. Hardy and J. E. Littlewood, The approximate functional equation in the theory of the zeta- function, with applications to the divisor-problems of Dirichlet and Piltz, Proc. London Math. Soc. (2) 21 (1923) 39-74. The Riemann Zeta-Function. A Ivic, WileyA.Ivic, The Riemann Zeta-Function, Wiley (1985). On the distribution of Pythagorean triangles. J Lambek, L Moser, Pacific J. Math. 5J. Lambek and L. Moser, On the distribution of Pythagorean triangles, Pacific J. Math. 5 (1955), 73-83. . D H Lehmer, Krishnaswami, Bull. Amer. Math. Soc. 54D. H. Lehmer, A conjecture of Krishnaswami, Bull. Amer. Math. Soc. 54 (1948), 1185-1190 The asymptotic behaviour of the mean-square of fractional part sums. M Kuhleitner, W G Nowak, Proc. Edinb. Math. Soc. 43M. Kuhleitner and W. G. Nowak, The asymptotic behaviour of the mean-square of fractional part sums, Proc. Edinb. Math. Soc. 43 (2000), 309-323. On the distribution of primitive Pythagorean triangles. Kui Liu, Acta Arith. 144Kui Liu, On the distribution of primitive Pythagorean triangles, Acta Arith. 144 (2010), 135-150. C D Pan, C B Pan, Foundations of the Analytic Number Theory. BeijingScience Pressin ChineseC. D. Pan and C. B. Pan, Foundations of the Analytic Number Theory, Science Press, Beijing, 1991 (in Chinese). Sur une fonction transcendante et ses applications a la sommation de quelques series. G F Voronoi, Ann Ecole Normale. 21G. F. Voronoi, Sur une fonction transcendante et ses applications a la sommation de quelques series. Ann Ecole Normale, 21: 207C268; 21: 459C534 On the error term in Weyls law for the Heisenberg. W G Zhai, Acta Arith. 134Acta ArithW. G. Zhai, On the error term in Weyls law for the Heisenberg , Acta Arith. 134. 3 (2008), 219-257 Acta Arith. . P. R. China E. Department of Mathematics, Qingdao Universitymail address: email: [email protected] of Mathematics, Qingdao University, Qingdao 266071, P. R. China E-mail address: email: [email protected]	[]
[ "Multivariate q -Pólya and inverse q -Pólya distributions", "Multivariate q -Pólya and inverse q -Pólya distributions" ]	[ "Charalambos A Charalambides \nDepartment of Mathematics\nUniversity of Athens\nGR-15784Panepistemiopolis, AthensGreece\n" ]	[ "Department of Mathematics\nUniversity of Athens\nGR-15784Panepistemiopolis, AthensGreece" ]	[]	An urn containing specified numbers of balls of distinct ordered colors is considered. A multiple q-Pólya urn model is introduced by assuming that the probability of q-drawing a ball of a specific color from the urn varies geometrically, with rate q, both with the number of drawings and the number of balls of the specific color, together with the total number of balls of the preceded colors, drawn in the previous q-drawings. Then, the joint distributions of the numbers of balls of distinct colors drawn (a) in a specific number of q-drawings and (b) until the occurrence of a specific number of balls of a certain color, are derived. These two distributions turned out to be q-analogues of the multivariate Pólya and inverse Pólya distributions, respectively. Also, the limiting distributions of the multivariate q-Pólya and inverse q-Pólya distributions, as the initial total number of balls in the urn tends to infinity, are shown to be q-multinomial and negative q-multinomial distributions, respectively.	10.1080/03610926.2020.1825740	[ "https://arxiv.org/pdf/2002.10162v1.pdf" ]	211,258,763	2002.10162	9515a0d196450a7c59f7caa3cb4d0e8b98dcb923	Multivariate q -Pólya and inverse q -Pólya distributions 24 Feb 2020 Charalambos A Charalambides Department of Mathematics University of Athens GR-15784Panepistemiopolis, AthensGreece Multivariate q -Pólya and inverse q -Pólya distributions 24 Feb 2020AMS(2000) subject classification Primary 60C05Secondary 05A30 Keywords and phrases: multivariate absorption distributionmultivariate inverse ab- sorption distributionmultivariate inverse q-hypergeometric distributionmultivariate q-hypergeometric distributionnegative q-multinomial distributionq-multinomial dis- tribution An urn containing specified numbers of balls of distinct ordered colors is considered. A multiple q-Pólya urn model is introduced by assuming that the probability of q-drawing a ball of a specific color from the urn varies geometrically, with rate q, both with the number of drawings and the number of balls of the specific color, together with the total number of balls of the preceded colors, drawn in the previous q-drawings. Then, the joint distributions of the numbers of balls of distinct colors drawn (a) in a specific number of q-drawings and (b) until the occurrence of a specific number of balls of a certain color, are derived. These two distributions turned out to be q-analogues of the multivariate Pólya and inverse Pólya distributions, respectively. Also, the limiting distributions of the multivariate q-Pólya and inverse q-Pólya distributions, as the initial total number of balls in the urn tends to infinity, are shown to be q-multinomial and negative q-multinomial distributions, respectively. Introduction A q-Pólya urn model is introduced in Charalambides (2012Charalambides ( , 2016 and q-Pólya and inverse q-Pólya distributions are studied. Moreover, their limiting distributions, as the number of balls in the urn, or the number of drawn balls of one of the two colors, tends to infinity are shown to be q-binomial and negative q-binomial distributions. Kupershmidt (2000) introduced a q-hypergeometric distribution and a q-Pólya distribution (under the name q-contagious distribution) and represented the corresponding random variable as a sum of two-valued dependent random variables. Kemp (2005) starting from the Chu-Vandermonde sum as a probability generating function obtained two q-confluent hypergeometric distributions. She also, deduced these distributions as steady-state birth and death Markov chains. The aim of this article is to introduced and studied in detail multivariate q-Pólya and inverse q-Pólya distributions and also, examined some of their limiting distributions. Section 2 is devoted to q-multinomial convolutions. Precisely, multivariate q-Vandermonde and inverse q-Vandermonde formulae are presented. Also, a closely connected multivariate q-Cauchy formula is deduced. In section 3, the q-Pólya urn model is extended to a multiple q-Pólya urn model by considering successive q-drawings of balls from an urn containing specified numbers of balls of different colors and assuming that the probability of randomly q-drawing a ball of a specific color from the urn varies geometrically, with rate q. Then, on the stochastic model of a sequence of a specific number of random q-drawings of balls, a multivariate q-Pólya distribution is introduced and examined. Furthermore, in section 4, and on the stochastic model of a sequence of random q-drawings that is terminated with the occurrence of a specific number of balls of a given color, a multivariate inverse q-Pólya distribution is discussed. q -Multinomial convolutions Let x and q be real numbers, with q = 1, and r be an integer. The function of x, with parameter q, [x] q = (1 − q x )/(1 − q) is called q-number and in particular [r] q is called q-integer. Also, the product [x] r,q = [x] q [x − 1] q · · · [x − r + 1] q , r = 1, 2, . . . , defines the q-factorial of x of order r. In particular, [r] q ! = [1] q [2] q · · · [r] q is the q-factorial of r. The notion of q-factorial is extended to zero order by [x] 0,q = 1 and to negative order by [x] −r,q = 1/[x + r] r,q , r = 1, 2, . . . . The q-multinomial coefficient is defined by x r 1 , r 2 , . . . , r k q = [x] r 1 +r 2 +···+r k ,q [r 1 ] q ![r 2 ] q ! · · · [r k ] q ! , r j = 0, 1, . . . , j = 1, 2, . . . , k. (2.1) Notice that a q −1 -number is readily expressed into a q-number by [x] q −1 = q −x+1 [x] q . Consequently, [x] r,q −1 = q −xr+( r+1 2 ) [x] r,q , [r] q −1 ! = q −( r 2 ) [r] q !. Furthermore, setting s j = j i=1 r i and m j = k i=j r i , for j = 1, 2, . . . , k, and using the expression −xs k + s k + 1 2 + k j=1 r j 2 = − k j=1 r j (x − m j ) = − k j=1 r j (x − s j ), it follows that x r 1 , r 2 , . . . , r k q −1 = q − k j=1 r j (x−m j ) x r 1 , r 2 , . . . , r k q = q − k j=1 r j (x−s j ) x r 1 , r 2 , . . . , r k q . (2.2) Therefore, a formula involving q-numbers, q-factorials, and q-multinomial coefficients in the base q, with 1 < q < ∞, can be converted, with respect to the base, into a similar formula in the base p = q −1 , with 0 < p < 1. Two versions of a recurrence relation for the q-multinomial coefficients, useful in the sequel, are quoted here for easy reference. The q-multinomial coefficient satisfies the recurrence relation x r 1 , r 2 , . . . , r k q = x − 1 r 1 , r 2 , . . . , r k q + q x−m 1 x − 1 r 1 − 1, r 2 , . . . , r k q + q x−m 2 x − 1 r 1 , r 2 − 1, . . . , r k q + · · · + q x−m k x − 1 r 1 , r 2 , . . . , r k − 1 q , (2.3) and alternatively, the recurrence relation x r 1 , r 2 , . . . , r k q = q s k x − 1 r 1 , r 2 , . . . , r k q + x − 1 r 1 − 1, r 2 , . . . , r k q + q s 1 x − 1 r 1 , r 2 − 1, . . . , r k q + · · · + q s k−1 x − 1 r 1 , r 2 , . . . , r k − 1 q , (2.4) for r j = 0, 1, . . . and j = 1, 2, . . . , k, with m j = k i=j r i and s j = j i=1 r i . Recurrence relations (2.3) and (2.4), by replacing the base q by q −1 , and using the first and the second expression in (2.2), respectively, are expressed as x r 1 , r 2 , . . . , r k q = q m 1 x − 1 r 1 , r 2 , . . . , r k q + q m 2 x − 1 r 1 − 1, r 2 , . . . , r k q + q m 3 x − 1 r 1 , r 2 − 1, . . . , r k q + · · · + x − 1 r 1 , r 2 , . . . , r k − 1 q . (2.5) and x r 1 , r 2 , . . . , r k q = x − 1 r 1 , r 2 , . . . , r k q + q x−s 1 x − 1 r 1 − 1, r 2 , . . . , r k q + q x−s 2 x − 1 r 1 , r 2 − 1, . . . , r k q + · · · + q x−s k x − 1 r 1 , r 2 , . . . , r k − 1 q , (2.6) respectively. It should be noticed that reversing the order of labeling the arguments of the q-multinomials, these expressions may be transformed to (2.4) and (2.3), respectively. Two versions of a multivariate q-Vandermonde formula are derived in the next theorem. Theorem 2.1. Let n be a positive integer and let x j , j = 1, 2, . . . , k + 1, and q be real numbers, with q = 1. Then, [x 1 + x 2 + · · · + x k+1 ] n,q = n r 1 , r 2 , . . . , r k q q k j=1 (n−s j )(x j −r j ) k+1 j=1 [x j ] r j ,q , (2.7) and, alternatively, [x 1 +x 2 +· · ·+x k+1 +n−1] n,q = n r 1 , r 2 , . . . , r k q q k j=1 r j z j k+1 j=1 [x j +r j −1] r j ,q . (2.8) Also, [x 1 + x 2 + · · · + x k+1 ] n,q = n r 1 , r 2 , . . . , r k q q k j=1 r j (z j −(n−s j )) k+1 j=1 [x j ] r j ,q , (2.9) and, alternatively, [x 1 +x 2 +· · ·+x k+1 +n−1] n,q = n r 1 , r 2 , . . . , r k q q k j=1 x j (n−s j ) k+1 j=1 [x j +r j −1] r j ,q , (2.10) where s j = j i=1 r i , z j = k+1 i=j+1 x i , j = 1, 2, . . . , k, and r k+1 = n − s k , and the summation, in all four sums, is extended over all r j = 0, 1, . . . , n, j = 1, 2, . . . , k, with k i=1 r i ≤ n. Proof. Consider the sequence of multiple sums s n (x 1 , x 2 , . . . , x k+1 ; q) = n r 1 , r 2 , . . . , r k q q k j=1 (n−s j )(x j −r j ) k+1 j=1 [x j ] r j ,q , for n = 1, 2 . . . , with initial value s 1 (x 1 , x 2 , . . . , x k+1 ; q) = [x 1 ] q + k+1 j=2 q j−1 i=1 x i [x j ] q = [x 1 + x 2 + · · · + x k+1 ] q . Using recurrence relation (2.6), with x = n, the sequence may be expressed as s n (x 1 , x 2 , . . . , x k+1 ; q) = n − 1 r 1 , r 2 , . . . , r k q q k j=1 (n−s j )(x j −r j ) k+1 j=1 [x j ] r j ,q + n − 1 r 1 − 1, r 2 , . . . , r k q q n−s 1 + k j=1 (n−s j )(x j −r j ) k+1 j=1 [x j ] r j ,q + · · · + n − 1 r 1 , r 2 , . . . , r k − 1 q q n−s k + k j=1 (n−s j )(x j −r j ) k+1 j=1 [x j ] r j ,q . Replacing r j − 1 by r j in the (j + 1)th multiple sum, for j = 1, 2, . . . , k, and then executing the multiplications, summations, and cancelations in the exponents of q, we get the expression s n (x 1 , x 2 , . . . , x k+1 ; q) = n − 1 r 1 , r 2 , . . . , r k q q k j=1 (n−1−s j )(x j −r j )+ k j=1 (x j −r j ) [x k+1 − r k+1 + 1] q k+1 j=1 [x j ] r j ,q + n − 1 r 1 , r 2 , . . . , r k q q k j=1 (n−1−s j )(x j −r j ) [x 1 − r 1 ] q k+1 j=1 [x j ] r j ,q + · · · + n − 1 r 1 , r 2 , . . . , r k q q k j=1 (n−1−s j )(x j −r j )+ k−1 j=1 (x j −r j ) [x k − r k ] q k+1 j=1 [x j ] r j ,q . which, since [x 1 − r 1 ] q + q x 1 −r 1 [x 2 − r 2 ] q + · · · + q k−1 j=1 (x j −r j ) [x k − r k ] q + q k j=1 (x j −r j ) [x k+1 − r k+1 + 1] q = [x 1 , x 2 , . . . , x k+1 − n + 1] q implies for the sequence s n (x 1 , x 2 , . . . , x k+1 ; q), n = 1, 2 . . . , the first-order recurrence relation s n (x 1 , x 2 , . . . , x k+1 ; q) = [x 1 , x 2 , . . . , x k+1 − n + 1] q s n−1 (x 1 , x 2 , . . . , x k+1 ; q), for n = 1, 2 . . . , with initial condition s 1 (x 1 , x 2 , . . . , x k+1 ; q) = [x 1 + x 2 + · · · + x k+1 ] q . Applying it successively, it follows that s n (x 1 , x 2 , . . . , x k+1 ; q) = [x 1 +x 2 +· · ·+x k+1 ] n,q , and so (2.7) is shown. Formula (2.9) may be derived by following the steps of the derivation of (2.7) and using recurrence relation (2.4), with x = n, and the expression [x k+1 − r k+1 + 1] q + q (x k+1 −r k+1 )+1 [x k − r k ] q + · · · + q k+1 j=3 (x j −r j )+1 [x 2 − r 2 ] q + q k+1 j=2 (x j −r j )+1 [x 1 − r 1 ] q = [x 1 , x 2 , . . . , x k+1 − n + 1] q . The alternative formulae (2.8) and (2.10) are deduced from (2.7) and (2.9), respectively, by replacing x j by −x j , for j = 1, 2, . . . , k + 1 and using the relations [−x] r,q = (−1) x q −xr−( r 2 ) [x + r − 1] r,q , n 2 = k+1 j=1 r j 2 + k j=1 r j (n − s j ). Two versions of a multivariate q-Cauchy formula, which by virtue of n r 1 , r 2 , . . . , r k q = [n] q ! [r 1 ] q ![r 2 ] q ! · · · [r k ] q ![r k+1 ] q ! , x j r j q = [x j ] r j ,q [r j ] q ! , constitute reformulations of the corresponding two versions of a multivariate q-Vandermonde formula, are stated in the following corollary of Theorem 2.1. Corollary 2.1. Let n be a positive integer and let x j , j = 1, 2, . . . , k + 1, and q be real numbers, with q = 1. Then, x 1 + x 2 + · · · + x k+1 n q = q k j=1 (n−s j )(x j −r j ) k+1 j=1 x j r j q (2.11) and, alternatively, x 1 + x 2 + · · · + x k+1 + n − 1 n q = q k j=1 r j z j k+1 j=1 x j + r j − 1 r j q . (2.12) Also, x 1 + x 2 + · · · + x k+1 n q = q k j=1 r j (z j −(n−s j )) k+1 j=1 x j r j q (2.13) and, alternatively, x 1 + x 2 + · · · + x k+1 + n − 1 n q = q k j=1 x j (n−s j ) k+1 j=1 x j + r j − 1 r j q , (2.14) where s j = j i=1 r i , z j = k+1 i=j+1 x i , j = 1, 2, . . . , k, and r k+1 = n − s k and the summation, in all four sums, is extended over all r j = 0, 1, . . . , n, j = 1, 2, . . . , k, with k i=1 r i ≤ n. Remark 2.1. Additional expressions of the multivariate q-Cauchy formulae. The alternative expressions (2.12) and (2.14), which are useful in probability theory, may be rewritten as r + k n + k q = q k j=1 (r j −x j )(n−y j +k−j+1) k+1 j=1 r j x j q (2.15) and r + k n + k q = q k j=1 (x j +1)(n−s j −r+y j ) k+1 j=1 r j x j q , (2.16) respectively, where s j = j i=1 r i , y j = j i=1 x i , j = 1, 2, . . . , k, r k+1 = r − s k , x k+1 = r − y k , and the summation, in both sums, is extended over all r j = x j , x j + 1, . . . , r, j = 1, 2, . . . , k, with k i=1 r i ≤ r. Indeed, replacing, the bound variable r j by r j −(x j −1) and the constant x j by x j + 1, for all j = 1, 2, . . . , k + 1, formulae (2.12) and (2.14), after some algebra, are transformed to (2.15) and (2.16), respectively. Two versions of a multivariate inverse q-Vandermonde formula are derived in the following theorem. Theorem 2.2. Let n be a positive integer and let x j , j = 1, 2, . . . , k + 1, and q be real numbers, with q = 1. Then, 1 [x k+1 ] n,q = n + s k − 1 r 1 , r 2 , . . . , r k q q k j=1 (n+s k −s j )(x j −r j ) k j=1 [x j ] r j ,q [x 1 + x 2 + · · · + x k+1 ] n+s k ,q ,(2.17) provided \|q −x k+1 \| < 1, and 1 [x k+1 ] n,q = n + s k − 1 r 1 , r 2 , . . . , r k q q k j=1 r j (z j −s k +s j −n+1) k j=1 [x j ] r j ,q [x 1 + x 2 + · · · + x k+1 ] n+s k ,q ,(2.18) provided \|q x k+1 \| < 1, where s j = j i=1 r i and z j = k+1 i=j+1 x i , for j = 1, 2, . . . , k, and the summation, in both sums, is extended over all r j = 0, 1, . . . , j = 1, 2, . . . , k. Proof. According to an inverse q-Vandermonde formula (Charalambides (2016), p. 14), it holds true 1 [x k+1 ] n,q = ∞ r k =0 n + r k − 1 r k q q n(x k −r k ) [x k ] r k ,q [x k + x k+1 ] n+r k ,q . Similarly, 1 [x k + x k+1 ] n+r k ,q = ∞ r k−1 =0 n + r k + r k−1 − 1 r k−1 q q (n+r k )(x k−1 −r k−1 ) [x k−1 ] r k−1 ,q [x k−1 + x k + x k+1 ] n+r k +r k−1 ,q and finally, 1 [x 2 + x 3 + · · · + x k+1 ] n+s k −s 1 ,q = ∞ r 1 =0 n + s k − 1 r 1 q q (n+s k −s 1 )(x 1 −r 1 ) [x 1 ] r 1 ,q [x 1 + x 2 + · · · + x k+1 ] n+s k ,q . Applying these k expansions, one after the other in the inner sum of each step, and using the relation n + r k − 1 r k q n + r k + r k−1 − 1 r k−1 q n + s k − 1 r 1 q = n + s k − 1 r 1 , r 2 , . . . , r k q , expansion (2.17) is obtained. The alternative expansion (2.18), is similarly deduced by using the following inverse q-Vandermonde expansions (Charalambides (2016), p. 14) 1 [x j+1 + · · · + x k+1 ] n+s k −s j ,q = ∞ r j =0 n + s k − s j−1 − 1 r j q q r j (z j −s k +s j −n+1) [x j ] r j ,q [x j + · · · + x k+1 ] n+s k −s j−1 ,q , for j = 1, 2, . . . , k, with s 0 = 0. Multivariate q -Pólya distribution A multiple q-Pólya urn model may be introduced, by first defining a q-analogue of the notion of a random drawing of a ball from an urn. Consider an urn containing r balls, {b 1 , b 2 , . . . , b r }, of k + 1 different ordered colors, with r ν distinct balls of color c ν , {b s ν−1 +1 , b s ν−1 +2 , . . . , b sν }, for ν = 1, 2, . . . , k + 1, where s 0 = 0, s ν = ν i=1 r i , for ν = 1, 2, . . . , k + 1, with s k+1 = r. A random q-drawing (or q-selection) of a ball from the urn is carried out as follows. Assume that the balls in the urn are forced to pass through a random mechanism, one by one, in the order (b 1 , b 2 , . . . , b r ) or in the reverse order (b r , b r−1 , . . . , b 1 ). Also, suppose that each passing ball may or may not be caught by the mechanism, with probabilities p = 1 − q and q, respectively. The first caught ball is drawn out of the urn. In the case all balls in the urn pass through the mechanism and no ball is caught, the ball passing procedure is repeated, with the same order. Clearly, the probability that ball b x is drawn from the urn is given by ∞ k=0 (1 − q)q (x−1)+rk = (1 − q)q x−1 ∞ k=0 q rk = q x−1 [r] q , or by ∞ k=0 (1 − q)q (r−x)+rk = q r−x [r] q = q −(x−1) [r] q −1 , where 0 < q < 1, according to whether the ball passing order is (b 1 , b 2 , . . . , b r ) or (b r , b r−1 , . . . , b 1 ). Consequently, the probability function of the number N r on the drawn ball is given by p r (x; q) = P (N r = x) = q x−1 [r] q , x = 1, 2, . . . , r, where 0 < q < 1 or 1 < q < ∞. Note that this is the probability function of the discrete q-uniform distribution on the set {1, 2, . . . , r}. Also, the probability P r (r ν ; q), that a ball of color c ν is drawn from the urn is given by P r (r ν ; q) = P (s ν−1 < N r ≤ s ν ) = q s ν−1 [r ν ] q [r] q = q −(r−sν ) [r ν ] q −1 [r] q −1 , for ν = 1, 2, . . . , k + 1, with s 0 = 0, where 0 < q < 1 or 1 < q < ∞. As expected, the sum of these probabilities, on using successively the relation [s] q + q s [r] q = [s + r] q , is obtained as k+1 ν=1 P r (r ν ; q) = 1 [r] q k+1 ν=1 q s ν−1 [r ν ] q = [r 1 + r 2 + · · · + r k+1 ] q [r] q = 1, where 0 < q < 1 or 1 < q < ∞. Finally, notice that a random q-drawing of a ball, for q → 1 and since lim q→1 P r (r ν ; q) = r ν r , ν = 1, 2, . . . , k + 1, reduces to the usual random drawing of a ball from the urn. Furthermore, assume that random q-drawings of balls are sequentially carried out, one after the other, from an urn, initially containing r balls of k + 1 different colors, with r ν distinct balls of color c ν , for ν = 1, 2, . . . , k + 1, according to the following scheme. After each q-drawing, the drawn ball is placed back in the urn together with m balls of the same color. Then, the conditional probability of drawing a ball of color c ν at the ith q-drawing, given that j ν − 1 balls of color c ν and a total of i ν−1 balls of colors c 1 , c 2 , . . . , c ν−1 are drawn in the previous i − 1 q-drawings, is given by p i,jν (i ν−1 ) = q s ν−1 +mi ν−1 1 − q rν +m(jν −1) 1 − q r+m(i−1) = q −m(β ν−1 −i ν−1 ) [α ν − j ν + 1] q −m [α − i + 1] q −m , (3.1) for j ν = 1, 2, . . . , i, i ν = 0, 1, . . . , i − 1, i = 1, 2, . . . , and ν = 1, 2, . . . , k + 1, with i 0 = 0, where 0 < q < 1 or 1 < q < ∞ and α = −r/m, α ν = −r ν /m, β ν = −s ν /m, ν = 1, 2, . . . , k + 1, β 0 = 0. Note that i ν = i ν−1 + j ν = j 1 + j 2 + · · · + j ν , for ν = 1, 2, . . . , k + 1. This model, which for q → 1 and since p i,jν = lim q→1 p i,jν (i ν−1 ) = r ν + m(j ν − 1) r + m(i − 1) = α ν − j ν + 1 α − i + 1 , α ν = − r ν m , α = − r m , for j ν = 1, 2, . . . , i, i = 1, 2, . . . , and ν = 1, 2, . . . , k + 1, reduces to the (classical) multiple Pólya urn model, may be called multiple q-Pólya urn model. Definition 3.1. Let X ν be the number of balls of color c ν drawn in n q-drawings in a multiple q-Pólya urn model, with conditional probability of drawing a ball of color c ν at the ith q-drawing, given that j ν − 1 balls of color c ν and a total of i ν−1 balls of colors c 1 , c 2 , . . . , c ν−1 are drawn in the previous i − 1 q-drawings, is given by (3.1), for ν = 1, 2, . . . , k. The distribution of the random vector (X 1 , X 2 , . . . , X k ) is called k-variate q-Pólya distribution, with parameters n, (α 1 , α 2 , . . . , α k ), α, and q. The probability function of the k-variate q-Pólya distribution is obtained in the following theorem. Theorem 3.1. The probability function of the k-variate q-Pólya distribution, with parameters n, (α 1 , α 2 , . . . , α k ), α, and q, is given by P (X 1 = x 1 , X 2 = x 2 , . . . , X k = x k ) = q −m k j=1 (n−y j )(α j −x j ) k+1 j=1 α j x j q −m α n q −m = n x 1 , x 2 , . . . , x k q −m q −m k j=1 (n−y j )(α j −x j ) k+1 j=1 [α j ] x j ,q −m [α] n,q −m , (3.2) for x j = 0, 1, . . . , n, j = 1, 2, . . . , k, with k j=1 x j ≤ n, and 0 < q < 1 or 1 < q < ∞, where x k+1 = n − k j=1 x j , α k+1 = α − k j=1 α j , and y j = j i=1 x i , for j = 1, 2, . . . , k. Proof. The probability function p n (x 1 , x 2 , . . . , x k ) = P (X 1 = x 1 , X 2 = x 2 , . . . , X k = x k ), on using the total probability theorem, satisfies the recurrence relation p n (x 1 , x 2 , . . . , x k ) = p n−1 (x 1 , x 2 , . . . , x k ) q −m(β k −y k ) [α k+1 − x k+1 + 1] q −m [α − n + 1] q −m + p n−1 (x 1 − 1, x 2 , . . . , x k ) [α 1 − x 1 + 1] q −m [α − n + 1] q −m + p n−1 (x 1 , x 2 − 1, . . . , x k ) q −m(β 1 −y 1 ) [α 2 − x 2 + 1] q −m [α − n + 1] q −m + · · · + p n−1 (x 1 , x 2 , . . . , x k − 1) q −m(β k−1 −y k−1 ) [α k − x k + 1] q −m [α − n + 1] q −m , for x j = 1, 2, . . . , n, j = 1, 2, . . . , k and n = 1, 2, . . . , with k j=1 x j ≤ n, x k+1 = n − k j=1 x j , and p 0 (0, 0, . . . , 0) = 1. Also, p n (0, 0, . . . , 0) = n i=1 q −mβ k [a k+1 − i + 1] q −m n i=1 [a − i + 1] q −m = q −mnβ k [a k+1 ] n,q −m [a] n,q −m . Clearly, the sequence c n (x 1 , x 2 , . . . , x k ) = q m k j=1 (n−y j )(α j −x j ) [α] n,q −m k+1 j=1 [α j ] x j ,q −m p n (x 1 , x 2 , . . . , x k ) (3.3) satisfies the recurrence relation c n (x 1 , x 2 , . . . , x k ) = c n−1 (x 1 , x 2 , . . . , x k ) + q −m(n−y 1 ) c n−1 (x 1 − 1, x 2 , . . . , x k ) + q −m(n−y 2 ) c n−1 (x 1 , x 2 − 1, . . . , x k ) + · · · + q −m(n−y k ) c n−1 (x 1 , x 2 , . . . , x k − 1), for x j = 1, 2, . . . , n, j = 1, 2, . . . , k and n = 1, 2, . . . , with k j=1 x j ≤ n, and c 0 (0, 0, . . . , 0) = 1. Since this recurrence relation, according to (2.6), uniquely determines the qmultinomial coefficient, c n (x 1 , x 2 , . . . , x k ) = n x 1 , x 2 , . . . , x k q −m , the second part of expression (3.2) is readily deduced from (3.3). The first part of (3.2), which is a reformulation of the second, is deduced by using the expressions n x 1 , x 2 , . . . , x k q = [n] q ! [x 1 ] q ![x 2 ] q ! · · · [x k ] q ![x k+1 ] q ! , α j x j q = [α j ] x j ,q [x j ] q ! . Note that the multivariate q-Vandermonde formula (2.7) and, equivalently, the multivariate q-Cauchy formula (2.11), guarantees that the probabilities (3.2) sum to unity. Certain (and not any) marginal and conditional distributions of a k-variate q-Pólya distribution are derived in the next theorem. Theorem 3.2. Suppose that the random vector (X 1 , X 2 , . . . , X k ) obeys a k-variate q-Pólya distribution, with parameters n, (α 1 , α 2 , . . . , α k ), α, and q. Then, (a) the marginal distribution of the random vector (X 1 , X 2 , . . . , X ν ) is a ν-variate q-Pólya, with parameters n, (α 1 , α 2 , . . . , α ν ), α, and q, for ν = 1, 2, . . . , k, and (b) the conditional distribution of the random vector (X ν , X ν+1 , . . . , X ν+κ−1 ), given that (X 1 , X 2 , . . . , X ν−1 ) = (x 1 , x 2 , . . . , x ν−1 ) is a κ-variate q-Pólya, with parameters n, (α ν , α ν+1 , . . . , α ν+κ−1 ), α − α 1 − α 2 − · · · − α ν−1 , and q, for κ = 1, 2, . . . , k − ν + 1 and ν = 1, 2, . . . , k. Proof. (a) Summing the probabilities (3.2) for x j = 0, 1, . . . , n − y ν , j = ν + 1, ν + 2, . . . , k, with x ν+1 + x ν+2 + · · · + x k ≤ n − y ν , and using (2.11), we get P (X 1 = x 1 , X 2 = x 2 , . . . , X ν = x ν ) = q −m ν j=1 (n−y j )(α j −x j ) ν j=1 α j x j q −m × q −m k j=ν+1 (n−y j )(α j −x j ) k j=ν+1 α j x j q −m α − α 1 − · · · − α k n − x 1 − · · · − x k q −m α n q −m = q −m ν j=1 (n−y j )(α j −x j ) ν j=1 α j x j q −m α − α 1 − · · · − α ν n − x 1 − · · · − x ν q −m α n q −m , which is the probability function of a ν-variate q-Pólya distribution, with parameters n, (α 1 , α 2 , . . . , α ν ), α, and q. (b) The probability function of the conditional distribution of the random vector (X ν , X ν+1 , . . . , X ν+κ−1 ), given that (X 1 , X 2 , . . . , X ν−1 ) = (x 1 , x 2 , . . . , x ν−1 ), using the expression P (X ν = x ν , . . . , X ν+κ−1 = x ν+κ−1 \|X 1 = x 1 , X 2 = x 2 , . . . , X ν−1 = x ν−1 ) = P (X 1 = x 1 , X 2 = x 2 , . . . , X ν+κ−1 = x ν+κ−1 ) P (X 1 = x 1 , X 2 = x 2 , . . . , X ν−1 = x ν−1 ) , is obtained as P (X ν = x ν , . . . , X ν+κ−1 = x ν+κ−1 \|X 1 = x 1 , X 2 = x 2 , . . . , X ν−1 = x ν−1 ) = q −m ν+κ−1 j=1 (n−y j )(α j −x j ) ν+κ−1 j=1 α j x j q −m α−α 1 −···−α ν+κ−1 n−x 1 −···−x ν+κ−1 q −m α n q −m q −m ν−1 j=1 (n−y j )(α j −x j ) ν−1 j=1 α j x j q −m α−α 1 −···−α ν−1 n−x 1 −···−x ν−1 q −m α n q −m = q −m ν+κ−1 j=ν (n−y j )(α j −x j ) ν+κ−1 j=ν α j x j q −m α − α ν − · · · − α ν+κ−1 n − x ν − · · · − x ν+κ−1 q −m α n q −m , which is the probability function of a κ-variate q-Pólya distribution, with parameters n, (α ν , α ν+1 , . . . , α ν+κ−1 ), α − α 1 − α 2 − · · · − α ν−1 , and q. The multivariate q-Pólya distribution, for large r, can be approximated by a qmultinomial distribution of the second kind, which is introduced and studied in Charalambides (2020). Specifically, the following limiting theorem is derived. Theorem 3.3. Consider the multivariate q-Pólya distribution, with probability function p n (r; m, q) = P (X 1 = x 1 , X 2 = x 2 , . . . , X k = x k ) given by (3.2). For 0 < q < 1, assume that lim r→∞ [r − s j ] q−1 [r − s j−1 ] q−1 = θ j , s j = j i=1 r i , j = 1, 2, . . . , k, s 0 = 0,(3. 4) and in the case of a negative integer m assume, in addition, that θ j < q −m(ν−1) , j = 1, 2, . . . , k, for some positive integer ν. Then, lim r→∞ p n (r; m, q) = n x 1 , x 2 , . . . , x k q m k j=1 θ n−y j j x j i j =1 (1 − θ j q m(i j −1) ),(3. 5) for x j = 0, 1, . . . , n, j = 1, 2, . . . , k, with x 1 + x 2 + · · · + x k ≤ n, where y j = j i=1 x i , 0 < q < 1 and 0 < θ j < 1, j = 1, 2, . . . , k, in the case m is a positive integer, or 0 < θ j < q −m(ν−1) , j = 1, 2, . . . , k, for some positive integer ν ≥ n, in the case m is a negative integer. Also, for 1 < q < ∞, assume that lim r→∞ [r j ] q [r − s j−1 ] q = λ j , s j = j i=1 r i , j = 1, 2, . . . , k, s 0 = 0, (3.6) and in the case of a negative integer m assume, in addition, that λ j < q m(ν−1) , j = 1, 2, . . . , k, for some positive integer ν. Then, lim r→∞ p n (r; m, q) = n x 1 , x 2 , . . . , x k q −m k j=1 λ x j j n−y j i j =1 (1 − λ j q −m(i j −1) ), (3.7) for x j = 0, 1, . . . , n, j = 1, 2, . . . , k, with x 1 + x 2 + · · · + x k ≤ n, where y j = j i=1 x i , 1 < q < ∞ and 0 < λ j < 1, in the case m is a positive integer, or 0 < λ j < q m(ν−1) , for some positive integer ν ≥ n, in the case m is a negative integer. Proof. For 0 < q < 1, the probability function (3.2), p n (r; m, q) = n x 1 , x 2 , . . . , x k q −m k j=1 q −m(n−y j )(α j −x j ) [α j ] x j ,q −m [α − β j ] n−y j ,q −m [α − β j−1 ] n−y j−1 ,q −m , using (2.2), may be written as p n (r; m, q) = n x 1 , x 2 , . . . , x k q m k j=1 x j i j =1 (q −r j −m(i j −1) − 1)q −r+s j +m(i j −1) n−y j i j =1 (q −r+s j −m(i j −1) − 1)q m(i j −1) n−y j−1 i j−1 =1 (q −r+s j−1 −m(i j−1 −1) − 1)q m(i−1) . Moreover, by the assumption (3.4), it follows that lim r→∞ (q −r j −m(i j −1) − 1)q −r+s j +m(i j −1) q −r+s j−1 − 1 = 1 − q m(i j −1) lim r→∞ q −r+s j − 1 q −r+s j−1 − 1 + lim r→∞ q m(i j −1) − 1 q −r+s j−1 − 1 = 1 − θ j q m(i j −1) , lim r→∞ (q −r+s j −m(i j −1) − 1)q m(i j −1) q −r+s j−1 − 1 = lim r→∞ q −r+s j − 1 q −r+s j−1 − 1 − lim r→∞ q m(i j −1) − 1 q −r+s j−1 − 1 = θ j , and lim r→∞ (q −r+s j−1 −m(i j −1) − 1)q m(i j −1) q −r+s j−1 − 1 = 1 − lim r→∞ q m(i j −1) − 1 q −r+s j−1 − 1 = 1, for j = 1, 2, . . . , k. Thus, dividing both the numerator and denominator of the jth factor in the last expression of the probability function (3.2) by (q −r+s j−1 − 1) n−y j−1 and taking the limits as r → ∞, the limiting expression (3.5) is readily deduced. For 1 < q < ∞, the probability function (3.2), p n (r; m, q) = n x 1 , x 2 , . . . , x k q −m k j=1 q −m(n−y j )(α j −x j ) [α j ] x j ,q −m [α − β j ] n−y j ,q −m [α − β j−1 ] n−y j−1 ,q −m , may be written as p n (r; m, q) = n x 1 , x 2 , . . . , x k q −m k j=1 x j i j =1 (1 − q r j +m(i j −1) )q −m(i j −1) × n−y j i j =1 (1 − q r−s j +m(i j −1) )q r j −m(i j −1) n−y j−1 i j−1 =1 (1 − q r−s j−1 +m(i j−1 ) )q −m(i j−1 −1) . Moreover, by the assumption (3.6), it follows that lim r→∞ (q r j +m(i j −1) − 1)q −m(i j −1) q r−s j−1 − 1 = lim r→∞ q r j − 1 q r−s j−1 − 1 − lim r→∞ q −m(i j −1) − 1 q r−s j−1 − 1 = λ j , lim r→∞ (q r−s j +m(i j −1) − 1)q r j −m(i j −1) q r−s j−1 − 1 = 1 − q −m(i j −1) lim r→∞ q r j − 1 q r−s j−1 − 1 − lim r→∞ q −m(i j −1) − 1 q r−s j−1 − 1 = 1 − λ j q −m(i j −1) , and lim r→∞ (q r−s j−1 +m(i j −1) − 1)q −m(i j −1) q r−s j−1 − 1 = 1 − lim r→∞ q −m(i j −1) − 1 q r−s j−1 − 1 = 1, for j = 1, 2, . . . , k. Thus, dividing both the numerator and denominator of the jth factor in the last expression of the probability function (3.2) by (q r−s j−1 − 1) n−y j−1 and taking the limits as r → ∞, the limiting expression (3.7) is readily deduced. The multiple q-Pólya urn model in the particular case m = 0 reduces to q-drawings with replacement and the distribution (3.2) reduces to the classical multinomial distribution with probability of success of the νth kind p ν = q s ν−1 [r ν ] q /[r] q , ν = 1, 2, . . . , k+1. Also, for m = −1, the case corresponds to q-drawings without replacement and the probability function (3.2) reduces to a P (X 1 = x 1 , X 2 = x 2 , . . . , X k = x k ) = q k j=1 (n−y j )(r j −x j ) k+1 j=1 r j x j q r n q = n x 1 , x 2 , . . . , x k q q k j=1 (n−y j )(r j −x j ) k+1 j=1 [r j ] x j ,q [r] n,q , (3.8) for x j = 0, 1, . . . , n, j = 1, 2, . . . , k, with k j=1 x j ≤ n, and 0 < q < 1 or 1 < q < ∞, where x k+1 = n − k j=1 x j , r k+1 = r − k j=1 r j , and y j = j i=1 x i , j = 1, 2, . . . , k. The distribution with probability function (3.8) may be called multivariate q-hypergeometric distribution. Furthermore, for m = 1, the case to q-drawings with replacement and addition of another ball of the same color. The particular probability function may be deduced from (3.2) by setting α j = −r j , for j = 1, 2 . . . , k, α k+1 = −r + s k and x k+1 = n − y k , as P (X 1 = x 1 , X 2 = x 2 , . . . , X k = x k ) = k j=1 q (n−y j )(r j +x j ) −r j x j q −1 −r + s k n − y k q −1 −r n q −1 . which, using the expression −r + s k n − y k q −1 −r n q −1 = k j=1 −r + s j n − y j q −1 −r + s j−1 n − y j−1 q −1 , with y 0 = 0, becomes P (X 1 = x 1 , . . . , X k = x k ) = k j=1 q (n−y j )(r j +x j ) −r j x j q −1 −r + s j n − y j q −1 −r + s j−1 n − y j−1 q −1 . Then, since −r j x j q −1 = (−1) x j q x j +( x j 2 ) r j + x j − 1 x j q , −r + s j n − y j q −1 = (−1) n−y j q n−y j +( n−y j 2 ) r − s j + n − y j − 1 n − y j q −1 , and n − y j−1 2 = x j + (n − y j ) 2 = x j 2 + n − y j 2 + x j (n − y j ), it takes the form P (X 1 = x 1 , X 2 = x 2 , . . . , X k = x k ) = k j=1 q r j (n−y j ) r j +x j −1 x j q r−s j +n−y j −1 n−y j q r−s j−1 +n−y j−1 −1 n−y j−1 q , which after cancelations, reduces to P (X 1 = x 1 , X 2 = x 2 , . . . , X k = x k ) = q k j=1 r j (n−y j ) k+1 j=1 r j + x j − 1 x j q r + n − 1 n q = n x 1 , x 2 , . . . , x k q q k j=1 r j (n−y j ) k+1 j=1 [r j + x j − 1] x j ,q [r + n − 1] n,q , (3.9) for x j = 0, 1, . . . , n, j = 1, 2, . . . , k, with k j=1 x j ≤ n, and 0 < q < 1 or 1 < q < ∞, where x k+1 = n − k j=1 x j , r k+1 = r − k j=1 r j , and y j = j i=1 x i , j = 1, 2, . . . , k. The distribution with probability function (3.9) may be called multivariate negative q-hypergeometric distribution. Note that the probabilities (3.9), according to (2.10), sum to unity. Example 3.1. Distribution of the numbers of errors in the chapters of a manuscript. Consider a manuscript of k + 1 chapters (sections, parts), with chapter c ν containing r ν typographical errors, {e s ν−1 +1 , e s ν−1 +2 , . . . , e sν }, for ν = 1, 2, . . . , k + 1, where s 0 = 0, s ν = ν i=1 r i , for ν = 1, 2, . . . , k + 1, with s k+1 = r. Assume that a proofreader reads the manuscript and when he/she finds an error corrects it and starts reading the manuscript from the beginning. Also, the proofreader starts reading the manuscript from the beginning when he/she reaches its end. Assume that the probability of finding any particular error is p = 1 − q. Clearly, the probability of finding an error in chapter c ν at the first scan is given by q s ν−1 [r ν ] q /[r] q , for ν = 1, 2, . . . , k + 1, with s 0 = 0. Then, the conditional probability of finding (and correcting) an error in chapter c ν at the ith scan, given that j ν − 1 errors of chapter c ν and a total of i ν−1 errors of chapters c 1 , c 2 , . . . , c ν−1 are found in the previous i − 1 scans, is given by p i,jν (i ν−1 ) = q s ν−1 −i ν−1 [r ν − j ν + 1] q [r − i + 1] q , for j ν = 1, 2, . . . , i, i ν = 0, 1, . . . , i − 1, i = 1, 2, . . . , and ν = 1, 2, . . . , k + 1, with s 0 = 0 and i 0 = 0, where 0 < q < 1. Clearly, the joint distributions of the numbers X ν of errors of chapter c ν found (and corrected) in n scans is the multivariate q-hypergeometric distribution, with probability function (3.8). Example 3.2. Random q-selection from a finite population. Consider a finite population of r people, classified into k +1 classes c j , j = 1, 2, . . . , k, with an unknown number of people in each class. Suppose that a sample of n people is randomly q-selected from this population, without replacement. Let x j be the number of people of class c j , for j = 1, 2, . . . , k, in the sample. We are interested in the probability that the number of people of the population who belong in class c j equals r j , for j = 1, 2, . . . , k. Let X j and R j be the numbers of people of class c j , for j = 1, 2, . . . , k, in the sample and the population, respectively. The conditional distribution of the random vector X = (X 1 , X 2 , . . . , X k ), given that the random vector R = (R 1 , R 2 , . . . , R k ) equals r = (r 1 , r 2 , . . . , r k ), is the multivariate q-hypergeometric distribution, with probability function (3.8), f X\|R (x 1 , x 2 , . . . , x k \|r 1 , r 2 , . . . , r k ) = q k j=1 (n−y j )(r j −x j ) k+1 j=1 r j x j q r n q , for x j = 0, 1, . . . , n, j = 1, 2, . . . , k, with k j=1 x j ≤ n, and 0 < q < 1 or 1 < q < ∞, where x k+1 = n − k j=1 x j , r k+1 = r − k j=1 r j , and y j = j i=1 x i , j = 1, 2, . . . , k. The required probability is given by the value of the conditional probability function of the random vector R = (R 1 , R 2 , . . . , R k ), given X = (X 1 , X 2 , . . . , X k ), at the point r = (r 1 , r 2 , . . . , r k ). This conditional probability function is given f R\|X (r 1 , r 2 , . . . , r k \|x 1 , x 2 , . . . , x k ) = f R,X (r 1 , r 2 , . . . , r k , x 1 , x 2 , . . . , x k ) f X (x 1 , x 2 , . . . , x k ) = f R (r 1 , r 2 , . . . , r k )f X\|R (x 1 , x 2 , . . . , x k \|r 1 , r 2 , . . . , r k ) f X (x 1 , x 2 , . . . , x k ) and the probability function of the random vector X = (X 1 , X 2 , . . . , X k ) is f X (x 1 , x 2 , . . . , x k ) = f R (r 1 , r 2 , . . . , r k )f X\|R (x 1 , x 2 , . . . , x k \|r 1 , r 2 , . . . , r k ), where the summation is extended over all r j = x j , x j + 1, . . . , r, j = 1, 2, . . . , k, with k j=1 r j ≤ r. Thus, for the calculation of the probability in question, the additional knowledge of the distribution of the random vector R = (R 1 , R 2 , . . . , R k ) is required. Assume that this distribution is the k-variate discrete q-uniform with probability function (Bose-Einstein q-stochastic model (q-statistic)) f R (r 1 , r 2 , . . . , r k ) = k+1 j=1 q (k−j+1)r j r + k k q , for r j = 0, 1, . . . , r, j = 1, 2, . . . , k, with k j=1 r j ≤ r, where r k+1 = r − k j=1 r j . Hence f X (x 1 , x 2 , . . . , x k ) = q k j=1 (n−y j )(r j −x j )+(k−j+1)r j k+1 j=1 r j x j q r+k k q r n q , where the summation is extended over all r j = x j , x j + 1, . . . , r, j = 1, 2, . . . , k, with k j=1 r j ≤ r, and 0 < q < 1 or 1 < q < ∞, where r k+1 = r − k j=1 r j , x k+1 = n − k j=1 x j , and y j = j i=1 x i . Then, using the q-Cauchy formula, q k j=1 (r j −x j )(n−y j +k−j+1) k+1 j=1 r j x j q = r + k n + k q where the summation is extended over all r j = x j , x j + 1, . . . , r, j = 1, 2, . . . , k, with k j=1 r j ≤ r, and 0 < q < 1 or 1 < q < ∞, where r k+1 = r − k j=1 r j , x k+1 = n − k j=1 x j , and y j = j i=1 x i , the probability function of the random vector X = (X 1 , X 2 , . . . , X k ), is deduced as f X (x 1 , x 2 , . . . , x k ) = k+1 j=1 q (k−j+1)x j r + k n + k q r + k k q r n q . Since r + k k q r n q = r + k n + k q n + k k q , the last expression reduces to f X (x 1 , x 2 , . . . , x k ) = k+1 j=1 q (k−j+1)x j n + k k q , for x j = 0, 1, . . . , n, j = 1, 2, . . . , k, with k j=1 r j ≤ r, and 0 < q < 1 or 1 < q < ∞, which is a k-variate discrete q-uniform probability function. Therefore, the required conditional probability function of the random vector R = (R 1 , R 2 , . . . , R k ), given that X = (X 1 , X 2 , . . . , X k ) is given by f R\|X (r 1 , r 2 , . . . , r k \|x 1 , x 2 , . . . , x k ) = q k j=1 (n−y j )(r j −x j ) k+1 j=1 r j x j q r + k n + k q , for r j = x j , x j +1, . . . , r, j = 1, 2, . . . , k, with k j=1 r j ≤ r, and 0 < q < 1 or 1 < q < ∞, where r k+1 = r − k j=1 r j , x k+1 = n − k j=1 x j , and y j = j i=1 x i . The multivariate q-hypergeometric distribution may be obtained as the conditional distribution of k independent q-binomial distributions of the first kind, given their sum with another q-binomial distribution of the first kind independent of them. Precisely, the following theorem is shown. Theorem 3.4. Consider a sequence of independent Bernoulli trials and assume that the probability of success at the ith trial is given by p i = θq i−1 1 + θq i−1 , i = 1, 2, . . . , 0 < q < 1 or 1 < q < ∞. Let X j be the number of successes after the (s j−1 )th trial and until the (s j )th trial, for j = 1, 2, . . . , k + 1, with s 0 = 0, s j = j i=1 r i , j = 1, 2, . . . , k + 1, and s k+1 = r. Then, the conditional probability function of the random vector (X 1 , X 2 , . . . , X k ), given that X 1 + X 2 + · · · + X k+1 = n, is the multivariate q-hypergeometric distribution with probability function (3.8). Proof. The random variables X j , j = 1, 2, . . . , k + 1, are independent, with probability function, according to Theorem 2.1 in Charalambides (2016), is given by P (X j = x j ) = r j x j q (θq s j−1 ) x j q ( x j 2 ) r j i=1 (1 + θq s j−1 +i−1 ) , x j = 0, 1, . . . , r j , j = 1, 2, . . . , k + 1. Similarly, the probability function of the sum Y = X 1 + X 2 + · · · + X k+1 , which is the number of successes in r trials, is P (Y = n) = r n q θ n q ( n 2 ) r i=1 (1 + θq i−1 ) , n = 0, 1, . . . , r. Then, the joint conditional probability function of the random vector (X 1 , X 2 , . . . , X k ), given that Y = n, P (X 1 = x 1 , . . . , X k = x k \|Y = n) = P (X 1 = x 1 ) · · · P (X k = x k )P (X k+1 = n − y k ) P (Y = n) , on using these expressions, is obtained as P (X 1 = x 1 , X 2 = x 2 , . . . , X k = x k \|Y = n) = q c k k+1 j=1 r j x j q r n q , where c k = k i=1 x i s i−1 + (n − y k )s k + k j=1 x j 2 + n − y k 2 − n 2 . Thus, after some algebraic manipulations, it reduces to c k = n k j=1 r j − k i=1 x i (s k − s i−1 ) + k j=1 x j 2 + y k + 1 2 − ny k = k j=1 r j (n − y j ) − k j=1 x j (n − y j ) = k j=1 (n − y j )(y j − x j ), and the derivation of (3.8) is completed. Furthermore, the multivariate negative q-hypergeometric distribution may be obtained as the conditional distribution of k independent negative q-binomial distributions of the second kind, given their sum with another negative q-binomial distribution of the second kind independent of them, according to the following theorem. Theorem 3.5. Consider a sequence of independent Bernoulli trials and assume that the conditional probability of success at a trial, given that j − 1 successes occur in the previous trials, is given by p j = 1 − θq j−1 , j = 1, 2, . . . , 0 < θ < 1, 0 < q < 1 or 1 < q < ∞, where, for 1 < q < ∞, the number j of successes is restricted by j ≤ m = − log θ/ log q. Let W j be the number of failures after the (s j−1 )th success and until the occurrence of the (s j )th success, for j = 1, 2, . . . , k + 1, with s 0 = 0, s j = j i=1 r i , j = 1, 2, . . . , k + 1, and s k+1 = r, where r ≤ m in the case 1 < q < ∞. Then, the conditional probability function of the random vector (W 1 , W 2 , . . . , W k ), given that W 1 +W 2 +· · ·+W k+1 = n, is the multivariate negative q-hypergeometric distribution with probability function (3.9). Proof. The random variables W j , j = 1, 2, . . . , k + 1, are independent, with probability function, according to Theorem 3.1 in Charalambides (2016), is given by P (W j = w j ) = r j + w j − 1 w j q (θq s j−1 ) w j r j i=1 (1 − θq s j−1 +i−1 ), w j = 0, 1, . . . , for all j = 1, 2, . . . , k + 1. Similarly, the probability function of the sum U = W 1 + W 2 + · · · + W k+1 , which is the number of failures until the occurrence of the rth success, is P (U = n) = r + n − 1 n q θ n r i=1 (1 − θq i−1 ), n = 0, 1, . . . . Then, the joint conditional probability function of the random vector (W 1 , W 2 , . . . , W k ), given that U = n, P (W 1 = w 1 , . . . , W k = w k \|U = n) = P (W 1 = w 1 ) · · · P (W k = w k )P (W k+1 = n − u k ) P (U = n) , on using these expressions, is obtained as P (W 1 = w 1 , . . . , W k = w k \|U = n) = q c k k+1 j=1 r j + w j − 1 w j q r + n − 1 n q , where c k = k i=1 w i s i−1 + (n − u k )s k , u j = j i=1 w i , j = 1, 2, . . . , k. Thus, after some algebra, it reduces to c k = n k j=1 r j − k i=1 w i (s k − s i−1 ) = n k j=1 r j − k j=1 r j u j = k j=1 r j (n − u j ) and the derivation of (3.9) is completed. Multivariate inverse q -Pólya distribution Consider again the multiple q-Pólya urn model. Specifically, assume that random qdrawings of balls are sequentially carried out, one after the other, from an urn, initially containing r ν balls of color c ν , for ν = 1, 2, . . . , k + 1, according to the following scheme. After each q-drawing the drawn ball is placed back in the urn together with k balls of the same color. Assume that the conditional probability of drawing a ball of color c ν at the ith q-drawing, given that j ν − 1 balls of color c ν and a total of i ν−1 balls of colors c 1 , c 2 , . . . , c ν−1 are drawn in the previous i − 1 q-drawings, is given by (3.1). In this section the interest is turned to the study of the particular numbers of balls of colors c 1 , c 2 , . . . , c k drawn until the nth ball of color c k+1 is drawn. For this reason, the following definition is introduced. Definition 4.1. Let W ν be the number of balls of color c ν drawn until the nth ball of color c k+1 is drawn in a multiple q-Pólya urn model, with the conditional probability of drawing a ball of color c ν at the ith q-drawing, given that j ν − 1 balls of color c ν and a total of i ν−1 balls of colors c 1 , c 2 , . . . , c ν−1 are drawn in the previous i − 1 qdrawings, given by (3.1), for ν = 1, 2, . . . , k. The distribution of the random vector (W 1 , W 2 , . . . , W k ) is called multivariate inverse q-Pólya distribution, with parameters n, (α 1 , α 2 , . . . , α k ), α, and q. The probability function of the k-variate inverse q-Pólya distribution is obtained in the following theorem. Theorem 4.1. The probability function of the k-variate inverse q-Pólya distribution, with parameters n, (α 1 , α 2 , . . . , α k ), α, and q, is given by P (W 1 = w 1 , W 2 = w 2 , . . . , W k = w k ) = n + u k − 1 w 1 , w 2 , . . . , w k q −m q −m k j=1 (n+u k −u j )(α j −w j ) k j=1 [α j ] w j ,q −m [α k+1 ] n,q −m [α] n+u k ,q −m , (4.1) for w j = 0, 1, . . . , j = 1, 2, . . . , k, and 0 < q < 1 or 1 < q < ∞, where α k+1 = α − k j=1 α j , and u j = j i=1 w i , for j = 1, 2, . . . , k. Proof. The probability function of the k-variate inverse q-Pólya distribution is closely connected to the probability function k-variate q-Pólya distribution. Specifically, P (W 1 = w 1 , W 2 = w 2 , . . . , W k = w k ) = p n+u k −1 (w 1 , w 2 , . . . , w k )p n+u k ,n , where p n+u k −1 (w 1 , w 2 , . . . , w k ) is the probability of drawing w ν balls of color c ν , for all ν = 1, 2, . . . , k, and n − 1 balls of color c k+1 in n + u k − 1 q-drawings and p n+u k ,n = q −m(β k −u k ) [a k+1 −n+1] q −m /[a−n−u k +1] q −m is the conditional probability of drawing a ball of color c k+1 at the (n + u k )th q-drawing, given that n − 1 balls of color c k+1 and a total of u k balls of colors c 1 , c 2 , . . . , c k are drawn in the previous n + u k − 1 q-drawings. Thus using (3.2), expression (4.1) is deduced. Note that the multivariate inverse q-Vandermonde formula (2.17) guarantees that the probabilities (4.1) sum to unity. The multivariate inverse q-Pólya distribution, for large r, can be approximated by a negative q-multinomial distribution of the second kind, which is introduced and studied in Charalambides (2020). Specifically, the following limiting theorem is derived. and, alternatively, as q n (r; m, q) = n + u k − 1 w 1 , w 2 , . . . , w k q −m k j=1 q −m(n+u k −u j )(α j −w j ) [α j ] w j ,q −m × [α − β j ] n+u k −u j ,q −m [α − β j−1 ] n+u k −u j−1 ,q −m = n + u k − 1 w 1 , w 2 , . . . , w k q −m k j=1 w j i j =1 (1 − q r j +m(i j −1) )q −m(i j −1) × n+u k −u j i j =1 (1 − q r−s j +m(i j −1) )q r j −m(i j −1) n+u k −u j−1 i j−1 =1 (1 − q r−s j−1 +m(i j−1 −1) )q −m(i−1) . Then, proceeding as in the derivation of Theorem 3.3, the required limiting expressions (4.2) and (4.3) are deduced. Theorem 4.2. Consider the multivariate inverse q-Pólya distribution, with probability function q n (r; m, q) = P (W 1 = w 1 , W 2 = w 2 , . . . , W k = w k ) given by (4.1).For 0 < q < 1, assume that the limiting expression (3.4) holds true. Then,for some positive integer ν ≥ n, in the case m is a negative integer.Also, for 1 < q < ∞, assume that the limiting expression (3.6) holds true. Then,(1 − λ j q −m(i j −1) ), (4.3)for w j = 0, 1, . . . , and j = 1, 2, . . . , k, where u j = j i=1 w i , 1 < q < ∞ and 0 < λ j < 1, j = 1, 2, . . . , k, in the case m is a positive integer, or 0 < λ j < q m(ν−1) , j = 1, 2, . . . , k, for some positive integer ν ≥ n, in the case m is a negative integer.Proof. The probability function (4.1), using (2.2), may be written as(q −r+s j −m(i j −1) − 1)q m(i j −1) n+u k −u j−1 i j−1 =1 (q −r+s j−1 −m(i j−1 −1) − 1)q m(i−1). A q-Pólya urn model and the q-Pólya and inverse q-Pólya distributions. Ch A Charalambides, J. Statist. Plann. Infer. 142Charalambides, Ch. A. (2012) A q-Pólya urn model and the q-Pólya and inverse q-Pólya distributions, J. Statist. Plann. Infer. 142, 279-288. Discrete q-Distributions. Ch A Charalambides, John Wiley & SonsHoboken, New JerseyCharalambides, Ch. A. (2016) Discrete q-Distributions, John Wiley & Sons, Hoboken, New Jersey. 2020) q-Multinomial and negative q-multinomial distributions, Accepted for publication in the. Ch A Charalambides, Comm. Statist. Theory Meth. Charalambides, Ch. A. (2020) q-Multinomial and negative q-multinomial distribu- tions, Accepted for publication in the Comm. Statist. Theory Meth.. Steady-state Markov chain models for certain q-confluent hypergeometric distributions. A W Kemp, J. Statist. Plann. Infer. 135Kemp, A. W. (2005) Steady-state Markov chain models for certain q-confluent hy- pergeometric distributions, J. Statist. Plann. Infer. 135, 107-120. q-Probability: I. Basic discrete distributions. B A Kupershmidt, J. Nonlinear Math. Phys. 7Kupershmidt, B. A. (2000) q-Probability: I. Basic discrete distributions, J. Nonlinear Math. Phys. 7, 73-93.	[]
[ "TapirXLA: Embedding Fork-Join Parallelism into the XLA Compiler in TensorFlow Using Tapir", "TapirXLA: Embedding Fork-Join Parallelism into the XLA Compiler in TensorFlow Using Tapir" ]	[ "Tao B Schardl \nMIT Computer Science\nArtificial Intelligence Laboratory\n32 Vassar Street Cambridge02139MA\n", "Siddharth Samsi \nMIT Lincoln Laboratory\n240 Wood Street Lexington02421MA\n" ]	[ "MIT Computer Science\nArtificial Intelligence Laboratory\n32 Vassar Street Cambridge02139MA", "MIT Lincoln Laboratory\n240 Wood Street Lexington02421MA" ]	[]	This work introduces TapirXLA, a replacement for TensorFlow's XLA compiler that embeds recursive fork-join parallelism into XLA's low-level representation of code. Machinelearning applications rely on efficient parallel processing to achieve performance, and they employ a variety of technologies to improve performance, including compiler technology. But compilers in machine-learning frameworks lack a deep understanding of parallelism, causing them to lose performance by missing optimizations on parallel computation. This work studies how Tapir, a compiler intermediate representation (IR) that embeds parallelism into a mainstream compiler IR, can be incorporated into a compiler for machine learning to remedy this problem.TapirXLA modifies the XLA compiler in TensorFlow to employ the Tapir/LLVM compiler to optimize low-level parallel computation. TapirXLA encodes the parallelism within high-level TensorFlow operations using Tapir's representation of fork-join parallelism. TapirXLA also exposes to the compiler implementations of linear-algebra library routines whose parallel operations are encoded using Tapir's representation. We compared the performance of TensorFlow using TapirXLA against TensorFlow using an unmodified XLA compiler. On four neural-network benchmarks, TapirXLA speeds up the parallel running time of the network by a geometric-mean multiplicative factor of 30% to 100%, across four CPU architectures.	10.1109/hpec43674.2020.9286232	[ "https://arxiv.org/pdf/1908.11338v1.pdf" ]	201,668,095	1908.11338	20cedf71878cadc6d07f4bc77f050fec52e2083e	TapirXLA: Embedding Fork-Join Parallelism into the XLA Compiler in TensorFlow Using Tapir Tao B Schardl MIT Computer Science Artificial Intelligence Laboratory 32 Vassar Street Cambridge02139MA Siddharth Samsi MIT Lincoln Laboratory 240 Wood Street Lexington02421MA TapirXLA: Embedding Fork-Join Parallelism into the XLA Compiler in TensorFlow Using Tapir This work introduces TapirXLA, a replacement for TensorFlow's XLA compiler that embeds recursive fork-join parallelism into XLA's low-level representation of code. Machinelearning applications rely on efficient parallel processing to achieve performance, and they employ a variety of technologies to improve performance, including compiler technology. But compilers in machine-learning frameworks lack a deep understanding of parallelism, causing them to lose performance by missing optimizations on parallel computation. This work studies how Tapir, a compiler intermediate representation (IR) that embeds parallelism into a mainstream compiler IR, can be incorporated into a compiler for machine learning to remedy this problem.TapirXLA modifies the XLA compiler in TensorFlow to employ the Tapir/LLVM compiler to optimize low-level parallel computation. TapirXLA encodes the parallelism within high-level TensorFlow operations using Tapir's representation of fork-join parallelism. TapirXLA also exposes to the compiler implementations of linear-algebra library routines whose parallel operations are encoded using Tapir's representation. We compared the performance of TensorFlow using TapirXLA against TensorFlow using an unmodified XLA compiler. On four neural-network benchmarks, TapirXLA speeds up the parallel running time of the network by a geometric-mean multiplicative factor of 30% to 100%, across four CPU architectures. I. INTRODUCTION Machine-learning (ML) frameworks, including Caffe [1], Flux [2], MXNet [3], PyTorch [4], TensorFlow [5], and Theano [6] have emerged as popular environments for developing ML applications. Because of the high demands of ML applications on computing resources, ML frameworks employ a variety of technologies to improve the performance of ML applications, including compiler technology for optimizing the computation within ML applications. In addition, ML applications typically exhibit substantial parallelism, which ML frameworks seek to exploit for performance using hardware accelerators, including GPUs and TPUs [7], and by invoking software libraries, such as Eigen [8], Intel's MKL-DNN [9], cuDNN [10], and cuBLAS [11], that implement linear-algebra routines that have been optimized to exploit parallel computing hardware. ML frameworks use all three technologies -compilers, highperformance software libraries, and hardware acceleratorsto execute ML applications efficiently. But compilers in ML frameworks struggle to effectively optimize low-level parallel computation within ML applications. For example, consider TensorFlow's XLA compiler [12], whose compilation pipeline for CPUs and GPUs is illustrated in Figure 1. As the figure shows, the XLA compiler compiles a TensorFlow graph -an ML network in TensorFlow -into executable machine code through a sequence of stages. The TensorFlow graph is first transformed into a high-level representation, called HLO IR, by a front-end, such as the xla.compile API [13]. Optimizations, such as operator fusion and common-subexpression elimination [14,Sec. 12.2], are performed on HLO IR before the graph is transformed into a lower-level representation, namely, LLVM's intermediate representation (IR) [15], for a target hardware architecture. LLVM performs additional optimizations on LLVM IR before it finally generates executable code from the optimized LLVM IR. Other ML compilers adopt a similar architecture, with multiple levels of IRs and a mainstream compiler, such as LLVM, operating on the lowest-level IR for final optimization and machine-code generation. The LLVM optimizer in XLA fails to perform many ef- The problem stems from the fact that parallel control flow is represented in LLVM IR as function calls into a parallel runtime system. Previous work [16] has observed that these function calls inhibit standard compiler optimizations, either because they are opaque -meaning the implementations of those functions are not exposed to the compiler -or they implement concurrency that compilers struggle to reason about. These missed optimizations can hurt the efficiency and parallel speedup of parallel code [16]. To mitigate this problem, as Figure 1 shows, XLA's highlevel optimizer not only performs high-level optimizations on a TensorFlow graph, but it also decides how to implement these operations for a particular parallel back-end, such as a multicore CPU or a GPU. In particular, the high-level codegenerator decides how to subdivide the computation into parallel tasks, in order to insert appropriate parallel runtime calls and low-level primitives to implement those parallel tasks for a given hardware back-end. These decisions are made before classic compiler optimizations in LLVM, such as loop unrolling or vectorization, have optimized the computation in each task. XLA uses heuristics to estimate how to subdivide the computation into parallel tasks, but subsequent compiler optimizations can upset these estimates, depending on how effectively different tasks are optimized. A similar problem arises when ML compilers insert calls to libraries that implement optimized parallel linear-algebra kernels. These library calls inhibit compiler optimizations for similar reasons. When the library itself is opaque to the compiler, then the compiler cannot perform optimizations, such as function inlining [17, p. 536], constant propagation [17, p. 632], or common-subexpression elimination, on library routines based on the context in which they are called. When the source of a library routine is exposed to the compiler, the parallel library routine itself implements complex concurrency or calls to a parallel runtime system. Either way, these library calls inhibit compiler optimizations on the parallel computation in the application itself. The TapirXLA compiler This work studies the effect of enabling optimizations on parallel ML computations by embedding parallelism into an ML compiler and exposing parallel linear-algebra libraries to the compiler. Analysis and optimization of general parallel programs is a long-standing hard problem for compilers (see, for example, [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]). However, ML computations and linear-algebra routines often exhibit structured parallelism, specifically, recursive fork-join parallelism [29,Sec. 3.3], which includes loop parallelism. Previous work on Tapir [16] embeds recursive fork-join parallelism into the IR of a mainstream compiler to enable effective optimization on parallel computations. This work introduces TapirXLA, which integrates the Tapir/LLVM compiler [16] in place of the LLVM component in XLA to enable effective compiler optimizations on parallel ML computations. Tapir supports a simple approach to compiling ML applications into efficient parallel code. Because the low-level Tapir/LLVM optimizer in TapirXLA optimizes parallel computation before inserting calls to a particular parallel runtime, the high-level optimizer need not use heuristics to decide how to partition the computation into parallel tasks. Instead, the highlevel optimizer represents all logical fork-join parallelism in the ML computation using Tapir. The Tapir/LLVM optimizer then decides how to schedule and load-balance the computation among parallel tasks after it performs other optimizations on the parallel tasks. TapirXLA exhibits substantial performance improvements compared to the XLA compiler in TensorFlow. Previous work [16] has shown that Tapir broadly improves the efficiency and parallel speedups of parallel programs, improving many benchmarks programs by factors of 5% to 25%. Although these are substantial performance gains for a compiler, TapirXLA demonstrates significantly larger performance improvements from applying Tapir to compile TensorFlow graphs. By embedding Tapir's representation of parallel computation into XLA and exposing the implementations of parallel linear-algebra kernels in Tapir, on four example networks, TapirXLA produces geometric-mean multiplicative speedups ranging from 30% to 100% across four multicore or manycore CPU architectures. Impact on machine-learning hardware Compiler technology, such as Tapir, for optimizing parallel computation has bearing on how ML frameworks utilize parallel hardware, including general-purpose multicore CPUs, GPUs, and specialized hardware accelerators such as TPUs [7]. Let us briefly survey the hardware currently used for machine learning. Currently, the most widely used hardware platforms for training deep neural networks include NVIDIA GPUs. While open source ML frameworks such as TensorFlow and PyTorch tend to be optimized for the most commonly available hardware, the popularity of GPUs for neural network training has also been facilitated by the availability of highly optimized libraries such as cuBLAS and cuDNN that work on all GPUs ranging from laptops and desktops to high end systems such as the NVIDA DGX-2 that can achieve a peak performance of 2 PetaFLOPS. While training neural networks in a High-Performance Computing center or in the cloud can leverage powerful GPUs, there are many applications where the availability of GPUs may be lacking due to power or form-factor constraints. For example, AI applications at the edge that run on IoT devices or on platforms with limited power capacity require specialized hardware for inference. GPUs also suffer from their limited memory capacity, compared to CPUs. The smaller amount of memory available on a GPU can lead to compromises such the use of small individual data sizes, down-sampling of training images, smaller than desired batch sizes, and changes to a model architecture. For example, a typical workflow may consist of extracting small image tiles from large, high resolution satellite or medical imaging data. Training a model then requires many data transfers between the CPU and the GPU, which is cost that can be minimized but not completely avoided. CPU-based systems can have significantly larger amounts of memory, from several hundred gigabytes to multiple terabytes, which can enable completely different kinds of applications. As a result, some networks, such as recurrent neural networks, have been observed to perform better on CPUs than GPUs [30]. While GPUs remain the predominant hardware platform used for training deep neural networks, CPUs have also shown to be effective on machine-learning tasks. You et al. [31] demonstrated the training of ResNet50 and AlexNet deep convolutional networks on the ImageNet [32] dataset on 2,048 Intel Xeon Platinum 8160 processors. Shen et al. [33] demonstrated how CPUs can outperform GPUs on inference tasks. In addition, CPUs are cheaper, have broad software support and remain more readily available in every data center, cloud platform and in deployed systems. Every GPU system continues to have a host CPU that has the potential to be used for DNN training and inference. In recent years, high core count CPUs such as the Intel Xeon Phi and the Intel Xeon Scalable Processors have been integrated into many of the systems in the Top500 list. For example, some of the largest HPC systems in the world are built on a combination of thousands of CPU cores in conjunction with GPUs. Table I illustrates the availability of CPUs in large scale HPC systems. There are many efforts underway in the academic [35] and commercial space to develop custom processors for training and inference applications [36]. Figure 2 shows an overview of the wide variety of hardware currently available as well as future hardware being developed in the academic and commercial world. The current trend is towards the development of custom hardware for dedicated applications. For example, the Google TPU v1 is an application specific integrated circuit (ASIC) designed primarily for inference applications [7]. Newer generations of the TPU are designed for both inference and training [37] and the Edge TPU is built for AI applications at the edge, where the inference applications are run at the point of data collection. Examples of such applications include autonomous vehicles and unmanned drones. Figure 2 graphs custom AI/ML architectures and capabilities, mapping peak performance vs. power usage. This work on TapirXLA complements ongoing develop- Moreover, Tapir's representation of parallel computation is not tied to a particular hardware architecture or parallel runtime system. Indeed, Tapir has been used for synthesizing efficient hardware for task-parallel programs [38]. By incorporating Tapir, an ML compiler can perform low-level optimizations on parallel computation that can benefit all hardware platforms and parallel runtime systems. Contributions This work explores the effectiveness of using Tapir to embed a parallelism into the compiler of an ML framework to enable low-level optimizations on ML applications. In particular, this paper makes the following contributions: • We introduce TapirXLA, which incorporates Tapir into Ten-sorFlow's XLA compiler. TapirXLA modifies XLA's highlevel optimizer to encode fork-join parallelism in higherlevel TensorFlow operations using Tapir. • We developed parallel linear-algebra libraries whose parallel implementations are encoded using Tapir and exposed to TapirXLA. • We evaluated TapirXLA on four neural networks and four multicore CPU systems. On these neural-network benchmarks, TapirXLA outperforms the XLA compiler in Ten-sorFlow by a geometric mean multiplicative factor of 30% to 100%, across the different CPU architectures. The remainder of the paper is organized as follows. Section II provides background on XLA and the model of recursive fork-join parallelism supported by Tapir. Section III describes how TapirXLA modifies the XLA compiler to incorporate Tapir and leverage its ability to optimize low-level parallel computation. Section IV presents our empirical evaluation of TensorFlow with the prototype TapirXLA compiler. Section V discusses related work, and Section VI offers some concluding remarks. II. BACKGROUND ON XLA AND TAPIR This section presents background on TensorFlow's XLA compiler [13] and the Tapir compiler intermediate representation (IR) [16]. We overview the design of TensorFlow's XLA compiler for CPUs and GPUs, which employs both a highlevel IR and LLVM IR. We review Tapir/LLVM [16], which embeds fork-join parallelism into LLVM IR [15]. TensorFlow's XLA compiler TensorFlow's XLA compiler [13] compiles TensorFlow graphs to run efficiently on different hardware architectures. Figure 1 illustrates the architecture of XLA, specifically, for targeting CPUs and GPUs. (XLA can also target TPUs [7] using a similar compilation pipeline.) As the figure shows, XLA optimizes TensorFlow graphs via transformations through multiple intermediate representations (IRs). XLA's high-level optimizer operates on HLO IR, which represents the TensorFlow graph as a data-flow graph of high-level operations on tensors. The high-level optimizer performs various optimizations on HLO IR, including operator fusion, before transforming the optimized HLO IR to LLVM IR. When it generates LLVM IR, XLA exploits parallelism within the operations in the HLO IR. XLA generates LLVM IR to implement each operation in the HLO IR for the chosen hardware back-end, e.g., using appropriate calls into appropriate parallel CPU or GPU runtime libraries. To generate efficient parallel code, during this translation process, XLA uses heuristics to decide how to subdivide computation among parallel tasks for the target hardware. XLA also inserts calls to parallel libraries for standard linear-algebra routines, such as matrix multiplication or 2D convolution, that have been optimized for the target hardware. By default, when targeting CPUs, XLA inserts calls to the multithreaded Eigen linearalgebra library [8]. Recursive fork-join parallelism and Tapir/LLVM In its implementation of HLO IR operations, XLA exploits (recursive) fork-join parallelism, which allows subroutines to be spawned recursively in parallel and iterations of a parallel loop to execute concurrently. The execution and synchronization of fork-join parallel tasks is managed "under the covers" by a runtime system, such as the Cilk Plus [39] or OpenMP runtime systems [40], [41]. Many computations can be parallelized efficiently using this simple model of parallelism. For some HLO IR operations, XLA parallelizes the operations directly using fork-join parallelism. For operations implemented in optimized linear-algebra libraries, XLA allows the library to use any model of parallelism to implement the operation. Although some multithreaded linear-algebra libraries do not use fork-join parallelism, the linear-algebra algorithms themselves can be parallelized efficiently using fork-join parallelism. Tapir/LLVM [16] embeds fork-join parallelism into the intermediate representation of the LLVM compiler. Tapir adds three instructions -detach, reattach, and sync -to LLVM IR to express fork-join parallel control flow. These instructions allow the existing compiler's analyses and optimizations for serial programs to effectively optimize parallel programs with only minimal modifications. Tapir also enables new compiler optimizations specifically designed for parallel code. The Tapir/LLVM compiler has shown to be effective at optimizing recursive fork-join parallel programs to improve their efficiency and parallel speedup. III. DESIGN OF TAPIRXLA This section describes how TapirXLA makes use of the Tapir/LLVM compiler in place of LLVM in the compilation pipeline, to enable optimizations on parallel machine-learning operations. We describe the simple strategy that TapirXLA employs for compiling and optimizing TensorFlow graphs. We describe the optimized parallel linear-algebra routines TapirXLA uses, whose parallel implementations are exposed to the compiler for optimization. To leverage the Tapir/LLVM compiler's ability to optimize parallel computation, TapirXLA consists of two main changes to the XLA compiler in TensorFlow. First, TapirXLA compiles operations in HLO IR into parallel implementations that use Tapir's instructions for fork-join parallelism. Second, rather than emit opaque calls to the Eigen linear-algebra library, as XLA does by default, TapirXLA emits calls to Tapirbased parallel implementations of these linear-algebra routines and exposes those implementations to Tapir/LLVM. Let us examine these two changes more closely. TapirXLA's compilation strategy By leveraging Tapir's ability to optimize parallel code, TapirXLA supports a straightforward and effective strategy to optimize TensorFlow graphs. The high-level optimizer identifies operations in HLO IR that can be parallelized using forkjoin parallelism. Typically, an operation in HLO IR can be implemented simply using nested parallel loops, e.g., over the dimensions of the operation's tensor input. For operations not handled by an optimized library, TapirXLA emits a parallel implementation of the operator using Tapir's constructs for fork-join parallelism. In particular, these parallelizable HLO IR operations are translated into Tapir's simple representation of a parallel loop [16]. Unlike XLA, TapirXLA does not attempt to optimize the parallel implementations of HLO IR operations, but instead emits a fork-join implementation of each operation that exposes all of its logical parallelism. TapirXLA thus relies on the Tapir/LLVM compiler to optimize these routines efficiently. Tapir/LLVM optimizes these operations using the full suite of LLVM optimization passes, which have been minimally modified to optimize parallel code [16]. Tapir/LLVM also performs optimizations that specifically target Tapir constructs, including parallel-loop strip-mining, loop spawning [16], and small-task serialization, which serializes the execution of a parallel task that performs too little work to overcome scheduling overheads. These Tapir/LLVM optimizations are not specialized to ML applications, but instead are generally applicable to fork-join parallel programs. Finally, Tapir/LLVM lowers Tapir's parallel constructs in the code to a parallel runtime system, meaning that it replaces Tapir instructions with appropriate calls to a parallel runtime library. Exposing parallel linear-algebra routines To uncover additional opportunities to optimize parallel code, TapirXLA incorporates the implementation of the parallel linear-algebra library. Hence, when TapirXLA inserts a call to this library, the Tapir/LLVM optimizer can subsequently optimize the library routine based on the context in which it is called. We ensured that the linear-algebra library routines were parallelized using Tapir to enable such optimization. We developed these parallel linear-algebra routines using Cilk [39] and compiled these routines using Tapir/LLVM [42] to an LLVM bitcode that uses Tapir instructions to encode the fork-join parallel control flow. When compiling a TensorFlow graph, TapirXLA incorporates this bitcode into the LLVM IR it produces for the TensorFlow graph. To facilitate optimization of these routines within the ML computation, these routines are optimized minimally when generating the bitcode file. IV. EVALUATION This section describes the evaluation of TapirXLA in comparison to the original XLA compiler in TensorFlow. We describe the implementation of TapirXLA and the experimental setup to perform a fair comparison TapirXLA against XLA. We evaluated TapirXLA on a variety of multicore and manycore CPUs on the MIT Supercloud system [43], a heterogenous supercomputing system consisting of compute nodes with a variety of multicore and manycore processors. Implementation of TapirXLA We implemented TapirXLA by modifying XLA in Tensor-Flow r1.13, the latest stable release of TensorFlow at the time of writing. We modified XLA to incorporate a version of Tapir/LLVM based on version 7.0 of the LLVM compiler [15]. We used the productivity tool suite integrated with Tapir/L-LVM [42], including the Cilksan nondeterminism detector, to verify the correctness of the implementation. To create a scientific control to compare against this implementation of TapirXLA, we built a version TensorFlow r1.13 with an unmodified XLA compiler from source that incorporates the same version of LLVM and use the same configuration settings. For the different test machines, our builds of TensorFlow support the subset of, AVX, AVX2, and the fused-multiply-add (FMA) operations supported on the target CPU. The TapirXLA implementation uses the Cilk Plus runtime [39] to execute parallel tasks. Although Tapir/LLVM contains prototype back-ends for other parallel runtime systems, the Tapir/LLVM back-end for the Cilk Plus runtime system is the most stable back-end at the time of writing. TapirXLA ensures that all parallel operations it compiles use the same backend parallel runtime system, including parallel operations in linear-algebra library routines. As a result, TapirXLA mitigates performance issues arising from multiple parallel runtime systems competing for processor cores at the same time. Performance comparison of TapirXLA versus XLA We evaluated TapirXLA on training four benchmark neural networks written in TensorFlow: a small convolutional neural network (CNN), two LSTMs, and a recommendation network (NCF) [44]. The two LSTM benchmarks represent LSTM networks for isolated digit recognition (referred to as LSTM1 in Table 3) and continuous speech recognition (LSTM2), based on implementations described in [45]. The recommendation network was obtained from the suite of TensorFlow official models [46] and was evaluated using the MovieLens 1million dataset [47]. All networks were compiled using the xla.compile API [13] to invoke TensorFlow's compiler. We evaluated TapirXLA and XLA on all networks on a variety of multicore and manycore CPUs in the MIT Supercloud system [43], including an Intel Xeon Gold, an Intel Xeon E5, an Intel Xeon Phi, and an AMD Opteron. We followed the TensorFlow guidelines [48] and to set the threads used for intra-and inter-op thread counts equal to the number of processor cores and sockets on the system, respectively. We used taskset to pin worker threads in the parallel runtime systems onto single processor chips (sockets) on the target system. We evaluated other settings of these parameters to verify that these settings yield the best parallel performance for TensorFlow using either XLA or TapirXLA. Figure 3 presents our performance results comparing TapirXLA and XLA on the benchmark ML networks and processor hardware. As the figure shows, TapirXLA consistently outperforms XLA across all networks and hardware systems. The magnitude of the performance improvement varies between networks and systems. TapirXLA yields a geometric mean multiplicative speedup of 2.1 on newer Intel Xeon E5 and Intel Xeon Gold processors, whereas on the older Intel Xeon Phi and AMD Opteron systems, TapirXLA yields a geometric mean multiplicative speedup of 1.3. TapirXLA appears to speed up different networks similarly, as the network that TapirXLA speeds up the most on a given processor differs between processors. V. RELATED WORK This section overviews related work on software technologies to optimize machine-learning applications. By and large, this software technology employs optimized libraries for machine-learning tasks as well as compiler technology to perform domain-specific optimizations on machine-learning applications. We discuss how this work on incorporating Tapir complements these other efforts and can, in principle, be used in conjunction with these technologies. Software libraries, such as Intel's MKL-DNN [9], cuBLAS [10], and cuDNN [11], have been developed that encode highly optimized implementations of common ML operations for different hardware architectures. ML frameworks can use these highly optimized library routines by calling them directly from the language in which the ML application is written. Alternatively, compilers in ML frameworks can insert calls to these libraries when they compile and optimize an ML application. The later approach is compatible with this work to integrate Tapir into an ML compiler. This work also shows that substantial performance improvements can be obtained when a call to a library routine is not opaque, and instead, the parallel routine is encoded in Tapir and exposed to the compiler for optimization. Many ML frameworks employ compiler technology to perform domain-specific optimizations on ML applications. TensorFlow's XLA compiler [12] performs operator fusion and common-subexpression elimination on high-level operations in a TensorFlow graph. PyTorch's Glow compiler [49] represents the ML computation through mutliple levels of IR to perform differentiation and a variety of optimizations, including domain-specific optimizations, memory optimizations, and quantization. DLVM [50] provides a compiler infrastructure to perform domain-specific optimizations on tensor computations, such as algebraic simplification and compute-kernel fusion. The TVM/NNVM compiler stack [51], [52] extends Halide [53] to perform loop optimizations on machine-learning networks. Flux and Zygote [54] employ the compiler technology in Julia to optimize ML applications and perform efficient reverse-mode automatic differentiation. All of these compilers employ LLVM late in their compilation pipelines to perform low-level optimizations and code generation. Hence, for any of these compilers, one can in principle apply Tapir/LLVM in place of LLVM to perform low-level optimizations on parallel computation, as this work does for the XLA compiler. Intel's nGraph [55] provides a unified compiler stack to perform optimizations on ML applications written in a variety of frameworks and to target a variety of hardware back-ends. It remains an open research topic how compiler technology such as Tapir that enables compiler optimizations on low-level parallel operations can be applied within nGraph and what performance benefits Tapir may provide. VI. CONCLUSION To conclude, this section discusses potential impact of Tapir on machine-learning frameworks. In principle, Tapir can make it easy for ML compilers to target new parallel hardware architectures or runtime systems. For example, to target a new architecture or runtime system, XLA's high-level code-generation stage must emit efficient implementations of HLO IR operations for that architecture or runtime system. Tapir/LLVM shifts this burden to developing a new back-end for lowering Tapir's three instructions for parallel control flow [16]. Such a back-end allows any compiler that incorporates Tapir/LLVM to use the new hardware or runtime system. Tapir also allows scheduling and load-balancing decisions for parallel code to be made after optimizations have been performed on that code. One compelling question considers how Tapir can optimize GPU code. ML frameworks often employ GPUs for training for neural networks, and many of the same issues explored in this paper concerning ML compilers and library calls pertain to GPUs as well as CPUs. Tapir seems to be a promising technology to overcome those challenges with GPU code, especially as Tapir's instructions are not tied to a particular hardware architecture or parallel runtime system. Developing a GPU back-end for Tapir remains an open research question. This material is based upon work supported by the Assistant Secretary of Defense for Research and Engineering under Air Force Contract No. (FA8721-05-C-0002 and/or FA8702-15-D-0001). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Assistant Secretary of Defense for Research and Engineering. Fig. 1 . 1Illustration of the high-level design of TensorFlow's XLA compiler. Ovals indicate the representation of the TensorFlow graph at different points in the compilation process. Rectangles denote stages of the compilation pipeline. Selected systems in the Top500 list[34] from November 2018.System CPU Cores Summit IBM Power9 2,397,824 Sierra IBM Power9 1,572,480 Tianhe 2A Intel Xeon E5 2692v2 10,649,600 Piz Daint Intel Xeon E5 2690v3 387,872 Trinity Intel Xeon E5 2698v3 and Xeon Phi 7250 979,072 Table I Fig. 2. Capabilities of current and planned processors for AI training and inference applications. The chart includes existing, commercially available hardware as well as processors being developed by academia. For more details, readers are referred to[36]. The variety of processors architectures available for AI applications makes it extremely challenging to optimize AI/ML frameworks for every possible processor architecture. ments with parallel ML hardware. Embedding parallelism into an ML compiler affects how the compiler can optimize computation for different parallel hardware and target new hardware architectures. TapirXLA's performance results have bearing directly on the utility of CPUs for ML, specifically, in making CPUs more efficient and cost effective for ML applications.Legend Computation Precision Int8 Int16 Float16 Float16 -> Float32 Float32 Float64 Form Factor Chip Card System Computation Type Inference Training Fig. 3. Performance comparison of TensorFlow using vanilla XLA compiler versus TapirXLA. Each column lists TensorFlow's CPU performance on a given benchmark network when using either vanilla XLA or TapirXLA. The performance of the CNN benchmark is measured in overall images per second, while the performance of the other networks is measured in total running time (seconds). The rows labeled "Ratio" give the ratio of performance improvement that TapirXLA exhibits over XLA. For the CNN benchmark, this ratio equals the images-per-second performance of TapirXLA divided by the that of XLA. For all other benchmarks, this ratio equals the running time of XLA divided by that of TapirXLA. All performance values represent the average of 10 runs.Processor Compiler CNN (img/s) LSTM1 (s) LSTM2 (s) NCF (s) Intel Xeon Gold 6252/N, 2.3 GHz, 24 Cores XLA 1765.18 208.41 2567.73 248.42 TapirXLA 3201.43 111.05 1182.14 148.64 Ratio 1.81 1.88 2.17 1.67 Intel Xeon E5-2683 v3, 2.00 GHz, 14 Cores XLA 423.15 443.19 4303.69 240.70 TapirXLA 1792.36 185.83 1764.95 195.36 Ratio 4.23 2.38 2.43 1.23 Intel Xeon Phi 7210, 1.30 GHz, 64 Cores XLA 450.99 949.37 7907.04 793.70 TapirXLA 649.65 635.51 4947.71 703.26 Ratio 1.32 1.49 1.59 1.12 AMD Opteron 6274, 2.20 GHz, 8 Cores XLA 496.17 1204.23 13,982.45 353.63 TapirXLA 669.48 800.96 10,885.32 305.17 Ratio 1.35 1.50 1.28 1.15 ACKNOWLEDGMENTSThe authors acknowledge the MIT Lincoln Laboratory Supercomputing Center for providing HPC resources that have contributed to the research results reported in this paper. 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[ "Improved estimation of cluster mass profiles from the cosmic microwave background", "Improved estimation of cluster mass profiles from the cosmic microwave background" ]	[ "Jaiyul Yoo \nHarvard-Smithsonian Center for Astrophysics\nHarvard University\n60 Garden Street02138CambridgeMA\n", "Matias Zaldarriaga \nHarvard-Smithsonian Center for Astrophysics\nHarvard University\n60 Garden Street02138CambridgeMA\n\nJefferson Physical Laboratory\nHarvard University\n17 Oxford Street02138CambridgeMA\n" ]	[ "Harvard-Smithsonian Center for Astrophysics\nHarvard University\n60 Garden Street02138CambridgeMA", "Harvard-Smithsonian Center for Astrophysics\nHarvard University\n60 Garden Street02138CambridgeMA", "Jefferson Physical Laboratory\nHarvard University\n17 Oxford Street02138CambridgeMA" ]	[]	We develop a new method for reconstructing cluster mass profiles and large-scale structure from the cosmic microwave background (CMB). By analyzing the likelihood of CMB lensing, we analytically prove that standard quadratic estimators for CMB lensing are unbiased and achieve the optimal condition only in the limit of no lensing; they become progressively biased and sub-optimal, when the lensing effect is large, especially for clusters that can be found by ongoing Sunyaev-Zel'dovich surveys. Adopting an alternative approach to the CMB likelihood, we construct a new maximum likelihood estimator that utilizes delensed CMB temperature fields based on an assumed model. We analytically show that this estimator asymptotically approaches the optimal condition as our assumed model is refined, and we numerically show that as we iteratively apply it to CMB maps our estimator quickly converges to the true model with a factor of ten less number of clusters than standard quadratic estimators need. For realistic CMB experiments, we demonstrate the applicability of the maximum likelihood estimator with tests against numerical simulations in the presence of CMB secondary contaminants. With significant improvement on the signal-to-noise ratio, our new maximum likelihood estimator can be used to measure the cluster-mass cross-correlation functions at different redshifts, probing the evolution of dark energy.	10.1103/physrevd.78.083002	[ "https://arxiv.org/pdf/0805.2155v2.pdf" ]	119,114,547	0805.2155	6a3ebd7fd093a4cd4138130a0969509416e26ebc	Improved estimation of cluster mass profiles from the cosmic microwave background 23 Sep 2008 Jaiyul Yoo Harvard-Smithsonian Center for Astrophysics Harvard University 60 Garden Street02138CambridgeMA Matias Zaldarriaga Harvard-Smithsonian Center for Astrophysics Harvard University 60 Garden Street02138CambridgeMA Jefferson Physical Laboratory Harvard University 17 Oxford Street02138CambridgeMA Improved estimation of cluster mass profiles from the cosmic microwave background 23 Sep 2008numbers: 9862Sb9870Vc9880Es We develop a new method for reconstructing cluster mass profiles and large-scale structure from the cosmic microwave background (CMB). By analyzing the likelihood of CMB lensing, we analytically prove that standard quadratic estimators for CMB lensing are unbiased and achieve the optimal condition only in the limit of no lensing; they become progressively biased and sub-optimal, when the lensing effect is large, especially for clusters that can be found by ongoing Sunyaev-Zel'dovich surveys. Adopting an alternative approach to the CMB likelihood, we construct a new maximum likelihood estimator that utilizes delensed CMB temperature fields based on an assumed model. We analytically show that this estimator asymptotically approaches the optimal condition as our assumed model is refined, and we numerically show that as we iteratively apply it to CMB maps our estimator quickly converges to the true model with a factor of ten less number of clusters than standard quadratic estimators need. For realistic CMB experiments, we demonstrate the applicability of the maximum likelihood estimator with tests against numerical simulations in the presence of CMB secondary contaminants. With significant improvement on the signal-to-noise ratio, our new maximum likelihood estimator can be used to measure the cluster-mass cross-correlation functions at different redshifts, probing the evolution of dark energy. I. INTRODUCTION As the most distant observable sources, the cosmic microwave background (CMB) anisotropies provide a unique channel to probe the universe after the cosmological recombination epoch. In particular, weak gravitational lensing of the CMB can be used to map the matter distribution in the universe at higher redshift than weak lensing of faint background galaxies can ever achieve. Recent work [1,2,3,4] has focused on measuring the lensing signature in the CMB by large-scale structure between the last scattering surface and the present universe, but relatively little attention has been paid to weak lensing of the CMB by clusters of galaxies. The abundance of massive clusters is exponentially sensitive to the growth of the underlying matter distribution, and hence it has been recognized as a powerful probe of the evolution of dark energy (e.g., [5]). However, the constraining power as a cosmological probe can be only realized, if the cluster masses are accurately measured. To achieve this goal, many cluster surveys are designed to detect massive clusters and measure their mass using the Sunyaev-Zel'dovich (SZ) effect, and some of the planned surveys are already operational using the South Pole Telescope (SPT), and the Atacama Cosmology Telescope (ACT). Weak lensing of the CMB can be applied to the same clusters found in the SZ surveys without additional observations, providing independent measurements of their mass. Furthermore, the CMB provides the highest redshift source plane with precision measurements of its distance, which can be combined with galaxy weak lensing measurements of the same lensing clusters to obtain angular diameter distance ratio estimates that are independent of the mass distribution, substantially increasing the leverage to constrain cosmological parameters [6]. * Electronic address: [email protected] Gravitational lensing by clusters imprints a unique signature in the CMB anisotropies. On arcminute scales, the primordial CMB anisotropies decay exponentially due to the photon diffusion from the baryon-photon fluid around the recombination epoch [7], and to a good approximation the CMB can be considered as a pure temperature gradient on small scales. Based on this approximation, Seljak and Zaldarriaga [8] showed that clusters create dipole-like wiggles in the CMB temperature by remapping the otherwise smooth gradient field, and this unique feature can be used to isolate the lensing effect by clusters and to reconstruct the deflection angle, once the temperature gradient is separately measured on large scales. Vale, Amblard, and White [9] and Holder and Kosowsky [10] used N -body simulations to model realistic lensing clusters, and they found that the mass reconstruction for individual clusters is compromised, since it is hard to measure the large-scale temperature gradient accurately and secondary anisotropies in the CMB can partially mimic the lensing signature. However, it has been realized that one can apply the same technique developed for reconstructing large-scale structure to clusters of galaxies, measuring the statistical properties of a sample of clusters. Unlike galaxy weak lensing, CMB anisotropies have no characteristic shape, even statistically, from which the deviation is a measure of the lensing effect. Gravitational lensing, however, gives rise to a deviation of the two-point correlation function of the CMB temperature anisotropies from statistical isotropy. The standard technique is to construct a lensing estimator that is quadratic in observed temperature anisotropies, measuring the correlation between different Fourier modes, which is directly proportional to the lensing effect [11]. This method is easy to implement in analyzing real data compared to the full likelihood analysis [12] and no separate measurement is required to obtain the large-scale temperature gradient. However, Maturi et al. [13] showed that standard quadratic estimators need a modification to be an unbiased es-timator in a region around massive clusters. Hu, DeDeo, and Vale [14] quantitatively demonstrated that standard quadratic estimators based on the linear approximation ignore higherorder terms in the lensing effect that coherently contribute to the lensing reconstruction, and hence the reconstruction is biased low when the lensing effect is large. Furthermore, they proposed modified quadratic estimators that remove the higher-order terms in violation of the linear approximation by low-pass filtering observed temperature fields, and they showed that the modified quadratic estimators recover cluster mass profiles with no significant bias. However, the cutoff scale of the low-pass filter is somewhat arbitrary and it depends on the lensing effect, which we want to measure with the estimators. Here we develop a new maximum likelihood estimator for reconstructing cluster mass profiles and large-scale structure by analyzing the likelihood of CMB lensing. Our approach is similar in making full use of the likelihood information to one advocated by Hirata and Seljak [12]. While they derive an analytic expression for a maximum likelihood estimator, it is impractical to apply to a realistic problem, because the solution is too general and computationally expensive. However, our maximum likelihood estimator is different from theirs and it is easy to use in practice, because we adopt an alternative approach to setting up the likelihood: it takes a similar form of standard quadratic estimators and it approaches the optimal condition as it is iteratively applied to CMB maps. Furthermore, we show that our maximum likelihood estimator can reconstruct cluster mass profiles with a factor of ten less number of clusters than standard or modified quadratic estimators need. The rest of the paper is organized as follows. We first derive a quadratic estimator, accounting for the telescope beam effect in Sec. II. This consideration makes a difference compared to the usual practice in the literature, where quadratic estimators are often applied to beam deconvolved CMB maps. In Sec. III we analytically show that the quadratic estimators are unbiased and optimal only when the lensing effect vanishes, and why the modified quadratic estimators outperform the standard quadratic estimators when the lensing effect is large. Based on this observation, we construct a delensed temperature field and derive a maximum likelihood estimator using the delensed temperature field. We demonstrate its applicability to realistic CMB experiments using numerical simulations in Sec. IV. We discuss the impact of the telescope beam and instrumental noise in the delensing process and we conclude in Sec. V. In this paper we will only consider lensing estimators based on CMB temperature anisotropies, since the planned surveys are not yet sensitive to CMB polarization anisotropies on arcminute scales. However, it is straightforward to extend our formalism to lensing estimators based on CMB polarization anisotropies. Throughout the paper we assume a flat ΛCDM universe with the matter density parameter Ω m h 2 = 0.127, the baryon density parameter Ω b h 2 = 0.0222, the Hubble constant h = 0.73, the spectral index n s = 0.95, the optical depth to the last scattering surface τ = 0.09, and the primordial curvature perturbation amplitude A s = 2.5 × 10 −9 (corresponding to the matter power spectrum normalization σ 8 = 0.75), consistent with the recent cosmological parameter estimation (e.g., [15,16,17]) II. FORMALISM Here we describe our notations for weak lensing of the CMB and derive a quadratic estimator for CMB lensing reconstruction. A. Weak Lensing of the CMB Gravitational lensing deflects light rays as they propagate through fluctuating gravitational fields, and the deflection vector d(n) at the angular positionn on the sky is related to the line-of-sight projection of the gravitational potential ψ as d(n) =∇φ(n), where the projected potential is φ(n) = −2 D⋆ 0 dD D ⋆ − D DD ⋆ ψ(Dn, D),(1) ∇ is the derivative with respect ton, and D ⋆ is the comoving angular diameter distance to the last scattering surface. Here we have assumed a flat universe and c ≡ 1. The projected potential is further related to the convergence κ as∇ 2 φ(n) = −2κ(n). Since gravitational lensing conserves the surface brightness of diffuse backgrounds, the lensed temperature fieldT (n) of the CMB is simply the intrinsic (unlensed) temperature field T (n) remapped by the deflection vector, T (n) = T n +∇φ(n) . (2) We will use notation with (or without) tilde to represent lensed (or unlensed) quantities. Note that we mainly work in the Rayleigh-Jeans tail and express the surface brightness in terms of temperature. In a sufficiently small patch of the sky, it significantly simplifies the manipulations to work in Fourier space [see 18,19,20, for all-sky formalism]. In Fourier space the lensed temperature is T l = d 2nT (n) e −il·n (3) = T l − d 2 l ′ (2π) 2 [(l − l ′ ) · l ′ ] T l ′ φ l−l ′ + · · · , where we Taylor expandedT l to the first order in φ l . We kept the same notation for Fourier components, while the functional dependence is indicated as a subscript (e.g., T (n) and T l are Fourier counterparts). The rms deflection angle d · d 1/2 is a few arcminutes and the deflection power peaks at a few degree scale, comparable to the angular sizes of clusters. However, the large-scale deflection field is coherent over the scales of the temperature fluctuations, resulting in an unobservable overall shift of the temperature field [21], and the linear approximation remains valid. In Sec. III we discuss the limitation of this approximation when the lensing effect is large in a region around massive clusters. Since the intrinsic CMB is Gaussian and isotropic, the statistical properties of the temperature field can be completely described by the power spectrum C l , T l1 T * l2 = (2π) 2 δ(l 1 − l 2 ) C l1 ,(4) where the asterisk represents complex conjugation and δ is the Dirac delta function. Analogously, we define the projected potential power spectrum C φφ l . Thus the deflection and the convergence power spectra are C dd l = l 2 C φφ l and C κκ l = l 4 C φφ l /4, respectively. Note that C φφ l can always be defined in this way, though it may be an incomplete description of the statistical properties of the projected potential when φ l is non-Gaussian. Finally, the power spectrum of the lensed temperature field is C l = 1 − l 2 R C l + d 2 l ′ (2π) 2 [(l − l ′ ) · l ′ ] 2 C l−l ′ C φφ l ′ ,(5) where R ≡ (1/4π) d ln l l 4 C φφ l is the half of the rms deflection angle [18,22]. In practice, the observed temperature field has two additional contributions: detector noise independent of the signal, and telescope beam convolving the signals from different patches of the sky. We assume that the detector noise is white, so that the noise power spectrum is constant, C N l ≡ ∆ 2 T = σ 2 pix 4πf sky N pix ,(6) where σ pix is the rms error in each pixel of the detector in units of µK, f sky is the fraction of the survey area on the sky, and N pix is the total number of detector pixels [23]. Convolution is simply a multiplication in Fourier space, and the beam factor for a simple Gaussian beam we consider is B l = exp − 1 2 l 2 σ 2 b . The beam width σ b is related to the full-width half-maximum (FWHM) as σ b = θ FWHM / √ 8 ln 2. The observed temperature field and its power spectrum are thenT obs l =T l e − 1 2 l 2 σ 2 b + T N l ,(7) C obs l =C l e −l 2 σ 2 b + C N l .(8) In reality, one needs to consider other contributions tõ T obs , such as residual foregrounds, point radio sources, and CMB secondary anisotropies. We will only consider secondary contributions in Sec. IV C. B. Quadratic Estimator Here we consider a convergence estimatorκ(n) that is quadratic in the observed temperature field, accounting for telescope beam and detector noise. 1 We require that the estimator be unbiased when averaged over an ensemble of CMB maps, κ(n) = κ(n). With these conditions, the estimator takes the general form in Fourier spacê κ L = N L 2 d 2 l 1 (2π) 2 F (l 1 , l 2 )T obs l1T obs l2 ,(9) where l 2 = L − l 1 and N L is a normalization coefficient, which only depends on L = \|L\|. The functional form of F (l 1 , l 2 ) can be obtained by minimizing the variance ofκ L and imposing the normalization condition F (l 1 , l 2 ) = [L · l 1 C l1 + L · l 2 C l2 ] 2C obs l1C obs l2 e − 1 2 l 2 1 σ 2 b e − 1 2 l 2 2 σ 2 b ,(10) and the normalization coefficient is 1 N L = 1 L 2 d 2 l 1 (2π) 2 [L · l 1 C l1 + L · l 2 C l2 ] 2 2C obs l1C obs l2 e −l 2 1 σ 2 b e −l 2 2 σ 2 b .(11) Finally, the variance of the estimator is κ Lκ * L ′ = (2π) 2 δ(L − L ′ )(C κκ L + N κκ L ),(12) where N κκ L = L 2 N L /4 is the noise power spectrum ofκ L . One can think of C κκ L /N κκ L as a signal-to-noise ratio, and the reconstruction becomes difficult at the angular scale L, where C κκ L ≃ N κκ L . Given experimental specifications, the noise power spectrum N κκ L , as a function of the intrinsic CMB power spectrum C L , becomes smallest, when there exists substantial power in C L at the scale of interest, with its shape deviating from the scale-invariance (L 2 C L =constant) [24]. 1 We will use quantities with hat to represent estimators of the quantities without hat, e.g., a convergence estimator is denoted asκ and a true convergence field is denoted as κ. However, this notational convention should not be confused with that used for temperature fields: T ,T ,T obs , andT represent the intrinsic (unlensed), the lensed [Eq. Our estimator recovers the general form of the standard quadratic estimators as σ b → 0, and N L corresponds to the noise power spectrum of a deflection estimatord L = 2Lκ L /L 2 used in the literature [11]. The estimator can be decomposed as two Wiener-filtered temperature functions in real space, which essentially correlates the gradient of the lensed temperature field with the unlensed temperature field to isolate the lensing effect, G(n) = d 2 l (2π) 2 ilT obs l C l C obs l e − 1 2 l 2 σ 2 b +il·n (13) W (n) = d 2 l (2π) 2T obs l 1 C obs l e − 1 2 l 2 σ 2 b +il·n ,(14) and the convergence estimator can be expressed in terms of G(n) and W (n) aŝ κ L = − N L 2 iL · d 2n G(n)W (n) e −iL·n .(15) This approach of using the two Wiener-filtered functions is more convenient for computingκ L by using Fast Fourier Transform (FFT) routines than by directly computing Eq. (9). Furthermore, it is more physically intuitive than the general derivation, though the latter has clear advantage in its transparency and understanding the uniqueness of the functional form F (l 1 , l 2 ). A modified quadratic estimator can be constructed by removing the signals in Eq. (13) at l ≥ l cut , while Eq. (14) remains unchanged. To better understand how quadratic estimators operate, we Fourier transform and rearrange Eq. (15) as 1 2∇ · [G(n)W (n)] = d 2 L (2π) 2 −κ L N L e iL·n (16) = d 2m H(m −n)κ(m). The divergence of the two Wiener-Filtered functions is a convolution of the convergence estimateκ(n) and the filter Figure 1 plots the filter H(θ) as a function of separation θ = \|n\| for experiments with σ pix = 5µK and 10µK, to which we apply quadratic estimators in Sec. IV. The filter peaks at the center and its width is ≃ 3 ′ , roughly set by the scale that the intrinsic CMB and detector noise power spectra become comparable. While the filter is highly oscillating at its tail, it is negligible at θ ≥ 10 ′ due to the large weight near the center. A factor of two change in σ pix has little impact on the width of the filter, because the crossing scale is already at the CMB damping tail. H(n) = d 2 L (2π) 2 −1 N L e iL·n .(17) III. MAXIMUM LIKELIHOOD ESTIMATOR In this section, we analyze the likelihood of CMB lensing by singular isothermal clusters. We first derive a quadratic estimator for singular isothermal clusters and compare the estimator to the optimal estimator from the likelihood. With the simple singular isothermal model, our analysis will be carried out analytically, showing that (1) the standard quadratic estimators are unbiased and optimal in the limit of no lensing, (2) they progressively become biased and sub-optimal when the lensing effect increases, and (3) why the modified quadratic estimators perform better than the standard quadratic estimators. Finally, we develop a unbiased maximum likelihood estimator to reconstruct cluster mass profiles as well as large-scale structure. We demonstrate its applicability to CMB experiments with tests against numerical simulations using more realistic cluster models in Sec. IV. A. Quadratic Estimator for a Singular Isothermal Cluster A singular isothermal cluster has a density profile ρ(r) ∝ r −2 and its enclosed mass increases with r, which requires truncation at some radius to be a viable model for real clusters. However, this model has advantage in its simplicity: its properties are described by one parameter, Einstein radius θ E = 4πσ 2 D ⋆ − D L D ⋆ ,(18) where σ is one-dimensional velocity dispersion of a cluster and D L is the comoving angular diameter distance to the lensing cluster. CMB lensing has a well-defined single plane of the source redshift and the comoving angular diameter distance to the last scattering surface D ⋆ = 14.12 Gpc is now measured with less than 1% uncertainty [17]. The convergence is κ(n) = θ E /2θ and the deflection vector is d(n) = −θ En given the angular separation θ = \|n\| from the origin in a cluster centric coordinate. When a virial radius R vir is defined as the radius inside which the mean density is 200 times the cosmic mean matter density, a singular isothermal cluster of mass M = 10 14 h −1 M ⊙ within the virial radius at z L = 1 has an Einstein radius θ E = 8. ′′ 0 and a velocity dispersion σ = 2.0 × 10 −3 (= 610 km s −1 ), and they scale as θ E ∝ M 2/3 and σ ∝ M 1/3 . A quadratic estimatorθ QE E for singular isothermal clusters can be readily derived using the method described in Sec. II B, but here we take an idealized approach for the purpose of comparison, where we assume σ pix = σ b = 0. Under the condition that the estimator is unbiased θ QE E = θ E and it has the minimum variance, the quadratic estimator iŝ θ QE E = 1 F d 2 l 1 (2π) 2 d 2 l 2 (2π) 2 (19) ×T l1Tl2 C l1Cl2 π(l 1 C l1 + l 2 C l2 ) · (l 1 + l 2 ) \|l 1 + l 2 \| 3 , with the normalization coefficient F = d 2 l 1 (2π) 2 d 2 l 2 (2π) 2 (20) × 2π 2 C l1Cl2 (l 1 C l1 + l 2 C l2 ) · (l 1 + l 2 ) \|l 1 + l 2 \| 3 2 . The variance of the estimator is (θ QE E − θ E )(θ QE E − θ E ) = 1/F . Here we Taylor expandedT l and kept terms only to the first order in θ E in derivingθ QE E . B. Relation to the Optimal Estimator The likelihood function P (T \|θ m E ) simply represents the probability that a singular isothermal model with θ m E can have the lensed temperature fieldT (n). Since the intrinsic CMB follows a Gaussian distribution and gravitational lensing only remaps the intrinsic CMB, the distribution ofT (n) is also Gaussian and its statistical properties are fully described by the covariance matrix ofT (n) C(n,n ′ ) = T (n)T (n ′ ) = d 2 l (2π) 2C l e il·(n−n ′ ) . (21) For convenience, we take a negative logarithm of P (T \|θ m E ) and call it likelihood, L(T \|θ m E ) ≡ − ln P (T \|θ m E ) (22) = 1 2T (n)C −1 (n,n ′ \|θ m E )T (n ′ ) + 1 2 ln detC(θ m E ), where the summation overn andn ′ is implicitly assumed and hereafter we will suppress the angular dependence for simplicity. In general, the likelihood is a functional with its argument of a scalar field, such as κ(n) or φ(n). However, in our case it reduces to a function with its argument of a scalar θ m E , substantially simplifying the manipulation. We take a derivative of L with respect to θ m E , ∂L ∂θ m E = − 1 2TC −1 ∂C ∂θ m EC −1T (23) = − d 2 l 1 (2π) 2 d 2 l 2 (2π) 2T l1Tl2 C l1Cl2 π(l 1 C l1 + l 2 C l2 ) · (l 1 + l 2 ) \|l 1 + l 2 \| 3 , where we computed the derivative to the first order in θ m E . Since gravitational lensing only redistributes the intrinsic CMB, the last term (log determinant) in Eq. (22) is independent of θ m E and hence the derivative with respect to θ m E vanishes in Eq. (23). However, in the presence of non-white instrumental noise, and/or other secondary contaminants, the derivative acquires a nonzero value but it is in general negligible compared to the quadratic term in Eq. (23). We will neglect this effect in the remainder of this paper. In the presence of significant contaminants from secondaries, the assumption that the likelihood function is Gaussian becomes invalid before the log determinant term becomes non-negligible. With the derivative of L, we can compute the Fisher information matrix F = ∂ 2 L ∂θ m2 E = ∂L ∂θ m E ∂L ∂θ m E(24) where for the second equality we used the normalization condition of the likelihood function 1 = dT P (T \|θ m E ) = dT e −L . Within the Gaussian approximation, F can be evaluated at any value of θ m E . Note that F is identical to the normalization coefficient in Eq. (20). In statistical parameter estimation, there exists a powerful theorem, known as the Cramér-Rao inequality that error bars in a parameter estimation have a definite lower bound σ(θ m E ) ≥ F −1/2 set by the Fisher matrix. Moreover, this theorem provides a necessary and sufficient condition for an estimator to saturate the Cramér-Rao inequality, i.e., to be an optimal estimatorθ opt E [25], ∂L ∂θ m E = F (θ m E −θ opt E ).(25) Now it is apparent that only in the limit of no lensing (the true Einstein radius θ E = θ m E = 0) does the quadratic estimatorθ QE E become an optimal estimatorθ opt E with the smallest variance attainable from the data. Conversely,θ QE E becomes progressively biased and sub-optimal as the lensing effect increases. This can be also understood by the validity of the linear approximation: since the quadratic estimator is constructed to be unbiased and to minimize the variance wheñ T l is expanded to the linear order in φ l , it is natural to expect that this condition breaks down when higher-order terms in φ l become dominant over the linear order term. The modified quadratic estimator, on the other hand, removes the angular modes of the signals at l ≥ l cut by explicitly setting the integrand zero in Eq. (19), where the linear approximation breaks down, and this process helps suppress the contributions from the higher-order terms in φ l because the higher-order terms are related to multiple integrals over the modes that are suppressed most. Precisely for this reason could the modified quadratic estimators be more robust than the standard quadratic estimators even when the lensing effect is large. However, the modified quadratic estimator requires a rather arbitrary choice of the cutoff scale l cut , which depends on the lensing effect, though it may be possible to calibrate against simulations [14]. Furthermore, the removal of the lensing signals at l ≥ l cut inevitably results in lower signal-to-noise ratio, making the reconstruction noisier. We discuss this issue with numerical simulations in Sec. IV B. C. Maximum Likelihood Estimator Given the Gaussian probability distribution of the CMB, the likelihood retains all the information of the observed data. Even when there exists no optimal estimator, one can always find an estimator, if not analytically, that maximizes the likelihood: the maximum likelihood estimatorθ ML E is the solution of ∂L ∂θ m E θ m E =θ ML E = 0.(26) However, this equation is highly non-linear in general and requires approximations to be solved even numerically. Equations (25) and (26) show that an optimal estimator is always the maximum likelihood estimator. However, note that while the converse is not true in general, the maximum likelihood estimator asymptotically approaches to the optimal condition. Having understood that the quadratic estimator becomes an optimal (and maximum likelihood) estimator in the limit of no lensing in Sec. III B, we present an alternative approach to modeling the likelihood and derive a new maximum likelihood estimator for singular isothermal clusters. We then gen-eralize this approach to clusters with arbitrary mass distributions. Consider a model with θ m E and its deflection field d m (n) = −θ m En . We construct a delensed temperature fieldT (n) by delensing the observedT (n) with d m (n), andT (n) is related to the intrinsic temperature field T (n) aŝ T (n) ≡T (n − d m ) (27) = T (n − d m + d) = T [(1 + ∆)n] , with ∆ = θ m E − θ E . Now we can write the likelihood in terms of the delensed temperature fieldT (n) L(T \|θ m E ) = 1 2T (θ m E ) C −1T (θ m E ) + 1 2 ln det C,(28) where we emphasized the dependence ofT (n) on θ m E , and C is the covariance matrix of T (n). Taking a derivative of L with respect to θ m E gives ∂L ∂θ m E = 1 2 ∂T ∂θ m E C −1T +T C −1 ∂T ∂θ m E (29) = − d 2 l 1 (2π) 2 d 2 l 2 (2π) 2 T l1 T l2 C l1 C l2 π(l 1 C l1 + l 2 C l2 ) · (l 1 + l 2 ) \|l 1 + l 2 \| 3 . The second equality is obtained by evaluating the derivative at ∆ = 0. Assuming that our initial model with θ ⋆ E is a good approximation to the true model with θ E (∆ ⋆ = θ ⋆ E −θ E ≃ 0), the likelihood can be expanded around ∆ ⋆ L = L ⋆ + ∂L ∂θ m E ⋆ (∆ − ∆ ⋆ )(30)+ 1 convergence ofθ ML E depends on the goodness of θ ⋆ E to θ E . Eq. (31) still involves computationally intensive evaluations of the second derivative, or the curvature matrix. We further simplifyθ ML E by replacing the curvature matrix with its ensemble average, Fisher matrix F = d 2 l 1 (2π) 2 d 2 l 2 (2π) 2 (32) × 2π 2 C l1 C l2 (l 1 C l1 + l 2 C l2 ) · (l 1 + l 2 ) \|l 1 + l 2 \| 3 2 , and by evaluating the derivatives at ∆ ⋆ = 0. Finally, our new maximum likelihood estimator iŝ θ ML E = θ ⋆ E + 1 F d 2 l 1 (2π) 2 d 2 l 2 (2π) 2 (33) ×T l1Tl2 C l1 C l2 π(l 1 C l1 + l 2 C l2 ) · (l 1 + l 2 ) \|l 1 + l 2 \| 3 . This equation is readily recognizable as the standard quadratic estimator in Eq. (19), exceptC l andT l replaced with C l andT l . The resemblance should not be surprising, and in hindsight one could have expected this outcome given the result in Sec. III B: the quadratic estimator becomes optimal when the lensing effect is vanishingly small; as we delensT (n) well enough thatT (n) is close to T (n), the residual lensing effect inT (n) is substantially reduced and therefore the maximum likelihood estimator takes the form of the quadratic estimator, returning diminishing change of the second term in Eq. (33), i.e.,θ ML E ≃ θ ⋆ E ≃ θ E . We want to emphasize that this new estimator in the form of quadratic estimators is derived by iteratively solving for the maximum likelihood in Eq. (26) and updating the initial model θ ⋆ E as in the standard Newton-Raphson method, i.e., it is a maximum likelihood estimator and is independent of the linear approximation, to which the validity of the standard quadratic estimator is limited. One may be concerned about replacing the curvature matrix with the Fisher matrix in Eq. (33) and obtaining a solution quadratic inT l instead of a solution rational inT l (quadratic inT l both in numerator and in denominator). However, both procedures guarantee that the correct solution of Eq. (26) is iteratively found reaching the same peak of the likelihood, while the error estimation of parameters is approximated by using the Fisher matrix, rather than the full curvature matrix. In Sec. IV we demonstrate that this is a good approximation and the initial model converges quickly to the true model. Given the nomenclature of the existing quadratic estimators, now let us call our new maximum likelihood estimator an improved quadratic estimator. 2 2 However, note that since our new estimator takes the result of the previous In practice we can use the standard quadratic estimators to obtain an initial model and then proceed with our improved quadratic estimator to refine the solution, even when the lensing effect is large. In general, the reconstruction of cluster mass profiles is too noisy to provide a good initial model. However, we can adopt an initial model for clusters from other observations (e.g., galaxy weak lensing and X-ray measurement) or theoretical expectations (e.g., Navarro-Frenk-White (NFW) profiles [26]). As opposed to the modified quadratic estimators, there is no arbitrary choice of l cut in our method. The toy model developed here can be readily generalized and our improved quadratic estimator can be used to reconstruct mass profiles of realistic clusters and large-scale structure. However, in the presence of the telescope beam and detector noise, the delensing process becomes non-optimal because it does not commute with the beam smoothing. In the absence of detector noise, one can deconvolve the beam factor, delens the temperature field, and convolve the beam again, which can solve the problem of non-commutativity. However, in the presence of detector noise, the beam deconvolved noise can produce unwanted power on all scales when it is delensed due to the non-white power below the beam scale. One can in principle filter out or remove these small scales before delensing to mitigate the problem [14], which however introduces additional ad hoc scale to the problem. The impact of telescope beam and detector noise is small in practice for surveys like SPT (∆ T ≃ 6µK-arcmin) and ACT (∆ T ≃ 10µK-arcmin) as we numerically demonstrate in Sec. IV. We explicitly show in Appendix A that the delensing process suppresses the beam effect by a factor of the average magnification by clusters, since it corresponds to a mapping from the image plane to the source plane. Non-white instrumental noise and boundary effect of detectors may affect the delensing process. However, compared to the survey area, the lensing signals are limited to a relatively small region around clusters where none of those effect is expected to be significant. IV. RECONSTRUCTING CLUSTER MASS PROFILES Here we use numerical simulations of the CMB and cluster lensing potential to demonstrate the applicability of our improved quadratic estimator to CMB experiments. First, we adopt a more realistic model for massive clusters and investigate the dependence of our improved quadratic estimator on assumed initial models in Sec. IV A. Then we reconstruct cluster mass profiles using the standard, modified, and improved quadratic estimators, and we compare their performance in Sec. IV B. Finally, we discuss the effects of contaminants and investigate the robustness of our improved quadratic iteration as an initial model, another iteration makes use ofT (n) that is constructed by using the initial model and this initial model is also a function ofT (n) in the previous iteration, which makes the estimator a rational function of temperature, instead of a quadratic function. Therefore, it is technically incorrect to call it a quadratic estimator. estimators in the presence of the Sunyaev-Zel'dovich (SZ) effects. A. Improved Quadratic Estimator A singular isothermal model used in Sec. III is useful in developing an analytic solution of the likelihood approach. However, it is rather an academic model than a realistic model for massive clusters. Recent numerical simulations show that there exist a universal mass profile for dark matter halos, NFW profiles [26] ρ(r) = ρ s r/r s (1 + r/r s ) 2 . The scale radius r s is described by the concentration parameter c = R vir /r s and the normalization coefficient ρ s is related to the mass of clusters M = 4πr 3 s ρ s [ln(1 + c) − c/(1 + c)]. We now use NFW profiles to model massive clusters. The convergence field κ(n) of NFW profiles can be obtained by the ratio of the projected mass density Σ(r) to the critical surface density Σ crit of the lensing cluster at z L , κ θ = r D L = Σ(r) Σ crit = 2 r s ρ s Σ crit P r r s (1 + z L ) 2 ,(35) where the functional form P (x) of the projected density is [27,28] P (x) = 1 x 2 − 1 1 − 2 √ 1 − x 2 tanh −1 1 − x 1 + x , (x < 1) = 1 3 , (x = 1) (36) = 1 x 2 − 1 1 − 2 √ x 2 − 1 tan −1 x − 1 x + 1 , (x > 1) and the critical surface density Σ −1 crit = 4πGD L (D ⋆ − D L )/D ⋆ (1 + z L ) is only a function of z L given the precise measurement of D ⋆ . Note that the convergence field κ of NFW profiles depend only on the angular separation θ = \|n\| due to spherical symmetry. The redshift dependence in Eq. (35) arises due to our use of comoving coordinates, reflecting higher densities of the universe at z L > 0. We use CMBFAST [29] to generate CMB temperature maps of 200 ′ × 200 ′ (1000 × 1000 pixels) and set the pixel scale 0. ′ 2 smaller than detector beam sizes. Given a cluster mass M and redshift z L , we first compute the convergence field κ(n) using Eq. (35). The lensing potential φ(n) and its deflection vector d(n) of the cluster are then computed in Fourier space, where their relations to κ(n) become a simple multiplication. The lensed temperature fieldT (n) is computed by displacing the intrinsic temperature field T (n) with d(n) according to Eq. (2). Finally, we smoothT (n) with a telescope beam and add detector noises to obtainT obs (n). Standard quadratic estimators can be used to reconstruct a convergence fieldκ(n) by using Eqs. (13), (14), and (15) withT obs (n), and so can modified quadratic estimators with a choice of l cut , beyond which the integrand in Eq. (13) is set zero. Similarly, our new estimation process begins with finding a solutionŝ to the delensing equationŝ =n +∇φ m (n) given the lensing potential φ m (n) of an assumed initial model. We then construct a delensed temperature fieldT (ŝ) =T obs (n) and use the same equations withT obs (n) replaced byT (ŝ) to reconstructκ L . Imposing a consistency condition between the assumed model and the estimation result can provide a criterion for the iteration convergence of our improved quadratic estimators. ACT and SPT will find ∼ 2 × 10 4 massive clusters mainly by the spectral distortion of the CMB arising from the inverse Compton scattering of hot electrons in clusters, so called the SZ effect [30,31], with roughly redshift-independent threshold mass M ≥ 2 × 10 14 h −1 M ⊙ . To test our improved quadratic estimators, we consider a typical cluster of M = 5 × 10 14 h −1 M ⊙ and c = 3. Figure 2 shows the reconstructedκ(n) of a massive cluster at z L = 1 in an ideal experiment with ∆ T = 0. Here we simply adopt a NFW profile with fixed concentration c = 3 for our initial model and allow mass M init of the model to vary. Even with fixed concentration, r s changes as a function of M init , and hence our assumption allows for changes in the shape as well as the scaling of initial mass models. However, note that while we use this parametrized model of clusters, our reconstruction is general and non-parametric, such that we recover 2-D structure of κ(n) at each pixel rather than obtain model parameters M and c (see [32,33] for reconstructing a parametrized cluster model). We assume that the cluster center is known from other observations with uncertainty less than our pixel scale 0. ′ 2. The upper panels show the reconstructedκ(n) from our improved quadratic estimator using an initial model of M init = 5 × 10 14 h −1 M ⊙ (left) and 1 × 10 14 h −1 M ⊙ (right), and the bottom panels show the residual after the true κ(n) is subtracted from the top panels. With the perfect initial model in the left panels, the delensed temperature fieldT (n) is identical to the intrinsic T (n), and our improved quadratic estimator returns no change on average to the initial model (bottom). However, there exist random noises inκ(n) over the map, arising from the fluctuations of the intrinsic temperature gradient, though they are evidently small and discernible from the massive cluster (top). In the right panels,T (n) is delensed with the imperfect initial model, so thatT (n) is not identical to T (n) but the lensing effect is significantly reduced. In this regime, quadratic estimators become asymptotically optimal and reconstruct κ(n) unbiased. The top panel exhibits small anisotropy and some residual remains in the bottom panel. In a single patchy of the sky, the CMB anisotropy has a gradient direction and gravitational lensing of the CMB makes no difference orthogonal to the gradient direction, in which reconstruction is completely degenerate, resulting in the asymmetry inκ(n). However, since the CMB has no preferred direction, this obstacle can be overcome by stacking clusters in different patches of the sky. In practice, this stacking process provides the average κ(n) of the clusters, or the cluster-mass cross-correlation function [14]. Hereafter we assume that identical clusters are stacked for simplicity. We now quantify the ability to reconstruct κ(n) with varying accuracy of assumed models. Figure 3 plots the reconstructed cluster mass profiles from 500 clusters (thin solid). The mass profiles are obtained by averaging reconstructed κ(n) over the annulus of each cluster, and the uncertainties in the mean mass profile are shown as shaded regions. Figure 3a shows that our improved quadratic estimator is unbiased when our assumed model is perfect; it recovers the true model (thick solid) with no bias. If an assumed initial model is significantly different from the true model in Fig. 3b, the improved quadratic estimator suffers from the same problem that the standard quadratic estimators have, and the reconstruction is again biased low when the residual lensing effect is large. However, the reconstructedκ(n) is inconsistent with our assumed model (dashed), implying that it has not converged to the correct solution. In Fig. 3c we take the reconstructedκ(n) as a new initial model and apply our improved quadratic estimator to the same clusters. The reconstructedκ(n) is now close to the true κ(n), but still inconsistent with the assumed model. We iterate once more in Fig. 3d and the reconstructed κ(n) is identical to the true κ(n). One more iteration results in no further change and the estimate is consistent with the assumed and also the true models, indicating the convergence of our estimates. Even with the imperfect initial model, the reconstruction quickly converges to the true κ(n) and no significant bias develops even beyond R vir (dotted). When the reconstructed κ(n) is inconsistent with the assumed model, one can in principle adopt a different initial model for a faster convergence before applying the estimator iteratively. Note that the asymmetry seen in Fig. 2 disappears and the reconstructedκ(n) restores symmetry, once many clusters are stacked. Furthermore, the uncertainties in the mean profile decrease as our assumed model converges to the true model, because it solely results from the intrinsic fluctuations of the CMB in the case of perfect delensing. B. Performance Comparison Before we assess the performance of the three lensing estimators in realistic experiments, we first compare our improved quadratic estimator to the standard quadratic estimator, when the lensing effect is small. Figure 4 plots the re- Fig. 3). 10,000 (left) and 1000 (right) clusters are used to obtain the mean profile, and the shaded region shows the uncertainties in the mean profile. Both estimators recover the true mass profiles within Rvir in the low mass regime. Approximately ten times more clusters are needed for sQE to achieve the same accuracy than for iQE. However, for comparison we plot the mean profile from 1000 clusters as the dot-dashed line in the left panel. constructed cluster mass profiles in the same format as Fig. 3. For clusters of M = 1 × 10 14 h −1 M ⊙ at z L = 0.3 (κ ≪ 1), the improved quadratic estimator recovers the true mass profile with no detectable bias after two iterations. With signals smaller by a factor of five than in Fig. 3, 1000 clusters are stacked to obtain the mean mass profile, while 10,000 clusters are required for the standard quadratic estimator. As we quantify the difference in the signal-to-noise ratio below, the standard quadratic estimator needs approximately ten times as many clusters as the improved quadratic estimator needs to achieve the same accuracy, but we show the mean profile (dot-dashed) obtained by applying the standard quadratic estimator to 1000 clusters for comparison. Once enough clusters are stacked, the standard quadratic estimator works well within R vir , though it shows some hint of deviation at the core. Thus, the standard quadratic estimator may be safely used to reconstruct mass profiles of clusters with M < 1 × 10 14 h −1 M ⊙ at z L = 0.3. However, given the source of the CMB at z ⋆ = 1090, the lensing effect becomes larger as z L increases, until Σ crit reaches the minimum at z L ≃ 2.5, where D L becomes a half of D ⋆ . Therefore, the standard quadratic estimator cannot be used to reconstruct unbiased mass profiles of clusters that are either at z L ≥ 0.3 or massive M ≥ 1 × 10 14 h −1 M ⊙ . Since ACT and SPT will find clusters of M ≥ 2 × 10 14 h −1 M ⊙ at higher redshift, modified or improved quadratic estimators are preferred to the standard quadratic estimator. Now we consider realistic experiments with σ pix = 5µK and compare the performance of the lensing estimators in Fig. 5. Since the reconstruction becomes noisier in the presence of detector noise and telescope beam, 10,000 clusters are stacked for the mean mass profiles when the standard or modified quadratic estimator is used, while the improved quadratic estimator is iteratively applied to only 1000 clusters. For clusters of M = 5 × 10 14 h −1 M ⊙ at z L = 1, Fig. 5a shows that the standard quadratic estimators become substantially biased in a region around massive clusters, consistent with the previous results [13,14]. Quadratic terms in φ l ignored in the linear approximation coherently contribute toκ l , and hence the reconstructedκ(n) is biased low where the linear approximation is violated [14]. Next we consider a modified quadratic estimator in Fig. 5b and adopt l cut = 1500. The modified quadratic estimator recovers the true mass profile within R vir but with small deviation beyond R vir . The modified quadratic estimators operate in the same way of the standard quadratic estimators, except signals are removed on small scales (l ≥ l cut ), where the linear approximation is violated. However, the choice of l cut is rather arbitrary and should be calibrated against simulations: lower l cut is needed for more massive clusters. Note that the modified quadratic estimator with l cut → ∞ exactly reduces to the standard quadratic estimator (in practice l cut > ∼ 10 4 can achieve this limit because of the Silk damping). In other words, a modified quadratic estimator with l cut ≃ 10 4 fails to reconstruct the mass profile (born out by Fig. 5a). Moreover, we had to adopt l cut = 1500 to reconstruct the mass profile in Fig. 5b and 5d, a more aggressive choice than l cut = 2000 proposed in [14], with which we cannot recover the mass profile. This reflects the sensitivity of the modified quadratic estimator to l cut as a function of cluster mass. Larger number of clusters are also required to reconstruct the true mean mass profile due to the reduction in the signal-to-noise ratio. Figure 5c shows the reconstruction by our improved quadratic estimator with M init = 1 × 10 14 h −1 M ⊙ . The improved quadratic estimator recovers the true mass profile with no significant bias in the presence of detector noise. After a few iterations, the estimates quickly converge to the true model and the scatter around the mean is greatly reduced compared to Fig. 5b. Note that we iteratively applied the improved quadratic estimator to the same 1000 clusters. In Fig. 5d and 5e, we consider the effect of telescope beam with θ FWHM = 0. ′ 5. Both estimators in Fig. 5d and 5e recover the true mass profile unbiased in the presence of detector beam, while there exist some deviations in both cases. However, note that we explicitly account for the beam effect using the formulas developed in Sec. II B, rather than deconvolve the beam before applying the lensing estimators. The latter approach often used in the literature suffers from deconvolved detector noise exponentiating on small scales. This problem requires a low-pass filtering of reconstructedκ(n), additionally removing the signals below the beam scale, which results in a distortion of its shape ofκ(n), making it hard to compare directly to theoretical predictions. However, in reality beam convolution suppresses detector noises (of course lensing signals as well), and it simply makes the reconstruction noisy below the beam scale. The dot-dashed line in Fig. 5d contrasts the reconstruction when we explicitly removeκ(n) at l ≥ 1/σ b , whereκ(n) is obtained by applying the modified quadratic estimator with beam-deconvolved data (the line is displaced to avoid confusion with other lines). Significant shape distortion inκ(n) complicates the interpretation. For a larger beam size comparable to the scale radius of the clusters (θ FWHM ≃ 1 ′ ), the reconstruction becomes more challenging: modified quadratic estimators cannot recover the cluster mass profile without significant shape distortion (dotdashed). The improved quadratic estimator in Fig. 5f recovers the true mass profile beyond R vir , while it develops small bias below the beam scale. Figure 6 plots the fractional difference between the lensing estimates and the true cluster mass profile in Fig. 5, comparing their uncertainty in the mean profile. The difference (lines) is computed from the mean mass profiles by stacking 10,000 clusters for both estimators, while the statistical uncertainty (gray bands) in the difference is scaled for 500 clusters for comparison. The left panel shows that both estimators recover the cluster mass profile at the 5% level or better in the absence of telescope beam, while the modified quadratic estimator may need fine-tuning of l cut to achieve better accuracy. However, the difference in their measurement uncertainty is in stark contrast: the improved quadratic estimator has a significantly higher signal-to-noise ratio than the modified quadratic estimator. While the reconstruction becomes harder especially beyond R vir in the presence of telescope beam shown in the right panel, this trend of signal-to-noise ratio difference persists. Note that due to the beam smoothing effect the uncertainty in the estimates at θ ≤ θ FWHM is reduced while it is highly correlated among adjacent bins. So far we have numerically demonstrated the performance of the lensing estimators in Figs. 5 and 6: standard quadratic estimators are significantly biased; modified and improved quadratic estimators recover the cluster mass profile with no bias, while they show substantial difference in the number of clusters that is required to obtain the mean mass profile. To quantify this difference, we evaluate ∆χ 2 of each lensing estimator ∆χ 2 = θ,θ ′ κ(θ) C −1 κ (θ, θ ′ ) κ(θ ′ ),(37) where the covariance matrix ofκ(θ) is Cκ(θ, θ ′ ) = [κ(θ) − κ(θ)] [κ(θ ′ ) − κ(θ ′ )] .(38) Sinceκ(n) is computed from the two Wiener-filtered functions of the CMB temperature anisotropies, the covariance matrix is non-diagonal. The finite width of the convolution filter H(n) in Eq. (17) also reflects that the lensing estimators are a non-local function of the CMB temperature anisotropies, and hence non-zero Cκ when θ = θ ′ . In the absence of telescope beam in Figs. 5b, 5c, and 6a, the ratio of ∆χ 2 for the modified quadratic estimator relative to the improved quadratic estimator is 8.1: a factor of eight larger number of clusters is required for the modified quadratic estimator to achieve the same level of accuracy than that for the improved quadratic estimator. In the presence of telescope beam in Figs. 5d, 5e, and 6b, beam smoothing substantially degrades the ability to recover the true cluster mass profile for both estimators, and its effect is relatively larger for the modified quadratic estimator, increasing the ratio to 10.4. C. Sunyaev-Zel'dovich Effects On small scales (l > 2000), the primordial CMB temperature anisotropies decay exponentially due to the Silk damping [7] and the dominant source of secondary anisotropies is the thermal Sunyaev-Zel'dovich (tSZ) effect, arising from scattering off hot electrons in massive clusters. However, the tSZ effect imprints a unique frequency dependence in the CMB temperature anisotropies, which in principle can be used to remove the tSZ signals. The same Compton scattering process also gives rise to a Doppler effect in the CMB temperature anisotropies due to the bulk motion of electron gas, or the kinetic Sunyaev-Zel'dovich (kSZ) effect (see [34,35] for recent reviews). These kSZ signals, albeit smaller than tSZ signals, are spectrally indistinguishable from the intrinsic CMB temperature anisotropies, introducing an artifact in the lensing reconstruction. Here we assume that the tSZ signals can be cleaned perfectly, and we investigate how the kSZ signals deteriorate the lensing reconstruction. For simplicity, we assume that the gas density traces the dark matter distribution in a massive cluster, with the same NFW profile. Given the line-of-sight velocity v los of the cluster, the kSZ effect results in temperature anisotropies ∆T (θ) = −v los τ (θ) T CMB ≡ −∆T kSZ Σ(θ) Σ(0) ,(39) where τ (θ) is the Thompson scattering optical depth, proportional to the projected density Σ(r = D L θ). We parametrized lensed temperature map (left) and its difference from the intrinsic temperature map (right). Bottom panels: assuming that the cluster is moving toward an observer, the kSZ effect is set ∆T kSZ = 3 (left) and 15µK (right) at the center. The color scales in each panel represent the same temperature except in the upper right panel, where the color represents the difference ranging from −5µK to 5µK. the product of v los and τ (0) as ∆T kSZ . Note that since the intrinsic CMB and kSZ induced anisotropies dilute in the same way as the universe expands, there is no (1 + z L ) factor in Eq. (39) and T CMB = 2.725K is the CMB temperature today. For a typical cluster with electron number density ∼ 0.01 cm −3 and core radius ∼ 100 kpc, the Thompson scattering optical depth is τ (0) = 2 × 10 −3 at the core. The rms velocity dispersion in linear theory is σ v = 1.3 × 10 −3 (= 390 km s −1 ) at z L = 1, and this results in the rms temperature fluctuation ∆T kSZ = 3.7µK at the core. We randomly draw ∆T (0) from a Gaussian distribution with zero mean and dispersion σ = ∆T kSZ , then we add ∆T (n) toT (n) for observations of each cluster. First, we compare the cluster lensing and kSZ effects on the CMB temperature field. Figure 7 plots a 6'×6' regions of CMB maps around a cluster of M = 5 × 10 14 h −1 M ⊙ (θ vir = 3. ′ 0) at z L = 1. The top panels show the lensed temperature field (left) and the difference from the intrinsic temperature field (right). Gravitational lensing imprints dipolelike wiggles in the CMB map on top of the smooth large-scale gradient field. Perpendicular to the gradient direction there exists no temperature change and hence lensing reconstruction is degenerate along the direction. The bottom panels show the kSZ effect with ∆T kSZ = 3µK (left) and 15µK (right). We assume that the cluster is moving toward the observer. With the small optical depth in the left panel, the kSZ effect is rel-FIG. 8: Impact of kinetic Sunyaev-Zel'dovich (kSZ) effects on the mass profile reconstruction. Assuming that the gas distribution traces the dark matter distribution in clusters, the kSZ effect is computed by assigning a Gaussian random velocity to each cluster with rms temperature change ∆T kSZ = 3 (left) and 15µK (right) at the center, respectively. atively small compared to the lensing effect. Larger optical depth in the right panel substantially enhances the kSZ effect, dominating over the lensing effect at the center. However, since the lensing effect is much less concentrated than the kSZ effect as the dipole-like wiggles peak at a few scale radii (top right), the reconstruction is still possible. Figure 8a shows that the kSZ effect with ∆T kSZ = 3µK has relatively little impact on the reconstruction: the kSZ effect becomes negligible beyond r s because the density profile declines r −3 (the gas density in reality would be steeper and more confined to the center than we assumed here). The lensing effect, on the other hand, is sensitive to the deflection field and remains strong beyond r s , declining less rapidly than the kSZ effect [29]. In Fig. 8b, we consider a larger kSZ effect with ∆T kSZ = 15µK, expected either from higher electron number density or from higher matter fluctuation normalization σ 8 ∝ σ v . With the temperature anisotropies comparable to the lensing effect, the reconstruction becomes difficult and it starts to develop bias around R vir as ∆T kSZ increases. Note that the bias at the center is largely due to the telescope beam effect. In the presence of contaminants such as residual foreground or tSZ effect, radio point sources, and large kSZ effect, the lensing estimators based on temperature anisotropies need to be complemented by using lensing estimators based on combination of temperature and E-and B-mode polarization [19], since there exists no significant source of contamination that mimics the intrinsic CMB polarization. Furthermore, the unique relation between the E-and B-mode polarization signals [36,37] can be used to provide a robust consistency check. However, measurements of the lensed polarization fields would require an experiment with higher angular resolution and sensitive detectors than experiments that are currently available. V. DISCUSSION Weak gravitational lensing of the CMB gives rise to a deviation of the two-point correlation function of the CMB temperature anisotropies from otherwise statistically isotropic function. Quadratic estimators [11] have been widely used to reconstruct cluster mass profiles and large-scale structure by measuring the induced anisotropies in the two-point correlation function. We have shown that standard quadratic estimators become optimal in the limit of no lensing, saturating the Cramér-Rao bound, while they become progressively biased and sub-optimal as the lensing effect increases. Especially for clusters that can be found by the ongoing SZ surveys like ACT and SPT, the standard quadratic estimators start to be biased at z L ≃ 0.3, and at higher redshift, where the lensing effect is larger, other estimators should be used to reconstruct cluster mass profiles. It is recently proposed [14] that this obstacle in the standard quadratic estimators can be overcome by explicitly removing the signals in the CMB temperature gradient field at l ≥ l cut , where the lensing effect is large in violation of the linear approximation. However, although these modified quadratic estimators recover cluster mass profiles with no significant bias, the choice of l cut is somewhat arbitrary and it depends on the lensing effect, which requires prior calibrations against numerical simulations before one can apply the modified quadratic estimators to CMB maps. We have developed a new maximum likelihood estimator for reconstructing cluster mass profiles and large-scale structure. We first construct a CMB temperature field by delensing the observed temperature field based on an assumed mass model. We have proved that the delensed temperature field is close to the unlensed temperature field with telescope beam smoothed and detector noise added, if the assumed mass model is a good approximation to the true mass model. The delensed temperature field can then be used to set up the likelihood of the CMB, and our new estimator that maximizes this likelihood takes a similar form of the standard quadratic estimators, because it approaches to an optimal estimator as the assumed model becomes the true model. Our maximum likelihood estimator can be iteratively applied as we update the assumed mass model, until it converges (to the true model) and the estimate is consistent with the assumed model. Our maximum likelihood estimator, named as an improved quadratic estimator, is easy to implement in practice and it has no free parameter. Our improved quadratic estimators quickly converge to the true mass model after a few iterations, even when an assumed initial model is significantly different from the true model. When the estimate is inconsistent with the assumed model, one can adopt another initial model for iterations for faster convergence of the improved quadratic estimators. The telescope beam and detector noise renders the reconstruction harder, but we have demonstrated that the improved quadratic estimators recover cluster mass profiles with a beam size comparable to the cluster scale radius. Furthermore, our new estimator significantly improves the signal-to-noise ratio over the standard or modified quadratic estimators by a factor of ten in number of clusters, because when an assumed model is close to the true mass model, the only source of noise for our estimator is the intrinsic fluctuations of the CMB temperature gradient. We have tested the robustness of the improved quadratic estimators in the presence of the kSZ effect. The kSZ distortion ∆T kSZ ≤ 15µK at the center results in relatively small bias in the reconstructed cluster mass profiles. However, since the optical depth is a function of electron number density in the clusters, it is related to the true mass profile. Therefore, we could take a more aggressive approach to modeling kSZ signals from an assumed initial mass model and subtract the kSZ contributions before applying improved quadratic estimators. Furthermore, this template for kSZ signals can also be iteratively refined as we update our assumed mass model. Since the reconstruction is non-parametric, it is not limited to spherical clusters, while stacking many clusters ensures that irregular shapes of individual clusters become irrelevant. Similar arguments can be applied to projection effects: each cluster can be located at a line-of-sight with overdense or underdense regions, but projection effects become negligible once many lines-of-sight are combined. Given a sample of clusters from SZ surveys, the average mass profile of stacked clusters would provide a cluster-mass cross-correlation function, which can be used to measure the growth rate of structure, probing the evolution of dark energy, instead of individual cluster mass profiles. However, in reality it would be harder to reconstruct cluster mass profiles than considered here, because there exist other contaminants such as point radio sources and residual foreground and/or tSZ effect, and other complications such as non-isolated clusters and internal bulk motion of gas in clusters. However, additional information from polarization measurements may be used to overcome some of the difficulties, given the unique relation between the E-and B-mode polarization signals and relatively negligible primary and secondary contaminants. Finally we mention that our improved quadratic estimators can be applied to reconstruct large-scale structure, while in this regime standard quadratic estimators can be used without significant bias. ( 2 ) 2], the observed [Eq. (7)], and the delensed [Eq. (27)] temperature fields, respectively. FIG. 1: Convolution filter H(θ) as a function of separation θ = \|n\| for CMB experiments with σpix = 5µK and 10µK in Sec. IV. The insets show details of H(θ) at the center (lef t) and its tail (right). FIG. 2 : 2(color online) Reconstructed convergence fields of a 30 ′ ×30 ′ region around a cluster at zL = 1 from an ideal experiment with ∆T = 0. Cluster mass is set M = 5 × 10 14 h −1 M⊙. Improved quadratic estimators are applied once with initial mass models of Minit = 5 × 10 14 h −1 M⊙ (left) and Minit = 1 × 10 14 h −1 M⊙ (right) to a single patch of sky. The bottom panels show the residual after the true cluster convergence field is subtracted from the top panels. For reference, D L = 850h −1 Mpc and 2400h −1 Mpc, and Σ crit = 2.8 × 10 3 hM ⊙ pc −2 and 1.8 × 10 3 hM ⊙ pc −2 for z L = 0.3 and 1, respectively. For clusters of M = 5 × 10 14 h −1 M ⊙ and 1 × 10 14 h −1 M ⊙ , R vir = 2.1h −1 Mpc and 1.2h −1 Mpc appear subtended by 3. ′ 0 and 4. ′ 9 on the sky at z L = 1 and 0.3. FIG. 3 : 3Dependence of reconstructed mass profiles on an initial mass model Minit. Thick and thin solid lines represent the true cluster mass profile and the mean of reconstructed mass profiles from 500 clusters. The mass profiles are obtained by averaging reconstructed convergence over the annulus of each cluster. The uncertainties in the mean profile are shown as shaded regions. Dashed lines show an assumed initial mass model and the cluster virial radius is shown as vertical dotted lines. In Panels (c) and (d), the initial mass models are taken as the mean mass profile from the previous iteration. The reconstruction quickly converges to the true mass profile in two iterations even with an incorrect choice of Minit = 1 × 10 14 h −1 M⊙, exhibiting no detectable bias in an ideal experiment. FIG. 4 : 4Mass profile reconstruction for low mass clusters of M = 1 × 10 14 h −1 M⊙ at zL = 0.3 from standard (sQE) and improved (iQE) quadratic estimators (in the same format as in FIG. 5 : 5Comparison of reconstructed mass profiles from standard (sQE), modified (mQE), and improved (iQE) quadratic estimators in realistic experiments with σpix = 5µK. The reconstruction is more difficult in the presence of detector noise and telescope beam. For the mean of reconstructed mass profiles, 10,000 clusters of M = 5 × 10 14 h −1 M⊙ at zL = 1 are stacked when sQE or mQE is used, while iQE is iteratively applied to only 1000 clusters. The shaded regions show the uncertainties in the mean profile. The dot-dashed line (panel d) shows the shape distortion inκ(n) when mQE is applied after beam-deconvolution, and the line is displaced to avoid confusion (see the text). With θFWHM = 1 ′ (panel f ), iQE can recover the mean mass profile with small bias below the beam scale. For comparison, we plot the reconstructed mass profile (dot-dashed) using mQE in Panel (f ). FIG. 6 : 6Fractional difference between the lensing estimates and the true cluster mass profile inFig. 5. The difference (lines) is computed from the mean mass profiles obtained by stacking 10,000 clusters for both estimators, while the statistical uncertainty (gray bands) in the difference is scaled for 500 clusters. The vertical dotted lines show the cluster virial radius. FIG. 7 : 7(color online) Cluster lensing and kinetic Sunyaev-Zel'dovich (kSZ) effects on the CMB. For comparison, we plot 6 ′ × 6 ′ regions of CMB temperature maps around a cluster of M = 5 × 10 14 h −1 M⊙ (θvir = 3. ′ 0) at zL = 1. Upper panels: Figure 8 8shows the impact of the kSZ effect on reconstructing mass profiles. For clusters of M = 5 × 10 14 h −1 M ⊙ at z L = 1 in an experiment with θ FWHM = 1 ′ and σ pix = 5µK, we iteratively use improved quadratic estimators with M init = 1 × 10 14 h −1 M ⊙ . The mean and the uncertainties are computed from 1000 clusters. FIG. 9 : 9Effects of telescope beam and detector noise on the delensing process. The top panel comparesT l (thin) with T l e ) in terms of their power spectrum, and the bottom panel shows the fractional deviations. The vertical dotted line represents the beam scale l = 1/σ b . CMB experiments with θFWHM = 1 ′ and σpix = 5µK are considered for clusters of M = 5×10 14 h −1 M⊙ at zL = 1. The noise only case is largely obscured by the solid line. AcknowledgmentsWe thank Oliver Zahn for useful discussion. J. Y. thanks C. K. Chan for technical help on FFTw routines. J. Y. is supported by the Harvard College Observatory under the Donald H. Menzel fund. M. Z. is supported by the David and Lucile Packard, the Alfred P. Sloan, and the John D. and Catherine T. MacArthur Foundations. This work was further supported by NSF grant AST 05-06556 and NASA ATP grant NNG 05GJ40G.APPENDIX A: DELENSED TEMPERATURE FIELDHere we derive a relationT l ≃ T l e − 1 2 l 2 σ 2 b + T N l in the presence of telescope beam and detector noise. Given the lensing potential φ m (n) of an assumed mass model, the lensing equation relates an image positionn to a source position s m =n +∇φ m (n). Here we keep the superscript m to indicate the relation to the assumed model. The true source position is thenŝ =n +∇φ(n), where φ(n) is the true lensing potential. Now we construct a delensed temperature fieldwhere B(m) is the telescope beam function. Since the lensing equation is not analytically invertible in general, we keep bothŝ m andn, but note that they are not independent variables. In Fourier space, the delensed temperature field iŝwith a contribution from the CMBand a contribution from the detector noisêThe lensed temperature isT (n) = T (ŝ) and its Fourier mode isTWith the linear approximation, one can expand the exponential term to the first order in φ l and this equation reduces to Eq. (3). However, we keep the equation as general as possible to be valid, even when the lensing effect is large. Substituting T l1 in Eq. (A3) and changing the integration variablen toŝ m giveŝThis is the final expression for the delensed temperature field. The first exponential term of the integrand controls the delensing process: when the assumed model is close to the true model after a few iterations (φ m (n) ≃ φ(n),ŝ m ≃ŝ), the integral becomes a Dirac delta function andT S l = T l , when the beam smoothing is negligible. The distortion matrix is close to the identity matrix beyond R vir and T S l ≃ T l e − 1 2 l 2 σ 2 b . Around massive clusters, the distortion matrix deviates from the identity matrix and its determinant becomes smaller than one, making the exponential factor unity. This reflects that the beam size is reduced by a mapping from the image plane to the source plane, and practicallŷFor a white detector noise, the delensed detector noise is simply the redistributed white noise. However, since the delensing process alters the unit area on the sky, it becomes nonwhite but its deviation is confined to relatively small region; the noise power spectrum isIt is the Jacobian of the distortion matrix that makes white noise non-white in a region around massive clusters. Outside R vir , where the Jacobian is near unity, the integral becomes a Dirac delta function and the noise is again white.Figure 9compares our delensing (T l : thin) and perfect delensing (T l e − 1 2 l 2 σ 2 b + T N l : thick) processes in terms of their power spectrum. In the absence of detector noise (dashed), T S l starts to deviate from T l e − 1 2 l 2 σ 2 b around the beam scale l ≃ 1/σ b (vertical dotted), declining less rapidly. On scales below the beam scale, our approximation (∆ ≪ 1) breaks down and M −1 (n) differs from the identity matrix, leading to the excess power. However, at this scale, signals are dominated by the detector noise (solid). 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[ "TWISTOR LINES ON NAGATA THREEFOLD", "TWISTOR LINES ON NAGATA THREEFOLD" ]	[ "Nobuhiro Honda " ]	[]	[]	We give an explicit description of rational curves in the product of three copies of complex projective lines, which are transformed into twistor lines in M. Nagata's example of non-projective complete algebraic variety, viewed as the twistor space of Eguchi-Hanson metric. In particular, we show that there exist two families of such curves and both of them are parameterized by mutually diffeomorphic, connected real 4-dimensional manifolds. We also give a relationship between these two families through a birational transformation naturally associated to the Nagata's example.	10.1215/kjm/1250692292	[ "https://arxiv.org/pdf/math/0608455v2.pdf" ]	13,395,079	math/0608455	049386bfe1e4f941c0de3517f3ba096f8a45acd4	TWISTOR LINES ON NAGATA THREEFOLD 4 Oct 2007 Nobuhiro Honda TWISTOR LINES ON NAGATA THREEFOLD 4 Oct 2007 We give an explicit description of rational curves in the product of three copies of complex projective lines, which are transformed into twistor lines in M. Nagata's example of non-projective complete algebraic variety, viewed as the twistor space of Eguchi-Hanson metric. In particular, we show that there exist two families of such curves and both of them are parameterized by mutually diffeomorphic, connected real 4-dimensional manifolds. We also give a relationship between these two families through a birational transformation naturally associated to the Nagata's example. Introduction. In 1958, M. Nagata [8] constructed a remarkable example of a compact complex threefold, which was the first example of a non-projective complete algebraic variety. A new aspect of this threefold was given by A. Fujiki [5], who showed that Nagata's example is a compactification of the twistor space of the famous Eguchi-Hanson metric on the cotangent bundle of P 1 [4]. Fujiki proved this result by investigating the Calabi family naturally associated to a hyper-Kähler metric. The most important geometric object in a twistor space is the twistor lines. Since Nagata's example is obtained from a product P 1 × P 1 × P 1 by applying a simple birational transformation, it seems natural to ask which curves in P 1 × P 1 × P 1 are transformed into twistor lines in Nagata's example. In this note we give an answer to this question by describing all of them explicitly. After recalling Nagata's construction, we first determine which curves in P 1 × P 1 × P 1 can be transformed into real lines in the Nagata threefold (Propositions 2.1 and 2.2). Here, a smooth rational curve in a threefold is called a real line if the normal bundle of the curve is isomorphic to O(1) ⊕ O(1) and it is invariant under an anti-holomorphic involution of the threefold. These are candidates of twistor lines. The parameter space of these candidates consists of two connected 4-manifolds, which will be denoted by M + and M − . There is a natural isomorphism between M + and M − which comes from an involution of P 1 × P 1 × P 1 of the same degree. Although M + and M − parameterize real curves in P 1 × P 1 × P 1 , both contain 2-dimensional submanifolds parameterizing reducible curves. We find that, as a consequence of Nagata's birational transformation, this situation is resolved for precisely one of M + and M − so that all its curves become irreducible. Then by using our explicit description of the candidates, we show that the resolved family actually becomes the set of twistor lines (Theorem 2.6). All irreducible curves of the other family are also transformed into real lines in the Nagata threefold. But the family cannot be the set of twistor lines since the reducible curves (parameterized by the 2-dimensional submanifold mentioned above) remain reducible in Nagata threefold (even after applying the birational transformation). We observe that all these reducible curves contain a common curve (which will be written by B + 0 + B + ∞ ) which is homologous to zero. We will also find that any members of this family can be deformed into a twistor line (keeping the reality and smoothness) if one allows the members to pass the situation that they have O(2) ⊕ O as their normal bundles in the Nagata threefold. In particular, all irreducible curves in the other family are homologous to actual twistor lines. Finally, we mention that in the paper [6] N. J. Hitchin gave an explicit description of the twistor spaces of multi-Eguchi-Hanson metrics (i. e. Gibbons-Hawking metrics) and their twistor lines. For the twistor spaces of multi-Eguchi-Hanson metrics, explicit birational transformation into a rational variety of a simple form seems unknown, except for the original Eguchi-Hanson metric, which is adapted in this papar. Also, in the paper [3], D. Burns explicitly constructed the twistor spaces of the contangent bundles of Hermitian symmetric spaces and their twistor lines. The author does not know how to identify these works with the explicit description in this paper. Acknowledgment The author would like to thank the referee for careful reading and invaluable comments. Also he would like to thank Shihoko Ishii for answering his question on a Hilbert scheme. Explicit description of real lines First of all, we recall the construction of Nagata threefold [8]. We write P = P 1 for the complex projective line throughout this paper. Let X = P × P × P be the product of three complex projective lines, and write C = P for the last factor which plays a special role. Let f : X → C be the projection, and 0 and ∞ two distinct points of C. We fix an isomorphism between the first and the second factors of X and let ∆ ⊂ P×P be the diagonal. Put Q = ∆×C, a divisor on X. For t ∈ C, we write ∆ t for the diagonal of X t := f −1 (t). Let µ : Y → X be the blowing-up along ∆ 0 ∪ ∆ ∞ , and X ′ 0 and X ′ ∞ the strict transforms of X 0 and X ∞ respectively. Because ∆ 0 ⊂ X 0 and ∆ ∞ ⊂ X ∞ , µ gives isomorphisms X ′ 0 ≃ X 0 and X ′ ∞ ≃ X ∞ . Moreover, N X ′ 0 /Y and N X ′ ∞ /Y , the normal bundles of X ′ 0 and X ′ ∞ in Y respectively, are isomorphic to O(−1, −1), where O(m, n) denotes the holomorphic line bundle on P × P of bidegree (m, n). Therefore X ′ 0 (≃ P × P) can be blown-down to P along each two projections, and the same thing holds for X ′ ∞ . Let ν + 0 : X ′ 0 → P be the projection to the first factor and ν + ∞ : X ′ ∞ → P the projection to the second factor. Let ν + : Y → Z + be the blowing down of Y inducing ν + 0 and ν + ∞ on X ′ 0 and X ′ ∞ respectively. If we interchange the role of the two factors, we obtain another threefold Z − . In this way we obtain two smooth threefolds Z + and Z − . Both Z + and Z − are called Nagata threefolds. The isomorphism of X interchanging the first and the second factors naturally induces a biholomorphic map i : Z + → Z − and hence these are naturally biholomorphic. We denote by f + : Z + → C for the projection naturally induced from f : X → C. This is a holomorphic submersion. The fibers f −1 + (0) and f −1 (2)) and all the other fibers are isomorphic to P × P. If we denote B + 0 = ν + (X ′ 0 ) and B + ∞ = ν + (X ′ ∞ ), B + 0 and B + ∞ are the minimal section of Σ 2 → P. It was shown in [8] that the effective curve B + 0 + B + ∞ is homologous to zero and hence Z + does not admit a Kähler metric. Finally, we define a divisors Q + in Z + as the bimeromorphic image of Q = ∆ × C into Z + . Similarly f − : Z − → C, B − 0 , B − ∞ and Q − are defined for Z − . We have i(Q + ) = Q − . Next we introduce real structures. Let σ 1 be an anti-podal map of P, which is an antiholomorphic involution without fixed points. σ 1 can be explicitly given by σ 1 (z 0 : z 1 ) = (−z 1 : z 0 ) using homogeneous coordinates. Then we define an anti-holomorphic involution σ on X by (1) σ(x, y, z) = (σ 1 (y), σ 1 (x), σ 1 (z)). + (∞) are isomorphic to Σ 2 = P(O ⊕ O Here we are using the identification of the first and the second factors. Clearly σ has no fixed points. Further σ survives under the above construction and induces real structures on the Nagata threefolds. We denote these real structures by σ + on Z + and σ − on Z − respectively. Then Q ± ⊂ Z ± are invariant under σ ± ; namely they are real. Moreover, it is immediate to see σ − i = iσ + and hence (Z + , σ + ) and (Z − , σ − ) are isomorphic as complex manifolds with real structures. So in the sequel we mainly consider (Z + , σ + ); corresponding results for (Z − , σ − ) are immediately obtained by using the isomorphism i. We are going to show that (Z + \Q + , σ + ) is the twistor space of a hyperKähler metric on the total space of the cotangent bundle of P, mainly by showing that the complex threefold Z + \Q + is actually foliated by real lines. As a first step, we show the following proposition describing the images of real lines in Z + (if any) under the above birational correspondence between Z + and X. To indicate curves in X, we use the following notation: by taking three factors of X as generators, we have a natural isomorphism H 2 (X, Z) ≃ Z 3 , and the homology class of an effective curve on X is determined by (k 1 , k 2 , k 3 ) ∈ Z 3 ; in other words, k i is the intersection number of the curve with fibers of the i-th projection from X to P. We call (k 1 , k 2 , k 3 ) the degree of a curve for simplicity. On the other hand, a real line in a threefold (with real structure) means a smooth rational curve which is invariant under the real structure and which has O(1) ⊕ O(1) as the normal bundle. The following proposition gives a necessary condition for a rational curve in X = P × P × P to be transformed into a real line in the Nagata threefold. Proposition 2.1. Let (Z + , σ + ) be the Nagata threefold equipped with the real structure given above, and suppose that L is a real line in Z + which is disjoint from the divisor Q + . Then L must be a section of f + : Z + → C. Further by the birational correspondence between Z + and X, L is transformed into a real curve in X satisfying the following. (i) If L does not go through B + 0 , then the degree of the corresponding curve in X is (1, 1, 1) and it goes through ∆ 0 (and hence ∆ ∞ also). (ii) If L goes through B + 0 (and hence B + ∞ also), then the degree of the corresponding curve in X is (0, 0, 1). Proof. Assume L is a real line disjoint from Q + . First we show that L is a section of f + . By standard calculations, we can show that the anti-canonical line bundle of Z + satisfies (2) − K Z + ≃ O(2Q + ) ⊗ f * + O(4) . On the other hand, by adjunction formula, we have −K Z + · L = 4. Hence since we have assumed that L is disjoint from Q + , we have f * + O(1) · L = 1 and thus L is a section of f + . Next to prove (i) suppose that L is a real line in Z + that is disjoint from B 0 (and B ∞ ), and let ν + : Y → Z + and µ : Y → X be as in the construction of Z + . L is a section of f + . Obviously ν + does not change a neighborhood of L in Z and therefore L still intersects X ′ 0 and X ′ ∞ once respectively in Y . Hence under the blowing-down µ : Y → X the image of the real line goes through ∆ 0 = µ(X ′ 0 ) and ∆ ∞ = µ(X ′ ∞ ). Next under the same assumption we show that the degree of the image of L in X is (1, 1, 1). Because the image remains to be a section of f : X → C, the degree must be of the form (k 1 , k 2 , 1) for non-negative integers k 1 and k 2 . Moreover, since the birational transformations keep the reality of L, and since the real structure on X interchanges the first and second factors as in (1), we have k 1 = k 2 . Moreover, the degree of the normal bundle must increase by two under the blowing-down µ : Y → X. Therefore it becomes 2 + 2 = 4. It is readily seen that this happens iff k 1 = (k 2 =)1. Thus we obtain (i) of the proposition. Next suppose that L is a real line intersecting B + 0 (and B + ∞ ). Then since L is a section of f + , L intersects f −1 + (0) and f −1 + (∞) transversally at a unique point respectively. Hence the degree of the normal bundle in Y becomes 2 − 2 = 0. Moreover, this time, the blowing down µ : Y → X makes no effect in a neighborhood of the curve. Hence the degree of the normal bundle of the image of L in X must be zero. This happens only when the degree is (0, 0, 1) and we obtain (ii) of the proposition. The real curves in (i) and (ii) of Proposition 2.1 can be easily written down explicitly. Let (x, y, t) be an affine coordinate on X so that the real structure σ is given by σ(x, y, t) = (−1/y, −1/x, −1/t). Then we have the following. Proposition 2.2. (i) A real irreducible curve of degree (1, 1, 1) in X going through the point (d, d, 0) ∈ ∆ 0 with d ∈ C is of the form (3) x = d − at 1 + adt , y = ad − t a + dt , for some a ∈ C * , whereas if d = ∞ it is of the form (4) x = 1 at , y = a t , for some a ∈ C * . (ii) A real curve in X of degree (0, 0, 1) is given by (5) is always disjoint from Q. (iv) By the birational correspondence between X and Z + , all the curves (5) are transformed into real lines in Z + , while the curves (3) and (4) are transformed into real curves whose normal bundles are O(2) ⊕ O, as long as \|a\| = 1. (5) x = d, y = − 1 d , where d ∈ C ∪ {∞}. (iii) If \|a\| = 1, the curves (3) and (4) are contained in Q. If \|a\| = 1, the curves intersect Q only at (d, d, 0) ∈ ∆ 0 and σ(d, d, 0) ∈ ∆ ∞ . The curve The final assertion of (iv) means that if \|a\| = 1 the curves (3) and (4) are not transformed into lines. On the other hand, it will be shown (in the proof of Theorem 2.6) that if \|a\| = 1 the curves (3) and (4) are transformed into real lines in Z + . Proof of Proposition 2.2. (i), (ii) and (iii) are elementary and we omit proofs. For (iv), let C ⊂ X be a curve defined by (5) (for some d ∈ C ∪ {∞}). Then we can readily find divisors D 1 and D 2 in Z + intersecting transversally along the image of C in Z + , to conclude that C is transformed into a smooth rational curve satisfying N ≃ O(1) ⊕ O(1). On the other hand, if C is a curve defined by (3) or (4) with \|a\| = 1, C is contained in Q as in (iii). It is readily seen that N C/Q ≃ O(2). If we write C + for the image of C in Z + , we obtain from the inclusions C + ⊂ Q + ⊂ Z + an exact sequence 0 −→ O(2) −→ N C + /Z + −→ N Q + /Z + \| C + −→ 0. Further, there is a natural isomorphism Q + ≃ Q ≃ P × P and we have N Q + /Z + ≃ O(2, −2), while C + is a (1, 1)-curve in Q + . Hence the restriction N Q + /Z + \| C + is trivial and we obtain N C + /Z + ≃ O(2) ⊕ O. Since the curve (3) is determined by two numbers d ∈ C and a ∈ C * , we denote it by L d,a . Also for d = ∞ and a ∈ C * , we denote L ∞,a for the curve (4), by abuse of notation. Of course, d ∈ C ∪ {∞} specifies the intersection point of the curve with ∆ 0 . Further for d ∈ C ∪ {∞} we write L d,0 for the curve (5) whose degree is (0, 0, 1). Then as a goes to 0 or ∞, L d,a degenerates into a reducible curve as follows: Lemma 2.3. (i) When the parameter a goes to zero, the curve L d,a degenerates into a reducible, connected curve L d,0 + C 0 + C ∞ , where C 0 (resp. C ∞ ) is the unique holomorphic curve of degree (1, 0, 0) (resp. (0, 1, 0)) in X going through the point (d, d, 0) (resp. (−1/d, −1/d, ∞)). (ii) When the parameter a goes to infinity, the curve L d,a degenerates into a reducible, connected curve L −1/d,0 + C ′ 0 + C ′ ∞ , where C ′ 0 (resp. C ′ ∞ ) is the unique holomorphic curve of degree (0, 1, 0) (resp. (1, 0, 0)) in X going through the point (d, d, 0) (resp. (−1/d, −1/d, ∞)). The involution of X interchanging the first and the second factors naturally induces an involution of M . It is explicitly given by (8) (d, a) → d, 1 a . and it follows that the involution of M fixes every points of K (which is obvious from the construction) and induces an isomorphism between M + and M − . Then the following lemma is important in proving our main result: Lemma 2.5. Let M , K and M \K = M + ∪ M − be as above, and fix t ∈ C with t = 0, ∞. Let u : M → X t = P × P be a (non-holomorphic) map defined by u(d, a) = d − at 1 + adt , ad − t a + dt . (Namely u assigns the intersection point L d,a ∩ X t for each (d, a) ∈ M .) Then (i) the map u is surjective, (ii) u(K) = ∆ t (= the diagonal of X t ) and u −1 (∆ t ) = K, (iii) u maps M + and M − diffeomorphically onto X t \ ∆ t . Proof. First we show u is surjective. By eliminating a from (3), we readily see that, if we fix d ∈ C and t ∈ C (t = 0, ∞), then the set of intersection points {L d,a ∩ X t \| a ∈ C * } is contained in an antiholomorphic curve C d,t in X t defined by (9) C d,t : y = d(1 + R)x + (R − \|d\| 2 ) (\|d\| 2 R − 1)x + d(1 + R) , where we set R = \|t\| 2 for simplicity, and (x, y) is an affine coordinate on X t = P × P as before. Note that (9) is the graph of (an anti-holomorphic) fractional transformation. It is easy to see that C d,t is not a graph of a constant function. On the other hand, viewing the x-factor of u as a fractional transformation of the variable a, the map P ∋ a → x(u(d, a)) ∈ P is readily verified to be surjective. It follows that the map P ∋ a → u(d, a) ∈ C d,t is surjective for any fixed d ∈ P and t = 0, ∞. Therefore, in order to prove the surjectivity of the map u : M → X t , it suffices to show that the map d → y defined by (9) is surjective, while x and t are fixed. To see this, we introduce a new variable c by setting c = d − x xd + 1 , which is a non-degenerate fractional transformation of d. Then the equation of C d,t becomes (10) y = − (1 − cx)R − c (c + x) (c + x)R + c (1 − cx) . It suffices to show that the map c → y defined by (10) is surjective. It is readily seen that (10) is the composition of two maps c → R c − c (=: b) and b → − b − x(1 + R) xb + (1 + R) . The former map contracts the circle \|c\| = √ R to a point {0} and maps the remaining two discs diffeomorphically onto P\{0}. The latter fractional transformation is readily verified to be biholomorphic on P, which maps the point {b = 0} to the point {x}. Thus, the map c → y is surjective, contracting the circle \|c\| = \|t\| to a point {x}. Equivalently, d → y is surjective, contracting the circle \|x − d\|/\|dx + 1\| = \|t\| to a point {x}. In particular we obtain (i) of the lemma. (ii) is immediate from Proposition 2.2 (iii). Finally we prove (iii). Since the Jacobian of u is calculated to be (1 + \|d\| 2 )(1 + \|t\| 2 )\|t\| 2 \|1 + adt\| 4 \|a + dt\| (The two extra components C 0 and C ∞ are contracted to points by ν + : Y → Z + . See Figure 1). • For reducible curves L d,0 + C ′ 0 + C ′ ∞ belonging to (C2), L d,0 is transformed into real line as in Proposition 2.2 (iv), and C ′ 0 and C ′ ∞ are transformed into B + 0 and B + ∞ respectively. (Note that B + 0 + B + ∞ is homologous to zero as mentioned in the first paragraph of this section.) We remark that if we consider ν − : Y → Z − instead of ν + : Y → Z + , the role of (C1) and (C2) in the final item are interchanged. Let L + (resp. L − ) be the family of curves in Z + whose members are the transformations of curves in X parameterized by M + (resp. M − ). Then the above 3 items mean that all members of L + are irreducible, while L − contains a 2-dimensional family of reducible members. Moreover, members of L + and L − are connected by a family of real curves which are not lines (parameterized by K). In particular, they are homologous in Z + . Then the following is the main result: Theorem 2.6. Let Z + and Q + ⊂ Z + be as before, and L + the family of real irreducible curves in Z + as above. Then the complement Z + \Q + has a structure of the twistor space of a hyperKähler metric on the cotangent bundle of P, which has L + as the set of twistor lines. By a result of Fujiki [5] the hyperKähler metric we obtain is precisely the Eguchi-Hanson metric [4]. Note that another family L − in Z + cannot be the set of twistor lines since it contains reducible members. Proof of Theorem 2.6. According to [7,Theorem 3.3], it suffices to show (a) Z + \Q + is foliated by members of L + ; namely Z + \Q + is covered by curves of L + , and that different curves of L + are always disjoint, (b) there is a non-vanishing holomorphic section of ∧ 2 T * Z/C ⊗ f * + O(2), (c) the parameter space M + of L + is diffeomorphic to the cotangent bundle of P. (Theorem 3.3 of [7] requires that the normal bundles of the curves in L + are isomorphic to O(1) ⊕ O(1). But this follows from (a). In fact it is easy to show that the normal bundles in question are isomorphic to O(1) ⊕ O(1) or O(2) ⊕ O. If there is a curve such that the latter holds, there has to exist a real 3-dimensional subfamily of L − such that every two members of the subfamily intersect at two points. This contradicts (a). ) We first verify (a). It is clear that Z + \(Z 0 ∪ Z ∞ ) is naturally biholomorphic to X\(X 0 ∪ X ∞ ) (≃ P × P × C * ), where Z t = f −1 + (t) and X t = f −1 (t) for t ∈ C. So any point z on Z + \(Z 0 ∪ Z ∞ ) can be specified by (x, y, t) for some t = 0, ∞ and (x, y) ∈ P × P. We have (Z + \Q + ) ∩ Z t ≃ X t \∆ t . Therefore, Lemma 2.5 directly implies that z = (x, y, t) ∈ Z + \Q + is passed by a unique member of L + . Next take any z ∈ Z 0 \Q + . By construction Z 0 is identified with the projectified normal bundle P(N ∆ 0 /X ) and we have N ∆ 0 /X ≃ N ∆ 0 /X 0 ⊕ N ∆ 0 /Q , where Q = ∆ × C ⊂ X as before. Thus the intersection point of a member of L + with Z 0 is represented by a tangent vector of the corresponding curve in X at its intersection point with X 0 . If d = ∞, by (3), a tangent vector v at the point is readily seen to be (11) v = −a 1 + \|d\| 2 , − 1 a 1 + \|d\| 2 , 1 . On the other hand, because ∆ 0 is defined by x − y = t = 0 in X, we can take (x − y, t) as a linear coordinate on the vector space T ξ X/T ξ ∆ 0 (≃ C 2 ), where ξ = (d, d, 0) denotes the intersection point. Thus the point Z 0 ∩ L d,a is represented by a vector (12) (1 + \|d\| 2 ) a − 1 a , 1 . It is easy to see that a map from C * to C defined by a → a − (1/a) has the following properties: it contracts U (1) = {\|a\| = 1} to the origin, and maps each connected component of C * \U (1) diffeomorphically onto C * . By (12) it follows that if we fix d ∈ C and allow the parameter a to move in the domain \|a\| < 1, then the intersection point with Z 0 takes each point of the fiber of Z 0 → ∆ 0 over ξ precisely once, except the two points corresponding to the two vectors represented by (1, 0) and (0, 1) in the above coordinate. The point corresponding to (0, 1) (which is represented by a trivial section of f : X → C) is then passed by the transformation of a curve of (C1). On the other hand, the point corresponding to (1, 0) (which is represented by a vector tangent to X 0 = f −1 (0)) is contained in Q + . So the point does not need to be passed. It remains to show that the fiber of Z 0 → ∆ 0 (outside Q + ) over (∞, ∞, 0) ∈ ∆ 0 is passed by some L ∞,a for a unique a with \|a\| < 1. But this can be also verified by easier calculation using (4) instead of (3). Furthermore, all points of Z ∞ \Q + are also passed by a unique member of L + , since Z ∞ = σ(Z 0 ) and all members of L + are real. Thus we have shown that Z + \Q + is foliated by members of L + , proving the assertion (a). Next we show the assertion (b). Since f + : Z + → C is a holomorphic submersion, there is an exact sequence (13) 0 −→ T Z + /C −→ T Z + −→ f * T C −→ 0. It follows that det T Z + ≃ ∧ 2 T Z + /C ⊗ f * O(2), and equivalently, K Z + ≃ ∧ 2 T * Z + /C ⊗ f * O(−2) . On the other hand, by (2), we have K Z + ≃ O(−2Q + ) ⊗ f * + O(−4) . From these we obtain ∧ 2 T * Z + /C ≃ O(−2Q + ) ⊗ f * O(−2). From this we conclude ∧ 2 T * Z + /C ⊗ f * O(2) is trivial over Q + . Thus we have shown (b). (c) is obvious since we have just seen that any point of Z 0 \Q + is passed by a unique member of L + and since Z 0 \Q + is isomorphic to the total space of O(2). This finishes a proof of Theorem 2.6. As in Z + , there are two families of real curves in Z − whose parameter spaces are M + and M − . If we again write them by L + and L − respectively, the complement Z − \Q − has a structure of the twistor space of a hyperKähler metric having L − as the set of twistor lines. Of course this is isomorphic to the hyperKähler metric obtained in Theorem 2.6. Finally we give some remarks on the automorphism group of Nagata threefold and its induced action on the 4-manifold M = M + ∪ K ∪ M − . The identity component of the holomorphic automorphism group Aut 0 (X) of X = P × P × P is of course the product of three copies of the Lie group P SL(2, C). An automorphism g = (g 1 , g 2 , g 3 ) ∈ Aut 0 (X) (g i ∈ P SL(2, C)) induces that on the Nagata threefold iff g preseves ∆ 0 and ∆ ∞ (which are the center of the blowing-up µ : Y → X). This condition is equivalent to the conditions g 1 = g 2 and g 3 ∈ C * , where C * is the subgroup consisting of automorphisms which fix 0 and ∞ ∈ P. Conversely, if an automorphism h of the Nagata threefold belongs to an identity component of the holomorphic automorphism group, h preserves the center of the blowing-down ν ± : Y → Z ± . Hence h always induces an automorphism on X, which necessarily belongs to the identity component. These mean that the identity component of the holomorphic automorphism group of the Nagata threefold is the subgroup P SL(2, C) × C * of Aut 0 (X) given by the injection (g 1 , g 3 ) → (g 1 , g 1 , g 3 ). Further, an automorphism g = (g 1 , g 1 , g 3 ) (g 1 ∈ P SL(2, C) and g 3 ∈ C * ) commutes with the real structure σ (defined by (1)) iff g 1 ∈ P SU (2) and g 3 ∈ U (1). This means that the identity component of the group of holomorphic automorphisms of the Nagata threefold commuting with the real structure is exactly P SU (2) × U (1). Since the action of P SU (2) × U (1) preserves ∆ 0 ∪ ∆ ∞ and commutes with the real structure, it naturally operates on the 4-manifold M . If g 1 ∈ P SU (2) is represented by the matrix α β −β α satisfying \|α\| 2 + \|β\| 2 = 1, then (g 1 , g 3 ) ∈ P SU (2) × U (1) maps the curve L d,a (defined by the equation (3)) to the curve L d ′ ,a ′ where d ′ and a ′ satisfy (14) d ′ = αd + β −βd + α , a ′ = α − βd α − βd · g 3 a This shows that the P SU (2)×U (1)-action on M preserves the decomposition M = M + ∪K ∪M − obtained in (7). Also it follows from (14) that the P SU (2) × U (1)-action on the U (1)-bundle K (over P) is transitive and its isotropy subgroup at the point (d, a) = (0, 1) is {id} × U (1). Thus K is diffeomorphic to P SU (2) = SO(3). Further, the U (1)-bundle map K → P (assigning d) is exactly obtained from the Hopf fibration S 3 → S 2 by dividing each fiber by multiplying −1 ∈ U (1). This means that K is identified with the unit circle bundle of the line bundle O(−2) → P. Therefore, the disk bundles M + → P and M − → P (also assigning d) are identified with a disk bundle of O(−2). Figure 1 . 1The limit curves (bold ones) as a → 0 and a → ∞ The proof is also straightforward from the expressions (3)-(5), if one keeps in mind that the degree of a curve is preserved even after taking limits. An important point in the lemma is that, due to the difference of degrees of the extra components (C 0 of degree (1, 0, 0) and C ′ 0 of degree (0, 1, 0), and also C ∞ of degree (0, 1, 0) and C ′ ∞ of degree (d, d ′ ∈ C ∪ {∞}. Thus we have obtained a family of real curves of degree (1, 1, 1) in X. It consists of the following three kinds of curves: (A) irreducible curves L d,a defined by (3) and (4), where d ∈ C ∪ {∞} and a ∈ C * with \|a\| = 1, (B) irreducible curves L d,a defined by(3)and(4), where d ∈ C ∪ {∞} and \|a\| = 1 (C) reducible curves L d,0+ C 0 + C ∞ as in (i) of Lemma 2.3, and L d ′ ,0 + C ′ 0 + C ′ ∞ as in (ii) of the lemma, where d, d ′ ∈ C ∪ {∞}.For explanation we divide (A) into two subfamilies (A1) and (A2), where (A1) consists of L d,a with \|a\| < 1 and (A2) consists of L d,a with \|a\| > 1. Similarly we define (C1) to be the family consisting of the former reducible curves L d,0 + C 0 + C ∞ in (C), and (C2) to be the family consisting of the latter reducible curves L d ′ ,0 + C ′ 0 + C ′ ∞ in (C). Then by Lemma 2.3, the union (A1) ∪ (C1) constitutes a connected family and (A2) ∪ (C2) constitutes (another) connected family. Note thanks to(6)these two connected families (A1) ∪ (C1) and (A2) ∪ (C2) contain no common curves. Let M + and M − be the parameter spaces of the former and the latter families of curves respectively. Let K be the parameter space of the curves (B). Since curves in (A) and (B) are clearly deformation equivalent, M + and M − are connected by K and we obtain a compact, connected 4-manifold (7) M := M + ∪ K ∪ M − . M parameterizes all curves in (A), (B) and (C). By the projection assigning d ∈ C ∪ {∞} = P, M + and M − have structures of disk bundles over P and K has a structure of a circle bundle over P. Consequently, M has a structure of an S 2 = P bundle over P. Remark 2.4. The 4-manifold M has a natural structure of a differential manifold which has (d, a), (1/d, a), (d, 1/a) and (1/d, 1/a) as complex local coordinates. From algebraic geometric point of view, M is a connected component of a real locus of a Hilbert scheme of curves in X = P × P × P whose degrees are (1, 1, 1) and which intersect both of ∆ 0 and ∆ ∞ . (This Hilbert scheme is readily shown to be 4-dimensional (over C).) fail to be locally diffeomorphic only on the set K = {\|a\| = 1}. Hence u : M \K → P×P\∆ is an unramified double covering. Moreover since P × P\∆ is simply connected, u : M \K = M + ∪ M − → P × P\∆ must be diffeomorphic on M + and M − respectively. This proves (iii) of the lemma. So far we have investigated a family of curves in X that can be transformed into real lines in Z + . Their parameter spaces are 4-manifolds M + and M − . Now we consider images of the curves under the birational transformation from X to Z + . The results are as follows: • All curves belonging to (A) and (B) are transformed into real irreducible curves in Z + . (Recall that curves in (B) are not transformed into lines by Proposition 2.2 (iv).) • Reducible curves belonging to (C1) are transformed into real irreducible curves in Z + . Department of Mathematics Graduate School of Science and Engineering Tokyo Institute of Technology 2-12-1, O-okayama, Meguro, 152-8551, JAPAN [email protected] Self-duality in four-dimensional Riemannian geometry. M Atiyah, N Hitchin, I Singer, Proc. Roy. Soc. London, Ser. A. 362M. Atiyah, N. Hitchin, I. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London, Ser. A 362 (1978), 425-461. O Biquard, Twisteurs des orbites coadjointes et métriques hyper-pseudo Kählériennes. 126O. Biquard, Twisteurs des orbites coadjointes et métriques hyper-pseudo Kählériennes, Bull. Soc. Math. France 126 (1998), 79-105. Some examples of the twistor constructions, in "Contributions to several complex variables: in honor of Wilhelm Stoll. D Burns, A. Howard and P. -M. WongVievegBraunschweigD. Burns, Some examples of the twistor constructions, in "Contributions to several complex variables: in honor of Wilhelm Stoll" (A. Howard and P. -M. Wong ed.) Vieveg, Braunschweig (1986), 51-67. Asymptotically flat solutions to Euclidean gravity. T Eguchi, A J Hanson, Phys. Lett. 74T. Eguchi, A.J. Hanson, Asymptotically flat solutions to Euclidean gravity, Phys. Lett. 74B (1978), 249-251. Nagata threefold and twistor space. A Fujiki, Quaternionic structures in mathematics and physics. Trieste; TriesteA. Fujiki, Nagata threefold and twistor space, Quaternionic structures in mathematics and physics (Trieste, 1994), 139-146 (electronic), Int. Sch. Adv. Stud. (SISSA), Trieste, 1998. Polygons and Gravitons. N Hitchin, Math. Proc. Cambridge Philos. Soc. 85N. Hitchin, Polygons and Gravitons, Math. Proc. Cambridge Philos. Soc. 85 (1979), 465-476. Hyper-Kähler metrics and supersymmetry. N J Hitchin, A Karlhede, U Lindström, M Roček, Comm. Math. Phys. 108N. J. Hitchin, A. Karlhede, U. Lindström, M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535-589. Existence theorems for non-projective complete algebraic varieties. M Nagata, Ill. J. Math. 2M. Nagata, Existence theorems for non-projective complete algebraic varieties, Ill. J. Math 2 (1958), 490-498.	[]
[ "Large Deviations for the Fleming-Viot process with neutral mutation and selection", "Large Deviations for the Fleming-Viot process with neutral mutation and selection" ]	[ "D A Dawson \nThe Fields Institute\nMcMaster University\n\n", "S Feng \nThe Fields Institute\nMcMaster University\n\n" ]	[ "The Fields Institute\nMcMaster University\n", "The Fields Institute\nMcMaster University\n" ]	[]	Large deviation principles are established for the Fleming-Viot processes with neutral mutation and selection, and the corresponding equilibrium measures as the sampling rate goes to 0. All results are rst proved for the nite allele model, and then generalized, through the projective limit technique, to the in nite allele model. Explicit expressions are obtained for the rate functions.Keywords: Fleming-Viot process, large deviations, relative entropy.AMS 1991 subject classi cations: Primary 60F10; secondary 92D10.(E)) is called non-degenerate if for every t in 0; T], (t) is non-degenerate.Remark: Any probability measure with support E is non-degenerate.De nition 3.2 A path ( ) 2 C( 0; T]; M pro 1 (E)) is called absolutely continuous if for every { 2 J , { ( )(t) is absolutely continuous as a multidimensional real valued function.	10.1016/s0304-4149(00)00070-3	[ "https://arxiv.org/pdf/math/9809203v1.pdf" ]	8,021,814	math/9809203	1bbe8365e3ef5330f98192842572e74ac987f225	Large Deviations for the Fleming-Viot process with neutral mutation and selection October 20, 1998 D A Dawson The Fields Institute McMaster University S Feng The Fields Institute McMaster University Large Deviations for the Fleming-Viot process with neutral mutation and selection October 20, 1998Fleming-Viot processlarge deviationsrelative entropy AMS 1991 subject classi cations: Primary 60F10; secondary 92D10 Large deviation principles are established for the Fleming-Viot processes with neutral mutation and selection, and the corresponding equilibrium measures as the sampling rate goes to 0. All results are rst proved for the nite allele model, and then generalized, through the projective limit technique, to the in nite allele model. Explicit expressions are obtained for the rate functions.Keywords: Fleming-Viot process, large deviations, relative entropy.AMS 1991 subject classi cations: Primary 60F10; secondary 92D10.(E)) is called non-degenerate if for every t in 0; T], (t) is non-degenerate.Remark: Any probability measure with support E is non-degenerate.De nition 3.2 A path ( ) 2 C( 0; T]; M pro 1 (E)) is called absolutely continuous if for every { 2 J , { ( )(t) is absolutely continuous as a multidimensional real valued function. Introduction The Fleming-Viot process is a measure-valued process describing the evolution of the distribution of genotypes in a population. In the case of two alleles it reduces to a one-dimensional di usion process that approximates the classical Wright-Fisher model. The standard model in population genetics involves mutation, replacement sampling, and selective advantages among various genotypes. Let E be a compact metric space, C(E) be the set of continuous functions on E, and M 1 (E) denote the space of all probability measures on E with the topology of weak convergence. Let "!0+ " 1 fF( + " x ) F( )g; 2 F( )= (x) (y) = lim " 1 !0+; " 2 !0+ (" 1 " 2 ) 1 fF( + " 1 x + " 2 y ) F( )g; Q( ; dx; dy) = (dx) x (dy) (dx) (dy); and x stands for the Dirac measure at x 2 E. The domain of L is D. E is called the type space, A is known as the mutation operator, and the last term in (1:1) describes the continuous sampling. If the mutation operator has the form of Af(x) = 2 R (f(y) f(x)) 0 (dy) with 0 2 M 1 (E), we call the process a Fleming-Viot process with neutral mutation. It is known that the Fleming-Viot process with neutral mutation has a unique reversible probability measure (cf. Ethier and Kurtz 6]). In the present article we will consider the limiting behavior of this process as ! 0: In the rst principal result we establish a large deviation principle (henceforth, LDP) for the sequence of reversible measures. It turns out that the sequence converges to the probability xed point 0 of the mutation operator exponentially fast and for = 1; 0 the deviation is characterized by the relative entropy de ned as H( 0 j ) = 8 < : R E h log hd if 0 1 otherwise, (1.2) where h is the Radon-Nikodym derivative of 0 with respect to . It is known, by Sanov's theorem, that the empirical measure f 1 n P n k=1 X k g n 1 of an i.i.d. sequence of random variables with common distribution 0 converges exponentially fast to 0 as n goes to in nity, and the deviation is characterized by the relative entropy H( j 0 ). Our example here may be the rst among the large deviation literature that has this \reversed" form of relative entropy as rate function. In Sanov's theorem, the in uence of sampling is dominant while in the Fleming-Viot case this in uence decreases to zero. This may be an explanation for the \reversed" expression of the two rate functions. The second principal result of this article establishes a path level LDP for the Fleming-Viot process with neutral mutation and with selection. This can be viewed as an Freidlin-Wentzell type result in in nite dimension. Furthermore, the existing results on large deviations for nite dimensional di usions usually assume either the di usion coe cient is non-degenerate or the square root of the di usion coe cient is uniformly Lipschitz continuous, but our model with nite alleles satis es neither of them. Hence our results also include an extension of the nite dimensional Freidlin-Wentzell theory. The sampling rate can be interpreted as the inverse population size and the large deviation results of this paper describe the deviations from the \in nite population" deterministic limit. The large deviation result for equilibrium measures is proved in Section 2 and the path level LDP is proved in Section 3. We will rst prove the LDP for the Fleming-Viot process with neutral mutation, and then, by the Cameron-Martin-Girsanov transformation, for the case with selection. For processes without selection we will rst prove the result for the nite allele model, and then generalize, through the projective limit technique, to the in nite allele model. LDP for Equilibrium Measures Let the following objects be given: X a Hausdor topological space, B X a -algebra of space X, fP " g ">0 a family of probability measures on (X; B X ), fa " g ">0 a family of positive numbers tending to zero as " goes to zero, and a function I : X ! 0; 1]. De nition 2.1 The function I : X ! 0; 1] is called a rate function if it is lower semicontinuous. A rate function is called the good rate function if for any r 0, the level set I (r) = fx 2 X : I(x) rg is compact. The constant a " is called the speed. De nition 2.2 fP " g satis es a LDP with the rate function (or good rate function) I if 1. for each B X -measurable open subset G of X lim inf "!0 a " log P " (G) inf x2G I(x); (2.1) 2. for each B X -measurable closed subset B of X lim sup "!0 a " log P " (B) inf x2B I(x); (2.2) Remark: The B X -measurable condition is needed when B is not the Borel -algebra and not all open or closed sets are B X -measurable. This situation may occur when the space X is not separable. When the space X is compact, all rate functions are good rate functions. Also the function I in the above de nition is unique when X is a regular topological space. For an excellent introduction to basic concepts and techniques of large deviations refer to 4]. In this section we will establish LDP for the equilibrium measures of the Fleming-Viot processes with neutral mutation and with selection. We will start with the case when the type space E is nite. Then, by using the projective limit technique, we obtain results for the case of E = 0; 1]. For any n 1, let E = f1; 2; ; ng. The space M 1 (E), the set of all probability measures on E, can be identi ed with the (n 1)-dimensional simplex n = fx = (x 1 ; ; x n ) : x i 0; i = 1; ; n; n X i=1 x i = 1g: The Fleming-Viot process with neutral mutation reduces to the neutral one-locus n-allele di usion process with generator A = 2 n X i;j=1 x i ( ij x j ) @ 2 @x i @x j + n X i=1 n X j=1 x j q ji @ @x i ; where q ji (j 6 = i) is the intensity of a mutation from allele j to allele i, and q jj = P i6 =j q ji : Let m denote the Lebesgue measure on n . If the in nitesimal matrix (q ij ) is irreducible, then this di usion has a unique stationary distribution which is absolutely continuous with respect to m. (cf. Shiga 11]) In the special case of parent-independent mutation, i.e., q ij = 2 p j > 0; 1 i 6 = j n; n X i=1 p i = 1; > 0: Wright 14] discovered that for p = (p 1 ; ; p n ), the unique stationary distribution ; ;p 2 M 1 ( n ) is the Dirichlet distribution with parameters p 1 ; ; p n given by where 0 < ; 1 ; ; n < 1 are some constants. For any " > 0, let C " = fx 2 C : min 1 i n x i "g. For any measurable function f on n , jjfjj L 1 denotes the L 1 norm of f with respect to measure m. Then we have for < min 1 i n f p i g, x 1 pn 1 n dx 1 dx n 1 ] = jjI C e P n i=1 ( p i ) log x i jj L 1 jjI A" e P n i=1 p i log x i jj L 1 ( 1 " ) n + m(C n C " ) jjI C e P n i=1 p i log x i jj L 1 ( 1 " ) + m(C n C " ): Letting ! 0, then " ! 0, we get lim !0 Z C x 1 p 1 1 1 x 1 pn 1 n dx 1 dx n 1 ] = ess supfI C e P n i=1 p i log x i : x 2 n g: (2.5) For any subset B of n , we have ess supfI B e P n i=1 p i log x i : x 2 n g e inf x2B P n i=1 p i log 1 x i : (2.6) On the other hand for any open subset G of n , ess supfI G e P n i=1 p i log x i : x 2 n g = ess supfe P n i=1 p i log x i : x 2 Gg x i + p i log p i )g: (2.9) Note that in the present situation the relative entropy H(pjx) of p with respect to x is given by H(pjx) = n X i=1 p i log p i x i ; and is non-negative, continuous. This combined with the compactness of n implies that all level sets of fx 2 n : H(pjx) rg are compact. Thus we have proved the following theorem. Theorem 2.1 When the mutation is parent independent, the family f ; ;p g >0 satis es a LDP on space n with the good rate function I p (x) = H(pjx) as goes to zero. Remark: It is well known that the relative entropy H(xjp) is the rate function describing the large deviations of the empirical measure of an i.i.d. sequence of random variables with common distribution p. In the case of = 1 the rate function in Theorem 2.1 has an \reversed" expression H(pjx). In Theorem 2.1, the probability measure p is positive for every type i in E. By making x i = 0 whenever p i = 0, the Dirichlet distribution can be de ned for any p 2 n as follows. Without loss of generality, let p = (p 1 ; ; p r ; 0; ; 0); r < n;p = (p 1 ; ; p r ). We de ne On the other hand, for any n 1, let h n (x) = n2 n X k=1 (k 1)=2 n I B k (x) + nI An (x); where B k = fz : (k 1)=2 n h(z) < k=2 n g, A n = fz : h(z) ng. The fact that h is integrable with respect to implies that h n is an increasing sequence of nonnegative simple functions converging to h almost surely with respect to . Let`= fB 1 ; ; B n2 n; A n g. Then we have sup \|2P H( \| ( )j \| ( )) H( `( )j `( )) Z 1 0 h n log h n d : Let n go to in nity, we get (2:14) by (1:2) and the monotone convergence theorem. 2 Now we are ready to prove large deviation results for equilibrium measures of some Fleming-Viot processes. A Fleming-Viot process with neutral mutation has a generator of the following form: For any symmetric bounded measurable function V (z; y) 2 B( 0; 1] 0; 1]) , let LF( ) = Z E A F( ) (z) (dz) + 2 Z E Z E 2 F( ) (z) (V ( ) = Z 1 0 Z 1 0 V (z; y) (dz) (dy); and h V ( ) ; F i = Z 1 0 Z 1 0 Z 1 0 F (z) V (z; y) V (y; w)] (dz) (dy) (dw): Then the generator of a Fleming-Viot process with neutral mutation and selection takes the form: L V F( ) = LF( ) + h V ( ) ; F i; (2.16) where V is called the tness function. It is known (cf. theorem 3.1 and theorem 3.2 in 6]) that the martingale problem associated with generators L and L V are well-posed. Shiga 12] shows that the Fleming-Viot process with generator L has a unique, reversible stationary distribution ; ; 0 2 M 1 (M 1 ( 0; 1])), which is the distribution of a M 1 ( 0; 1])-valued random variable characterized by the property that whenever r 2 and B 1 ; ; B r is a partition of 0; 1], ( (B 1 ); ; (B r )) has the Dirichlet distribution with parameters 1 0 (B 1 ); ; 1 0 (B r ). (Note that zero parameters can be removed to create a Dirichlet distribution on a simplex with dimension less than r.) The Fleming-Viot process with neutral mutation and selection also has a unique, reversible stationary distribution given by On the other hand, any element of X can be viewed as a nitely additive measure on 0; 1]. For any f 2 C( 0; 1]) and 2 X, we de ne the following "abstract integral" of f with respect to : h ; fi = lim n!1 2 n 1 X k=0 f(2 n k)x B k ; where B 0 = 0; 2 n ), B k = (2 n k; 2 n (k + 1)] for k = 1; ; 2 n 1. The existence of this limit is guaranteed since P 2 n 1 k=0 f(2 n k)x B k is a bounded Cauchy sequence. From this de nition and the fact that any decreasing sequence f n 2 C( 0; 1]) which converges to zero pointwisely converges to zero uniformly (Dini's theorem), we conclude that h ; fi is a linear form on space C( 0; 1]) satisfying: 3 Path Level LDP In this section we will establish three LDPs at the path level: LDP for nite type (or allele) model, LDP for the Fleming-Viot process with neutral mutation, and LDP for the Fleming-Viot process with neutral mutation and selection. The novelty of our result for nite type model is that the di usion coe cient of the corresponding di usion process is degenerate and the square root of the di usion coe cient is non-Lipschitz. This is an extension of the Freidlin-Wentzell theory. LDP For Finite Allele Model De ne S n = fx = (x 1 ; ; x n 1 ) : x i 0; i = 1; ; n 1; n 1 X i=1 x i 1g: The Fleming-Viot process with nite allele and neutral mutation is a nite dimensional di usion process described by the following system of stochastic di erential equations dx " k (t) = b k (x " (t))dt + " n 1 X l=1 kl (x " (t))dB l (t); 1 k n 1; (3.1) where x " (t) = (x " 1 (t); ; x " n 1 (t)), b k (x " (t)) = 2 (p k x " k (t)), and (x " (t)) = ( kl (x " (t))) 1 k;l n 1 is given by (x " (t)) 0 (x " (t)) = D(x " (t)) = (x " k (t)( kl x " l (t))) 1 k;l n 1 ; and " 2 = ,p k = 0 (k) > 0, B l (t); 1 l n 1 are independent Brownian motions. x fK c a g a: (3.3) The lemma follows since K a is a compact set in C( 0; T]; S n ). 2 Let @S n and S n denote the boundary and interior of S n , respectively. By direct calculation we get det(A(x)) = x 1 x n 1 (1 P n 1 i=1 x i ): Thus for any x 2 S n , D(x) is invertible and the inverse is given by By using the explicit expression of D 1 (x), we get the following for functions '(t) whose pathes are completely contained inside S n . D 1 (x) = 0 B B B B B B B B B B B B B B @ 1 P 1 C C C C C C C C C C C C C C A_ '(t) b('(t))]D 1 ('(t)) _ '(t) b('(t))] 0 (3.6) = n 1 X i;j=1;i6 =j _ ' i (t) b i ('(t))] _ ' j (t) b j ('(t))] 1 n 1 X k=1 ' k (t) 1 + n 1 X i=1 _ ' i (t) b i ('(t))] 2 1 P n 1 k=1;k6 =i ' k (t) ' i (t)(1 P n 1 k=1 ' k (t)) = n X i=1 ( _ ' i (t) b i ('(t))) 2 ' i (t) ; where ' n (t) = 1 P n 1 i=1 ' i (t). Lemma 3.2 If ' hits the boundary @S n , then I x (') = 1. Proof: We will prove this result by contradiction. Assume there is a ' such that I x (') < 1 and ' hits the boundary. Let t 0 2 (0; T] be the rst time that ' hits the boundary. Without loss of generality we further assume that the hitting occurs on the rst coordinate ' 1 . Now choose 0 < t 1 < t 2 < t 0 such that inf t2 t 1 ;t 2 ] fb 1 (' 1 (t))g > 0, and log(' 1 (t)) is absolutely continuous on t 1 ; t 2 ]. By direct calculation we get 2 Z t 2 t 1 _ ' 1 (t)b 1 ('(t)) ' 1 (t) dt Z t 2 t 1 ( _ ' 1 (t) b 1 ('(t))) 2 ' 1 (t) dt I x (') < 1: On the other hand, performing integration by parts twice, we get 2 Z t 2 t 1 _ ' 1 (t)b 1 ('(t)) ' 1 (t) dt = 2fb 1 ('(t 2 )) log(' 1 (t 2 )) b 1 ('(t 1 )) log(' 1 (t 1 )) + 2 Z t 2 t 1 log(' 1 (t)) _ ' 1 (t) dtg = 2fb 1 ('(t 2 )) log(' 1 (t 2 )) b 1 ('(t 1 )) log(' 1 (t 1 )) + 2 ' 1 (t 2 )(log(' 1 (t 2 )) 1) ' 1 (t 1 )(log(' 1 (t 1 )) 1)]g: Letting t 2 goes to t 0 , the above will go to in nity because the rst term goes to in nity while all other terms are bounded. This certainly contradicts the assumption of niteness of I x ('). 2 Hence if we de ne H x 0 = H x 1 \ C( 0; T]; S n ), then I x (') = 8 < : 1 2 R T 0 P n i=1 ( _ ' i (t) b i ('(t))) 2 ' i (t) dt; ' 2 H x 0 1; ' 6 2 H x 0 (3.7) Theorem 3.3 For any x 2 S n , the family fP " x g ">0 satis es a LDP on space C( 0; T]; S n ) with the good rate function I x ( ) and speed " 2 . Remark: If we introduce a map between spaces S n and n such that (x 1 ; ; x n 1 ) = (x 1 ; ; x n 1 ; 1 n 1 X i=1 x i ); and for simplicity, let fP " x g ">0 denote its image probability on space C( 0; T]; n ) under the map , then by contraction principle we have that for any x 2 n , the family fP " x g ">0 satis es a large deviation principle on space C( 0; T]; n ) with the good rate function I x ( ) and speed " 2 . Proof: By Corollary 3.4 in Pukhalskii 10] and Lemma 3.1, it su ces to show that for every ' 2 C x ( 0; T]; S n ); lim !0 lim inf "!0 " 2 log P " x f sup jx(t) f(t)j g (3.9) = lim !0 lim sup "!0 " 2 logP " x f sup t2 0;T ] jx(t) f(t)j g = Ĩ x (f): Replacing f with ' in (3:9), we end up with (3:8) because the corresponding probabilities in both equations are the same for < 0 . Now we are ready to prove (3:8) for paths that hit the boundary of S n . Let ' be such a path. The following is trivially true. lim !0 lim inf "!0 " 2 log P " x f sup t2 0;T ] jx(t) '(t)j g I x (') = 1: (3.10) On the other hand, let t 0 > 0 be the rst time when ' hits the boundary. Then for any t 2 (0; t 0 ), ' will not hit the boundary on 0; t]. By using an argument similar to that used in the derivation of (3:9) we get lim !0 lim sup "!0 " 2 logP " x f sup LDP for Fleming-Viot Processes with Neutral Mutation The Fleming-Viot process with neutral mutation has many nice properties. One of them is called the partition property, namely, given any nite partition of the type space E = K i=1 E i , then fX(t; E i ) : i = 1; ; Kg is a nite dimensional di usion process as in section 3.1.(cf. Ethier and Kurtz 7].) By using this property and the projective limit technique, we will establish a LDP for the Fleming-Viot process with neutral mutation. This kind of result can be viewed as an in nite dimensional generalization of the Freidlin-Wentzell theory. In the remainder of this article we will have E = 0; 1]. Consider the following family of partitions of space E J = ff 0; t 1 ]; (t 1 ; t 2 ]; ; (t n ; 1]g : 0 < t 1 < t 2 < < t n < 1; n = 1; 2; g; with the same partial ordering as in P of (2:12). It is clear that J P. Thus we will denote generic element of J by {; \|; etc., and for any {; \| 2 J , the space X { and the mappings {\| ; { are de ned accordingly. By an argument similar to that used in Section 2, we can show that the projective limit of (X { ; {\| ) {;\|2J is M 1 (E) and projective topology is stronger than the weak topology but weaker than the -topology. We will use M pro 1 (E) to denote the space M 1 (E) with this projective limit topology. Let C \| = C( 0; T]; X \| ) be equipped with the usual uniform topology. For any \| = fB 1 ; ; B r g { = fC 1 ; ; C l g, de ne a map p {\| between spaces C \| and C { such that For any 2 M 1 (E), let P ; ; 0 be the unique solution of the martingale problem associated with generator L in (2:15) starting at . p {\| : C \| ! C { ; (x B 1 (t); ; x Br (t)) ! ( X B k C 1 x B k (t); ; X B k C l x B k (t) De nition 3.1 A probability measure 2 M 1 (E) is called non-degenerate if for any \| 2 J , \| ( ) has no zero component. A path ( ) 2 C( 0; T]; M pro Theorem 3.4 For any non-degenerate 2 M 1 (E), the family fP ; ; 0 g satis es a LDP on space C( 0; T]; M pro 1 (E)) as goes to zero with the good rate function I ( ( )) = sup \|2J I \|( ) (p \| ( ( ))): (3.14) Remark: Let (t) be a path of the Fleming-Viot process with neutral mutation. Then for every t > 0, the support of (t) is a subset of the support of 0 . Therefore the essential part of the type space is the support of 0 . If we choose E to be the support of 0 , then the result still holds. Because of this we assume in the sequel, without loss of generality, that the support of 0 is E = 0; 1]. Proof: Since the path of Fleming-Viot process is continuous in the -topology (cf. Shiga 12]), we have that P ; ; 0 fC( 0; T]; M pro 1 (E))g = 1. On the other hand for any \| 2 J , by the nondegeneracy of , Theorem 3.3 and the remark following it, we have that p \| (P ; ; 0 ) satis es a LDP on space C \| with the good rate function I \|( ) ( ). Applying Theorem 3.3 in Dawson and G artner 3] we get the result. (3.22) where A is the formal adjoint of A de ne through the equality hA ( ); fi = h ; Afi. We defer the proof of this theorem to Appendix. LDP for Fleming-Viot Processes with Selection Finally we turn to prove the LDP of the Fleming-Viot process with selection. The generator of the process is given in (2:16). We will assume that the tness function V (x; y) is continuous on E 2 in the sequel. It is not hard to check that space (L 2 ( s; t] E)=L; jj jj) is a pre-Hilbert space. Let L 2 ( s; t] E) be the completion of space L 2 ( s; t] E)=L (cf. for existence refer to page 56 of Yosida 15]). Then (L 2 ( s; t] E); jj jj) becomes a Hilbert space. Let L 2 sub ( s; t] E) denote the closure in L 2 ( s; t] E) of the linear span of the set C 1;0 ( s; t] E). By an argument similar to Dawson and G artner 3] on pages 277-280, and the Hahn-Banach Extension Theorem (cf. page 106 of Yosida 15]) we have that l s;t is a bounded linear functional A be the generator of a Markov process on E with domain D(A). De ne D = fF : F( ) = f(h ; i)) (y)Q( ; dx; dy); Research supported by the Natural Science and Engineering Research Council of Canada where F( )= (x) = lim Next we consider the case when the type space E = 0; 1]. Let M 1 ( 0; 1]) be the collection of all probability measures on 0; 1] with the topology of weak convergence. Another topology on M 1 ( 0; 1]) needed in the sequel is called the topology which is the smallest topology such that for each bounded measurable function f on 0; 1], the map ! R 1 0 f(z) (dz) is continuous. The -topology on M 1 ( 0; 1]) is stronger than the weak topology. But when the type space E is nite, the two topologies coincide with each other. We use M 1 ( 0; 1]) to denote the space of all probability measures on 0; 1] equipped with the topology. Note that for each x 2 0; 1], let V x = f 2 M 1 ( 0; 1]) : (x) > 3=4g. Then each V x is an nonempty open set in the topology and for di erent x; y 2 0; 1], V x and V y are disjoint. Since there are uncountable number of such open sets, the space M 1 ( 0; 1]) is not separable. The -algebra B on M 1 ( 0; 1]) is de ned to be the smallest -algebra such that for every bounded, measurable function f on 0; 1], the map ! R 1 0 f(x)d (x) is measurable. It is known (cf. 4])that the Borel -algebra of space M 1 ( 0; 1]) is the same as B . We use B to denote this common -algebra throughout the remainder of this section. Let P = ffB 1 ; ; B r g : r 1; B 1 ; ; B r is a partition of 0; 1] by Borel measurable setsg(2.12) be the collection of all nite partitions of 0; 1]. For any 2 ), let H( j ) be the relative entropy of with respect to de ned in (1:2). Then it is known (cf. 5]If is not absolutely continuous with respect to , then both side of (2:14) are in nity. 13), we get H( \| ( )j \| ( )) H( j ) which implies that sup \|2P H( \| ( )j \| ( )) H( j ). ; dz; dy) = (dz) z (dy) (dz) (dy): First note that the set of all nite partitions P, partially ordered by \| { i \| is ner than {, is a partially ordered right-ltering set. For every \| = ( Daniell-Stone theorem (see e.g. page 197 of Bauer 1]), is a probability measure on the Borel -algebra of 0; 1]. The projective topology obtained is just the -topology. Hence we have X = M 1 ( 0; 1]). By Theorem 2.2 and Theorem 3.3 of Dawson and G artner 3], we have that the family f ; ; 0 g satis es a LDP on space X with the good rate Since C( ; 0 ; V ) is a constant, V ( ) is bounded continuous in the -topology, any level set of function I ;V 0 ( ) is a -closed subset of a level set of function I 0 ( ). Hence I ;V 0 ( ) is a good rate function in the -topology, and thus in the weak topology too. ) is compact and thus the family ; ; 0 ;V is exponential tight, by Bryc's inverse Varadhan Lemma (cf. section 4.4 of Dembo and Zeitouni 4]), we get that the family f ; For a xed T > 0 2 02and x 2 S n , let C( 0; T]; S n ) be the space of all S n -valued continuous functions on 0; T] endowed with the uniform topology, and P "x denote the law of x " ( ) starting at x. Lemma 3.1 The family fP " x g ">0 is exponentially tight on C( 0; T]; S n ) for all x replacing (x) with~ (x), we de ne the di usionx " (t), the lawP " x , the matrixÃ(x), and the functionĨ x ( ) respectively. It is easy to see that I x (') =Ĩ x ('). By using Theorem 5.6. with (3:10) and (3:13) implies that (3:8) holds for all ' 2 C( 0; T]; S n ). 2 clear that p {\| is continuous, and for any` \| {, p {`= p {\| p \|`. Now let C be the projective limit of the family f(C { ; p {\| ); {; \| 2 J g and p \| : C ! C \| be the corresponding projection. )) is a subset of C. On the other hand, let f \| ( ) : \| 2 J g be any element of C. Then for any t 2 0; T], f \| (t) : \| 2 J g can be identi ed as a unique element (t) of M pro 1 (E) and thus f \| ( ) : \| 2 J g can be identi ed as ( ). For any (a; b] E, by the de nition of projective limit, (t)((a; b]) is continuous in t. Hence ( It is clear that I ( ( )) = 1 for any ( ) not in H . The next theorem gives an variational form of the rate function I ( ( (t; y) 2 C 1;0 ( 0; T] E), let b 0 = 0 and \| (f)(t; y) = f(t; b k ); for y 2 B k ; k = 0; ; f)(s; x) \| (f)(s; y) Q( (s); dx; dy) dsg; where the inequality holds because g can be approximated pointwise by functions in the set f \| (together with (3:18), implies that I \|( ) (p \| ( ( )last equality follows from integration by parts (3:17). E) be the family of all continuous functions on E possessing continuous derivatives of all order. For any linear functional # on space C 1 (E) For any 2 2M 1 (E), let P ; ;V; 0 be the unique solution of the martingale problem associated with generator L V started at . For simplicity we will not distinguish between P ; ; 0 , P ; ;Vy; z) (dz)]Q( ; dx; dy):Then we have the following theorem. Theorem 3.7 For any non-degenerate 2 M 1 (E), the family fP ; ;V; 0 g satis es the following local LDP on space C w ( 0; T]; M pro 1 (E)) as goes to zero with the good rate function I ;V ( ( 25) for paths that are not absolutely continuous.Let ( ) be an arbitrary absolutely continuous path in C w ( 0; T]x; z 1 )V (y; z 2 ) (dz 1 ) (dz 2 ) (dx) (dy);which is also continuous in the topology generated by %. ( (s); x) V ( (s); x))(V ( (s); y) V ( (s); y))Q( (s); dx; dy)dsj < ": argument similar to that used in the derivation of (3:38) and H older'( (s); y) V ( (s); y))Q( (s); dx; dy)ds]gdP ; ; 0 ) = expf 1 V ( ( )) ( 3 2 + 2 )"]gP ; ; 0 fB ( ( ); )g ; in (3:25) will follow if we can show that for absolutely continuous path ( is the linear subspace of L 2 ( s; t] E) of all functions which are constant in space variable x. Let L 2 ( s; t] E)=L be the quotient space of L 2 ( s; t] E) module L. We introduce on L 2 ( s; t] i n 1;i6 =1 x i x 1 xn 1 xn 1 xn 1 xn 1 P 1 i n 1;i6 =2 x i x 2 xn 1 xn 1 xn 1 xn 1 P 1 i n 1;i6 =n 1 x i x n 1 xn (E)). By the Cameron-Martin-Girsanov transformation (seeDawson 2]) we have that, restricted on C( 0; T]; M pro 1 (E)), dP ; ; 0 dP ; ;V; 0 = Z V (T) = exp 1 G V ( ( ))] > 0;(3.23) where (E)) equipped with the subspace topology of C( 0; T]; M 1 (E)). For any ( ); ( ) 2 C w ( 0; T]; M pro 1 (E)), let Acknowledgements.We thank an anonymous referee for comments and suggestions that helped improve the presentation of this paper. Thus by the Riesz Representation Theorem, one can nd a function h in L 2 sub. on L 2 sub ( s; t] E)on L 2 sub ( s; t] E). Thus by the Riesz Representation Theorem, one can nd a function h in L 2 sub ( 0 Probability Theory and Elements of Measure Theory. H Bauer, Academic PressLondonBauer,H. Probability Theory and Elements of Measure Theory. Academic Press, London 1981. Geostochastic calculus. D A Dawson, Canadian Journal of Statistics. 6Dawson, D.A.(1978). Geostochastic calculus. Canadian Journal of Statistics, 6:143{168. Large deviations from the McKean-Vlasov limit for weakly interacting di usions. D A Dawson, J Artner, Stochastics. 20Dawson, D.A. and G artner, J.(1987). Large deviations from the McKean-Vlasov limit for weakly interacting di usions. Stochastics, 20:247{308. Large Deviations and Applications. A Dembo, O Zeitouni, Jones and Bartlett PublishersBostonDembo, A. and Zeitouni, O. Large Deviations and Applications. Jones and Bartlett Publishers, Boston 1993. Asymptotic evaluation of certain Markov process expectations for large time. M D Donsker, S R S Varadhan, I. Comm. Pure Appl. Math. 28Donsker, M.D. and Varadhan, S.R.S.(1975). Asymptotic evaluation of certain Markov process expectations for large time, I. Comm. Pure Appl. Math., 28, pp. 1-47. Fleming-Viot processes in population genetics. S N Ethier, T G Kurtz, SIAM J. Control and Optimization. 312Ethier, S.N. and Kurtz, T.G.(1993). Fleming-Viot processes in population genetics. SIAM J. Control and Optimization , 31, N.2:345{386. Convergence to Fleming-Viot processes in the weak atomic topology. S N Ethier, T G Kurtz, Stoch. Proc. Appl. 54Ethier, S.N. and Kurtz, T.G.(1994). Convergence to Fleming-Viot processes in the weak atomic topology. Stoch. Proc. Appl. , 54, 1{27. Random Perturbations of Dynamical Systems. M I Freidlin, A D Wentzell, Springer-VerlagNew YorkFreidlin, M.I. and Wentzell, A.D. Random Perturbations of Dynamical Systems. Springer- Verlag, New York 1984. Principles of Population Genetics. D L Hartl, A G Clark, Sinauer Associates, IncSunderland, Massachusetts2nd edHartl, D.L. and Clark, A.G. Principles of Population Genetics, 2nd ed. Sinauer Associates, Inc., Sunderland, Massachusetts, 1989. On functional principle of large deviations. A A Pukhalskii, Sazonov and T. Shervashidze, VSP Moks'las. MoskvaNew Trends in Probability and StatisticsPukhalskii, A.A.(1991). On functional principle of large deviations. in New Trends in Proba- bility and Statistics, ed. V. Sazonov and T. Shervashidze, VSP Moks'las, Moskva. Di usion processes in population genetics. T Shiga, J. Math. and Kyoto Univ. 21Shiga, T.(1981). Di usion processes in population genetics. J. Math. and Kyoto Univ., 21:133{ 151. A stochastic equation based on a Poisson system for a class of measure-valued di usion processes. T Shiga, J. Math. and Kyoto Univ. 30Shiga, T.(1990). A stochastic equation based on a Poisson system for a class of measure-valued di usion processes. J. Math. and Kyoto Univ., 30:245{279. Multidimensional Di usion Processes. D W Stroock, S R S Varadhan, Springer-VerlagBerlinStroock, D.W. and Varadhan, S.R.S. Multidimensional Di usion Processes. Springer-Verlag, Berlin 1979. Adaptation and selection. S Wright, Genetics, Paleontology, and Evolution. G.L. Jepson et alPrinceton Univ. PressWright, S.(1949). Adaptation and selection. in Genetics, Paleontology, and Evolution. ed. G.L. Jepson et al, 365-389, Princeton Univ. Press. Functional Analysis, Sixth Ed. K Yosida, Springer-VerlagBerlinYosida, K. Functional Analysis, Sixth Ed.. Springer-Verlag, Berlin 1980. Dawson The Fields Institute 222 College Street Toronto, Ontario Canada M5T 3J1 e-mail: don@ elds. A Donald, Donald A. Dawson The Fields Institute 222 College Street Toronto, Ontario Canada M5T 3J1 e-mail: don@ elds. elds.utoronto.ca	[]
[ "IEEE TRANSACTIONS ON IMAGE PROCESSING 1 Unsupervised active speaker detection in media content using cross-modal information", "IEEE TRANSACTIONS ON IMAGE PROCESSING 1 Unsupervised active speaker detection in media content using cross-modal information" ]	[ "Rahul Sharma ", "Fellow, IEEEShrikanth Narayanan " ]	[]	[]	We present a cross-modal unsupervised framework for active speaker detection in media content such as TV shows and movies. Machine learning advances have enabled impressive performance in identifying individuals from speech and facial images. We leverage speaker identity information from speech and faces, and formulate active speaker detection as a speechface assignment task such that the active speaker's face and the underlying speech identify the same person (character). We express the speech segments in terms of their associated speaker identity distances, from all other speech segments, to capture a relative identity structure for the video. Then we assign an active speaker's face to each speech segment from the concurrently appearing faces such that the obtained set of active speaker faces displays a similar relative identity structure. Furthermore, we propose a simple and effective approach to address speech segments where speakers are present off-screen. We evaluate the proposed system on three benchmark datasets-Visual Person Clustering dataset, AVA-active speaker dataset, and Columbia dataset-consisting of videos from entertainment and broadcast media, and show competitive performance to state-of-the-art fully supervised methods.	10.48550/arxiv.2209.11896	[ "https://export.arxiv.org/pdf/2209.11896v1.pdf" ]	252,531,425	2209.11896	660e2af3ab67fac8b27645d7a67c6f447f25cb7f	IEEE TRANSACTIONS ON IMAGE PROCESSING 1 Unsupervised active speaker detection in media content using cross-modal information Rahul Sharma Fellow, IEEEShrikanth Narayanan IEEE TRANSACTIONS ON IMAGE PROCESSING 1 Unsupervised active speaker detection in media content using cross-modal information Index Terms-unsupervisedcross-modalspeaker identityface recognitionactive speaker detectionentertainment media We present a cross-modal unsupervised framework for active speaker detection in media content such as TV shows and movies. Machine learning advances have enabled impressive performance in identifying individuals from speech and facial images. We leverage speaker identity information from speech and faces, and formulate active speaker detection as a speechface assignment task such that the active speaker's face and the underlying speech identify the same person (character). We express the speech segments in terms of their associated speaker identity distances, from all other speech segments, to capture a relative identity structure for the video. Then we assign an active speaker's face to each speech segment from the concurrently appearing faces such that the obtained set of active speaker faces displays a similar relative identity structure. Furthermore, we propose a simple and effective approach to address speech segments where speakers are present off-screen. We evaluate the proposed system on three benchmark datasets-Visual Person Clustering dataset, AVA-active speaker dataset, and Columbia dataset-consisting of videos from entertainment and broadcast media, and show competitive performance to state-of-the-art fully supervised methods. I. INTRODUCTION S PEECH-related activity in a video manifests spatiotemporally both in audio and visual modalities. Mapping speech activity in the temporal domain related to when, in time, someone is speaking is often approached from the audio modality. On the other hand, in the spatial domain, determining where in a frame the speaker is present relies on the visual signal. In this work, we use the speech activity information obtained from the audio modality to find the corresponding speech activity in the visual modality. We address the problem of finding the speaking face (if any) from the set of candidate faces appearing in the video. We refer to this task as active speaker detection (ASD). ASD plays a crucial role in computational media intelligence (CMI) [1], enabling the study of representation and portrayals of characters portrayed in media and their societal impact. Specifically, ASD is a crucial component of tools to discern who is speaking when? in a video, such as for primary character detection, background character detection [2], speaker diarization [3]- [5], and other humancomputer-interaction applications. This work focuses on ASD in entertainment media content, notably Hollywood movies and TV shows. Rahul Sharma ([email protected]) and Shrikanth Narayanan ([email protected]) is with Electrical and Computer Engineering, University of Southern California, USA ASD in entertainment media content is a challenging task due to the unknown and varying number of speakers and faces in the visual frames. The speaking faces appear in diverse poses because of the rich dynamics in character interactions. The face detector fails when the speakers appear in extreme poses including looking away from the camera or moving out of the frame. Non-ideal conditions in the audio modality, such as background (including off-screen) speakers and noisy speech, also contribute to problem's complexity. The inherent multimodal nature of speech activity has motivated numerous approaches for active speaker detection to use the connections between the visual and audio modality. The idea of localizing the visual actions in the video frames responsible for the audio activity, notably speech, dominates these approaches. Audio-visual speech modeling typically involves capturing the activity in the lip region of the detected faces in the visual frames, explicitly [3], [6] or implicitly [7], [8], and studying its correlation with the speech waveform to discern the active speaker. Most of these approaches share the same drawbacks: i) These methods employ computationally expensive 3D Convolutional neural networks and Recurrent neural networks to model the visual activity. ii) They train the models in a full-supervised manner requiring manual annotations, which are tedious and expensive to acquire. iii) A severe limitation is that the visual activity in the facial gestures concerning other activities, such as laughing, chewing, etc., can be easily confused with the speech activity, thus adding false positives. To complement the noisy visual activity information, we propose to use the speaker identity information for active speaker face-voice assignment. Using the knowledge that face and voice both possess information about a speaker's identity, we present an unsupervised framework that exploits the fact that both the active speaker's face in the visual modality and the concurrent voice in the audio modality refer to the same speaker. We use well researched and readily available face and speech recognition systems to represent the faces and speech, respectively, so that both capture the speaker's identity. Using the obtained representations, we construct a speechidentity distance matrix, capturing the relative identity structure of the speech segments through the entire video. Figure 1a shows an example of such a distance matrix. We assign an active speaker face to each speech segment such that the obtained set of corresponding active speaker faces displays a similar relative identity structure in form of a face-identity distance matrix, an example is shown in Figure 1b. Figure 2 shows an overview of the proposed strategy. Furthermore, studying the similarity between the two distance matrices (speech and face), we present a simple strategy to address the speech segments with speakers present off-screen. We evaluate the system's performance for active speaker detection on three benchmark datasets: i) AVA active speaker [9] and ii) Visual Person Clustering [10] datasets, consisting of videos of TV shows and movies, and iii) Columbia [11] dataset, consisting of a panel discussion video. The implementation code and the required dependencies to reproduce all the results in this work will be publicly available. The contributions of this work are as follows: 1) We present a novel framework to harness the speaker identity information, present within the speech in the audio modality and the speaker's face in the visual modality, for the speech-face assignment task. This framework can complement state-of-the-art visual-activitybased active speaker detection systems as the acquired information does not depend solely on the visual activity present in a frame. 2) We formulate the active speaker detection task as a cross-modal unsupervised optimization task deriving information from pretrained speaker recognition and face recognition models. This eliminates the need for manual annotations and does not require additional training of any expensive models. 3) We present an extensive performance evaluation of the system, validating the strategy's applicability for various challenging domains, including videos from TV shows, Hollywood movies, international movies, and panel discussions. Furthermore, we present ablation studies to study the performance of the proposed system and to highlight potential limitations. II. RELATED WORK An intuitive solution for detecting an active speaker is to study the activity in the lip region of faces in a video; this idea has led to a wide range of methods, from using just the visual signal [6] to posing it as multi-modal [6], [7], [12]- [14] and cross-modal [11], [15]- [17] formulations. One of the earlier works proposed explicitly detecting the lip region and quantifying the observed activity to classify faces as talking or non-talking in TV shows [6]. Along similar lines, sync-net [13] has proposed jointly modeling the activity in the explicitly detected lip region and the concurrent audio stream using a ConvNet architecture and training it to detect a speech-face synchronization leading to a self-supervised active speaker detection system. Following the success of modeling audiovisual synchronization, various self-supervised methods have emerged targeting the task of sound source localization [7], [8], [12], [18]. These methods were qualitatively shown to detect an active speaker's face as the location for the speech sound event. The nature of speech activity, interleaving the audio and visual modalities, has led to various cross-modal approaches, predicting primarily audio-centric tasks such as voice activity detection [11], [15]- [17] using the visual input. Earlier works [11], [15] have modeled the facial activity using HoG features and trained SVM-based classifiers using weak supervision from audio VAD labels. More recently, the authors in [16], [17] introduced a more sophisticated 3D ConvNets architecture to model the visual frames and trained a weaklysupervised system for active speaker detection for the more challenging Hollywood movie video domain. Lately, the introduction of large-scale audio-visual datasets, such as AVAactive speaker [9] and Active Speaker in Wild [19] consisting of manual active speaker annotations, has led to the emergence of a series of fully-supervised methods. These methods rely on the large-scale nature of labeled data to model the audio-visual activity by training train a wide variety of massive ConvNet architectures, ranging from 3D convolutions [14], [20] to graph convolutional networks [21], [22]. Unlike the extensively explored idea of modeling the visual activity, efforts in the direction of utilizing the speaker identity information for active speaker detection is relatively limited. The work of Hoover et al. [23] proposed to cluster the speech and faces separately to contain instances of a single speaker and then derive the information from the temporal co-occurrence of speech and face clusters to match speech and face for the same speaker. However, for the audio-visual diarization problem [24], the speaker's identity information from speech and faces is widely explored [3], [4], [25]. For instance, using the temporal co-occurrence of the speech and the face identities (obtained using face clusters) to derive the speech identities [25]. Inspired by the same, in this work, we propose to use the well-established speaker recognition [26], [27] and face recognition [28], [29] systems to capture the speaker's identity information from speech and faces and rely on speech and corresponding active speaker's face identifying the same character to derive speech-face associations. III. METHODOLOGY A. Problem Formulation The aim is to generate a speech-face association such that the associated face, if there is one, for every speech activity in the audio modality, is the source of the underlying speech. We formulate the problem definition under the following assumptions: i) that the video content under consideration is shot to capture the speaker's face, which stands valid, especially in the case of entertainment media videos. ii) that at Face-identity distance matrices for possible active-speaker faces sequences We gather temporally overlapping faces for each speaker-homogeneous speech segment. Using the speaker recognition embeddings, we construct a speech distance matrix. From the set of possible sequences of corresponding active speaker faces, we select the sequence of faces such that the face-identity distance matrix, obtained using the face-recognition features, displays a high resemblance with the speech-identity distance matrix. Table I. We observe that the fraction is insignificant for all the videos, thus validating the assumption. We use the contiguous segments of speech activity in the audio modality of one speaker as the fundamental unit for our analysis. By definition, such a voiced segment consists of voices from only one speaker, called a speaker-homogeneous voiced segment. We denote the set of all the speaker-homogeneous segments through a video as S all ≡ {s 1 . . . s n , . . . s N }. Similarly, we use a face-track (locations of one face appearing in consecutive frames) as the smallest unit for the visual modality. For each s n , we gather the temporally overlapping face-tracks available in visual modality and denote them as F n ≡ {f 1 , . . . f k , . . . f K }. A B C A B A B B C B B A We call f k as the active speaker face for s n , denoted as (s n ←→ f k ), if face-track f k in video modality is the source of speech segment s n in audio modality. However, a speech segment, s n , may have no face tracks available in the visual modality in its course, F n = φ; therein, the speaker is offscreen. Such segments are not relevant to this work, and hence purged from S all . We only consider the speech segments which have one or more face-tracks overlapping in time, be the active-speaker's face is on-screen (S on ) or off-screen (S off ), denoted in equation (1) and (2) S on ≡ {s n \| ∃f k ∈ F n : (s n ←→ f k )} (1) S off ≡ {s n \| F n = φ and f k ∈ F n : (s n ←→ f k )} (2) We formulate the task of active speaker detection (ASD) as the problem of finding an active speaker face-track for all the speaker-homogeneous speech segments, if there exists one. We denote the set of active speaker faces, A, as: A ≡ {a n \| a n = f k and f k ∈ F n and (s n ←→ f k )} (3) Such a formulation using the speaker-homogeneous speech segments helps reduce the ASD task to choosing one face track out of temporally overlapping tracks of (possibly multiple) faces for each speech segment. It also satisfies the constraint that active speaker faces can only be present when there is speech activity in the audio modality. At the same time, using a face track as one entity, rather than each face, helps facilitate that there can be at most one active speaker face in a frame which aligns with our assumption of non-overlapping speakers. This formulation intends to find an active speaker face for each speech segment, s n , but as mentioned earlier, the speech segments may have off-screen speakers (s n ∈ S off ), which may introduce false positives. Since the information separating the speech segments into the sets S on and S off is not readily available we propose a two-stage process. We first devise a system that provides an associated active speaker face-track for all speech segments in S a ≡ S on ∪ S off and call it stage1. This is then followed by a system to remove the speech segments with off-screen speakers from the set S all , to correct the introduced errors and call it stage2. B. Stage1: Speech-face Assignment Speech in the audio and face in the visual modality is widely known to possess an individual character's identity information. Utilizing it, we propose to select an active speaker face track for each speech segment s n from the set of temporally overlapping face tracks, a n ∈ F n , imposing that the selected face and the underlying speech depict the same character identity. Since the identity captured in the audio modality (using speech) can not be directly compared with the captured identity in the visual modality (using faces), we compute an identity distance matrix for the two modalities independently, bringing them into a shared space of relative identities. We start with initializing the set of active speaker face tracks A by randomly selecting a face track for each speech segment from the set of face tracks temporally overlapping with the speech segment, a n = f k \| f k ∈ F n ∀s n ∈ S all . Then we construct representations for the speech segments and the active speaker faces to capture the character identity information. For speech segments, we extract the speaker recognition embeddings using a ResNet34 [30] pretrained on VoxCeleb2 [27] with angular learning objective [26]. We represent the faces f k using SENet-50 [29], pretrained on MS-Celeb-1M [28], widely used in face recognition systems. Using the obtained representations, we construct a distance matrix for each modality. In the audio modality, we represent the speech segments, s n in terms of its distances from all other speech segments. We compute the distance between two speech segments s i and s j as the cosine distance between their speaker recognition embeddings. We call this matrix as the speech-identity distance matrix, SD and show it in equation 4. Similarly, in the visual modality, we construct a face-identity distance matrix, F D representing each active speaker face, a n in terms of its distance from all other active speaker faces. We compute the distances between two active speaker faces a i and a j as the cosine distance between their obtained face recognition embeddings, as shown in equation 4. While computing these distance matrices, we preserve the order of the speech segments and corresponding active speaker faces, implying that the i th row of SD and F D represents the speech segment, s i , and the corresponding active speaker face, a i , respectively. SD[i, j] = s i · s j s i s j F D[i, j] = a i · a j a i a j(4) The use of identity-capturing speaker recognition representations for speech segment, s n enables the distance matrix SD to capture the relative identity structure with context from the entire video. For instance, speech segments s i and s j from the same speaker will show a smaller value in SD[i, j] than those from different speakers. Similarly, the face-identity distance matrix F D constructed using the face recognition representations captures the relative identity of the active speaker faces in the visual modality. Figure 1 shows the SD and F D for the Hollywood movie Hidden Figures, where we have constructed the matrix F D using the ground truth active speaker faces. For the visualization purposes, we gather the speech segments of each speaker together. Since the active speaker's face and the concurrent speech must identify the same person, we hypothesize that the relative identity structure captured by F D should have a high resemblance with SD. We observe this similarity in Figure 1, in the form of similar low-distance square formations along the diagonals of both the matrices. The i th row in both the matrices depicted as SD[i] and F D[i] represents the identity of the underlying person (character) in terms of its distance from all other characters in the video. Since the components of the two rows come from distances computed in two independent dimensions (the audio and visual modality), they may differ in scale and thus may not be directly comparable. However, as the components of each row vector show the distance relative to other identities and all the identities are supposed to be the same among the two matrices, we expect the row vectors from the two matrices to show a linear relationship. So we propose to use Pearson's correlation as the measure to quantify the similarity between the two matrices. We formalize the ASD task as an optimization problem where we select the active speaker faces for each speech segment from the set of temporally overlapping face tracks, such that the Pearson's correlation between the speech-identity matrix and face-identity matrix, computed as an average of the row-wise correlations, is maximized. The formulation is denoted below in equation 5, where SD[i] and F D[i] denotes the vector corresponding to the i th rows of SD and F D respectively and N is the number of rows. Corr(F D) = 1 N i (SD[i] − SD[i])(F D[i] − F D[i]) (SD[i] − SD[i]) 2 (F D[i] − F D[i]) 2 (5) A ≡ arg max {an\|an=f k and f k ∈Fn} Corr(F D)(6) The notable point is that the face-identity distance matrix, F D is symmetric; the rows are not independent. The change in the active speaker's face for i th speech segment implies the change in the i th row of F D, which reflects a change in all the other rows of F D in the form of the altered i th column. Changed F D further reflects a change in the objective function Corr(F D). Thus, in the ideal case scenario, the active speaker faces for all the speech segments should be jointly selected to maximize the Corr(F D). Without loss of generality, let us consider that there are N speech segments in a video and, on average, k faces that temporally overlap each speech segment. The complexity for computing the objective function Corr(F D) for a given set of active speaker faces, A would be O(N 2 ), and there would exist k N such possible sets, A (all possible combinations of potential active speaker faces). Thus maximizing the Corr(F D) jointly for all active speaker faces will cost O(k N N 2 ). For a typical TV Hollywood movie, k ≈ 3 and N ≈ 900 bring the cost of jointly optimizing high and out of the scope of current computing standards. We propose to approximate the optimization process by iteratively finding the active speaker's face corresponding to one speech segment at one time to maximize the objective function Corr(F D), keeping the active speaker faces for all other speech segments fixed. Once iterating through all the speech segments (an epoch), we repeat the process until the Corr(F D) converges. This approximation reduces the complexity of the optimization from an exponential to a cubic order of N , O(EkN 3 ), where E is the number of epochs, kN is the number of times the Corr(F D) is computed in each epoch and O(N 2 ) is complexity for computing Corr(F D). Furthermore, assuming that even a portion of a movie has enough information to disambiguate between the speech and speaker's faces, we propose to split the number of speech segments into p partitions and optimize them separately. This helps further reduce the complexity of the process to O(Ek N 3 p 2 ) where p > 1. The process is detailed in Algorithm 1. Algorithm 1: Stage1: Speech-face assignation Random initialization:A ≡ {a n \| a n ∈ F n }∀s n ∈ S all ; Compute SD and F D ; // using eq: (4) objective ← Corr(F D) ; // using eq: (5) while objective increases do for each a i ∈ A do a i = arg max f k ∈Fn Corr(F D) ; // i th row Update F D; Update A; end objective ← Corr(F D); end C. Stage2: Off-screen speaker correction Maximizing the objective function, Corr(F D), we assign a corresponding active speaker face for all the speech segment under consideration, i.e., s n ∈ S all , which includes an active speaker face even for the speech segments having off-screen speakers, s n ∈ S Off . This incorrect assignment of speakers' faces introduces false positives. The underlying hypothesis to maximize the correlation between the speech-identity and the face-identity distance matrices is that since the active speaker's faces and the speech segments must identify the same person, the relative identity patterns from the speech-distance matrix (SD) and the facedistance matrix (F D) should show a high resemblance. In contrast, in the case of speech segments with speaker present off-screen, none of the face tracks temporally overlapping with the speech segment corresponds to the active speaker's face. Thus the speech segment represents an identity different from any of the concurrent face tracks. We hypothesize that this discrepancy in the identity leads to a relatively lower similarity between the identity representations obtained using the underlying speech segment and any of the temporally overlapping faces for the speech segments with off-screen speakers. Formally we classify the speech segment, s i , as the speech segment with an off-screen speaker if it displays a low enough row-wise correlation between the identity representations obtained from the speech-distance matrix, SD[i] and any of the temporally overlapping face tracks, F D[i], denoted in Eq 7. Furthermore, we remove the face tracks, earlier incorrectly classified as active speaker faces corresponding to such speech segments with off-screen speakers. S off ≡    s i (SD[i] − SD[i])(F D[i] − F D[i]) (SD[i] − SD[i]) 2 (F D[i] − F D[i]) 2 < τ   (7) IV. PERFORMANCE EVALUATION A. Implementation details We first obtain the active voice regions in a video using an off-the-shelf speech segmentation tool called pyannote [31]. As the proposed system requires speaker homogeneous speech segments, we use naive heuristics to segment further the obtained voice active regions instead of employing a sophisticated speaker change detection system. We partition the voiced regions at the shot boundaries, which we gather using PyScenedetect 1 . The partitioning at the shot boundary is motivated by speaker change being a prominent movie cut attribute [32], thus decreasing the likelihood of observing a speaker change in the obtained segments. We further partition the obtained segments to have a maximum duration of 1sec, effectively decreasing the fraction of speech segments consisting of a speaker change. We represent the speech segments using a ResNet-34 [26] model pretrained on Voxceleb2 [27] dataset for the speaker recognition task. Within the gathered shot boundaries, we use the Reti-naFace [33] to obtain face detections and track them using the SORT tracker [34] to construct face tracks. We extract identity representations for each face detection box using a SENet-50 [29], pretrained on MS-Celeb-1M [28] for the face recognition task, and average them over all the constituting faces to construct a representation for a face track. Once the representations for all the speaker homogeneous speech segments and all the faces are gathered, we initialize the active speaker face for each speech segment with a face randomly selected from a set of temporally overlapping faces. We partition the set of speech segments to contain at most 500 speech segments (L=500), grouped in temporal order, which helps reduce the optimization process's time complexity. We then employ the Algorithm 1 to assign an appropriate active speaker face to each of the speech segments for each partition (stage1) and further improve the predictions by discovering the speech segments with off-screen speakers (stage2). To evaluate the performance of the proposed system, we use three benchmark datasets: i) Visual person clustering dataset (VPCD) [10], ii) AVA-active speaker dataset [9] and iii) Columbia dataset [11]. The AVA-active speaker dataset consists of face-wise active speaker annotations for 161 international movies, freely available on YouTube. We use the publicly available validation split, consisting of 33 movies, to report and compare the performance of the proposed strategy against the state-of-the-art systems. VPCD is more exhaustive, consisting of character identity information in addition to the active speaker annotations for videos from several widely watched episodes of TV shows (Friends, Sherlock and TBBT) and 2 Hollywood feature films (Hidden Figures and About Last Night). It accompanies bounding boxes for the appearing faces and their corresponding character identities, along with the information about which character is speaking at any time. One of the challenges of working with VPCD is that the video files are not freely available. For purposes of this research, we acquired the video files using the DVDs and aligned the annotations in VPCD with the acquired videos. The Columbia dataset consists of manually obtained active speaker annotations for an 85 minutes long video of a panel discussion with six speakers. The panel discussion scenario is relatively more controlled than the movies in terms of frontal face poses and noise-free speech. B. Stage1: Speech-face assignation The hypothesis underlying the Algorithm 1 is that the relative identity patterns captured by the face-identity distance matrix F D, constructed using the active speaker faces, must show a high resemblance with the speech-identity distance matrix SD. In this section, we present the qualitative and quantitative validation of the hypothesis using videos from the VPCD. We compare the predictions for each face-bounding box against the ground truth and report the F1-score in Table II. In Figure 3 we show the evolution of the F D at various stages for the movie Hidden Figures. Using ground truth speaker identities, we order the speech segments in SD, and the corresponding predicted active speaker faces in F D to bring the speech segments from each speaker together. This makes the matrices interpretable and brings out the visible square patterns along the diagonal in SD in Figure 3a. Figure 3b shows the F D for the random initialization case with a low value of the Corr(F D), displayed in the form of minutely visible similarity with SD. However small, the visible similarity signifies the correct predictions for the trivial case of speech segments with just the speaker's face visible in the frames. The performance of the random chance system is quantified in terms of F1-scores in Table II. Figure 3c shows the F D post stage1 speech-face assignments, maximizing the Corr(F D) employing Algorithm 1 and displays significantly higher resemblance with SD, quantified by the higher value of Corr(F D). Figure 4 compares the objective function value (Corr(F D)) at various stages for the videos in VPCD, and shows the higher value fro stage1 than the random initialization case, consistently for all the videos. The higher resemblance translates to the enhanced active speaker detection performance, reported in terms of a higher F1-score in Table II, compared to the random chance system. We consistently observe a substantially higher F1score than the random chance baseline for all the videos in the VPCD. Although post optimization, the F D is visibly similar to SD (comparing Figure 3a&c), the amount of noise is still notable when compared against the F D obtained using the ground truth active speaker faces, shown in Figure 3e. The same is quantitatively visible in the difference between respective values of Corr(F D) and is due to the false positives introduced by the system for the speech segments with off-screen speakers. C. Stage2: Off-screen speaker correction The proposed strategy to segregate the speech segments with off-screen speakers relies on the hypothesis that for such speech segments, since the speaker's identity from the speech is different from any of the faces appearing in the frames, the row-wise correlations between SD and F D are relatively lower. To validate this hypothesis, we first divide the set of speech segments, using the ground truth, into S On (on-screen) and S Off (off-screen) and study the distribution of row-wise correlations between SD and F D for the two groups. We assign ground truth active speaker face for s i ∈ S On and a face randomly selected from the set of temporally overlapping face tracks for s i ∈ S Off and then construct the SD and F D using all speech segments s i ∈ S On ∪ S Off . In Figure 5, we show the distributions of row-wise correlations between SD and F D for the two sets of speech segments: i) Off-screen, S Off and ii) On-screen, S On , for the movie Hidden Figures. We observe a notable difference in the two distributions, which we quantify using the non-parametric Mann-Whitney U [35] test and obtain a p-value of 10 −70 verifying that the difference between the two distributions is statistically significant. We performed the same test for all other videos in VPCD and found that the hypothesis held for all of them. In Table III we report the fractional duration of speech segments having off-screen speakers (S Off : S all ) for each video in the VPCD and observe that the fraction can be significant for some videos and thus validating the need to address the induced errors in active speaker detection by the stage1 speech-face assignations. Using various values of τ in equation 7, we compute an ROC curve to evaluate the efficacy of the proposed strategy to classify each speech segment into S Off and S On . We report the area under ROC (auROC) curve in Table III for all the videos and notice the good performance of the simple strategy to discover speech segments with offscreen speakers. For all the videos in VPCD, we use τ = 0.1 to gather the speech segments with off-screen speakers and classify the faces, earlier incorrectly marked as active speaker faces in stage1, as non-speaking faces. In Figure 3d obtained F D for the movie, Hidden Figures and observe a visibly higher similarity with the speech-identity distance matrix (SD), further quantified in attained a higher value of Corr(F D), compared to F D post stage1. We further observe that the F D post stage2 looks even more similar to ground truth F D. In Figure 4, we show that the objective function Corr(F D) attains a higher value, after correcting the speech-face assignments for speech segments with off-screen speakers, compared to stage1 assignments for all the videos in VPCD. We report the F1-score for the active speaker detection using the corrected predictions in Table II (stage2) and compare them with the predictions from the stage1. We observe that offscreen speaker correction increases the F1-score consistently for all the videos. In Figure 6, we compare the performance of the two stages on a precision-recall curve and observe that the system's precision improves significantly with a slight drop in the recall. The enhanced precision is associated with removing the false positives introduced by assigning an active speaker face to speech segments with off-screen speakers (stage1), while the decline in the recall is due to the incorrect classification of a speech segment to have an off-screen speaker in stage2. D. Comparisons with state-of-the-art We evaluate the performance of the proposed strategy involving finding an active speaker face track for each speech segment and then removing the speech segments with offscreen speakers on the videos from the publicly provided validation split of the AVA dataset. We score the face boxes of the predicted active speaker face tracks with the rowwise correlation (correlation(SD[i], F D[i])) values where a higher correlation value signifies a better match in identities represented by the face and the speech. All other face boxes are scored -1, the minimum possible correlation value. We then use the AVA official implementation to evaluate the performance and report the mean average precision (mAP) values aggregated over all the videos in Table IV. We also report the performance of various recent state-of-the-art techniques in Table IV. We observe that the proposed unsupervised strategy of using the identity information performs well, although not as well as other fully supervised methods using videos from the AVA train set to model the visual activity in the frames. The primary reason for the observed lower performance lies with the nature of videos in the AVA dataset which consists of adverse audio conditions. Nearly 65% of the speech segments with a visible speaker overlap with either noise or music [9]. The presence of such noise conditions hinders the ability of the employed speaker recognition models to capture the identity information from speech segments, which naturally translates into poor active speaker detection performance. We further elaborate on the effect of speaker recognition performance on active speaker detection in section IV-E3. Using the same method as AVA, we report mAP values for the videos in the VPCD dataset (averaged over episodes for TV shows) in Table II. We observe that the performance for videos of TV shows, specifically Friends and TBBT, is notably higher than the movies (Hidden Figures and About Last Night). The scenes in Friends and TBBT predominantly consist of just primary characters' faces (characters who actively speak in the video), unlike the movies, which have a significant portion of scenes consisting of faces of background characters (i.e., characters who do not speak anytime in the video). The presence of primary characters' faces provides useful disambiguating information since we can gather their audio-visual identity information from the parts of videos where they speak. While the background characters' faces contribute to increased confusion since there is no way we can acquire their identity information. This leads to the displayed difference in the performances for the TV shows and the movies. Additionally, a higher fraction of off-screen speakers for the movies than the TV shows further contributes to the lower performance for movies (Table III). Selected video clips, presenting the system's active speaker predictions for several videos from VPCD are present in supplementary material. E. Ablation studies 1) Effects of VAD performance: The availability of the speaker-homogeneous speech segments is fundamental to the proposed approach. To obtain the speech segments, we use an off-the-shelf voice activity detector [31]. To make the obtained speech segments speaker-homogeneous, we use a simple approximation involving partitioning the speech segments by the scene boundaries and ensuring a maximum duration of 1sec. Since the number of speaker changes is constant in a video, splitting the speech segments into a larger number of partitions (obtained by making the partition length smaller) effectively decreases the fraction of speech segments consisting of a speaker change. Furthermore, the errors introduced by the speech segments with a speaker change can be the most for 1sec duration. To evaluate the effect of the proposed simple proxy for speaker-homogeneous speech segments, we compare the system's performance, on videos from VPCD, with the ideal case scenario. The VPCD annotations consist of characterwise speech segments, enabling to obtain ideal case speakerhomogeneous speech segments. We further segment the speech segments using the scene boundaries and ensuring a maximum duration of 1 sec. As the speech recognition methods' performance is susceptible to the duration of segments, splitting to maximum duration of 1 sec eliminates the impact of the difference in durations of speech segments in the two cases. In Figure 7 we compare the performance in terms of mAP for all the videos in VPCD for the two cases: i) System VAD using the proposed speaker-homogeneous proxy and ii) Oracle VAD using the ground truth speaker-homogeneous speech segments. We observe that the proposed setup using the VAD system and the speaker-homogeneous proxy approximates the ideal case scenario quite well, reflected in nearly equivalent mAP values for the two cases for most videos. On average, the system's performance attains nearly 91% of the performance of the ideal case. 2) Time complexity vs performance: In section III-B we proposed to split the video into p partitions and maximize the objective function, Corr(F D) for each partition independently to reduce the time complexity of the system to O(Ek N 3 p 2 ). It assumes that the obtained partitions have enough information to disambiguate the active speakers. For implementation purposes, we partition a video to have at maximum L speech segments in each partition. Using p = N L , we represent the time complexity of the system as O(EkN L 2 ), making it a linear function of the video's length, N and dependent on the parameter L. The length of a partition L signifies the context available to the video split to solve the speech-face assignments, implying that the larger values of L likely provide more disambiguating information and are ideal for the setup. In contrast, selecting a larger L increases the time complexity of the system, thus trading off the time for performance. In Figure 8 we compare the performance of the system (reported in mAP) and the computational time for various values of L = [200, 500, 800] and for using total video length at once (L = N ). For visualization purposes, we plot the computational time on a logarithmic scale. We observe a significant performance increase for L = 500 compared with L = 200 consistently for all the videos in VPCD. Further increasing L, we see marginal improvement in the performance along with a significant increase in computational time. Therefore we use L = 500 for all the videos providing enough context to the video partitions with reasonable computational time. Interestingly we also observe a marginal decrease in performance for the movies, Hidden Figures and About Last Night, and the TV show Sherlock when using the total video length (L = N ). This drop in performance is due to a more significant amount of noise in the form of off-screen speakers (Table III) compared to other videos, which accumulate while using a larger context. 3) Effects of speech and face recognition quality: The central idea of this work relies on the fact that speech and face both possess identity information and harness the certitude that the speaker's face and the underlying speech identify the same character. It makes capturing the identity information from speech and face fundamental to the system. Here we study the effect of the quality of captured speech and face identity information on the system's performance. We use two stateof-the-art speech recognition systems to construct different speech identity representations: i) ResNet-34 (default to the system) pretrained on Voxceleb2 dataset displaying 1.2% equal error rate (EER) for speaker recognition task on Voxceleb1 test set. ii) X-vector embeddings extracted from a TDNN model trained on Voxceleb1+Voxceleb2 training data and displays an EER of 3.2% on the same Voxceleb1 test set. Note that the lower the EER better is the speaker recognition performance; thus, ResNet-34 performs better for capturing speaker identity information from speech than x-vectors. In Figure 9, we show the performance of the active speaker detection system for videos from VPCD, utilizing the two speaker recognition systems. We observe that the performance displayed by the sub-optimal x-vectors embeddings is consistently lower for all the videos than the ResNet-derived embeddings. Using a non-parametric ManWhitneyU test, we noticed that the difference between the performance of the two systems is statistically significant. This implies that the suboptimal speaker recognition features lead to degraded active speaker recognition performance. It further explains that the low performance for the AVA validation set (reported in Table IV) is due to the sub-optimal speaker identity embeddings attributed to the noisy speech conditions. Similarly, to capture the impact of the face recognition quality, we use two state-of-the-art face recognition systems to construct the face identity representations: i) ResNet-50 trained on VGGFace2 (default to the system) and ii) Facenet trained on VGGFace2, compared on IJB-c displaying 0.95 and 0.66 True acceptance rate (TAR) respectively. Higher TAR implies better face recognition performance, pointing out that the Resnet-50 is a superior face recognition system to the Facenet. In Figure 9 we present the active speaker detection performance comparison of systems employing the two different face recognition systems and the same speaker recognition system (ResNet-34). We observe that the difference in performance is marginal and shows no statistical significance. It suggests that the system is more tolerable to degradation in face recognition performance. F. Discussion The proposed framework relies on the assumption that the video under consideration has enough information to disambiguate the characters in the video. The presence of scenes with just the speaking character in the frames is one of the simple examples of such character disambiguating information, as it provides an instant speech-face association. The videos intending to capture the speaking characters will likely have relevant information to derive the speech-face association based on the co-occurrence of the speech and faces through the video. On the contrary, one of the adverse limiting cases for the system is a video with a fixed set of faces appearing in the frames at all times. A simple example can be a panel discussion shot with a static camera, capturing all the panelists at all times. In such a video, the system will have no way to correctly associate the identity from the speech with one of the faces on the screen. Columbia dataset [11]: a video of a panel discussion is very similar to such a scenario. Even though the video is 85 min long, the annotations are available for 35 min duration, comprising six speakers. We evaluate the performance of the proposed system on the Columbia dataset and report the speaker-wise weighted F1-score, a widely used metric for evaluating this dataset, in Table V, along with performance reported by various other methods. We observe that the F1 score is pretty low for three speakers (Lieb, Long, and Sick), while for the other three, it is at par with other systems. We further investigated and observed that the face distance matrix F D obtained using the predicted active speaker faces attains a high value of the objective function Corr(F D). The speech (SD) and the face distance (F D) matrices are shown in Figure 10a & c. The obtained Corr(F D) is close to the one displayed by the ground truth active speaker faces, shown in Figure 10b, suggesting the intended working for the proposed system. On visual inspection of the predicted active speaker faces, we observed that the system incorrectly associates Long's speech with Sick's face and Lieb's with Long's face. The system selects Sick's face as an active speaker whenever Long speaks and similarly for the other two speakers. Relevant frames from the video with the system's predictions (green box: active speaker) and manually marked speakers' identities Figure 11a. We further observed that Sick always appears with Long throughout the video, providing the system with insufficient information to resolve the speech-face association. Similarly, Lieb always appears with Long, limiting the system to solve the discrepancy. To close the loop on the system's performance on the Columbia dataset, we simulate a setup by manually forcing disambiguating information into the video, which assists the system in solving the speech-face association. We randomly select 15% of the speech segments of each speaker and initialize the corresponding active speaker faces using the ground truth active speaker faces. While maximizing the objective function Corr(F D) (using Algorithm 1 in stage1), we forbid updating the active speaker face assignment for the selected speech segments. It replicates the trivial case of a scene with just the speaker's face visible in frames, thus synthetically injecting information to resolve speech-face associations. We show the face distance matrix (F D) for the obtained active speaker faces in Figure 10d and note that the objective function Corr(F D) is nearly equivalent to the one shown by the ground truth active speaker faces (Figure 10b). We report speaker-wise F1-score in Table V and observe that the assisted system performs similar to state-of-the-art methods for all the speakers. In Figure 11b we show the frames with corrected speech-face associations. Video clips highlighting the system's initial incorrect active speaker predictions and the later obtained corrected predictions using the assistance from the ground truth are present in supplementary material. We point out that this easy way of incorporating external information can be extended to integrate complementary information from visual-activity-based systems to generate a comprehensive solution to active speaker detection. V. SUMMARY AND CONCLUSION We presented a cross-modal unsupervised framework for active speaker detection in videos, harnessing the character identity information in the speech and the faces of the speakers. We evaluated the proposed system on three benchmark datasets, consisting of various videos from entertainment media (TV shows and movies) and broadcast media (panel discussion), and showed competitive performance compared to state-ofthe-art fully supervised methods. The framework utilizes offthe-shelf speech and face identity representation systems and shares their limitations when speech accompanies noise and music. The framework provides an easy way to inject external information, enabling it to collaborate with other methods. It can incorporate high-confidence predictions of other state-ofthe-art methods and even inputs from expert humans in the loop, enabling a provision to provide supervision at various degrees. One of the future directions can be to include the complementary set of information from visual-activity-based methods providing more disambiguating information to the system and thus further enhance the performance. Fig. 1 . 1a) Speech-identity distance matrix (SD) and b) Face-identity Distance matrix (F D) for the movie HiddenFigure.The active speaker faces are acquired form the ground truth. The speech segments are ordered to gather the speech segments of each speaker together. Fig. 2 . 2The overview of the proposed framework: 58 Fig. 3 . 583: Speech-face assignations Corr(FD) = 0.23 d) Stage2: Off-screen correction Corr(FD) = 0.49 e) GT active speaker faces Corr (FD) = 0.Distance matrices for the movie Hidden Figures at various stages of the system along with the value of objective function, Corr(F D). a) Speechidentity distance matrix (SD) b)Random: Face-identity distance matrix (F D) for randomly initialized ASD. c) Stage1: F D with speech-face assignation, maximizing Corr(F D). d) Stage2: F D post removing the speech segments with off-screen speaker. e) Ground truth: F D obtained using ground truth active speaker faces. Fig. 4 . 4Evolution of the objective function Corr(F D) at different stages for the videos in VPCD. Fig. 5 . 5Plot showing the individual contribution (corr i ) to O(Dv) vs the change in contribution (∆ i ) when substituted with other character faces, for speech segments a) having background speakers (S b ), b) having on-screen speakers (So). Fig. 6 . 6Increase in precision with a slight drop in recall introduced by correcting the speech-face assignments for speech segments with off-screen speakers (stage2). Fig. 7 . 7Comparison of the system's performance against the scenario with ideal case speaker-homogeneous speech segments, reported in terms of mAP for video from VPCD. The system's performance relative to the ideal case scenario is shown beneath each video. Fig. 8 . 8Computational time (in logarithmic scale) for the system against the performance (reported in mAP) for various values of the partition length L for the videos in VPCD. a) Speech distance matrix (SD) b) Ground truth FD (Corr(FD) = 0.79) c) System FD (Corr(FD) = 0.75) d) Assisted system FD (Corr(FD) = 0.79) Fig. 10. Distance matrices for Columbia dataset with Corr(F D). a) Speech distance matrix SD. b) F D with ground truth active speaker faces. c) F D post stage2 for proposed system. d) F D post stage 2 for the system assisted with 15% ground truth active speaker faces. Fig. 11 . 11system predictions: Confused speech-face for Lieb, Long and Sick b) Assisted system predictions: Correct speech-face assignations. Sample frames from Columbia dataset with manually marked identities of the faces and the name of the current speaker. a) The frames shows the incorrect speech-face associations by the proposed system. b) The frames shows the corrected speech-face associations when assisted with 15% ground truth faces are shown in TABLE I FRACTION IOF OVERLAPPING SPEECH (%) FOR VARIOUS VIDEOSVideos Fraction of overlapping speech (%) Friends 1.39 TBBT 4.42 Sherlock 0.79 Hidden Figures 0.44 About Last Night 0.21 AVA active speaker 2.31 any point in time, there can be at most one person speaking; thus, no overlapping speech. To investigate the validity of assuming no overlapping speech, we tabulate the fraction of overlapping-speech occurrences for several videos from VPCD and AVA active speaker datasets, in TABLE II F1 II-SCORES AT VARIOUS STAGES FOR THE VIDEOS FROM VPCD. REPORTED F1 SCORES ARE AVERAGED OVER ALL THE EPISODES FOR THE TV SHOWS (FRIENDS, TBBT, AND SHERLOCK).Videos Random initialization Stage 1 speech-face Stage 2 Off-screen speakers Stage 2 (mAP) Friends (25 episodes) 0.54 0.78 0.80 0.82 TBBT (6 episodes) 0.60 0.83 0.84 0.86 Sherlock (3 episodes) 0.55 0.64 0.66 0.70 Hidden Figures 0.45 0.62 0.67 0.66 About Last Night 0.51 0.59 0.62 0.57 TABLE III PERFORMANCE IIIFOR THE OFF-SCREEN SPEECH SEGMENT CLASSIFICATION FOR THE VIDEOS IN VPCD, IN TERMS OF AREA UNDER THE ROC CURVE (AUROC).Videos off-screen speech segments (%) Classification performance (auROC) Friends 0.1 0.8074 TBBT 0.1 0.8057 Sherlock 0.18 0.6927 Hidden Figures 0.21 0.8259 About Last Night 0.17 0.6897 TABLE IV COMPARISON IVWITH THE STATE-OF-THE-ART METHODS ON AVA ACTIVE SPEAKER VALIDATION SET IN TERMS OF MEAN AVERAGE PRECISION.Methods Stragetgy mAP (%) Roth et al. [9] Supervised 79.2 Zhang et al. [36] Supervised 84.0 Alxazar et al. [21] Supervised 87.1 Chung et al [14] Supervised 87.8 MAAS-TAN [22] Supervised 88.8 Proposed Unsupervised 71.40 Fig. 9. Comparison of active speaker detection performance for systems using different speech/face recognition architectures, on videos from the VPCD dataset. For speech embeddings, Resnet is superior to x-vectors and for face Resnet is superior to the Facenet.Friends TBBT Sherlock Hidden Figures About Last Night Videos 0.0 0.2 0.4 0.6 0.8 Mean average precision 0.75 0.81 0.62 0.55 0.49 0.82 0.86 0.70 0.66 0.58 0.82 0.86 0.69 0.69 0.58 x-vectors/Resnet Resnet/Resnet Resnet/Facenet TABLE V COMPARISON VOF THE SPEAKER-WISE WEIGHTED F1 SCORES FOR ALL THE SPEAKERS IN COLUMBIA DATASET.Methods Abbas Bell Boll Lieb Long Sick Avg Chakravarty et al. [11] - 82.9 65.8 73.6 86.9 81.8 78.2 Shahid et al [37] - 89.2 88.8 85.8 81.4 86 86.2 Chung et al. [13] - 93.7 83.4 86.8 97.7 86.1 89.5 Afouras et al. 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[ "DIOPHANTINE PROPERTY OF MATRICES AND ATTRACTORS OF PROJECTIVE ITERATED FUNCTION SYSTEMS IN RP 1", "DIOPHANTINE PROPERTY OF MATRICES AND ATTRACTORS OF PROJECTIVE ITERATED FUNCTION SYSTEMS IN RP 1" ]	[ "Boris Solomyak ", "Yuki Takahashi " ]	[]	[]	We prove that almost every finite collection of matrices in GL d (R) and SL d (R) with positive entries is Diophantine. Next we restrict ourselves to the case d = 2. A finite set of SL 2 (R) matrices induces a (generalized) iterated function system on the projective line RP 1 . Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.	10.1093/imrn/rnz309	[ "https://arxiv.org/pdf/1902.11059v2.pdf" ]	119,157,990	1902.11059	5cf33d755b9676d6c98d612b96d10ab17de98168	DIOPHANTINE PROPERTY OF MATRICES AND ATTRACTORS OF PROJECTIVE ITERATED FUNCTION SYSTEMS IN RP 1 28 Feb 2019 Boris Solomyak Yuki Takahashi DIOPHANTINE PROPERTY OF MATRICES AND ATTRACTORS OF PROJECTIVE ITERATED FUNCTION SYSTEMS IN RP 1 28 Feb 2019 We prove that almost every finite collection of matrices in GL d (R) and SL d (R) with positive entries is Diophantine. Next we restrict ourselves to the case d = 2. A finite set of SL 2 (R) matrices induces a (generalized) iterated function system on the projective line RP 1 . Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent. Introduction and main results Diophantine property of matrices. Recently there has been interest in Diophantine properties in non-Abelian groups. The following is a variant of [14,Definition 4.2]. Definition 1.1. Let A = {A i } i∈Λ be a finite subset of a topological group G equipped with a metric ̺. Write A i = A i1 · · · A in for i = i 1 . . . i n . We say that the set A is Diophantine if there exists a constant c > 0 such that for every n ∈ N, we have (1.1) i, j ∈ Λ n , A i = A j =⇒ ̺(A i , A j ) > c n . The set A is strongly Diophantine if there exists c > 0 such that for all n ∈ N, (1.2) i, j ∈ Λ n , i = j =⇒ ̺(A i , A j ) > c n . Clearly, A is strongly Diophantine if and only if it is Diophantine and generates a free semigroup. Gamburd, Jacobson, and Sarnak [14,Definition 4.2] gave a definition of a Diophantine set, which is equivalent to ours, except that they always consider symmetric sets (that is, g ∈ A ⇒ g −1 ∈ A). Diophantine-type questions in groups arise in connection with spectral gap estimates, see [14,8]. See [1,2] for a recent discussion of Diophantine properties in groups and related problems. In [2] a Lie group G is called Diophantine, if almost every k elements of G, chosen independently at random according to the Haar measure, together with their inverses, form a Diophantine set in G. Gamburd et al. [14] conjectured that SU 2 (R) is Diophantine. More generally, it is conjectured that semi-simple Lie groups are Diophantine. Kaloshin and Rodnianski [18] proved a weaker Diophantine-type property: for a.e. (A, B) ∈ SO 3 (R) × SO 3 (R), there exists c > 0 such that for any n 1 and any two distinct words W 1 , W 2 over the set A = {A, B, A −1 , B −1 } of length n, W 1 − W 2 c n 2 . It is mentioned in [18] that their method is general, and applies to SU 2 (R) as well, and also to m-tuples of matrices for any m 2. Next we state our first result. For any collection of linearly independent vectors v 1 , . . . , v d in R d consider the simplicial cone (1.3) Σ = Σ v1,...,v d = {x 1 v 1 + · · · + x d v d : x 1 , . . . , x d 0}. If a matrix A ∈ GL d (R) satisfies A(Σ {0}) ⊂ Σ • , we say that Σ is strictly invariant for A. Given a cone Σ = Σ v1,...,v d , denote by X Σ,m (respectively, Y Σ,m ) the set of all GL d (R) (respectively, SL d (R)) m-tuples of matrices for which Σ is strictly invariant. We consider X Σ,m as an open subset of R d 2 m and Y Σ,m as a (d 2 − 1)m-dimensional manifold. (i) For a.e. A ∈ X Σ,m , the m-tuple A is strongly Diophantine. In particular, a.e. m-tuple of positive GL d (R) matrices is strongly Diophantine. (ii) For a.e. A ∈ Y Σ,m , the m-tuple A is strongly Diophantine. In particular, a.e. m-tuple of positive SL d (R) matrices is strongly Diophantine. Remark 1.3. 1. Unfortunately, our results do no cover any example of a symmetric set, since the strict invariance property cannot hold for a matrix A and A −1 simultaneously. 2. Every m-tuple of matrices with algebraic entries is Diophantine (but not necessarily strongly Diophantine), see, e.g., [14,Prop. 4.3]. 3. It is well-known that Diophantine numbers in R form a set of full measure, which is, however, meagre in Baire category sense (its complement contains a dense G δ set). Baire category genericity of non-Diophantine m-tuples in SU 2 (R) has been pointed out in [14]. In G = SL d ( (R) is open, and a standard argument shows that a generic m-tuple that contains an elliptic matrix is not Diophantine. The scheme of the proof of Theorem 1.2 is as follows. We consider the induced action of the matrices on the projective space, and show that, given a non-degenerate family of m-tuples strictly preserving an open set, depending on a parameter real-analytically, for all parameters outside an exceptional set of zero Hausdorff dimension, the induced iterated function system (IFS) satisfies a version of the "exponential separation condition". This property implies the strong Diophantine condition for the matrices. We then locally foliate the space of m-tuples of matrices and apply Fubini's Theorem. The result on the zero-Hausdorff dimensional set of exceptions uses the notion of order-k transversality, which is a modified version of that which appeared in the work of Hochman [15,16]. The strict open set preservation property is needed to ensure that the induced IFS is contracting (uniformly hyperbolic). Projective IFS and linear cocycles. Let A = {A i } i∈Λ be a finite collection of SL d (R) matrices. The linear action of SL d (R) on R d induces an action on the projective space RP d−1 , and thus A defines an IFS Φ A = {ϕ A } A∈A on RP d−1 , called a (real) projective IFS. Such IFS were studied by Barnsley and Vince [4], and by De Leo [11,10]. Following [4], we say that the IFS Φ A has an attractor K if for every nonempty compact set B in a neighborhood of K, we have lim k→∞ Φ k A (B) = K in the Hausdorff metric, where Φ A (B) = A∈A ϕ A (B). Let A = {A i } i∈Λ be a finite collection of GL d (R) matrices and Φ A the associated IFS on RP d−1 . An alternative, but closely related viewpoint, is to consider the linear cocycle A : Λ Z → SL d (R) over the shift on Λ Z , defined by A(i) = A i1 . Strict contractivity of the projective IFS turns out to be equivalent to uniform hyperbolicity of the cocycle [6]. Here we restrict ourselves to the case of d = 2, which was investigated in great detail by Avila, Bochi, and Yoccoz [3]. There is a natural identification between [0, π) and the projective space RP 1 . Below we use this identification freely, and whenever necessary we view [0, π) as R/πZ. For A ∈ GL 2 (R) denote the action of A on [0, π) ∼ = RP 1 by the symbol ϕ A . Denote by d P the metric on RP 1 induced from the identification with R/πZ. Below we work with m-tuples of SL 2 (R)-matrices, since the action of GL 2 (R) factors through the SL 2 (R) action in the obvious way, via A → (det A) −1 A. In the following theorem we extracted the results relevant for us from [3,4] (note that [4] considers real projective IFS of any dimension). Theorem 1.5 ( [3,4]). Let A = {A i } i∈Λ be a family of SL 2 (R) matrices and let Φ A be the associated IFS on RP 1 . The following are equivalent: (i) the IFS Φ A has an attractor K = RP 1 ; (ii) the associated linear cocycle over Λ Z is uniformly hyperbolic; (iii) there is a multicone U , such that Φ A (U ) ⊂ U ; (iv) there is nonempty open set V ⊂ RP 1 such that Φ A is contractive on V , with respect to a metric equivalent to d P . Following [3], we will call a multicone satisfying Φ A (U ) ⊂ U , a strictly invariant multicone for the family of matrices and for the IFS. There are examples, see [3], which show that one may need a multicone having k components, for any given k 2, even for a pair of SL 2 (R) matrices {A 1 , A 2 }. Our next result concerns the dimension of the attractor. Following De Leo [11], consider the ζ-function ζ A (t) = n 1 i∈Λ n A i −t , and define the critical exponent of A by (1.4) s A = sup t 0 {t : ζ A (t) = ∞}. Theorem 1.6. Let A = {A i } i∈Λ be a finite set of SL 2 (R) matrices which has a strictly invariant multicone (or satisfies any of the equivalent conditions from Theorem 1.5), and let K be the attractor of the associated IFS Φ A on RP 1 . Assume that at least two of the maps ϕ Ai have distinct attracting fixed points. If A is strongly Diophantine, then dim H (K) = min{1, 1 2 s A }, where s A is the critical exponent (1.4). In the special case when the IFS Φ A satisfies the Open Set Condition, this result is due to De Leo [11,Th.4]. Recall that the strong Diophantine condition holds, in particular, when A generates a free semigroup and all the entries of A i are algebraic. Remark 1.7. It is further shown in [11] that for A hyperbolic (and in some parabolic cases), s A = lim r→∞ N A (r) log r , where N A (r) is the number of elements of norm r of the semigroup generated by A. An analogy is pointed out with the classical results on Kleinian and Fuchsian groups, see, e.g., [26]. Let Φ = Φ A . An alternative way to express the dimension, and one we actually use in the proof, is in terms of Bowen's pressure formula (1.5) P Φ (s) = 0, where P Φ (·) is the pressure function associated with the IFS Φ. Throughout the paper we use the notation ϕ i = ϕ i1 . . . ϕ in . The pressure is defined by (1.6) P Φ (t) = lim n→∞ 1 n log i∈Λ n ϕ ′ i t , where · is the supremum norm on U . As will be clear from the Bounded Distortion Property, the definition of P Φ (t) does not depend on the choice of strictly invariant multicone U , and moreover, (1.7) 2s = s A . It is a classical result, going back to Bowen [9] and Ruelle [22], see also [12], that if {ϕ i } i∈Λ is a hyperbolic IFS on R of smoothness C 1+ε , satisfying the Open Set Condition, then the dimension of the attractor K is given by the Bowen's equation. In the case that the maps ϕ i are affine, s > 0 is the unique solution of i∈Λ r s i = 1, where r i ∈ (0, 1) is the contraction ratio of ϕ i . For an IFS with overlaps this is not necessarily true. In [24], Simon, Solomyak, and Urbański showed that for a one-parameter family of nonlinear IFS with overlaps (hyperbolic and some parabolic) satisfying the order-1 transversality condition, for Lebesgue-a.e. parameter the dimension of the attractor is given by (1.8) dim H (K) = min{1, s}, where s is from (1.5) and the pressure is given by (1.6). Definition 1.8. Let F = {f i } i∈Λ be an IFS on a metric space (X, ̺), that is, f i : X → X. We say that F satisfies the exponential separation condition on a set J ⊂ X if there exists c > 0 such that for all n ∈ N we have (1.9) sup x∈J ̺(f i (x), f j (x)) > c n , for all i, j ∈ Λ n with i 1 = j 1 and f i ≡ f j . If, in addition, the semigroup generated by F is free, that is, f i ≡ f j ⇐⇒ i = j, we say that F satisfies the strong exponential separation condition. If these properties hold for infinitely many n, then we say that F satisfies the (strong) exponential separation condition on J along a subsequence. It is rather straightforward to show that the (strong) Diophantine condition for an m-tuple in SL 2 (R) matrices is equivalent to the (strong) exponential separation condition for the associated projective IFS (see Lemma 3.6 below). In [15, Cor. 1.2], Hochman proved (1.8) for an affine IFS F = {f i } i∈Λ satisfying the exponential separation condition on J = {0} along a subsequence. Thus our Theorem 1.6 is, in a sense, a generalization of Hochman's result to the case of contractive projective IFS. Remark 1.9. In fact, Hochman [15] used the condition (1.9) without the require- ment i 1 = j 1 . However, for an IFS {f i } i∈Λ on an interval J ⊂ R, such that inf x∈J,i∈Λ \|f ′ i (x)\| r min > 0, requiring i 1 = j 1 in (1.9) does not weaken the exponential separation conditionit only affects the constant c. This follows from the estimate \|f i (x) − f j (x)\| = \|f (i∧j)u (x) − f (i∧j)v (x)\| r n min \|f u (x) − f v (x)\|, i, j ∈ Λ n , where i ∧ j is the common initial segment of i and j, so that u 1 = v 1 . IFS of linear fractional transformations. It is well-known that the action of GL 2 (R) on RP 1 can be expressed in terms of linear fractional transformations. For A = a b c d ∈ GL 2 (R), let f A (x) = (ax + b)/(cx + d), and define ψ : [0, π) → R * by ψ(θ) = cos θ/ sin θ, where R * = R ∪ {∞}. It is easy to see that the following diagram commutes: [0, π) [0, π) R * R * ψ ϕ A f A ψ Observe that ψ is smooth, and on any compact subset of (0, π) the derivatives of ψ and ψ −1 are bounded. The following is then an immediate corollary of Theorem 1.6. f i (U ) ⊂ U for all i ∈ Λ. If F satisfies the strong exponential separation condition on U , then we have dim H (K) = min{1, s}, where s > 0 is the unique zero of the pressure function P F . 1.4. Furstenberg measure. Let A = {A i } i∈Λ be a finite collection of SL 2 (R) matrices, and let p = (p i ) i∈Λ be a probability vector. Assume that p i > 0 for all i ∈ Λ (we always assume this for any probability vector). We consider the finitely supported probability measure µ on SL 2 (R): (1.10) µ = i∈Λ p i δ Ai . Our standing assumption is that A generates an unbounded and totally irreducible subgroup (i.e., does not preserve any finite set in RP 1 ). Then there exists a unique probability measure ν on RP 1 satisfying µ · ν = ν, that is, (1.11) ν = i∈Λ p i A i ν, where A i ν is the push-forward of ν under the action of A i , see [13]. The measure ν is the stationary measure, or the Furstenberg measure, for the random matrix product A in · · · A i1 where the matrices are chosen i.i.d. from A according to the probability vector p. The properties of the Furstenberg measure for SL 2 (R) random matrix products, such as absolute continuity, singularity, Hausdorff dimension, etc., were studied by many authors, including [19,7]. In [21,20,25] this investigation was linked with the study of IFS consisting of linear fractional transformations. The reader is referred to [17] for a discussion of more recent applications. We will recall the main result of [17], since it will be the main tool in proving Theorem 1.6. Let χ A,p be the Lyapunov exponent, which is the almost sure value of the limit (1.12) lim n→∞ 1 n log A i1···in , where i 1 , i 2 , · · · ∈ Λ is a sequence chosen randomly according to the probability vector p = (p i ) i∈Λ . The Lyapunov exponent is usually defined as the almost sure value of the limit (1.13) lim n→∞ 1 n log A in···i1 , but it is easy to see that (1.12) and (1.13) define the same value (e.g., by Egorov's Theorem). Under the standing assumptions, the limit exists almost surely and is positive [13]. The Hausdorff dimension of a measure ν is defined by dim H (ν) = inf{dim H (E) : ν(E c ) = 0}. For a probability vector p = (p i ) i∈Λ , we denote the entropy H(p) by H(p) = − i∈Λ p i log p i . Theorem 1.11 ([17]). Let A = {A i } i∈Λ be a finite collection of SL 2 (R) matrices. Assume that A is strongly Diophantine and generates an unbounded and totally irreducible subgroup. Let p = (p i ) i∈Λ be a probability vector, and let ν be the associated Furstenberg measure. Then we have (1.14) dim H (ν) = min 1, H(p) 2χ A,p . Theorem 1.2 implies, in particular, that the dimension formula (1.14) holds for the Furstenberg measure associated with a.e. finite family of positive matrices (independent of the probability vector). Next we address the question: what is the Hausdorff dimension of the support of the Furstenberg measure? Sometimes, the support is all of RP 1 , in which case the answer is trivially one. The definition (1.11) implies that the support is invariant under the IFS Φ induced by A. Thus, Theorem 1.6 has the following immediate corollary: Corollary 1.12. Let A = {A i } i∈Λ be a Diophantine set of SL 2 (R) matrices which has a strictly invariant multicone, µ a finitely supported measure defined by (1.10), and ν the associated Furstenberg measure. Then dim H (supp ν) = min{1, 1 2 s A }, where s A is the critical exponent of A. Denote by H m the set of m-tuples in SL 2 (R) which have a strictly invariant multicone. Avila (see [27,Prop.6]) proved that the interior of the complement of H m in (SL 2 (R)) m is E m , where E m is the set of m-tuples which generate a semigroup containing an elliptic matrix. Observe that if an elliptic matrix is conjugate to an irrational rotation, then certainly the invariant set (support of the Furstenberg measure) is all of RP 1 . On the other hand, if it is conjugate to a rational rotation, then the semigroup generated by A contains the identity and the strong Diophantine property fails. We expect that our methods can be extended to cover strongly Diophantine families on the boundary of H m , which include parabolic systems. 1.5. Structure of the paper. The rest of the paper is organized as follows. In the next section we prove Theorem 1.2. In Section 3 we consider projective IFS and prove Theorem 1.6. Finally, in Section 4 we include proofs of some standard technical results for the reader's convenience. 2. Diophantine property of GL d+1 (R) and SL d+1 (R) matrices For notational reasons it is convenient to consider GL d+1 (R) instead of GL d (R). 2.1. GL d+1 (R) actions. Let A ∈ GL d+1 (R) be a matrix that strictly preserves a cone Σ = Σ v1,...,v d+1 ⊂ R d+1 . Without loss of generality, we can assume that Σ {0} is contained in the halfspace {x ∈ R d+1 : x d+1 > 0}. It is convenient to represent the induced action of A on RP d on the affine hyperplane {x ∈ R d+1 : x d+1 = 1}, and consider the corresponding action on R d . To be precise, for x = (x 1 , . . . , x d ) ∈ R d , we consider (x, 1) = (x 1 , . . . , x d , 1) ∈ R d+1 and let f A (x) = P d A(x, 1) A(x, 1) d+1 , when A(x, 1) d+1 = 0, where P d is the projection onto the first d coordinates. The components of f A are rational functions, which are, of course, real-analytic on their domain. Consider V := P d (Σ ∩ {x ∈ R d+1 : x d+1 = 1}). By assumption, f A is well-defined on V , and we have f A (V ) ⊂ V . We will also consider the action of A on the unit sphere, given by ϕ A (x) := A · x = Ax x , for a unit vector x ∈ S d . Consider also U , the intersection of Σ with the upper hemisphere. We have ϕ A (U ) ⊂ U . Lines through the origin provide a 1-to-1 correspondence between U and V , which is bi-Lipschitz in view of the assumption Σ {0} ⊂ {x ∈ R d+1 : x d+1 > 0}. It is well-known [5] (see also [4,Section 9]) that strictly preserving a cone implies that ϕ A is a strict contraction in the Hilbert metric on U , which is by-Lipschitz with the round metric. We thus obtain the following: Lemma 2.1. Suppose that the finite family A = {A i } i∈Λ ⊂ GL d (R) strictly pre- serves a simplicial cone Σ = Σ v1,...,v d+1 ⊂ {x ∈ R d+1 : x d+1 > 0} ∪ {0}. Then the associated IFS F A = {f A } A∈A is real-analytic and uniformly hyperbolic on V ⊂ R d , in the sense that there exist C > 0 and γ ∈ (0, 1) such that max x∈V f ′ i (x) Cγ n , for all i ∈ Λ n , where f i = f Ai 1 • · · · f Ai n and f ′ i (x) is the operator norm of the differential at the point x. 2.2. From exponential separation to the Diophantine property. Recall the strong exponential separation condition (Definition 1.8). Proposition 2.2. Let A be a finite family of GL d+1 (R) matrices, and let Φ A be the induced IFS on S d . If Φ A satisfies the strong exponential separation on a nonempty set, then A is strongly Diophantine. Proof. Let C 1 = max i∈Λ {1, A i } and C 2 = max i∈Λ {1, A −1 i }. Suppose that i = j in Λ n . Let us write i = (i ∧ j)u, j = (i ∧ j)v, where i ∧ j is the common initial segment of i and j, so that u = u 1 . . . u k , v = v 1 . . . v k for some k n, with u 1 = v 1 . We have (2.1) A i − A j A −1 i∧j −1 A u − A v C −n 2 A u − A v . Lemma 2.3. For any A, B ∈ GL d+1 (R) and any unit vector x ∈ R d+1 , we have A · x − B · x A −1 1 + B B −1 · A − B . Proof. We have A · x − B · x = Ax Ax − Bx Bx Ax Ax − Bx Ax + Bx Ax − Bx Bx =: R 1 + R 2 . Since 1 = A −1 (Ax) A −1 Ax , we have Ax −1 A −1 . Therefore, R 1 A − B · A −1 . Similarly, R 2 B · Ax − Bx · Ax −1 Bx −1 B · A − B · A −1 B −1 , and the desired estimate follows. Applying the lemma to A u and A v yields, in view of A w C n 1 , A −1 w C n 2 for any w ∈ Λ k , k n: (2.2) A u − A v 2 −n C −n 1 C −2n 2 A u · x − A v · x . Now we continue with the proof of the lemma. By assumption, Φ A satisfies the exponential separation condition on a nonempty set. Let c ∈ (0, 1) be the constant from the definition (1.9). It follows that there exists x ∈ S d such that A u · x − A v · x c k c n . Combining this inequality with (2.2) and (2.1) yields A i − A j 2 −n C −n 1 C −3n 2 c n , confirming the strong Diophantine property. Dimension of exceptions for one-parameter families. We consider a one-parameter family of real-analytic IFS on a compact subset of R d , and show that under some mild assumptions it satisfies the exponential separation condition outside of a Hausdorff dimension zero set. This section is based on [15,Section 5.4] and [16, Section 6.6], but we had to make a substantial number of modifications in the definitions and proofs. Let J be a compact interval in R and V an open set in R d . For each i ∈ Λ and t ∈ J , we assume that f i,t (·) : V → V is a continuously differentiable function. Assume also that t → f i,t (x) is real-analytic on a neighborhood of J for any i ∈ Λ and x ∈ V . Denote F t = {f i,t } i∈Λ . Further, assume that the IFS is uniformly hyperbolic in the following sense: there exist C > 0 and 0 < γ < 1, such that (2.3) f ′ i,t (x) Cγ n , for all i ∈ Λ n , x ∈ V, t ∈ J . Fix x 0 ∈ V . For any finite sequence i ∈ Λ n we define F i (t) = f i,t (x 0 ). Of course, this depends on x 0 , but we suppress it from notation. For i ∈ Λ N we have (2.4) Π t (i) = F i (t) := lim n→∞ F i\|n (t), where Π t : Λ N → R d is the natural projection corresponding to the IFS F t and i\| n = i 1 . . . i n . Notice that this is already independent of x 0 . The proof of the next Claim is standard and follows from uniform hyperbolicity (2.3). Claim. If i (k) ∈ Λ n(k) is a sequence of words, such that i (k) → i ∈ Λ N , in the sense that for any N ∈ N we have i (k) \| N = i\| N for k sufficiently large, then F i (k) (·) → F i (·) uniformly on J . In particular, F i\|n (·) → F i (·) uniformly on J for all i ∈ Λ N . Thus F i (·) is real-analytic on J , for any i ∈ Λ N . Next, for i, j ∈ ∞ n=1 Λ n ∪ Λ N , let ∆ i,j (t) = F i (t) − F j (t) ∈ R d . For any ε > 0, let E ε = ∞ N =1 n>N i,j∈Λ n ,i1 =j1 (∆ i,j ) −1 B ε n and (2.5) E = ε>0 E ε , where B ε n = {x ∈ R d : x ε n }. It is easy to see that if t / ∈ E then F t satisfies the strong exponential separation condition. Remark 2.4. In [15,16] Hochman considered the case where F t is an affine IFS. He defined the sets E ′ ε and E ′ as follows: E ′ ε = ∞ N =1 n>N i,j∈Λ n ,i =j (∆ i,j ) −1 B ε n and (2.6) E ′ = ε>0 E ′ ε . If t / ∈ E ′ then F t satisfies the strong exponential separation condition along a subsequence. Definition 2.5. For a family of IFS F t , t ∈ J , as above, and for i, j ∈ Λ N let ∆ i,j (t) = F i (t) − F j (t). We say that the family is non-degenerate if (2.7) ∆ i,j (·) ≡ 0 ⇐⇒ i = j for all i, j ∈ Λ N . We next prove the following: Theorem 2.6. Suppose that the family of IFS F t , t ∈ J , is non-degenerate. Then the set E from (2.5) has Hausdorff dimension zero, and therefore, F t satisfies the strong exponential separation condition on J , outside of a set of zero Hausdorff dimension. Corollary 2.7. For a family of IFS F t , t ∈ J , as above, assume that there exists t 0 ∈ J such that the sets {f i,t0 (V )} i∈Λ are pairwise disjoint. Then (2.7) holds, and hence the set E from (2.5) has Hausdorff dimension zero. Hochman [15,16] proved, for a non-degenerate family of affine IFS, with a realanalytic dependence on parameter, that the set E ′ from (2.6) has packing dimension zero. For any smooth function F : J → R d , denote F (p) (t) = d p dt p F (t). Definition 2.8. The family {F t } t∈J is said to be transverse of order k if there exists c > 0 such that for all n ∈ N and i, j ∈ Λ n , with i 1 = j 1 , we have ∀t ∈ J ∃p ∈ {0, · · · , k} s.t. ∆ (p) i,j (t) > c. Here the norm · is simply the Euclidean norm in R d . Remark 2.9. The above definition is different from [15] and it simplifies the proof of Theorem 2.6. Lemma 2.10. Suppose that the non-degeneracy condition (2.7) holds. Then {F t } t∈J is transverse of order k for some k ∈ N. Proof. Suppose that for all k ∈ N the family {F t } t∈J is not transverse of order k. Then by assumption, for {c k } with c k < 1/k, we can choose n(k), i (k) , j (k) ∈ Λ n(k) with i (k) 1 = j (k) 1 and a point t k ∈ J such that ∆ (p) i (k) ,j (k) (t k ) < c k for 0 p k. Passing to a subsequence {k l }, we can assume that t k l → t 0 ∈ J , i (k l ) → i ∈ Λ N and j (k l ) → j ∈ Λ N , with i 1 = j 1 . Note that ∆ i (k l ) ,j (k l ) → ∆ i,j uniformly on J and the same holds for p-th derivatives by real-analyticity. Hence for all p 0, we have ∆ (p) i,j (t 0 ) = lim l→∞ ∆ (p) i (k l ), j (k l ) (t k l ) = 0. Since ∆ i,j is real-analytic, the vanishing of its derivatives implies ∆ i,j ≡ 0 on J , contradicting (2.7), since i = j by construction. For a C k -smooth function F : J → R, write F J ,k = max p∈{0,··· ,k} max t∈J \|F (p) (t)\|. Lemma 2.11 (Lemma 5.8 in [15]). Let k ∈ N and let F : J → R be a k times continuously differentiable function. Let M = F J ,k , and let 0 < c < 1 be such that for every t ∈ J there is p ∈ {0, · · · , k} with \|F (p) (t)\| > c. Then there exists C = C c,M,\|J\| 1 such that for every 0 < ρ < (c/2) 2 k , the set F −1 (−ρ, ρ) ∩ J can be covered by C k intervals of length 2(ρ/c) 1/2 k each. Lemma 2.12. If {F t } t∈J is transverse of order k 1 on the compact interval J , then the set E from (2.5) has Hausdorff dimension zero. Proof. Extending the real-analytic functions to the complex plane, by Cauchy's formula, since sup n sup i,j∈Λ n ,i1 =j1 ∆ i,j V,0 < ∞ on a neighborhood V of J , and ∆ i,j (·) is real-analytic for all i, j ∈ Λ n , we have (2.8) M := sup n sup i,j∈Λ n ,i1 =j1 ∆ i,j J ,k < ∞. Let (2.9) E ε,n = i,j∈Λ n ,i1 =j1 (∆ i,j ) −1 (B ε n ). Then (2.10) E ε = ∞ N =1 n>N E ε,n . Let i, j ∈ Λ n , with i 1 = j 1 , and assume that ∆ i,j (t) < ε n . By Lemma 2.11 applied to a component of ∆ i,j , for ε sufficiently small, the set (∆ i,j ) −1 (B ε n ) may be covered by C k intervals of length 2(ε n · c −1 ) 1/2 k . It follows that the set E ε,n from (2.9) may be covered by O(\|Λ\| 2n · C k ) intervals of length (ε n · c −1 ) 1/2 k . Fix s > 0 and write H s for the s-dimensional Hausdorff measure. We obtain from (2.10) that H s (E ε ) O(1) · n 1 \|Λ\| 2n C k ε n · c −1 s/2 k < ∞ for ε sufficiently small. It follows that H s (E) = 0. Proof of Theorem 2.6. This is now immediate from Lemmas 2.10 and 2.12. 2.4. Proof of Theorem 1.2. The next lemma follows by an application of Fubini's Theorem. Lemma 2.13. Let F ⊂ R n and let v ∈ R n be a nonzero vector. Assume that for every x 0 ∈ R n , the set {x 0 + tv : t ∈ R} ∩ F has 1-dimensional Lebesgue measure 0. Then the set F has n-dimensional Lebesgue measure 0. Proof of Theorem 1.2. (i) Let Σ = Σ v1,...,v d+1 be a simplicial cone in R d+1 . Let U ⊂ X Σ,m be a small open set in (GL d+1 (R)) m of m-tuples of matrices for which Σ is strictly invariant. Choose vectors w i ∈ R d+1 (i ∈ Λ), with distinct directions, in such a way that (2.11) w i ∈ Σ Aiv1,...,Aiv d+1 for all (A i ) i∈Λ ∈ U. This is possible when U is sufficiently small. Let (A i ) i∈Λ ∈ U, and for each t 0 and i ∈ Λ let A i,t be such that A i,t v j = A i v j + tw i , j = 1, . . . , d + 1. Condition (2.11) guarantees that {A i,t v j } d+1 i=1 is linearly independent, and hence A i,t ∈ GL d+1 (R) for all t > 0. This is a consequence of the following elementary claim. Claim. Let y 1 , . . . , y d+1 ∈ R d+1 be linearly independent, and suppose that w = d+1 k=1 a k y k for some a k 0. Then the family {y 1 + w, . . . , y d+1 + w} is linearly independent as well. Proof of the Claim. We have d+1 j=1 c j y j + d+1 k=1 a k y k = 0 =⇒ d+1 j=1 c j + a j d+1 k=1 c k y j = 0, hence c j + a j d+1 k=1 c k = 0 for all j. If d+1 k=1 c k = 0, we obtain a contradiction, in view of a j 0, j = 1, . . . , d + 1; thus c j = 0, j = 1, . . . , d + 1, as claimed. Let A t = {A i,t } i∈Λ be the family of matrices defined above, for t 0, and let F t = F At be the corresponding one-parameter family of IFS on the set V ⊂ R d obtained by projection of Σ ∩ {x ∈ R d+1 : x d = 1} onto R d . Notice that the cone Σ is strictly preserved by all A t , t 0, by construction, hence by Lemma 2.1, these IFS are all uniformly hyperbolic. It is easy to see that the dependence on t is real-analytic, since the IFS is given by rational functions. Condition (2.3) holds for t ∈ [0, M ], for any M < ∞, by uniform hyperbolicity and compactness. Finally, observe that, given ε > 0, for t sufficiently large, we have f i,t (V ) ⊂ P d Σ ε (w i ) ∩ {x d+1 = 1} , where Σ ε (w i ) is the cone of vectors ε-close to w i in direction. By construction, w i are all distinct, hence Corollary 2.7 applies. We obtain that for all t ∈ [0, ∞) outside a set of Hausdorff dimension zero, the IFS F t satisfies the exponential separation condition, and then Proposition 2.2 implies that the m-tuple of matrices (A i,t ) i∈Λ is Diophantine for all t outside of a zero-dimensional set, so certainly for Lebesgue-a.e. t. Now Lemma 2.13 yields the desired claim. (ii) We consider (SL d+1 (R)) m as a codimension-m submanifold of (GL d+1 (R)) m ⊂ R (d+1) 2 m . In the proof of part (i) we showed that for a.e. (A i ) i∈Λ ∈ X Σ,m , the induced IFS on a subset of R d satisfies the strong exponential separation condition. Suppose that there is a positive measure subset E ⊂ Y Σ,m for which the strong Diophantine condition is violated. Then for every (A i ) i∈Λ ∈ E, the induced IFS Φ does not have strong exponential separation, by another application of Proposition 2.2. However, (A i ) i∈Λ ∈ Y Σ,m and (c i A i ) i∈Λ ∈ X Σ,m , for any c i > 0, induce the same IFS on the projective space, and we get a set of positive measure in X Σ,m for which the strong Diophantine condition does not hold. This is a contradiction, and the theorem is proved completely. Dimension of the attractor Let A ∈ SL 2 (R). It is easy to see that A * A has eigenvalues A 2 , A −2 . Let (cos t A , sin t A ) t be the unit eigenvector corresponding to the eigenvalue A −2 , where t A ∈ [0, π). We recall some basic properties of the map ϕ A . For more details see sections 2.2, 2.3 and 2.4 in [17]. The following simple lemma is [17,Section 2.4]. Lemma 3.1. Let A ∈ SL 2 (R). Then the induced map ϕ A expands by at most A 2 and contracts by at most A −2 . Furthermore, for any ε > 0 there exists C ε > 1 such that A −2 \|ϕ ′ A (x)\| < C ε A −2 for all x ∈ [0, π) (t A − ε, t A + ε). The following lemma is now immediate. Lemma 3.2. Let U (0, π) be an open set. Then, for every ε > 0 there exists C ε > 1 such that for any A ∈ SL 2 (R) with (t A − ε, t A + ε) ⊂ U , we have π − C ε A −2 < \|ϕ A (U )\| < π. Lemma 3.2 implies the following: Lemma 3.3. Let U (0, π) be an open set. Then, for every ε > 0 there exists M = M (ε) > 0 such that the following holds: for any A ∈ SL 2 (R) that satisfies ϕ A (U ) ⊂ U and A > M , we have (t A − ε, t A + ε) ⊂ U . Let A = {A i } i∈Λ be a finite collection of SL 2 (R) matrices and let Φ = {ϕ A } A∈A be the corresponding IFS on [0, π) ∼ = RP 1 . Recall the notation: Φ(E) = A∈A ϕ A (E). Assume that there is a strictly invariant multicone U ⊂ [0, π), that is, a nonempty open set having finitely many connected components with disjoint closures, such that U = RP 1 and Φ(U ) ⊂ U . By Theorem 1.5, the associated cocycle is uniformly hyperbolic, which implies that there exist c > 0 and λ > 1 such that (3.1) A i cλ n for all i ∈ Λ n , n ∈ N, see [27] and [3, Theorem 2.2]. Fix ε > 0 such that the (2ε)-neighborhood of Φ(U ) is contained in U , and let M = M (ε) from Lemma 3.3. By (3.1), there exists n 0 ∈ N such that A i > M for i ∈ Λ n , n n 0 . Lemma 3.3 implies that (t A i − ε, t A i + ε) ⊂ U , hence (t A i − ε, t A i + ε) ∩ Φ(U ) = ∅, for all i ∈ Λ n , n n 0 . Hence, by Lemmas 3.1 and 3.2 we obtain (3.2) A i −2 \|ϕ ′ i (x)\| C ε A i −2 , for all x ∈ U , i ∈ Λ n , n n 0 . Thus we obtain Lemma 3.4. (i) The Bounded Distortion Property holds for Φ on U : there exists C ′ > 1 such that (3.3) 1 C ′ \|ϕ ′ i (x)\| \|ϕ ′ i (y)\| C ′ for all x, y ∈ U , i ∈ Λ n , n ∈ N. (ii) The IFS Φ k is contractive on U in the metric d P for sufficiently large k. More precisely, there exists C ′′ > 0 such that (3.4) ϕ ′ i U C ′′ λ −2n , where λ > 1 is from (3.1). (iii) We have s = s A /2, where s is the solution of the Bowen's equation P Φ (s) = 0, with the pressure given by (1.6) and s A is the critical exponent, given by (1.4). Let p = (p i ) i∈Λ be a probability vector, and let x 0 ∈ U . Let χ Φ,p be the almost sure value of the limit (3.5) lim n→∞ − 1 n log \|(ϕ i1···in ) ′ (x 0 )\|, where i 1 , i 2 , · · · ∈ Λ is a sequence chosen randomly according to the probability vector p = (p i ) i∈Λ . The equations (3.2) and (1.12) imply Lemma 3.5. We have χ Φ,p = 2χ A,p . By the Birkhoff Ergodic Theorem, applied to the shift transformation on Λ N with the measure µ = p N , in view of the bounded distortion (3.3), we have (3.6) χ Φ,p = lim n→∞ − 1 n log (ϕ i1···in ) ′ (Π(i)) = − Λ N log ϕ ′ i1 (Π(i)) dµ(i), where Π : Λ N → RP 1 is the natural projection corresponding to Φ. Lemma 3.6. (i) Let A be a finite set of matrices in GL 2 (R), and let Φ be the IFS induced by A on the projective line RP 1 . If Φ satisfies the strong exponential separation condition on a nonempty set, then A is strongly Diophantine. (ii) Let A be a finite set of matrices in SL 2 (R), and let Φ be the IFS induced by A on the projective line RP 1 . Then Φ satisfies the strong exponential separation condition on a set containing at least three points if and only if A is strongly Diophantine. Proof. (i) This is a special case of Proposition 2.2, since exponential separation for the induced action on a subset of RP 1 is equivalent to that for the induced action on a subset of the circle. (ii) One direction, that the strong exponential separation for Φ implies the strong Diophantine property for A, follows from (i). For the converse, we refer to [17,Lemma 2.5], which says that SL 2 (R) is quantitatively separated by the action on three points of RP 1 . Proof of Theorem 1.6. Recall that A is a finite set of SL 2 (R) matrices satisfying the strong Diophantine condition and having a strictly invariant multicone U , and Φ = Φ A is the associated IFS on U . Then Φ has a compact attractor K, and our goal is to show that dim H (K) = s, where P Φ (s) = 0 and P Φ is given by (1.6). It is known that For convenience of the reader, we include the proof in the appendix, following [23]. Let p (n) = (p (n) i ) i∈Λ n be the probability vector such that p (n) i = \|U i \| dn . Let η (n) be the invariant probability measure for the IFS Φ n on U , corresponding to p (n) . Since η (n) is supported on K, we have dim η (n) dim H (K). We claim that A satisfies the assumptions of Theorem 1.11. Indeed, the existence of a strictly invariant multicone is known to imply that all the matrices in A are hyperbolic, hence the group generated by A is unbounded. Further, we assumed that not all attracting fixed points of A are the same, hence this group is totally irreducible. Thus the Furstenberg measure for (A n , p (n) ) is unique, and it coincides with η (n) . Since A is Diophantine, we have that A n is Diophantine as well. Now, by Theorem 1.11 and Lemma 3.5 we have H(p (n) ) χ Φ n ,p (n) dim H (K). We claim that there exists C > 0 such that (3.9) χ Φ n ,p (n) − i∈Λ n \|U i \| dn log \|U i \| + C for all n ∈ N. Indeed, by (3.6), we have χ Φ n ,χ Φ n ,p (n) i∈Λ n \|U i \| dn log C ′ \|U \| \|U i \| = − i∈Λ n \|U i \| dn log \|U i \| + log C ′ \|U \|. By (3.9), we have H(p (n) ) χ Φ n ,p (n) > − i∈Λ n \|U i \| dn log \|U i \| dn − i∈Λ n \|U i \| dn log \|U i \| + C = d n 1 + C − i∈Λ n \|U i \| dn log \|U i \| −1 . Since lim n→∞ d n = s and lim n→∞ − i∈Λ n \|U i \| dn log \|U i \| = ∞, we obtain s dim H (K), as desired. Finally, s = s A by Lemma 3.4(iii). 4. Appendix: the proof of (3.7) and (3.8) 4.1. Proof of (3.8) [23]. We have a projective IFS Φ = {ϕ i } i∈Λ on a strictly invariant multicone U . Observe that P Φ (t) = lim n→∞ 1 n log i∈Λ n ϕ ′ i t = lim n→∞ 1 n log i∈Λ n \|U i \| t , by the Bounded Distortion Property (3.3). Let Q n = 1 n log i∈Λ n \|U i \| s . Since P Φ (s) = 0, we have lim n→∞ Q n = 0. Let r 1 > 0 be such that r 1 \|ϕ ′ i (x)\| for all i ∈ Λ and x ∈ U . Recall (3.4), which says that ϕ ′ i U C ′′ λ −2n , for C ′′ > 0 and λ > 1. Then r n 1 \|U \| \|U i \| < C ′′ λ −2n \|U \| for i ∈ Λ n , and hence we have (r n 1 \|U \|) s−dn · \|U i \| dn < \|U i \| s < C ′′ (λ −2n \|U \|) s−dn · \|U i \| dn . In view of i∈Λ n \|U i \| dn = 1, we have 1 n log(r n 1 \|U \|) s−dn < Q n < 1 n log C ′′ + (s − d n )(log \|U \| − 2n log λ) , and it follows that Q n − (log C ′′ )/n −2 log λ + (log \|U \|)/n < s − d n < Q n log r 1 + (log \|U \|)/n , which implies d n → s, as desired. 4.2. Proof of (3.7). Fix ε > 0. Then for sufficiently large n we have d n < s + ε/2. Thus i∈Λ n \|U i \| s+ε < i∈Λ n \|U i \| dn+ε/2 < (r n 2 \|U \|) ε/2 → 0, as n → ∞. Therefore, the (s+ε)-dimensional Hausdorff measure of K is zero. By the definition of the Hausdorff dimension, this proves (3.7). Date: March 1, 2019. Both authors were supported by the Israel Science Foundation grant 396/15 (PI. B. Solomyak). Theorem 1. 2 . 2Let Σ = Σ v1,...,v d be a simplicial cone in R d and m 2. Definition 1.4. A multicone is a proper nonempty open subset U of RP 1 , having finitely many connected components with disjoint closures. Corollary 1. 10 . 10Let F = {f i } i∈Λ be a finite collection of linear fractional transformations with real coefficients. Assume that there exists U ⊂ R, a finite union of bounded open intervals with disjoint closures, such that H (K) s, see the appendix for a short proof. Let us show the opposite inequality. Let d n > 0 be the solution of the equation i∈Λ n \|U i \| dn = 1,where U i = ϕ i (U ) and \| · \| denotes the Lebesgue measure on [0, π) ∼ = RP 1 . R) the situation is different, since there are, for example, open sets of m-tuples in G × G which satisfy (1.2). For instance, if R d + is mapped by A, B into closed cones that are disjoint, except at the origin, then (1.2) holds for {A, B}. On the other hand, there are open sets in (SL d (R)) m in which non-Diophantine pairs are dense. For instance, the set of elliptic matrices in SL 2 p (n) where [i] is the cylinder set of sequences starting with i. Now µ([i]) = \|U \| dn , and \|U i \| C ′ \|U \| , by the Bounded Distortion Property (3.3). Therefore,i∈Λ n µ([i]) · log min x∈U \|ϕ ′ i (x)\| −1 , min x∈U \|ϕ ′ i (x)\| AcknowledgementsThe authors would like to acknowledge Balazs Barańy for helpful comments and for telling us about the papers[3,6]. Diophantine properties of nilpotent Lie groups. Menny Aka, Emmanuel Breuillard, Lior Rosenzweig, Nicolas De Saxcé, Compos. Math. 1516Menny Aka, Emmanuel Breuillard, Lior Rosenzweig, and Nicolas de Saxcé. 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[ "Broadband near-unidirectional absorption enabled by phonon-polariton resonances in SiC micropyramid arrays", "Broadband near-unidirectional absorption enabled by phonon-polariton resonances in SiC micropyramid arrays" ]	[ "G C R Devarapu \nSchool of Physics\nCollege of Engineering, Mathematics and Physical Sciences (CEMPS)\nUniversity of Exeter\nEX4 4QLExeterUnited Kingdom\n\nSchool of Physics and Astronomy\nUniversity of St\nAndrews\n\nNorth Haugh\nKY16 9SSSt AndrewsUnited Kingdom\n\nCork Institute of Technology\nT12 P928BishoptwonCorkIreland.\n", "S Foteinopoulou \nSchool of Physics\nCollege of Engineering, Mathematics and Physical Sciences (CEMPS)\nUniversity of Exeter\nEX4 4QLExeterUnited Kingdom\n\nCenter for High Technology Materials\nUniversity of New Mexico\n1313 Goddard St. SE87106AlbuquerqueNew MexicoUSA\n" ]	[ "School of Physics\nCollege of Engineering, Mathematics and Physical Sciences (CEMPS)\nUniversity of Exeter\nEX4 4QLExeterUnited Kingdom", "School of Physics and Astronomy\nUniversity of St\nAndrews", "North Haugh\nKY16 9SSSt AndrewsUnited Kingdom", "Cork Institute of Technology\nT12 P928BishoptwonCorkIreland.", "School of Physics\nCollege of Engineering, Mathematics and Physical Sciences (CEMPS)\nUniversity of Exeter\nEX4 4QLExeterUnited Kingdom", "Center for High Technology Materials\nUniversity of New Mexico\n1313 Goddard St. SE87106AlbuquerqueNew MexicoUSA" ]	[]	Inspired by moth eyes, nature's most powerful antireflex, we present a sub-wavelength SiC micropyramid design, which operates in the Reststrahlen band of SiC, namely the spectral band of strong phonon-photon coupling in the SiC material. While within this band SiC repels EM waves, we observe here a broad low-reflectivity window with unique attributes, with distinct characteristics different from typical dielectric moth-eye-like structures. To be specific, while the latter systems are entirely symmetric, the reflection response of our SiC micropyramid system can be highly asymmetric. In particular, the SiC micropyramid system can be near-reflectionless for light impinging from the tip side of the micropyramids and exhibits more than 90% reflection for light impinging from the base side of the micropyramids, over a broad wavelength range in the SiC Reststrahlen band. This strongly asymmetric reflection response emanates from the cascaded coupling of vortex-like cavity modes at each of the SiC blocks comprising the micropyramids and translates into a strongly uni-directional absorber response. We discuss how, by virtue of Kirchhoff's law, this strongly uni-directional superabsorber behavior implies a strongly uni-directional emission profile that is important for one-way infrared sources and passive radiative-cooling systems.	10.1103/physrevapplied.7.034001	[ "https://arxiv.org/pdf/1710.06977v1.pdf" ]	119,211,103	1710.06977	cfcc38209b064c599dfbfc8f0e6cb7203af3bec1	Broadband near-unidirectional absorption enabled by phonon-polariton resonances in SiC micropyramid arrays 19 Oct 2017 G C R Devarapu School of Physics College of Engineering, Mathematics and Physical Sciences (CEMPS) University of Exeter EX4 4QLExeterUnited Kingdom School of Physics and Astronomy University of St Andrews North Haugh KY16 9SSSt AndrewsUnited Kingdom Cork Institute of Technology T12 P928BishoptwonCorkIreland. S Foteinopoulou School of Physics College of Engineering, Mathematics and Physical Sciences (CEMPS) University of Exeter EX4 4QLExeterUnited Kingdom Center for High Technology Materials University of New Mexico 1313 Goddard St. SE87106AlbuquerqueNew MexicoUSA Broadband near-unidirectional absorption enabled by phonon-polariton resonances in SiC micropyramid arrays 19 Oct 20171phonon-polaritonsasymmetric reflectorssuperabsorbersemittersphotonic crystalsMid-IRdetectors 2 Inspired by moth eyes, nature's most powerful antireflex, we present a sub-wavelength SiC micropyramid design, which operates in the Reststrahlen band of SiC, namely the spectral band of strong phonon-photon coupling in the SiC material. While within this band SiC repels EM waves, we observe here a broad low-reflectivity window with unique attributes, with distinct characteristics different from typical dielectric moth-eye-like structures. To be specific, while the latter systems are entirely symmetric, the reflection response of our SiC micropyramid system can be highly asymmetric. In particular, the SiC micropyramid system can be near-reflectionless for light impinging from the tip side of the micropyramids and exhibits more than 90% reflection for light impinging from the base side of the micropyramids, over a broad wavelength range in the SiC Reststrahlen band. This strongly asymmetric reflection response emanates from the cascaded coupling of vortex-like cavity modes at each of the SiC blocks comprising the micropyramids and translates into a strongly uni-directional absorber response. We discuss how, by virtue of Kirchhoff's law, this strongly uni-directional superabsorber behavior implies a strongly uni-directional emission profile that is important for one-way infrared sources and passive radiative-cooling systems. I. INTRODUCTION Across the EM spectrum, recent intense research efforts aim to uncover new mechanisms for extreme absorption control that go beyond the traditional three-component platforms with the absorber sandwiched between an anti-reflex coating and a backreflector. The driving force behind these investigations are photovoltaic devices [1][2][3][4] as well as infrared/THz detectors [5][6][7][8][9], modulators [9,10] and thermal-emitters [7,9,11,12]. The strategies underpinning these current absorption management efforts could be grouped into five general approaches. In particular, new superabsorber architectures that have been proposed may utilize plasmonic resonances that enhance the EM field in the vicinity of the absorber [1,2,13], involve photonic metamaterials [14,15] either as impedance matchers [5,16,17] or enhancers of dark EM field components by virtue of a characteristic hyperbolic photonic dispersion [18], employ lossy material bilayers with destructive Fabry-Perot interference suppressing reflection [9,19], tailor resonant coupling to waveguide/cavity modes in the absorber structure [4,12], or manipulate a near-reflectionless coupling to the near band-edge Floquet-Bloch mode of a lossy photonic crystal [20,21]. Of the aforementioned avenues to extra-ordinary absorption management the last two are particularly attractive as they essentially represent one-step super-absorber platforms. That is to say the functional performance of these structures relies only on a single kind of absorbing medium; it is not needed to incorporate other type of metallic or absorber material that may make fabrication more complicated, lead to parasitic absorption [3,5] or compromise structural stability due to overheating. This is not the case for any of the other schemes, where together with the absorber of interest, additional patterned absorber/metallic material are integrated to enable impedance-matching [5,19,22], EM-field enhancement [1,3], or additional light passes [22,23]. Now, the valuable insight obtained for absorption with plasmonic nanostructures in the visible spectrum [2] can be transferred in the mid-IR spectral regime, albeit not with the use of metals. Metals in the infrared spectrum have a very small skin depth; thus they repel the EM field and do not respond with the familiar plasmonic resonances existing in the visible spectrum. However, "plasmonic effects" in the mid-IR can be mimicked if we instead consider microstructures comprising material with a strong phonon-photon coupling. Such phonon-polariton materials can exhibit a negative permittivity response in the mid-IR with a larger skin depth. They so emulate field-confinement effects with infrared light similar to those of plasmonic nanostructures [24] operating in the visible spectrum. Thus, for mid-IR absorption platforms, SiC would be an excellent material with its phonon-polariton gap spectrum (Reststrahlen band) between 10.3 to 12.6 µm [25]. This spectral regime is of high interest for detectors and sources as it encompasses the absorption fingerprints of many biomolecules [26]. Moreover, this spectrum is also highly relevant to passive radiative-cooling devices [27], since it lies within the atmospheric transparency window [28,29]. In addition, SiC is a refractory material, which is crucially important for the structural stability of absorber/emitter devices [30]. In this paper, we investigate SiC-based designs with the objective to achieve mid-IR sub-wavelength broadband superabsorption [2,[31][32][33] that is also highly uni-directional. Designing and controlling a highly unidirectional mid-IR absorption response, has not been thus-far investigated but can benefit various applications. In particular, highly unidirectional absorbers would efficiently absorb light that impinges from their top side while they would reflect, and thus not absorb, almost all light, that impinges from their bottom side. Kirchhoff's law then prescribes [34,35], that these systems would emit light predominantly towards the direction where they are strong absorbers if light was incident from that direction, in this manner acting as near-unidirectional emitters. Uni-directional emission is a highly attractive feature not only for the design of one-way infrared sources but also for simpler integrated components for passive-radiative-cooling devices to reduce temperature in buildings [27] or electronic circuits. In order to achieve this goal we will bring together two of the aforementioned explored avenues for harnessing absorption, i.e. invoking cavity resonances and photonic crystal (PC) effects and combine them with intuition obtained from the anti-reflex properties of moth-eye-like structures. This allows to consider SiC-only platforms without other auxiliary absorbing material structures that could compromise the resilience of the device under high temperatures. This paper is organized as follows: In Sec. II we present the SiC-based platform we envision for strong absorption control within the Reststrahlen band of SiC. In Sec. III we present our results for an initial design demonstrating enhanced broadband absorption. We discuss the origin of such absorption enhancement with respect to bulk SiC in the same frequency region in Sec. IV. In Sec. V we analyze the effects of a stronger interaction between the micropyramid building-blocks towards a broadband superabsorber [2,[31][32][33] behavior. In Sec. VI we quantify the absorption enhancement and demonstrate the robustness under misalignment of the micropyramid superabsorber platform. In Sec. VII we investigate the asymmetric absorber response for light illuminating from the tip-side and base-side of the micropyramid-array platform. In Sec. VIII we discuss how this asymmetry can be harnessed further towards a near-unidirectional absorber/emitter behavior. Finally, In Sec. IX we present our conclusion. II. THE SILICON CARBIDE PLATFORM FOR BROADBAND SUPERABSORP-TION In spite of Silicon Carbide's high loss tangent within its Reststrahlen band, which makes it potentially attractive for absorber/emitter systems, it is actually extremely difficult to couple light inside the SiC material in this spectrum; most light gets reflected upon hitting the SiC surface. This low in-coupling poses a serious bottleneck in SiC-based absorber/emitter platforms. However, recent results with a compact one-dimensional SiC judicious PC design [21] show that it is possible to suppress such reflection and achieve near-perfect absorption within a narrow spectral band. This band can be extended to cover most of the SiC Reststrahlen spectral region, by employing a very thick, several wavelengths long, PC [36]. However, in practical applications it is desirable to have compact designs exhibiting a broadband super-absorption [31][32][33]. In addition, we seek such super-absorber merit to be highly asymmetric, i.e. to exist only for light impinging from one side of the structure, a possibility that has not been hitherto explored. In order to achieve our target of a broadband absorption in SiC within its Reststrahlen band with a compact structure, we employ a design with a SiC micropyramid as its elementary building block arranged in a one-dimensional photonic crystal array. We draw our inspiration for such a design from the conical shape of corneal nipples in moth eyes [37,38], which are known for their superior antireflection properties. The moth-eye corneal nipples do not constitute an optically dense material. So, naturally one may wonder whether designs of a similar shape but made from a material with extreme optical properties, such as SiC in the Reststrahlen band, would retain these extra-ordinary antireflex properties. Antireflection behavior has been demonstrated [39,40] in the optical spectrum with nanostructured cones or pyramids mimicking moth eyes comprising a high-refractive-index material, such as GaP or Si, which have a purely dielectric optical response. These results provide a promising basis to expand such antireflex behavior in a platform made of SiC, whose material optical response in the Reststrahlen band involves both a negative permittivity and an optical loss tangent (see bottom-left inset in Fig. 1), due to the photon-phonon coupling resonance in this spectrum. Thus we proceed with our intuitive choice of SiC micropyramid-based designs which we investigate further in the following. We show the design under consideration in Fig. 1. We consider a stepped two-dimensional (2D) pyramid, with translational symmetry in the third dimension. The translational symmetry ensures full polarization decoupling [41], allowing for polarization selective emission suitable for constructing polarized infrared sources. Moreover, we are considering a stepped micropyramid structure, whose realization is possible with certain fabrication techniques, such as direct wafer-bonding [42] or layer-by-layer fabrication with e-beam patterning [43] and inductively coupled plasma reactive-ion etching (ICP-RIE) [44]. Each block of the micropyramid is expected to behave as a resonant cavity [45] microantenna with emission/absorption spectra in the mid-IR regime. By combining blocks of different sizes we aim to bring together different resonant frequencies, thus constructing a broadband response for the pyramid microantenna. Then by arranging the pyramid blocks in a moth-eye structure we aim to minimize reflection thus enabling a strong EM incoupling to each of the participant block microresonators. That is to say that our proposed design is conceived with a vision to operate simultaneously as a moth-eye antireflector and a broadband micro-antenna. The sizes of the participating SiC blocks in the pyramid of Fig. 1 are chosen so that a gradual size variation from one block to the next one is maintained, with a small number of layers -which is desirable from the fabrication point of view-. At the same time, we aim for a compact structure of sub-wavelength total thickness, i.e. thickness in the order of half the free space wavelength. In Fig. 1 we label each of the blocks comprising the micropyramid with 1, 2, 3 and 4. Their corresponding sizes (in the x-and y-direction respectively) are: w x1 = 0.5 µm, w y1 = 1 µm, w x2 = 1 µm, w y2 = 1 µm, w x3 = 2 µm, w y3 = 1 µm, w x4 = 4 µm, w y4 = 1 µm. These micropyramids are arranged periodically in an array with a spacing, a. We will vary this inter-spacing, a, so we can control the system's collective response from a regime where each micropyramid responds independently to the impinging light to a regime where the resonances between neighboring micropyramids are interacting. In our initial investigation, we will ignore the presence of a required substrate, seen in Fig. 1, in order to capture and understand the response of our proposed microstructure better. We will then consider the effect of the substrate afterward. So, unless a presence of a substrate is explicitly stated, all results correspond to the platform of Fig. 1, but without the depicted substrate. III. ENHANCED ABSORPTION WITH THE SIC-MICROPYRAMID SYSTEM We first study the response of the micropyramid-array system, where each micro-pyramid building block would respond independently to the impinging light. For this purpose the micropyramids are placed sparsely, about one free-space wavelength apart, with an interspacing of a = 10 µm. We consider normally incident light and both polarization cases, where the impinging light has its electric field aligned with the micropyramid block-axis z (TE-polarization) or its magnetic field aligned with the micropyramid block-axis z (THpolarization). We use a two-dimensional implementation of the Finite-Difference Time-Domain (FDTD) method with the Lumerical FDTD simulator [46] to investigate the response of the micropyramid-array system. In the FDTD simulator, periodic boundary conditions are applied in the x -direction as the micropyramid building block repeat themselves periodically in this direction. Open (absorbing) boundary conditions are applied in the y-direction to emulate an unbounded domain above and below the micropyramid. The micropyramid structure is discretized in the simulation with a mesh size of dx = dy = 10 nm. In our calculations, we consider a SiC permittivity function characteristic of the phonon-polariton resonant response in the Reststrahlen band, i.e.: ε(ω) = ε ∞ (1 + ω 2 L − ω 2 T ω 2 T − ω 2 − iωΓ )(1) The high frequency response, ε ∞ , the longitudinal and transverse optical phonon frequencies, ω L and ω T , respectively, and the intrinsic damping parameter Γ are determined by fitting to experimental optical data [25]. From Ref. 25 we take: ε ∞ = 6.7, ω L = 2π × 29.07 THz, ω T = 2π × 23.79 THz, and Γ = 2π × 0.1428 THz. The SiC permittivity spectral function corresponding to Eq. 1 is depicted in the bottom-left inset of Fig. 1. We clearly observe in Fig. 2 that the micropyramid-array system responds very differently to TE and TH light. For TE light, the system shows both very little reflection and very little absorption [solid-black lines of panel (a) and (b) respectively]. This is because the electric field drives the phonon-polariton resonance along the z-direction, where the SiC material is unbounded. Thus, the EM field behaves as in the vicinity of bulk SiC, and only penetrates very little the SiC micropyramid. However, since the micropyramid array is arranged sparsely the EM field can flow around the structure yielding a very low absorption and high transparency. Indeed, we confirmed that as we bring the micropyramids closer the reflective response of the micropyramid system to TE light becomes similar to that of bulk SiC. Clearly, the SiC-micropyramid array structure is a weak absorber/emitter for TE-polarized light. On the other hand, for TH-polarized light we observe in Fig . We note that a second window that we observe with relatively low reflection, appearing close to the red bound of the Reststrahlen band, is not accompanied by a high absorption. Clearly, we see in Fig. 2(b) that absorption is negligible within that range. We note in passing, that these two aforementioned spectral windows are separated by a spectrum with a high reflection response. In this intermittent regime, reflection reaches a maximum of about 80% at an impinging wavelength of 11.27µm, with an asymmetric peak that is typical of Fano resonances [50]. IV. UNDERSTANDING THE ABSORBER BEHAVIOR OF THE SIC MICROPY- RAMID SYSTEM Clearly, the results in Fig. 2 demonstrate that the design of Fig. 1 has a strong potential as a super-absorber for TH-polarized light, which we will explore further in the following. Therefore, we attempt to understand better the response of the micropyramid array by looking at the spectral response of each element it comprises. In other words, we will look at the spectral response of a periodic array comprised only of the top block, the second block, the third block and the fourth block of the micropyramid respectively, as designated in the bottom panel of Fig It is interesting to observe that the arrays comprising the third or fourth block individually demonstrate a clear asymmetric reflection peak as we see in Fig. 3(a), which separates the spectral regime where localized resonant modes exist within the block, -as for example we see from the electric field inset for the fourth block at free space wavelength M1, from the spectral regime where void modes are excited in the interstitial region between the neighboring blocks, -as for example we see from the electric field inset for the fourth block at free space wavelength M3-. In these void-modes the electric field is near-zero within the block. It is the former spectral regime, i.e. the regime of block-cavity resonances, that is evidently of interest with respect to achieving superabsorption. This is because in order to harness a strong absorption response, the strong electric fields must spatially overlap with the absorbing matter. This follows directly from Poynting's theorem for power dissipation [ see also eq. A3]. In passing, we also note that the presence of the spectrally asymmetric reflection peak observed in Fig. 3, suggests the occurrence of a Fano-type interference at that frequency, that arises from the interference between the localized, resonant, block modes, that are present at the shorter-wavelength side of the Reststrahlen band, and the extended, non-resonant [52], void modes, that are present at the longer-wavelength side of the Reststrahlen band. However, the spectral region we focus on here is the frequency regime near the blue bound of the Reststrahlen band. This is the frequency regime, where the localized block-cavity modes are excited, yielding the strong electric field required to achieve a strong absorption. In other words, the SiC blocks behave as microantennas in this spectral region. have observed in Fig. 2(b). Indeed, the cavity-mode behavior as evidenced by the vortices in the EM streamlines leads to strongly enhanced fields within the micropyramid that we show in Fig. 4(d). For a more clear view of these fields we also show the field values only within the micropyramid, with a saturated color-map, with red signifying an electric field intensity enhancement of ten or larger value [see Fig. 4(g)]. On the other hand, at free-space wavelength denoted M3, which is 12.0 µm, the incident EM wave essentially just streams downwards, completely avoiding the pyramid [see Accordingly, since almost-all impinging EM energy is injected into the micropyramid array system in all cases, the key to the respective absorption performance lies in the electric field re-connfiguration that results from the interaction between the cavity modes in the adjacent micropyramids. Since the power dissipation rate per volume is proportional to the electric field intensity [see eq. A3], we can infer that the higher the electric field intensity the better the absorption performance. However, we should not quickly deduce from this fact, that the integrated electric field intensity within the micropyramid, in the new closer-spaced arrays, needs to be higher in comparison with the one of the original micropyramid array. It only needs to be higher than the ratio of the micropyramid interspacing, a, in the newly considered arrays over the respective value of the original array. This is because, for example an array with half the micropyramid interspacing of the original array, has twice as much SiC material underneath the same illumination area in comparison to the original array. This can be clearly understood in the appendix, where we have derived an analytic expression that explicitly relates the absorption by the SiC micropyramid array with the integrated normalized electric field intensity within the SiC material and the array's micropyramid interspacing, a (see eq. A.6 in appendix). In order to get a feeling for the electric field reconnfiguration emanating from the strong coupling between the adjacent cavity modes, we plot the ratio between the electric field Indeed, we observe a strong electric-field reconnfiguration as a result of the strong coupling between the adjacent micropyramid modes. We observe, that there are locations where the electric field intensity is even enhanced; however in other areas we see that electric field intensity is weakened. If we now calculate the integrated value of the normalized electric field intensity, E norm [47], at a free-space wavelength of 10.5 µm, over the extend of the SiC micropyramid structure, for both the new arrays, we find that: \|E norm \| 2 dxdy a=7.5µm \|E norm \| 2 dxdy a=10µm ∼ 86.1%,(2) and \|E norm \| 2 dxdy a=5µm \|E norm \| 2 dxdy a=10µm ∼ 68.5%. From Eqs. (2) and (3) we expect that at the free-space wavelength of 10.5 µm, in the new arrays with a=7.5 µm and a=5.0 µm, the absorption should be enhanced by respective factors of ∼ 1.15 and ∼ 1.35 [54] with respect to the absorption of the original array with a=10 µm. The expected absorption enhancement factors due to the electric field reconnfiguration, as quoted above, are in excellent agreement with the observed absorption enhancement that we observe in Fig. 5(b). The closer-spaced micropyramid array, with interspacing of a=5.0 µm, demonstrates an extra-ordinary,-more than 80% absorptance-, over a broad wavelength range between roughly 10.4 µm and 11 µm, thus behaving as a superabsorber [31]. Note that such superabsorber behavior represents an extra-ordinary absorption enhancement in this range with respect to the absorption achieved by a bulk SiC slab that is about a wavelength thick [see We observe in Fig. 7(c) the characteristic vortex-like EM circulation along the sides or corners of the individual blocks. These are similar to the ones seen in Fig. 7(d) for the corresponding symmetric micropyramid, or the modes we observed before in Fig. 4(a) (for the micropyramid array with interspacing a=10 µm). What is different here for the misaligned-pyramid design is that the mode field/energy landscape seizes to show the mirror symmetry with respect to the center axis of each individual block, which is the case for the Fig. 8(b)]. In other words, the micropyramid array is only a powerful absorber for light impinging from the pyramids' tip-side. We stress that such interesting strongly asymmetric response is not a mere outcome of the y-axis-asymmetric shape of the micropyramids. For example, an identical pyramid made from a non-lossy optical material would not exhibit any asymmetric reflection response at all. In such non-lossy structure, Lorentz reciprocity [55,56], which mandates transmission, T, to be symmetric, i.e. the same both for tip-to-base and base-to-tip incidence, mandates also reflection, R to be symmetric. This is because in the absence of optical loss, R=1-T. Now, in the lossy SiC micropyramid array we saw that reflection can be highly asymmetrical. However, this does not mean that the system becomes non-reciprocal. Indeed, the system is Lorentz reciprocal with the transmission being the same for both tip-side and base-side incidence. Now, since R=1-T-A both reflection, R and absorptance, A, can become highly asymmetric while transmission, T is symmetric. In other words, material optical loss is necessary to obtain any asymmetry effects in reflection/absorption. Although necessary, optical material loss by itself is not sufficient to produce strongly asymmetric effects in reflection/absorption. The highly asymmetric To our knowledge, such asymmetric reflection/absorption behavior of Fig. 8 in the frequency regime designated with the green shaded area is the first report of a strongly asymmetric reflection/absorption in passive systems with Reststrahlen-band materials [51]. We note in passing that asymmetric absorption/reflection effects, albeit much weaker in comparison with the ones shown in Fig. 8, have been also reported with plasmonic systems in the visible range [57][58][59]. Also, although not originally studied within this context, plasmonic metasurface structures that are impedance matched with vacuum, connected to a ground plane via a dielectric spacer [60][61][62], would be expected to respond with a highly asymmetric absorption/reflection. However, in this class of systems the plasmonic absorber material is separated by a dielectric spacer with a lower thermal conductivity which may limit their functionality as thermal emitters. In contrast, our proposed SiC micropyramid design involves a single-kind of connected absorber material of high thermal conductivity. In the following section, we explore whether this observed asymmetric absorption response of the SiC micropyramid system can be further enhanced to achieve a near uni-directional absorption/emission behavior. Our objective will be to harness reflection to achieve as close as possible the following target: zero reflection for tip-to-base incidence and unity reflection for base-to-tip incidence, with transmission being zero for both incidences. It is important to have as close as possible to a zero transmission, i.e. an opaque behavior. Actually, opacity is a necessary condition to perfect emissivity, as a non-zero transmission makes a medium less emissive. Note, in the extreme case of a transparent system (i.e. with transmission one) emissivity is always zero [63]. A zero reflection for tip-to-base incidence and unity reflection for base-to-tip incidence, with opacity, implies a unity absorption for tip-to-base incidence and a zero absorption for base-to-tip incidence. Then Kirchhoff's law states [34,35] that at a certain temperature, T, and wavelength, λ, the emissivity, e, of an opaque structure, towards a certain direction equals with the absorptance, A, for the same structure for light incident from that direction. In other words: e(λ, T ; base-to-tip) = A(λ, T ; tip-to-base) = 1 − R(λ, T ; tip-to-base)(4) e(λ, T ; tip-to-base) = A(λ, T ; base-to-tip) = 1 − R(λ, T ; base-to-tip)(5) Eqs. (4) and (5) make it clear why an opaque micropyramid system, with perfectly asymmetric reflection, zero and one respectively for base-to-tip and tip-to-base incidence, would emit only in the direction from the base to the tip of the micropyramid. In the following we investigate enhancing the observed asymmetry in absorption/reflection of the system of VIII. NEAR UNI-DIRECTIONAL SUPERABSORBER/SUPER-EMITTER BE- HAVIOR OF THE MICROPYRAMID ARRAY Firstly, in order to strengthen the asymmetric reflection/absorption response of the micropyramid array, we explore increasing the thickness of the fourth block, while keeping the entire micropyramid thickness sub-wavelength (about half the free space wavelength). Therefore, we increase the thickness of the micropyramid's base block from 1 µm to 3 µm and refer to this modified micropyramid-array design as design B. In this section we focus only around the frequency regime where cavity modes are excited. As we have discussed in Sec. VI this is the frequency region with the potential for a strongly asymmetric response. Fig. 9(a) depicts the spectral reflection response while Fig. 9(b) depicts the spectral absorption response of the design B micropyramid-array. The black-solid lines represent the result for tip-to-base incidence while the red-solid lines represent the result for base-to-tip incidence. The results of the original design of Fig. 8, which we will refer to as design A from thereon, are also included in Fig. 9 for comparison. The green-dotted lines designate the response of the original design A array for tip-to-base incidence, while the blue-dotted lines designate the corresponding response for base-to-tip incidence. We observe an increased reflection for the base-to-tip incidence for the modified design B array which results in a reduced absorption when compared with the original, design A, array. We attribute this to the larger size of the SiC block which comes at first contact with the impinging light for the base-to-tip incidence case. At the same time, for tip-to-base incidence we do not see any significant changes in the reflection response. This is because for such case, the first three blocks that the incident EM encounters are identical with the original, design A, array. However, the absorption ends-up being larger in the modified design B array for tip-to-base incidence. This is because the same EM energy that gets sequentially coupled to the last block, now interacts with a larger volume of lossy matter. The combined effect of absorption decrease for base-to-tip incidence with absorption increase for tip-to-base incidence leads to a stronger asymmetry in the absorption response. Secondly, we explore the effect of bringing the micropyramids closer. Therefore we start from the original design, design A, and bring each micropyramid building block closer at an interspacing of 4.2 µm. We refer to this modified design as design C. We show the results for the reflection and absorption response in Figs. 9(c) and 9(d) respectively. Here also, the black-solid lines designate the results for tip-to-base incidence, while the red-solid lines designate the results for base-to-tip incidence. Like in the cases of Figs. 9(a) and 9(b), the original design A results are depicted as well for comparison purposes. Indeed, as expected this closer-spaced array shows a larger reflection and so a smaller absorption for base-to-tip incidence. This is because for such case the wave encounters thick closely-spaced SiC blocks resulting in a weaker coupling to the cavity modes. On the other hand, the tip-to-base incidence case is not affected much by bringing the micropyramids closer, as the wave still firstly encounters relatively sparsely-spaced small SiC blocks. Accordingly, since absorption does not change for tip-to-base incidence but decreases for base-to-tip incidence when compared with the original design A array, the net effect is that the asymmetry in the absorption response becomes stronger for the modified design C micro-array. We then explore if both effects of asymmetry enhancement in the response of the two aforementioned designs can work in synergy when combined into one micropyramid-array design. We therefore explore a design where the micropyramids are more closely spaced at 4.2 µm, and have also their base block thicker (3 µm versus 1 µm of the original design of Fig. 8) [design BC]. We show the results for design BC in Fig. 10(a), for the reflection response and Fig. 10(b), for the absorption response. Black-solid and red-solid lines represent the results for tip-to-base and base-to-tip incidence respectively. Indeed, we find that the asymmetry in the absorption response gets further enhanced. In particular, in the frequency region represented with the green shading in Fig. 10 we observe that the absorption is near-unity for tip-to-base incidence and very small (less than 20%) for base-to-tip incidence. Essentially, the micropyramid array is an effective absorber only for light incident from the tip-side of the micropyramid. Moreover, in the green-shaded region the micropyramid system is near-opaque with transmission less than 2%. Accordingly, Eqs. (4) and (5) suggest that the micropyramid system of Fig. 10 would emit predominantly only in the direction from the micropyramid's base towards the tip. To be specific, the emission in the base-to-tip direction would be in the range of five to fifteen times stronger in the wavelength spectrum designated with the green shaded area in Fig. 10 with respect to the emission in the tip-to-base direction. In other words, the micropyramid array of Fig. 10 behaves as a near-unidirectional absorber/emitter. IX. CONCLUSIONS We have presented a stepped micropyramid SiC array structure with super-absorber capabilities over a broad spectral range within the SiC phonon-polariton gap (Reststrahlen band). For TH-polarized light, this structure acts simultaneously as a moth-eye antireflector, allowing almost all light to couple inside, and a broad-band microantenna. The superabsorber capabilities emanate from the cascaded coupling of corner/side vortex modes at each of the SiC blocks comprising the micropyramid. The cascaded coupling of such modes from block-to-block of the lossy SiC micropyramid system along with the asymmetric incoupling to these modes are the key protagonists that enable a highly uni-directional reflection/absorption response. Specifically, the micropyramid system is a strong absorber only for light impinging from the tips to the bases of the micropyramids. These results and physical insight for the underpinning mechanisms provide transferable design principles towards achieving a strongly asymmetric broadband reflection/absorption in other systems. Furthermore, we discussed how the SiC micropyramid system, when near-opaque with a highly asymmetric reflection/absorption, would behave as a near-unidirectional emitter by vitue of Kirchhoff's law [34,35]. Our proposed platform could be applied in improving the efficiency and directionality of infrared globar-type of sources [64]. Moreover, the operational bandwidth of this system falls within the atmospheric transparency window [28,29]. At the same time SiC is a weakly absorbing material for most of the solar radiation spectrum. These two facts, in combination, suggest that the micropyramid-array platform can be highly relevant to the emerging area of passive radiative cooling, a promising avenue for cooling buildings and vehicles [27,65]. The highly unidirectional emission characteristics of our proposed platform may inspire designs for temperature management of electronic and plasmonic devices [67], which is crucial to their resilience and functionality. Moreover, our system studied here may inspire structures comprising other infrared active materials that could improve the sensitivity of infrared cameras and detectors [6]. Appendix: Absorptance and electric field intensity in the micro-pyramid array In the following, we derive an expression yielding the absorptance through the micropyramid array, A, versus the electric field intensity within the bounds of the micropyramid building block in each unit cell of the array. The total absorption through the stepped micropyramid array, should be the sum of the respective absorption, A(i), provided by each of the constituent blocks of the SiC pyramid structure, i.e: A = 4 i=1 A(i) (A.1) The fraction of incident power that gets absorbed within the i th SiC building block of the micropyramid, A(i) can be obtained as: A(i) = P loss,i P inc , (A.2) where P loss,i represents the time-averaged power dissipation in the i th SiC building block in the elementary unit cell of the micropyramid array. Conversely, P inc represents the timeaveraged power incident on the elementary unit cell of the micropyramid array. As there is translational symmetry along the z direction, we consider the EM power that impinges through an area of A inc = a · l z , with l z being the length of an arbitrary segment along the z-direction of the SiC block, and a, being the micropyramid's interspacing (see figure A.1). From Poynting's theorem [66], we have that the time-averaged power dissipation per unit volume of a material, P loss,v is given by: P loss,v = ωε 0 ε r (ω) 2 \|E\| 2 , (A.3) where ε 0 is the vacuum permittivity and ε r (ω) the imaginary part of the relative permittivity of the material at the frequency ω of the impinging wave. Therefore, the power dissipated within the volume, V i = w xi · w yi · l z in the i th SiC micropyramid block in the elementary unit cell in the array, P loss,i will be: P loss,i = ωε 0 ε r l z 2 w xi 0 w yi 0 \|E(x, y)\| 2 dxdy, (A.4) where the widths, w xi , w yi are depicted in the schematics of P inc = al z 2cµ 0 \|E inc \| 2 , (A.5) where µ 0 is the vacuum permeability and c the vacuum speed of light. Now with the use of Eqs. A.4 and A.5 Eq. A.2 we obtain: where \|E norm (x, y)\| = \|E(x, y)\| \|E inc \| represents the electric field, within the SiC block, normalized by the incident electric field, occurring when an incident wave of frequency ω is incident on the structure. In other words, \|E norm (x, y)\| represents the electric field enhancement, at a certain location (x, y) within the i th SiC block of the micropyramid. Eq. (A6) underlines the importance of obtaining an enhanced electric field in areas within the micropyramid (as we have seen in Fig. 4), in order to obtain an enhanced absorption. This in turn, stresses on the key role of the block-cavity modes on obtaining a strong absorption response. The total absorption through the entire micropyramid can be then obtained from the summation of the respective absorptions, A(i), in each of the SiC blocks. A(i)= ε r (ω)ε 0 µ 0 ωc a\|E inc \| 2 w xi . 2 a wide low-reflectivity window within the SiC Reststrahlen band between approximately 10.3 and 11.1 µm [dashedred line in (a)], in spite of bulk SiC being nearly-perfectly reflecting in this range [dottedgreen line in Fig. 2(a)]. This low-reflectivity window translates into a broad absorption window with absorption exceeding 50% for the most part and reaching values as high as 80% [dashed-red line inFig. 2(b)]. This highly efficient absorption represents extraordinary absorption-enhancement factors, as high as 16, with respect to the absorption capabilities of bulk SiC [dotted-green line inFig. 2(b)] . 1 . 1We show these results for the reflection, R, and absorption A, in Figs. 3(a) and 3(b) respectively, with a dotted-red line (top-block array), solid-black line (second-block array), dashed-green line (third-block array) and dot-dashed-blue line (fourthblock array). The arrays comprising either of the first two blocks have near-zero reflective properties. These block sizes are deep sub-wavelength, thus only minimally disturb the path of the incident light. However, clear resonances still exist as we can observe by the peaks in the absorption spectrum and the enhanced electric field within the SiC block. For example, see the inset depicting the electric field intensity enhancement within the first block for the free space wavelength M1.The behavior is similar for the arrays comprised of the either the third or fourth block, as we can see from the peaks in the absorption spectrum and the electric field enhancement inside the blocks (for example see the inset for the fourth block for the free space wavelength M1). All blocks show a resonant response over a relatively broad spectrum, with the larger blocks sustaining resonances through longer wavelengths. We observe inFig. 3that for the large blocks the resonances are characterized by strong fields at the corners while for the smaller block the field is more uniform, with stronger fields at the sides of the block.The resonances and their corresponding electric field morphology arises from the surface phonon-polaritons at the facets of the SiC blocks and their interaction. Note that it is the strong phonon-photon coupling in SiC that yields a negative permittivity to SiC, thus enabling surface bound modes similar to those of metals at optical frequencies[51]. As a result of such surface bound modes, resonant optical trapping occurs within the blocks as one can observe from the EM energy circulation plots (white steamlines) in panels (c) and (d) ofFig. 3for the cases of the first and fourth block respectively. We can clearly identify vortices in the EM circulation, in either the sides [case seen inFig. 3(c)] or in the vicinity of the corners [case seen inFig. 3(d)]. These vortices signify the existence of cavity-like optical trapping behavior[49]. On the other hand, for the lower frequencies near the red bound of the Reststrahlen band, we do not find any block cavity modes, as the field goes around the block [for example see the insets inFig. 3for blocks 1 and 4 for frequency M3]. Now, let us compare the spectral reflection response of the entire pyramid array with that of the individual-blocks arrays. We can observe that the micropyramid array has a hybrid response borrowing characteristics from the reflection response of the different individual-block arrays. In particular, the reflection response of the micropyramid array for the spectral regions of the Fano interference and void resonances is dominated by the reflection response of the largest constituent block. On the other hand, the respective reflection response for the spectral region where the localized cavity modes are excited mimics the near-zero reflection characteristics of the smaller-sized blocks. This essentially means, that the pyramid arrangement of the individual microantenna blocks facilitates an efficient cascaded coupling[53] of EM energy to each individual microantenna blocks. In this manner, the entire micropyramid acts like an efficient broadband micro-resonator, causing the trapped EM energy to get absorbed by the SiC material over a broad frequency range.This effect can be visualized inFig. 4, where we depict with white streamlines the EM energy circulation for three selected characteristic free space wavelengths, -denoted as M1, M2 and M3-. These characteristic free-space wavelengths were previously designated inFig. 2, in panels (a)-(c). At the free space wavelength denoted M1, which is 10.5 µm [seeFig. 2]we can identify clear vortices around the sides or corners in all the SiC blocks comprising the micropyramid. These vortices in the EM energy streamlines signify the existence of trapped cavity-type of EM modes[49] in all the micropyramid's SiC blocks, similar to the ones we observed in the case of the individual blocks on their own inFig. 3[see panel (c) for the first block and panel (d) for the fourth block]. This means that the cavity-modes in each block of the pyramid, synergistically contribute toward an increased interaction between the EM field and the SiC matter in the entire pyramid that yields the strong absorption response we Fig. 4 ( 4c)]. In this case, the electric field intensity inside the micropyramid is much weaker than that of the incident light and essentially near-zero for most of the micropyramid [see Figs. 4(f) and 4(i)]. As a result, the micropyramid does respond with a low reflection, albeit with little absorption, as the incident light does not interact with the SiC absorbing matter.For completeness, we also depict the situation at the free-space wavelength M2, which is 11.2 µm. This represents the transition from the block-cavity resonant response [for the free space wavelengths in the spectrum around M1], where the micropyramid is a strong absorber, to the EM down-streaming via the voids [for the free space wavelengths in the spectrum around M3], where the micropyramid is quite transparent. At this transitional regime we neither observe strong vortices, nor a down-streaming around the pyramid [seeFig. 4(b)], and the electric field intensity enhancement within the micropyramid is moderate [see Figs. 4(e) and 4(h)]. This situation actually represents the Fano interference, between the localized type of modes, existing in the spectrum around M1, and the extended modes, existing in the spectrum around M3, that results in the pronounced reflection peak we have observed in Fig. 2.The focus of our discussion, from here there on, would be only within the spectral regime around free-space wavelength M1, where the block-cavity modes responsible for the absorption enhancement are excited. Thus far, we have discussed the broadband absorption enhancement capabilities for sparse micropyramid arrays, with a micropyramid separation of 10 µm. This means, that the cavity modes in each micropyramid, interact very weakly since they are spatially apart by about a wavelength. It would be therefore interesting to explore how the cavity modes are affected when they can interact more strongly, and whether such stronger interaction can yield a stronger absorption-enhancement phenomenon.We therefore calculate the spectral reflection and absorption response of arrays with closer-spaced micropyramids. We show the results inFig. 5for interspacings of 7.5 µmand 5.0 µm with dashed-red and solid-blue lines respectively, along with the original micropyramid array for comparison [black dotted lines]. We observe in Fig. 5(a) that the moth-eye-like SiC micropyramid building blocks continue to possess the broad-spectrum reflectionless capability, even when placed very closely. In particular, the array of micropyramid interspacings of 5.0 µm, leaves only small, one-micron-wide, voids at the base of the pyramid for light to squeeze through; this is about one-tenth the impinging light's wavelength. Yet, impressively it still possesses a near-reflectionless spectral response between 10.3 and 11 µm. We attribute this to the efficient cascaded coupling to each of the micro-antenna cavity modes of the individual blocks. intensity in each location in the vicinity of the SiC micropyramid in the new arrays and the respective value in the original micropyramid array. The results are shown in Figs. 5(c) and (d) for the respective cases of micropyramid interspacings of a=7.5 µm and a=5.0 µm. Fig. 6 ( 6a)]. Such broadband superabsorber behavior is not affected by the presence of a transparent material substrate. We observe in Fig. 6(b) that the absorptance remains unaffected if the micropyramid array is placed on a 10 µm thick BaF 2 substrate (dotted-blue line in the figure). Note that BaF 2 is a fairly transparent material in the mid-IR wavelength range. For the calculations of Fig. 6(b) we have taken the refractive index of BaF 2 to be equal to 1.36. Moreover, we find that the micropyramid design does not show sensitivity with respect to the alignment of the individual blocks. We show indicatively in Figs. 7(a) and 7(b) with a solid black line the results for the reflectance, R, and the absorptance, A, respectively, for the case of a misaligned-micropyramid array. For comparison the results for the symmetric micropyramid design are also depicted with dotted green lines. The off-center shift of the misaligned-pyramid design (see schematic in the top right panel of Fig. 7) is 150 nm, for the first block, 100 nm for the second block and 250 nm for the third block. We also show the EM energy circulation around the misaligned micropyramid in Fig. 7(c), along with the result for the corresponding symmetric design, i.e. the design without misalignment [seen in Fig. 7(d)]. The depicted energy circulation is for wavelength M1, designated with the orange vertical line in the figure, for which cavity-localized modes are excited in the individual blocks. x-axis symmetric pyramid designs of Figs. 4(a) and7(d). In fact, coupling to a certain side and/or corner appears stronger than the other. However, we see that the cascaded coupling from block to block is not obstructed. Therefore, the micropyramid still shows a low reflection/high absorption response in the wavelength regime around M1 (shaded green area in the figure), as we observe in Figs. 7(a) and 7(b).VII. STRONGLY ASYMMETRIC RESPONSE IN THE SUPERABSORBER BE-HAVIOR OF THE MICROPYRAMID ARRAY In all the results we have shown thus far, light is incident from the tip side of the micropyramid. It would be interesting to see what would happen if we reversed the micropyramid, so as light would be incident from the base side toward the tip of the micropyramid. We show the results for the reflection and absorption response of the reversely-oriented micropyramid in Figs. 8(a) and 8(b) respectively. It is impressive to observe that the broadband superabsorption behavior of the micropyramid strongly depends on the direction the light is coming from. In fact, we see that if we reverse the micropyramids in the array, absorption decreases dramatically [see red-dashed lines in behavior of the micropyramid array in the reflection and absorption is a result of the coupling to the cavity modes of the block encountered by the impinging light and subsequent cascaded coupling to the cavity modes of the remaining SiC blocks [see Figs. 8(c) and 8(e)]. Note, that for the same SiC-micropyramid array at the frequency regime where the block cavity modes are not excited (i.e. outside the green shaded area in Fig. 8) [see Figs. 8(d) and 8(f)], the reflection and absorption are similar for both tip-side and base-side incidence.This stresses that a high dielectric loss factor in an asymmetrically shaped structure does not necessarily imply a strongly asymmetric reflection/absorption response. Our results indicate that the key protagonists to an asymmetric absorption/reflection response are cascaded resonances with highly asymmetric in-coupling. This new physical insight uncovered by our work establishes a transferable design principle towards achieving a strongly asymmetric reflection/absorption in other systems. Fig. 8 , 8as close as possible to the aforementioned perfect condition in order to enable a near-unidirectional emitter behavior for the micropyramid array. ACKNOWLEDGEMENTS Financial support for the Ph.D. studentship of G. C. R. Devarapu by the College of Engineering, Mathematics and Physical Sciences (CEMPS) University of Exeter is acknowledged. Fig. A.1. Note, that in the above integration the (x, y) Cartesian coordinates have been off-set to be zero at the bottom left corner of the i th SiC block in the micropyramid. On the other hand, the time-averaged incident power P inc through the area, A inc , depicted in the schematics of Fig. A.1 is: \|E norm (x, y)\| 2 dxdy, (A.6) FIG. 1 : 1The 2D SiC moth-eye superabsorber (the structure is assumed to be infinite in the z-direction). The bottom-right panel represents a front view of the SiC stepped micropyramid, where we designate with a numeral each participating block. We also show in the bottom-left panel the permittivity, ε, of bulk SiC [real (black-solid line) and imaginary (dashed-green line) parts] in the Reststrahlen band (yellow shaded region), where the strong photon-phonon coupling yields a negative permittivity. FIG. 2: Spectral response versus the free space wavelength, λ free of the SiC-micropyramid array under normal incidence for the TE(TH) polarization case [solid-black (dashed-red) lines]. In (a) [(b)] the reflectance, R [absorptance, A] is shown. In addition, the reflection [absorption] from bulk SiC is shown in (a) [(b)] for comparison with dotted-green lines. We designate with M 1 , M 2 and M 3 selected modes, each falling in one of the three characteristic spectral regimes: the broadband reflectionless region, the region near the reflection maximum, and the second low-reflectivity window past the occurrence of the reflection peak. FIG. 3: Spectral reflection response [in (a)] and absorption response [in (b)] of the different periodic arrays made of the respective individual blocks comprising the pyramid of Fig. 1. The two characteristic frequencies/free space wavelengths, in the vicinity of the cavity and void resonances, marked as M1 and M3 respectively, are designated with the gray vertical lines. The insets depict the electric field intensity around the SiC block [47], for the respective arrays comprising the first or fourth of the micropyramids blocks for these two frequencies. Note, the corresponding color-map on the top right of the figure is in logarithmic scale. In addition, we show in panels (c) and (d) the energy circulation (white streamlines with arrows) around the first and fourth block respectively for the frequency M1. Note in (c) and (d), the background color-map represents the y-component of the time-averaged Poynting vector,S y [48]; hence negative values represent a downward EM flow. FIG. 8: Strongly asymmetric reflection/absorption response of the micropyramid array system with a= 5µm. The reflection, R, and absorption, A, for EM waves incident from the tip side of the micropyramid are shown with solid black lines in panels (a) and (b) respectively. Conversely, R, and A, are shown with dashed-red lines in the same figures, for EM waves incident from the base side of the micropyramid a.The green shaded area designates the spectral regime with a strong asymmetry in the reflection/absorption response between the tip-to-base and base-to-tip incidence. (c)-(d) EM energy circulation for wavelengths M1 and M3 for the case of incidence from the tip side of the micropyramid (e)-(f) EM energy circulation for wavelengths M1 and M3 for the case of incidence from the base side of the micropyramid. The wavelengths M1 and M3 are designated in(a)-(b) with gray vertical lines. Note that as in Figs. 4(a)-(c), the background in (c) through (f) represents the y-component of the time-averaged Poynting vector [48].FIG. 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Infrared emission spectrum of silicon carbide heating elements. J E Stewart, J C Richmond, J. of Research of the National Bureau of Standards. 592810J. E. Stewart and J. C. Richmond, Infrared emission spectrum of silicon carbide heating elements, J. of Research of the National Bureau of Standards 59, 2810 (1957). Ultrabroadband photonic structures to achieve highperformance daytime radiative cooling. E Raphaelli, A Raman, S H Fan, Nano Letters. 131457E. Raphaelli, A. Raman, and S. H. Fan, Ultrabroadband photonic structures to achieve high- performance daytime radiative cooling, Nano Letters 13, 1457 (2013). J D Jackson, Classical Electrodynamics. HobokenJohn Wiley and SonsThird editionJ. D. Jackson, Classical Electrodynamics (Third edition, John Wiley and Sons, Hoboken, 1998). FIG. 4: (a)-(c) Energy circulation (white streamlines with arrows) around the micropyramid array building blocks for three characteristic free-space wavelengths. S V Boriskina, J K Tong, W.-C Hsu, L Weinstein, X Huang, J Loomis, Y Xu, G Chen, designated in the corresponding spectral reflectance response of Fig. 2; M1: coupling to cascaded cavity resonances in the SiC blocks, M2: Fano interference M3: EM energy streams downwards through the voids. Note, the background color-map represents the y-component of the time-averaged Poynting vector, S [48S. V. Boriskina, J. K. Tong, W.-C. Hsu, L. Weinstein, X. Huang, J. Loomis, Y. Xu, G. Chen, FIG. 4: (a)-(c) Energy circulation (white streamlines with arrows) around the micropyramid array building blocks for three characteristic free-space wavelengths, designated in the corresponding spectral reflectance response of Fig. 2; M1: coupling to cascaded cavity resonances in the SiC blocks, M2: Fano interference M3: EM energy streams downwards through the voids. Note, the background color-map represents the y-component of the time-averaged Poynting vector, S [48]; -(f) Corresponding electric-field intensity enhancement (g)-(i) Same as in (d)-(f) but only the electric-field intensity enhancement inside the micropyramid is shown with a saturated colormap. hence negative values represent a downward EM flow. (d). red is used for any electric-field intensity enhancement that is higher than the maximum value of the associated colormaphence negative values represent a downward EM flow. (d)-(f) Corresponding electric-field intensity enhancement (g)-(i) Same as in (d)-(f) but only the electric-field intensity enhancement inside the micropyramid is shown with a saturated colormap [red is used for any electric-field intensity enhancement that is higher than the maximum value of the associated colormap] a=7.5FIG. 5: (a) Spectral reflectance, R, for the new arrays with micropyramid interspacings. FIG. 5: (a) Spectral reflectance, R, for the new arrays with micropyramid interspacings, a=7.5 The reflectance of the original array with interspacing a=10 µm (dotted-black lines) is also shown for comparison. (b) Same as in (a) but for the absorptance, A. (c) Electric field intensity for the array with interspacing, a=7.5 µm, with respect to the corresponding values of the original micropyramid array with a= 10 µm. µm (dashed-red lines) and a=5 µm (solid-blue lines. d) Same as in (c) but for the array with interspacing, a=5 µmµm (dashed-red lines) and a=5 µm (solid-blue lines). The reflectance of the original array with interspacing a=10 µm (dotted-black lines) is also shown for comparison. (b) Same as in (a) but for the absorptance, A. (c) Electric field intensity for the array with interspacing, a=7.5 µm, with respect to the corresponding values of the original micropyramid array with a= 10 µm. (d) Same as in (c) but for the array with interspacing, a=5 µm. (a) Absorption enhancement, A enha , exhibited by the micropyramid-array superabsorber of interspacing a=5 µm. (b)The influence of a BaF 2 substrate of thickness d=10 µm, on the absorptance, A, is shown with a dotted-blue line. For comparison, the absorptance for the array without the substrate is shown with a solid-black line. The green shaded area designates the spectrum where the micropyramid-array with. interspacing a=5 µm behaves as a superabsorber [31FIG. 6: (a) Absorption enhancement, A enha , exhibited by the micropyramid-array superabsorber of interspacing a=5 µm. (b)The influence of a BaF 2 substrate of thickness d=10 µm, on the absorptance, A, is shown with a dotted-blue line. For comparison, the absorptance for the array without the substrate is shown with a solid-black line. The green shaded area designates the spectrum where the micropyramid-array with interspacing a=5 µm behaves as a superabsorber [31]. Results are shown with a solid black for the case of a misaligned-pyramid array. Spectral response of reflectance. in (a)] and absorptance [in (b). see schematic depicted at the right of figFIG. 7: Spectral response of reflectance [in (a)] and absorptance [in (b)]. Results are shown with a solid black for the case of a misaligned-pyramid array [see schematic depicted at the right of fig. The corresponding results for the symmetric micropyramid design. depicted at the right of figThe corresponding results for the symmetric micropyramid design [depicted at the right of fig. The shaded green area represents the spectrum where localized cavity resonances are excited in the individual blocks. The corresponding EM energy circulation for wavelength M1, designated with the vertical line in (a)-(b), is shown in (c) for the misaligned-pyramid array and in (d) for the corresponding symmetric one. Note that as in Figs. 4c), the background in (c) and (d) represents the y-component of the time-averaged Poynting vector [48are also shown with a dotted-green line for comparison. The shaded green area represents the spectrum where localized cavity resonances are excited in the individual blocks. The corresponding EM energy circulation for wavelength M1, designated with the vertical line in (a)-(b), is shown in (c) for the misaligned-pyramid array and in (d) for the corresponding symmetric one. Note that as in Figs. 4(a)-(c), the background in (c) and (d) represents the y-component of the time-averaged Poynting vector [48]. or by bringing the micropyramids closer at an interspacing of 4.2 µm (design C). The reflectance, R, (top panels) and absorptance, A (bottom panels) are shown for each design with black-solid lines (red-solid lines) for tip-to-base (base-to-tip) incidence. For comparison the respective results for the design of Fig. 8 (design A) are shown with green-dotted lines (tip-to-base incidence) and blue-dotted-lines (base-to-tip incidence). The green shading designates the frequency region of strong asymmetry in the reflection/absorption response between tip-to-base incidence and base-totip incidence FIG. 10: The micropyramid as a near-unidirectional absorber/emitter. Enhancing the asymmetry in the reflection/absorption response between tip-to-base and base-to-tip incidence by considering a thicker, 3 µm-thick block for the pyramid base (design B). 9Reflection, R, [(a)] and absorption, A, [(b)] versus free space wavelength are shown for a micropyramid array design (design BC. with interspacing a=4.2 µm and micropyramid block sizes same as the design of Figs. 2/4FIG. 9: Enhancing the asymmetry in the reflection/absorption response between tip-to-base and base-to-tip incidence by considering a thicker, 3 µm-thick block for the pyramid base (design B), or by bringing the micropyramids closer at an interspacing of 4.2 µm (design C). The reflectance, R, (top panels) and absorptance, A (bottom panels) are shown for each design with black-solid lines (red-solid lines) for tip-to-base (base-to-tip) incidence. For comparison the respective results for the design of Fig. 8 (design A) are shown with green-dotted lines (tip-to-base incidence) and blue-dotted-lines (base-to-tip incidence). The green shading designates the frequency region of strong asymmetry in the reflection/absorption response between tip-to-base incidence and base-to- tip incidence FIG. 10: The micropyramid as a near-unidirectional absorber/emitter. Reflection, R, [(a)] and absorption, A, [(b)] versus free space wavelength are shown for a micropyramid array design (design BC) with interspacing a=4.2 µm and micropyramid block sizes same as the design of Figs. 2/4 The green-shading designates a frequency regime where the micropyramid absorbs with near 100% efficiency when light incident from the tip-side of the micropyramid while it absorbs weakly. except for the fourth block being thicker (3 µm thick. less than 20%, when light is incident from the base-side of the micropyramid in this manner acting as a uni-directional absorber/emitterexcept for the fourth block being thicker (3 µm thick). The green-shading designates a frequency regime where the micropyramid absorbs with near 100% efficiency when light incident from the tip-side of the micropyramid while it absorbs weakly, less than 20%, when light is incident from the base-side of the micropyramid in this manner acting as a uni-directional absorber/emitter.	[]
[ "Conformally Sequestered SUSY Breaking in Vector-like Gauge Theories", "Conformally Sequestered SUSY Breaking in Vector-like Gauge Theories" ]	[ "M Ibe \nDepartment of Physics\nUniversity of Tokyo\n113-0033TokyoJapan\n", "Izawa K ", "-I ", "Y Nakayama \nDepartment of Physics\nUniversity of Tokyo\n113-0033TokyoJapan\n\nResearch Center for the Early Universe\nUniversity of Tokyo\n113-0033TokyoJapan\n", "Y Shinbara \nDepartment of Physics\nUniversity of Tokyo\n113-0033TokyoJapan\n", "T Yanagida \nDepartment of Physics\nUniversity of Tokyo\n113-0033TokyoJapan\n\nResearch Center for the Early Universe\nUniversity of Tokyo\n113-0033TokyoJapan\n" ]	[ "Department of Physics\nUniversity of Tokyo\n113-0033TokyoJapan", "Department of Physics\nUniversity of Tokyo\n113-0033TokyoJapan", "Research Center for the Early Universe\nUniversity of Tokyo\n113-0033TokyoJapan", "Department of Physics\nUniversity of Tokyo\n113-0033TokyoJapan", "Department of Physics\nUniversity of Tokyo\n113-0033TokyoJapan", "Research Center for the Early Universe\nUniversity of Tokyo\n113-0033TokyoJapan" ]	[]	We provide, in a framework of vector-like gauge theories, concrete models for conformal sequestering of dynamical supersymmetry (SUSY) breaking in the hidden sector. If the sequestering is sufficiently strong, anomaly mediation of the SUSY breaking may give dominant contributions to the mass spectrum of SUSY standardmodel particles, leading to negative slepton masses squared. Thus, we also consider a model with gravitational gauge mediation to circumvent the tachyonic slepton problem in pure anomaly mediation models. 7 Similar situations of the conformal fixed point with non-trivial Yukawa interactions are discussed in Refs.[14].8 Without loss of generality, we adopt the convention of the holomorphic gauge coupling in Ref.[2].	10.1103/physrevd.73.015004	[ "https://arxiv.org/pdf/hep-ph/0506023v2.pdf" ]	1,658,007	hep-ph/0506023	9575aef8dc98721d10abc30dec5a95e2d667ac64	Conformally Sequestered SUSY Breaking in Vector-like Gauge Theories 21 Feb 2006 M Ibe Department of Physics University of Tokyo 113-0033TokyoJapan Izawa K -I Y Nakayama Department of Physics University of Tokyo 113-0033TokyoJapan Research Center for the Early Universe University of Tokyo 113-0033TokyoJapan Y Shinbara Department of Physics University of Tokyo 113-0033TokyoJapan T Yanagida Department of Physics University of Tokyo 113-0033TokyoJapan Research Center for the Early Universe University of Tokyo 113-0033TokyoJapan Conformally Sequestered SUSY Breaking in Vector-like Gauge Theories 21 Feb 2006 We provide, in a framework of vector-like gauge theories, concrete models for conformal sequestering of dynamical supersymmetry (SUSY) breaking in the hidden sector. If the sequestering is sufficiently strong, anomaly mediation of the SUSY breaking may give dominant contributions to the mass spectrum of SUSY standardmodel particles, leading to negative slepton masses squared. Thus, we also consider a model with gravitational gauge mediation to circumvent the tachyonic slepton problem in pure anomaly mediation models. 7 Similar situations of the conformal fixed point with non-trivial Yukawa interactions are discussed in Refs.[14].8 Without loss of generality, we adopt the convention of the holomorphic gauge coupling in Ref.[2]. Introduction It is widely believed that conformal field theory is dynamically realized in a large class of non-abelian gauge theories with a certain number of matter multiplets (see Ref. [1]). Conformal gauge theory is very attractive in the phenomenological point of view, since if it includes a SUSY-breaking sector, conformal sequestering [2,3] of the SUSY breaking may occur, providing a solution to the flavor-changing neutral current (FCNC) problem in the supersymmetric standard model (SSM). 1 It is tempting to consider vector-like gauge theories for the SUSY breaking, since they are naturally incorporated into vector-like superconformal gauge theories, which are relatively well understood. In this paper we extend vector-like gauge theories for the SUSY breaking [5] by adding massive hyperquarks to turn the full high-energy theory above the mass threshold into conformal gauge theory. We find, however, that this simple extension does not achieve the conformal sequestering due to the presence of an unwanted global U(1) symmetry. To eliminate the unwanted global symmetry we introduce non-abelian gauge interactions acting on the additional massive hyperquarks. We find various examples realizing the sequestering. We first discuss SP (3N + 1) × SP (N) 6 gauge theories where all gauge coupling constants at the infrared fixed point are small for N > 1 and perturbative calculations are applicable. We show by an explicit one-loop calculation that the theories have non-trivial fixed points and the sequestering of the SUSY-breaking effects indeed occurs. However, the sequestering is too mild to be applied to the phenomenology, since all the couplings are weak. Therefore, we dwell on strongly coupled conformal gauge theory such as an SP (3) × SP (1) 2 theory in this paper. 2 We also propose a Planck-suppressed gauge mediation which circumvents the tachyonic mass problem for sleptons in anomaly mediation. 3 Owing to the gravitational nature of this gauge mediation, the size of the gauge-mediated SUSY breaking is at most comparable 1 See Ref. [4] for some other phenomenological applications of superconformal dynamics. 2 We are unable to prove explicitly that such a theory has a non-trivial infrared fixed point and the required sequestering is obtained, since gauge couplings are all strong. We only state, in this paper, why we expect that is the case. 3 This construction is essentially independent of the above model of conformal SUSY breaking and serves as a generic way to make anomaly mediation phenomenologically viable. to the anomaly-mediation effects. For the lowest messenger scale, the total model provides a hybrid scheme [6] of anomaly [7] and gauge [8] mediations of SUSY breaking. Conformal SUSY breaking The IYIT model [5] for SUSY breaking is based on an SP (N) gauge theory with 2(N + 1) chiral superfields (hyperquarks), Q i α , in the fundamental (2N-dimensional) representation. 4 Here, α = 1, · · · , 2N and i = 1, · · · , 2(N + 1). We introduce (N + 1)(2N + 1) gauge singlet chiral superfields, S ij (= −S ji ), and impose the flavor SU(2N + 2) symmetry in the superpotential, W = λS ij Q i Q j ,(1) where S ij are assumed to transform as an antisymmetric (N + 1)(2N + 1) representation of the flavor SU(2N + 2) and we omit the color indices for simplicity. The reason why we impose the SU(2N + 2) symmetry becomes clear in the next section. The effective low-energy superpotential is given by W eff = X(PfV ij − Λ 2(N +1) ) + λS ij V ij ,(2) in terms of gauge invariant low-energy degrees of freedom V ij ∼ Q i Q j . Here, X is an additional chiral superfield and Λ denotes a dynamical scale of the SP (N) gauge interaction. We see that the superfields S ij have non-vanishing F terms in the vacuum and the SUSY is spontaneously broken. Notice here that the model possesses a U(1) R symmetry in addition to the flavor SU(2N + 2). conformality Now let us introduce 2n F massive hyperquarks, Q ′k , where k = 1, · · · , 2n F . The mass term is written as W mass = i mQ ′i Q ′i+n F .(3) Here, i runs from 1 to n F . Above this mass scale, the high-energy theory is an SP (N) gauge theory with N F = 2(N + 1) + 2n F hyperquarks. The SUSY SP (N) gauge theory with N F hyperquarks is expected to be scale-invariant in the infrared for 3(N + 1) < N F < 6(N + 1) [9]. We check, in the following, that the theory with the superpotential Eq.(1) can also be scale-invariant in the infrared. The NSVZ beta function [10] relates the running of the canonical gauge coupling constant to the anomalous dimension factors, γ Q and γ Q ′ , of the hyperquarks, Q and Q ′ , as µ d dµ α g = −α 2 g 3(N + 1) − (N + 1)(1 − γ Q ) − n F (1 − γ Q ′ ) 2π − (N + 1)α g ,(4) where α g is defined in terms of the gauge coupling constant g of SP (N) as α g = g 2 /(4π) and µ denotes the renormalization scale. Here and hereafter in this section, we neglect the masses of the hyperquarks Q ′k . The beta function of the Yukawa coupling constant in Eq.(1) is also given in terms of the anomalous dimension factors of the hyperquarks, γ Q , and of the singlet chiral fields, γ S , by µ d dµ α λ = α λ (γ S + 2γ Q ),(5) where α λ is defined in terms of the Yukawa coupling constant λ as α λ = λ 2 /(4π). When the theory is scale-invariant with non-vanishing coupling constants, the beta functions in Eqs. (4) and (5) vanish. That is, we have, at the infrared fixed point, 3(N + 1) − (N + 1)(1 − γ Q ) − n F (1 − γ ′ Q ) = 0,(6)γ S + 2γ Q = 0.(7) These conditions determine the anomalous dimensions at the fixed point. The anomalous dimensions at the fixed point are consistent with the unitarity of the theory for − 1 ≤ γ Q , −1 ≤ γ Q ′ , 0 ≤ γ S ,(8) which comes from the restriction for unitary representation of the superconformal algebra [11]: 5 the above anomalous dimensions are consistent with the unitarity conditions for any gauge-singlet chiral multiplets such as QQ, Q ′ Q ′ , and S. Notice that the vanishing of the NSVZ beta function is consistent with the existence of the anomaly free U(1) R symmetry that enters in the superconformal algebra with the charges of the matter fields given by R i = (2 + γ i )/3; i = Q, Q ′ , S [11]. In this simple extension of the IYIT model, the anomalous dimensions cannot be determined uniquely from Eqs. (6) and (7), and hence, the charge assignment of the U(1) R is not determined only with this information. Now, we show by a perturbative calculation that the fixed point is infrared stable. We first see that the gauge and Yukawa coupling constants at the infrared fixed point are small if N F is just below 6(N + 1), as in the case of the Banks-Zaks fixed point [12]. In this case we can obtain the anomalous dimensions at the one-loop level as γ Q = 2N + 1 2π α λ − 2N + 1 4π α g ,(9)γ Q ′ = − 2N + 1 4π α g ,(10)γ S = 2N 2π α λ .(11) For n F = 2(N + 1) − ε, we determine the coupling constants at the fixed point from Eqs. (6) and (7), as α * g = 4πε 7N 2 + 9N + 2 3N + 1 2N + 1 1 + O ε N ,(12)α * λ = 2πε 7N 2 + 9N + 2 1 + O ε N ,(13) and the one-loop approximation is justified a posteriori for small ε/N. 6 We can explicitly examine the infrared stability of the fixed point by considering the renormalizaiton group (RG) evolutions near the fixed point. The RG equations of the small deviations, ∆α g ≡ α g − α * g , ∆α λ ≡ α λ − α * λ ,(14) are given by µ d dµ ∆α g = ∂β g ∂α g * ∆α g + ∂β g ∂α λ * ∆α λ ,(15)µ d dµ ∆α λ = ∂β λ ∂α g * ∆α g + ∂β λ ∂α λ * ∆α λ ,(16) where β G and β λ denote the beta functions of α g and α λ given by Eqs. (4) and (5), respectively, and the values with the subscript " * " are evaluated at the fixed point. By using Eqs.(9)-(13), we find that all the eigenvalues of the coefficient matrix {∂β k /∂α l } are positive at the fixed point in Eqs. (12) and (13), where k, l = g, λ. Therefore, the fixed point in Eqs. (12) and (13) is infrared stable at least against small deviations from the fixed point. 7 non-sequestering We are at the point to show that the sequestering of the SUSY breaking does not occur due to an unwanted global U(1) symmetry in this simple extension. By following Luty and Sundrum [2], we consider the RG evolutions of the wave function renormalization factors near the fixed point, d dt ∆ ln Z i = −γ i + γ * i , ∆ ln Z i ≡ ln Z i + γ * i t, t ≡ ln(µ/M * ),(17) where i = Q, Q ′ , S and γ * i are the anomalous dimensions at the fixed point given by Eqs.(9)- (11). Here, M * denotes the scale where the theory enters the conformal regime below the reduced Planck scale M G ≃ 2.4 ×10 18 GeV. The deviations from the fixed point can be parameterized by ∆α g and ∆α λ which, in turn, can be expressed as 8 ∆α g = α 2 g 2π − (N + 1)α g * ((N + 1)∆ ln Z Q + n F ∆ ln Z Q ′ ),(18)∆α λ = −α * λ (2∆ ln Z Q + ∆ ln Z S ).(19) By using the above expressions, we rewrite the RG equation Eq.(17) as d dt ∆ ln Z i = − ∂γ i ∂α g * ∆α g − ∂γ i ∂α λ * ∆α λ (20) = − ∂γ i ∂α g α 2 g 2π − (N + 1)α g * ((N + 1)∆ ln Z Q + n F ∆ ln Z Q ′ ) + ∂γ i ∂α λ α λ * (2∆ ln Z Q + ∆ ln Z S ),(21) and we define the coefficient matrix L ij by d dt ∆ ln Z i = j=Q,Q ′ ,S L ij ∆ ln Z j .(22) When all the eigenvalues of L are positive, all ∆ ln Z i go to zero as t → −∞ (the infrared limit) and hence the SUSY breaking is sequestered [2,3]. Unfortunately, we find that the coefficient matrix L has a zero eigenvalue. Thus, one linear combination of ∆ ln Z i is constant in the course of the RG evolution and it is not suppressed at the infrared fixed point. We call it as ∆ lnZ. Since the vanishing eigenvalue corresponds to the eigenvector (∆ ln Z Q , ∆ ln Z Q ′ , ∆ ln Z S ) = (1, −(N + 1)/n F , −2), we find that the solution to the Eq.(22) in the infrared limit is ∆ ln Z Q ∝ (∆ lnZ) 0 ,(23)∆ ln Z Q ′ ∝ − N + 1 n F (∆ lnZ) 0 ,(24)∆ ln Z S ∝ −2(∆ lnZ) 0 ,(25) with an O(1) proportionality factor, where (∆ lnZ) 0 denotes the value at t = 0. In general, the initial value (∆ lnZ) 0 contains visible sector superfields q i as weakly coupled spectators such as (∆ lnZ) 0 ⊃ κ ab M 2 G q † a q b ,(26) where κ ab denote O(1) coefficients. Therefore, from Eq.(25), we find that the SUSY breaking effects to the visible sector are not sequestered. 9 The reason of our failure can be traced to the existence of a global U(1) symmetry [2,3] under which the SUSY breaking superfield S ij transforms non-trivially. In general, when an anomaly-free (non-R) U(1) symmetry exists, the charge assignment ω i determines the eigenvector of the coefficient matrix L ij for a vanishing eigenvalue: j L ij ω j = 0.(27) In the present case, a linear combination ∆ lnZ (of ∆ ln Z i ) remains constant in the infrared limit and the SUSY breaking effects are not sequestered if the SUSY breaking SP (N) SP (N ′ ) SP (N ′ ) SP (N ′ ) SP (N ′ ) SP (N ′ ) SP (N ′ ) Q × 2(N + 1) 2N 1 1 1 1 1 1 Q ′ 2N 2N ′ 1 1 1 1 1 Q ′ 2N 1 2N ′ 1 1 1 1 Q ′ 2N 1 1 2N ′ 1 1 1 Q ′ 2N 1 1 1 2N ′ 1 1 Q ′ 2N 1 1 1 1 2N ′ 1 Q ′ 2N 1 1 1 1 1 2N ′ S ij 1 1 1 1 1 1 1SP (N) × SP (N ′ ) 6 , (N = 3N ′ +1). Here, the subscripts of the fundamental representations denote the dimensions of the representations. In terms of the SP (N) gauge theory, the number of the fundamental representation is given by N F = 2(N + 1) + 6 × 2N ′ = 6N − 2, while the number of the fundamental representation of each SP (N ′ ) gauge theory is given by N ′ F = 2N = 6N ′ + 2. superfields have non-vanishing charges. The eigenvector we have found above corresponds to the charge assignment (ω Q , ω Q ′ , ω S ) = (1, −(N + 1)/n F , −2) of an anomaly-free U(1) symmetry. Thus, in order to realize the sequestering, we should violate the global U (1) symmetry under which the SUSY breaking superfields transform non-trivially, provided we do not take (∆ lnZ) 0 = 0 by fine tuning. In the next section, we introduce additional gauge symmetries, where the unwanted U(1) symmetry is broken by anomaly due to the new gauge interactions. Conformally sequestered extensions We introduce gauge interactions acting on the massive hyperquarks, Q ′k , where the unwanted global U(1) symmetry is broken by anomaly due to the new gauge interactions. We deal, in this section, with SP (N) × SP (N ′ ) 6 , (N = 3N ′ + 1) gauge theory, where the former SP (N) corresponds to the gauge group for the SUSY breaking and the latter SP (N ′ ) 6 gauge group is introduced to break the unwanted U(1) symmetry. We list all the matter contents in Table 1. We take such a large gauge group to see explicitly by a perturbative calculation that the conformal sequestering occurs. Indeed all the couplings at the infrared fixed point are weak for N ′ > 1 in the present model. conformality Now, we check that the theory with the extended gauge symmetry can be scale-invariant in the infrared. In this model, the beta functions of the SP (N) gauge coupling constant α g , the SP (N ′ ) gauge coupling constant α g ′ , and the Yukawa coupling constant α λ in Eq.(1) are given by µ d dµ α g = −α 2 g 3(N + 1) − (N + 1)(1 − γ Q ) − 6N ′ (1 − γ Q ′ ) 2π − (N + 1)α g ,(28)µ d dµ α g ′ = −α 2 g ′ 3(N ′ + 1) − N(1 − γ Q ′ ) 2π − (N ′ + 1)α g ′ ,(29)µ d dµ α λ = α λ (γ S + 2γ Q ),(30) where we have assumed that all the SP (N ′ ) sectors are equivalent. Namely, we have imposed an exchange symmetry between any two SP (N ′ )'s in the SP (N ′ ) 6 so that the SP (N ′ ) 6 has a common gauge coupling constant α g ′ . Then, by requiring all the beta functions to vanish, we determine the anomalous dimensions uniquely as γ Q = − 2(N(N + 1) − 9N ′ (N ′ + 1)) N(N + 1) = − 4 9N ′2 + 9N ′ + 2 ,(31)γ Q ′ = N − 3(N ′ + 1) N = − 2 3N ′ + 1 ,(32)γ S = −2γ Q .(33) Here, we have neglected the masses of the hyperquarks Q ′ . We also determine the coupling constants at the infrared fixed point by a perturbative calculation. The anomalous dimensions at the one-loop level are given by γ Q = 2N + 1 2π α λ − 2N + 1 4π α g ,(34)γ Q ′ = − 2N ′ + 1 4π α g ′ − 2N + 1 4π α g ,(35)γ S = 2N 2π α λ .(36) Then, Eqs.(31)-(36) determine the coupling constants at the infrared fixed point by Table 2: Stability of the infrared fixed point. a g,g ′ ,λ denote the coupling constant, N + 1 2π α * g = 8(9N ′ + 4) 3(2N ′ + 1)(3N ′ + 1) 2 ,(37){a * g , a * g ′ , a * λ } eigenvalues of M N ′ = 2 {0.2, 0.07, 0.1} {0.5, 0.1, 0.002} N ′ = 3 {0.1, 0.1, 0.07} {0.2, 0.05, 0.002} N ′ = 4 {0.07, 0.1, 0.04} {0.1, 0.03, 0.001} N ′ = 5 {0.05, 0.09, 0.03} {0.1, 0.02, 0.0006}a * g = α * g (N + 1)/(2π), a * g ′ = α * g ′ (N ′ + 1)/(2π) , and a * λ = α * λ N/π. All the eigenvalues of the coefficient matrix M are positive for N ′ > 1. N ′ + 1 2π α * g ′ = 12(N ′ + 1)(3N ′2 − 3N ′ − 2) (2N ′ + 1)(3N ′ + 2)(3N ′ + 1) 2 ,(38)N π α * λ = 8 (3N ′ + 2)(3N ′ + 1) .(39) We see that all the coupling constants are small and the perturbative calculation is reliable for N ′ > 1. 10 The above result enables us to explicitly analyze the infrared stability of the fixed point in the same way as done in the previous section. The RG equations of the small deviations ∆α k ≡ α k − α * k , (k = g, g ′ , λ) are given by µ d dµ ∆α k = l=g,g ′ ,λ M kl ∆α l ,(40) where the coefficient matrix M is defined by M kl = ∂β k ∂α l * .(41) sequestering We now discuss the sequestering of the SUSY breaking. The RG equations of the wave function renormalization factors Z i near the fixed point are given by d dt ∆ ln Z i = − k=g,g ′ ,λ ∂γ i ∂α k * ∆α k (42) = i=Q,Q ′ ,S L ij ∆ ln Z j .(43) Here, the coefficient matrix L in the second line is given by using the following relations: ∆α g = α 2 g 2π − (N + 1)α g * ((N + 1)∆ ln Z Q + 6N ′ ∆ ln Z Q ′ ),(44)∆α g ′ = α 2 g ′ 2π − (N ′ + 1)α g ′ * N∆ ln Z Q ′ ,(45)∆α λ = −α * λ (2∆ ln Z Q + ∆ ln Z S ).(46) The sequestering of the SUSY breaking is realized when all the eigenvalues of L are positive. Interestingly, as we show below, the coefficient matrix L has the same eigenvalues as A kj γ j = b k ,(47) where the coefficient matrix A and the vector b can be read off from Eqs.(28)-(30) and k = g, g ′ , λ. Then, we see the following relations: M kl = j=Q,Q ′ ,S A kj Γ jl ,(48)L ij = k=g,g ′ ,λ Γ ik A kj ,(49) where we have defined Γ ik ≡ ∂γ i ∂α k * .(50) Since the coefficient matrix A is invertible, the coefficient matrices M and L are similar to each other, so that they have the same eigenvalues. Therefore, the sequestering occurs automatically when the anomalous dimensions are uniquely determined by the conditions for the vanishing beta functions (i.e. A is invertible) and the fixed point is infrared stable Fortunately, we can make such non-sequestered combinations vanishing by imposing the flavor SU(2N + 2) symmetry (or a sufficiently large discrete subgroup thereof) at high energies so that the conformal sequestering of the SUSY breaking is realized. Namely, by assuming that the Kähler potential inducing soft masses for squarks and sleptons is restricted by the flavor SU(2N + 2) symmetry as κ ab M 2 G ij S † ij S ij q †a q b ,(51) we can set the linear combinations of ∆ ln Z's which are not sequestered to be zero. Then, as we have discussed, the remaining combinations of ∆ ln Z's are sequestered and the squared masses of the sfermions from Eq.(51) are suppressed at the infrared fixed point. This is the reason why we have imposed the flavor SU(2N + 2) symmetry in the SUSY-breaking sector. Finally, in the rest of this section, we show that the sequestering is too mild in the present model to solve the FCNC problem. In view of the Table 2, the smallest eigenvalue β ′ * of the coefficient matrix L (or equivalently M) is of the order of 10 −3 for N ′ ≥ 2. Thus, the linear combination of ∆ ln Z i that corresponds to the smallest eigenvalue approaches to the fixed point very slowly, which, in turn, prevents ∆ ln Z i from getting up to the fixed point immediately. That is, in the infrared regime (t ≪ 0), we find ∆ ln Z S (t) ∼ e β ′ * t ∆(lnZ) 0 ,(52)(∆ lnZ) 0 = c S (∆ ln Z S ) 0 + c Q (∆ ln Z Q ) 0 + c Q ′ (∆ ln Z Q ′ ) 0 ,(53) where ∆ lnZ corresponds to the eigenvector for the smallest eigenvalue, the subscript "0" indicates the value at t = 0, and c i denote numerical coefficients. By explicit calculation, we find that the coefficients c i are typically O(0.1 − 0.01) in our perturbative models. In order to solve the FCNC problem by sequestering, we should require ∆ ln Z S < ∼ 10 −7 at the SUSY-breaking scale [15]. 11 Thus, without fine tuning among ln Z i , we should require e β ′ * t < ∼ 10 −7 . However, since β ′ * is of the order of 10 −3 , it takes too long to achieve the sufficient sequestering. Therefore, we find that the sufficient sequestering cannot be expected in our perturbative models. In the perturbative examples, we have seen that the size of the "sequestering speed" β ′ * is not larger than the anomalous dimensions at the fixed point. Thus, in order to realize the sufficient sequestering (i.e. β ′ * = O(1)), we should require that the anomalous dimensions at the fixed point are of the order one. 12 This means that we must consider a strongly coupled conformal gauge theory. 13 In the next section, we discuss such a strongly 11 Here, we assume that the flavor diagonal masses of the sfermions are of the order of 1 TeV, which are suppressed compared to the gravitino mass of the order of 100 TeV (see discussions in section 5). 12 It is based on a naive expectation that the speed of the sequestering, β ′ * ∼ (∂γ/∂α)α\| * or (∂γ/∂α)α 2 \| * , is not so far from γ * even in the strongly coupled case (see Eqs.(43)-(46)). 13 Unfortunately, the strongly coupled case N ′ = 1 also seems inadequate since the anomalous dimensions are not sufficiently large. coupled theory and present the reason why we consider the sequestering might be also realized there, although the perturbative calculation is not applicable. SP (N) SP (N ′ ) SP (N ′ ) Q × 2(N + 1) 2N 1 1 Q ′ 2N 2N ′ 1 Q ′ 2N 1 2N ′ S ij 1 1 1 Strongly coupled SP (3) × SP (1) 2 model In this section, we discuss SP (3) × SP (1) 2 gauge theory as an example, where SP (3) corresponds to the gauge group for the SUSY breaking and the SP (1) 2 gauge group acts on massive hyperquarks Q ′k . We list the matter contents in Table 3. We assume that such a gauge theory with the Yukawa interaction in Eq.(1) has a non-trivial fixed point. Then, the anomalous dimensions for Q, Q ′ , and S at the fixed point are determined as γ Q = −1, γ Q ′ = −1, γ S = 2,(54) which sit on a boundary of the unitarity bound Eq.(8). 14 In 14 The reason we take this example is only for simplicity. In the phenomenological point of view, we only require a large size of the anomalous dimensions which satisfy the unitarity bound Eq. (8). 15 We know no calculable example that has an infrared unstable (non-trivial) fixed point in the present Table 4: Fixed point in the SP (N) × SP (N ′ ) theory. a * g,g ′ ,λ denote the coupling constant, { a * g ,a * g ′ ,a * λ } {γ Q , γ Q ′ , γ S } eigenvalues of M SP (3) × SP (1) 2 non-perturbative {-1, -1, 2} non-perturbative SP (5) × SP (2) 2 non-perturbative {-a g = α * g (N + 1)/(2π), a * g ′ = α * g ′ (N ′ + 1)/(2π), and a * λ = α * λ N/π.µ M G < ∼ 10 − 7 β ′ * .(55) When the RG scale µ comes close to the physical mass scale m phys of Q ′ , the theory ceases to be scale invariant and effectively becomes an asymptotically free SP (3) gauge theory of strong coupling with 8 hyperquarks Q, and finally SUSY is broken dynamically at µ < ∼ m phys . Here, the physical mass is given by m phys = (mM −γ Q ′ G ) 1 1−γ Q ′ = mM G ,(56) where the last equality results from the unitarity boundary value γ Q ′ = −1 in the present model. Thus, the above condition Eq.(55) for the sufficient sequestering can be rewritten class of SP (N ) × SP (N ′ ) n , (n = 1, 2, · · ·) gauge theories. 16 In what follows, we assume that the theory is in the vicinity of the conformal fixed point at M G , that is, we assume M * ≃ M G (see below Eq. (17)). in terms of the mass scale of Q ′ or the SUSY breaking scale Λ as Λ M G < ∼ m phys M G < ∼ 10 − 7 β ′ * .(57) As discussed in the next section, we are interested in the case where the gravitino mass is of the order of 100 TeV, which implies Λ ∼ 10 11−12 GeV. Thus, we claim that the above conditions can be satisfied for β ′ * = O(1), and hence, the FCNC can be suppressed in the present strongly coupled model. Circumventing the tachyonic slepton problem If the sequestering of SUSY breaking occurs sufficiently, the SUSY-breaking masses for the squarks and sleptons become negligibly small at low energies. In this situation we must invoke some mechanism to transmit sizable SUSY-breaking effects to the visible sector of the SSM. The most natural candidate is anomaly mediation [7]. This mechanism is not only theoretically interesting, but also phenomenologically attractive. This is because the gravitino mass is expected at O(100) TeV, which provides us with a solution to the gravitino problem [16,17]. However, the anomaly mediation mechanism suffers from the tachyonic slepton mass problem [7]. In this section we consider a Planck-suppressed gauge mediation to remedy this phenomenological defect of the anomaly mediation. 17 Let us introduce a messenger sector which consists of chiral superfields ψ,ψ, ψ ′ , andψ ′ . Here, ψ ,ψ and ψ ′ ,ψ ′ transform as vector-like representations under the gauge group of the SSM, and we take them to fit in complete SU(5) GUT representations, 5 + 5 * , for simplicity. Our additional superpotential terms are given by δW = h M 2 G S ij Q i Q j ψψ + m m ψψ ′ + m m ψ ′ψ ,(58) which let the SUSY breaking intact for a sufficiently large mass parameter m m . 18 Here, h denotes a coupling constant of order one and the combination S ij Q i Q j stands just for a SUSY-breaking superpotential term which has a non-vanishing F component (see Eq.(2)). 17 In the Appendix B, we also provide a renormalizable setup for such a remedy. 18 In fact, m m > ∼ m 3/2 is required, where m 3/2 denotes the gravitino mass. The SUSY-breaking effects are transmitted to the sfermions and Higgs bosons by the SSM gauge interactions (see Ref. [18]). In the SUSY-breaking dynamics, we expect \| S \| < ∼ Λ [19], which yields only Plancksuppressed R breaking effects in the gauge mediation. In this case, gauginos do not obtain sizable SUSY-breaking masses via the gauge mediation and the gaugino spectrum is virtually the same as in the purely anomaly-mediated one. 19 On the other hand, the scalar field φ obtains the mass squared via the gauge mediation for m m < m phys as m 2 φ ≃ 2 a=1,2,3 C φ a α a 4π 2 \|hF S \| 2 m 2 m Λ M G 4 ,(59)≃ 18 a=1,2,3 C φ a α a 4π 2 \|h\|m 3/2 \|λ\|m m 2 m 2 3/2 ,(60) where C φ a (a = 1, 2, 3) is the quadratic Casimir invariant for each gauge group relevant to the scalar φ. 20 In the above equation, we have used √ F S ≃ √ λΛ and the gravitino mass m 3/2 given by m 3/2 ≃ \|F S \| √ 3M G ≃ \|λ\|Λ 2 √ 3M G .(61) can overwhelm the negative contribution of the anomaly-mediated mass squared, 19 Since the superpotential has the constant term which is required to obtain the flat universe, we may as well introduce an interaction term between ψψ and the constant term. Then, the R breaking effects in the gauge mediation possibly become sizable [18], which may result in the gaugino spectrum different from the one in the pure anomaly mediation. The expression of the gauge mediated mass squared in Eq.(60) may also be altered. 20 Here, we have neglected RG effects from the MSSM couplings. 21 For m m < ∼ m 3/2 , even if the total SUSY-breaking were kept intact, messenger scalar particles would become tachyonic. Hence we restrict ourselves to m m > ∼ m 3/2 . The choice m m ∼ m 3/2 realizes the lowestscale model of gauge mediation (see Ref. [6,20]) for m 3/2 of order 100 TeV. We note that m m ∼ m 3/2 is realized by a relation m m ∼ m of the Lagrangian parameters in view of Eq.(56). Therefore, we conclude that the tachyonic slepton problem is resolved in the total model of anomaly and gauge mediation hybrid by tunning scales of these two mediations with each other. m 2 e A.M. ≃ − 6 5 33 5 α 1 4π 2 m 2 3/2 .(63) Note that the newly added superpotential term in Eq.(58) does not violate the global SU(2N + 2) symmetry which is relevant for the conformal sequestering. Hence, we can apply the above mechanism to the conformally sequestered models. However, we should note that in the case of conformally sequestered models, the first term in Eq.(58) is also sequestered in the course of the RG evolution from M * to m phys . 22 Thus, in order to realize the sizable gauge mediation effects as in Eq.(62), we need to compensate the sequestering effects by preparing the additional superpotential terms M * m phys β ′ * h M 2 G S ij Q i Q j ψψ + m m ψψ ′ + m m ψ ′ψ(64) at the scale M * which effectively realize the Eq.(58) after the conformal sequestering. This implies that the higher-dimensional term stems from integrating out an intermediate matter of mass (m phys /M * ) β ′ * M G with Planck-suppressed coupling to S ij Q i Q j . 23 In the above analysis, we have simply used Eq.(58) as a resultant effective superpotential at the scale m phys for the conformally sequestered models. Finally, we comment on the cosmologcal aspects of this class of models. Since the relic density of the lightest messenger particle is too much to be consistent with the observation, we should require that they decay into the SSM particles at early stage of the universe. This is implemented by introducing small mixings between the messenger particles and the SSM particles. In addition, it should also be noted that there are Goldstone bosons in the SUSY breaking sector, which correspond to the spontaneous breaking of the global 22 The sequestering can be seen through a field redefinitionS ij = (1 + λ −1 hψψ/M 2 G )S ij , which turns the effects of the superpotential coupling h into those of the Kähler couplings appearing as perturbations to the renormalization factors. 23 For example, we may consider a concrete model by introducing extra singlet supermultiplets X and X with a superpotential h 1 M G XSQQ + M X XX + h 2X ψψ,(65) at the scale M * . Here, h 1,2 denote coupling constants, M X the mass parameter of X. Then, after integrating out X andX, we can effectively obtain the first term in Eq.(58) at the scale m phys for M X ∼ M G (m phys /M * ) β ′ * ((h 1 h 2 )/h). SU(2N + 2) symmetry. However, those massless particles are decoupled from the thermal bath since they only couple with the SSM particles via the Planck-supressed operator, and hence they do not affect the history of the universe. 24 Therefore, we find that the present hybrid scheme yields also a consistent scenario from the cosmological point of view. with γ Q ′ = 1 − 3(N + 1) − (N + 1)(1 − x) 2(N + 1) − ε ,(69) where ε = 2(N + 1) − n F . The claim is that among these one-parameter R currents, the conformal one maximizes the anomaly a, which is obtained as follows: a = 2N(2N + 2) 3(R Q − 1) 3 − (R Q − 1) +2(2N + 2 − ε)2N 3(R Q ′ − 1) 3 − (R Q ′ − 1) +(N + 1)(2N + 1) 3(R S − 1) 3 − (R S − 1) ,(70) where we note that the R charges appearing in a are those of fermions (i.e. R ψ Q = R Q −1) because only fermions contribute to the anomaly. By maximizing a with respect to x, we can determine x * = γ Q \| * . The unique local maximum is achieved by setting x * = − ε 2 (2 + 3N) − 4ε(1 + N)(2 + 3N) + (1 + N) 2 (8 + 13N) −1 A; A ≡ 4 − 4ε + ε 2 + 22N − 16εN + 3ε 2 N + 32N 2 − 12εN 2 + 14N 3 + (ε − 2(1 + N))B, B ≡ ε 2 (1 + 2N)(1 + 6N) − 4ε(1 + N)(1 + 2N)(1 + 6N) + (2 + 9N + 7N 2 ) 2 . (71) To compare this rather complicated expression with the perturbative results, we expand Eq.(71) in terms of ε. Remarkably, the first order approximation is given by x 1 = − N 2 + 9N + 7N 2 ε,(72) which completely agrees with our Banks-Zaks-like calculation. Furthermore we can systematically study higher order corrections. 25 It is quite intriguing that the a-maximization determines all-order loop effects only from the one-loop result Eq.(70). We can also study a of the gauged version of extended IYIT model in section 3, which leads to conformal sequestering. Since the gauging enforces yet another constraint on the anomaly free R charge, we obtain a unique R charge assignment without using the a-maximization procedure. It is important to realize, following the general argument of 25 For instance, the two loop contribution should be x 2 = − 3N (1 + N (7 + 11N )) (1 + N ) 2 (2 + 7N ) 3 ε 2 . the monotonically decreasing a, that a gauged is less than a ungauged . This is obvious when x gauged * is sufficiently close to x ungauged * , since x ungauged * yields the local maximum of a(x). For example, we can show by a direct computation that a of the gauged extended IYIT model presented in section 3 is always less than that of the ungaged version presented in section 2 for a fixed gauge group. This result is consistent with the fact that our conformal fixed point is a stable one. In particular, it is worthwhile to notice that this is even true for N = 1 case, which cannot be treated in the one-loop approximation. Finally, it would be an interesting but challenging problem to obtain the speed of the conformal sequestering from the interpolating a-function. In Ref. [21], the off-shell a-function is proposed as solving a-maximization condition with a Lagrange multiplier ξ that enforces the constraint on the R charge: a(R(ξ), ξ) = Tr 3R 3 − R + ξ(constraint) , where R(ξ) is obtained by maximizing a with respect to R for fixed ξ, and the constraint is either ABJ anomaly free condition or the requirement that the superpotential be marginal. As was observed in [21], the first derivative of a(ξ) is related to the β function of the coupling constant. 26 Furthermore, the second derivative (Hessian) of a(ξ) at a fixed point ξ * is proportional to the slope of the β function ∂ 2 a(ξ) ∂ξ i ∂ξ j * ∝ ∂β i (ξ) ∂ξ j * . Consequently there is a chance to read the conformal sequestering matrix without performing the explicit loop calculation even for a strongly coupled theory. Unfortunately, we do not know the proportionality factor (related to the denominator of the NSVZ beta function evaluated at the fixed point) and the transformation matrix {∂ξ i /∂g j } nonperturbatively, so we cannot determine the conformal sequestering matrix. Since the conformal sequestering matrix is a physical renormalization invariant quantity while a(ξ) is not, we need an off-shell scheme-independent a-function for our purposes. B Another example of the hybrid scheme In section 5, we have considered the hybrid model of the anomaly and gauge mediated SUSY breaking, which solves the tachyonic slepton problem in a pure anomaly mediation model. In this appendix, we propose another example of the hybrid model which can be constructed with renormalizable interactions between the SUSY breaking sector and the messenger sector. That is, in addition to the anomaly mediated SUSY breaking, we consider the gauge mediated SUSY breaking discussed in Ref. [18], where messenger sector consists of N m flavors of chiral superfields ψ i andψ j (i, j = 1, · · · N m ) and N m flavors of chiral superfields ψ ′ i andψ ′ j (i, j = 1, · · · N m ). Here, ψ i , ψ ′ i andψ j ,ψ ′ j transform as 5 and 5 * of SU(5) GUT, respectively. Then, with the superpotential, hS ij ψ iψj + m m ψ iψ ′ i + m m ψ ′ iψ i ,(75) the SUSY-breaking effects are transmitted to the sfermions and Higgs bosons by the gauge interactions. Here, h denotes the coupling constant, m m the mass parameter, and we assume that h = h 0 of order one at M * < ∼ M G . As discussed in section 3, we impose the global SU(2N +2) symmetry to the SUSY breaking sector, and in order for the interaction in Eq.(75) to respect this symmetry, we assume that N m = 2(N + 1) and ψ,ψ and ψ ′ ,ψ ′ transform as 2N + 2 and 2N + 2 representations, respectively, of the SU(2N + 2). 27 In this case, the scalar field φ obtains the mass squared via the gauge mediation, and at the messenger scale, it is given by, m 2 φ ≃ 2 a=1,2,3 C φ a α a 4π 2 \|hF S \| 2 m 2 Here, " * " indicates the values evaluated at the fixed point. If all the eigenvalues of the coefficient matrix are positive, the fixed point in Eqs.(37)-(39) is infrared stable. In the coefficient matrix M in Eq.(41). Therefore, the sequestering occurs automatically, if the infrared fixed point determined in Eqs.(31)-(33) is stable. To prove that, we rewrite the conditions for the vanishing beta functions as j=Q,Q ′ ,S (i.e. all the eigenvalues of M are positive). Notice that this is no accident: the conformal sequestering originates from nothing but the attractor structure of the infrared fixed point.In our SP (3N ′ +1)×SP (N ′ ) 6 model, we have shown that the fixed point is determined from the conditions of vanishing beta functions and the fixed point is infrared stable for N ′ > 1. Thus, we have found that the sequestering is realized in our model. Notice that the relation between the infrared stability and the sequestering holds independently of the perturbative calculation. Therefore, even if perturbative analysis is not applicable, we may argue that the sequestering occurs, if the fixed point is expected to be infrared stable.It should be noted here that the unwanted global U(1) symmetry discussed in theprevious section is broken by anomalies of the SP (N ′ ) 6 gauge interactions and hence there is no conserved U(1) current. This is the reason why the matrix M does not have a zero eigenvalue. In addition to the above global U(1) symmetry, there are many unbroken global U(1)'s acting on the gauge singlet superfields S ij , which consist of the U(1) subgroups of the flavor SU(2N + 2) of the hyperquarks Q i . Thus, there are many linear combinations of the wave function renormalization factors which are not sequestered in the course of the RG evolutions to the infrared fixed point. For example, a linear combination ∆ ln Z S 12 − ∆ ln Z S 34is not sequestered, since this corresponds to the global U(1) ⊂ SU(2N + 2) symmetry. 3 : 3The matter contents in strongly coupled model SP (N) × SP (N ′ ) 2 , (N = 3, N ′ = 1). Here, the subscripts of the fundamental representations denote the dimensions of the representations. In terms of the SP (N) gauge theory, the number of the fundamental representation is given by N F = 2(N + 1) + 2 × 2N ′ = 12, while the number of the fundamental representation of each SP (N ′ ) gauge theories is given by N ′ F = 2N = 6. For the calculable examples, all the eigenvalues of the coefficient matrix M are positive, and hence the fixed points are infrared stable. Now, we discuss the sequestering of the SUSY breaking effects in the strongly coupled SP (3) × SP (1) 2 model. As argued in the previous section, if the anomalous dimensions are determined uniquely from the conditions for the vanishing beta functions, then the sequestering is equivalent to the infrared stability of the fixed point. Hence, once we assume that the fixed point with Eq.(54) is infrared stable, the sequestering is guaranteed. By assuming that the "sequestering speed" β ′ * is not so far from the values of γ i (see Eqs.(43)-(46)), we expect β ′ * = O(1) in our strongly coupled model. The flavor-changing soft masses are sufficiently suppressed by sequestering 16 at the energy scale µ as high as As a result, we find that the gauge-mediated masses squared are comparable to the anomaly mediated ones for m m ∼ m 3/2 . 21 In particular, the positive contributions to the slepton masses squared in Eq. Table 1 : 1The matter contents in our perturbative example, Table 2 , 2we show numerical results on the eigenvalues of the matrix M for the case of N ′ > 1. ¿From the table, we see that eigenvalues are all positive for N ′ > 1. Therefore, we find that the SP (3N ′ + 1) × SP (N ′ ) 6 gauge theory has the stable infrared fixed pointin Eqs.(31)-(33). Table Table 4 , 4we list some other examples of the SP (N) × SP (N ′ ) 2 gauge theories, which include the cases where the perturbative analysis is marginally applicable. In suchexamples with gauge symmetry structures similar to SP (3) × SP (1) 2 , we can explicitly check that the fixed points are infrared stable. Thus, based on these results (i.e. consis- tency with the unitarity and the presence of similar but calculable examples), we expect that the fixed point with Eq.(54) is infrared stable, although it is hard to check it by an explicit calculation. 15 We adopt the notation where SP (1) = SU (2). Combining Eqs.(7) and(8), we also obtain γ Q ≤ 0 and γ S ≤ 2. The asymptotic freedom of SP (N ), namely, n F < 2(N + 1), results in γ Q ′ < −γ Q /2 ≤ 1/2 from Eqs.(6) and(8). A non-perturbative determination of the coupling constants through a-maximization[13] is given in the Appendix A. In Eqs.(23)-(25), we assume that other eigenvalues of L are positive. Even if it is not the case, the conclusion is not changed. For N ′ = 1, although the anomalous dimensions in Eq.(33) satisfy the unitarity bound Eq.(8), the gauge coupling constants of SP (N ′ ) in Eq.(37) turns out to be negative, which implies that the perturbative description is invalid. Here, we assume that the inflaton decays dominantly to the SSM particles. Since the number of the Lagrange multipliers ξ agrees with that of marginal deformations, it is conjectured that ξ can be regarded as a coupling constant in a certain scheme. Acknowledgements M.I. and Y.N thank the Japan Society for the Promotion of Science for financial support.The authors acknowledges the referee for useful comments.A Anomalous dimensions from a-maximizationRecently, Intriligator and Wecht proposed a powerful technique to compute the conformalR current in a certain class of conformal field theories in four dimensions and hence the anomalous dimensions thereof[13]. In this appendix we use this so-called a-maximization method to determine the anomalous dimensions of the fields in the conformally extended IYIT model beyond the Banks-Zaks approximation presented in section 3.The a-maximization method simply states that the conformal R current appearing in the superconformal algebra maximizes a particular t'Hooft anomalywhich is related to the conformal anomaly on a curved spacetimeIn our model of the SP (N) gauge theory, the candidate of the conformal R current contains one free parameter x = γ Q , from which the corresponding R charges are determined by Eqs.(6)and(7)as . N Seiberg, arXiv:hep-th/9411149N. Seiberg, arXiv:hep-th/9411149. . M Luty, R Sundrum, arXiv:hep-th/0105137arXiv:hep-th/0111231M. Luty and R. Sundrum, arXiv:hep-th/0105137; arXiv:hep-th/0111231. . M Dine, P J Fox, E Gorbatov, Y Shadmi, Y Shirman, S Thomas, arXiv:hep-ph/0405159M. Dine, P.J. Fox, E. Gorbatov, Y. Shadmi, Y. Shirman and S. Thomas, arXiv:hep-ph/0405159; . R Sundrum, arXiv:hep-th/0406012R. Sundrum, arXiv:hep-th/0406012. . A E Nelson, M J Strassler, arXiv:hep-ph/0006251arXiv:hep-ph/0104051A.E. Nelson and M.J. Strassler, arXiv:hep-ph/0006251; arXiv:hep-ph/0104051; . T Kobayashi, H Terao, arXiv:hep-ph/0103028T. Kobayashi and H. Terao, arXiv:hep-ph/0103028; Here, we have neglected RG effects from the MSSM couplings. Here, we have neglected RG effects from the MSSM couplings. the SSM remain perturbative up to Grand Unification scale M GUT ≃ 2 × 10 16 GeV for m m > ∼ 10 13 GeV. In this model, the gauge coupling constants inIn this model, the gauge coupling constants in the SSM remain perturbative up to Grand Unification scale M GUT ≃ 2 × 10 16 GeV for m m > ∼ 10 13 GeV. 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[ "Distribution-Free One-Pass Learning", "Distribution-Free One-Pass Learning" ]	[ "Peng Zhao \nNational Key Laboratory for Novel Software Technology\nNanjing University\n210093NanjingChina\n", "Zhi-Hua Zhou \nNational Key Laboratory for Novel Software Technology\nNanjing University\n210093NanjingChina\n" ]	[ "National Key Laboratory for Novel Software Technology\nNanjing University\n210093NanjingChina", "National Key Laboratory for Novel Software Technology\nNanjing University\n210093NanjingChina" ]	[]	In many large-scale machine learning applications, data are accumulated with time, and thus, an appropriate model should be able to update in an online paradigm. Moreover, as the whole data volume is unknown when constructing the model, it is desired to scan each data item only once with a storage independent with the data volume. It is also noteworthy that the distribution underlying may change during the data accumulation procedure. To handle such tasks, in this paper we propose DFOP, a distribution-free one-pass learning approach. This approach works well when distribution change occurs during data accumulation, without requiring prior knowledge about the change. Every data item can be discarded once it has been scanned.Besides, theoretical guarantee shows that the estimate error, under a mild assumption, decreases until convergence with high probability. The performance of DFOP for both regression and classification are validated in experiments.	10.1109/tkde.2019.2937078	[ "https://arxiv.org/pdf/1706.02471v1.pdf" ]	24,907,339	1706.02471	2526419f337d588c40927524e0b49864f649d439	Distribution-Free One-Pass Learning Peng Zhao National Key Laboratory for Novel Software Technology Nanjing University 210093NanjingChina Zhi-Hua Zhou National Key Laboratory for Novel Software Technology Nanjing University 210093NanjingChina Distribution-Free One-Pass Learning data streamnon-stationary learningone-pass learningtime-series In many large-scale machine learning applications, data are accumulated with time, and thus, an appropriate model should be able to update in an online paradigm. Moreover, as the whole data volume is unknown when constructing the model, it is desired to scan each data item only once with a storage independent with the data volume. It is also noteworthy that the distribution underlying may change during the data accumulation procedure. To handle such tasks, in this paper we propose DFOP, a distribution-free one-pass learning approach. This approach works well when distribution change occurs during data accumulation, without requiring prior knowledge about the change. Every data item can be discarded once it has been scanned.Besides, theoretical guarantee shows that the estimate error, under a mild assumption, decreases until convergence with high probability. The performance of DFOP for both regression and classification are validated in experiments. Introduction With a rapid growth in collecting data, the volume of data generated makes it a challenge for traditional machine learning approaches. The main challenges are multi-faced, in general, accumulating and evolving properties are two of the most troublesome issues. For the accumulating property, apparently, it's impractical to store all the data entirely due to the limitation of memory and computation resources. Hence, an offline approach is not suitable any more in such tasks. Only in an online processing paradigm can prediction models be trained and updated incrementally. It's also worthy to mention that an online streaming approach is called one-pass when it requires going through the training data only once without storing the entire dataset. The reason to pursue one-pass property is due to the fact that sometimes the raw data is discarded or no longer accessible after being processed. A one-pass approach guarantees the learning process be independent with the volume of data stream. Apparently, it is much more demanding and difficult. Furthermore, the evolving nature of data stream also makes it challenging to directly apply traditional machine learning approaches because it's not reasonable to assume that the current and future data are coming from the same distribution any more. In a non-stationary environment, which is very common in data generation process, the distribution underlying is likely to dynamically change over time. For instance, the clicking information collected in recommendation system is certainly evolving because customers' interests probably change when looking through the product pages. Another example is credit scoring, the criteria of credit granting should properly alter since a changing economic conditions would have a great influence on people's manner. To simultaneously address these two issues, in this work, we propose DFOP, a distribution free one-pass learning approach to deal with data stream emerging distribution change under onepass constraints. The advantages of our approach are following: firstly, by recursively solving the target, we guarantee that only one time would data stream be gone through. Secondly, based on a forgetting mechanism, the loss of older data is discounted without explicitly modelling the dynamics or assuming prior information about distribution change. In streaming regression, a theoretical guarantee is presented showing that the estimate error of dynamic concept would decrease until convergence with high probability. Meanwhile, empirical experiments on both synthesis and real-world datasets indicate the effectiveness and practicability of DFOP in both regression and classification scenarios. The rest of this paper is organized as follows. In Section 2, we briefly review of some related work. Then, static scenario model is introduced in preliminaries. Next, in the non-stationary environment, DFOP is presented to handle dynamics in regression and classification scenarios. Both theoretical guarantee and empirical effectiveness have been examined. Finally, we conclude the paper. Related Work Online and One-Pass Algorithms. With a rapid growth of data volumn and velocity, it's no longer practicable to adopt offline mode algorithm for streaming data learning tasks. Hence, online style algorithms become gradually attractive which update the current model with the most recent examples. In general, they are error driven, updating the current model depending on whether the current example is misclassified [GZB + 14]. Representative algorithms include Perceptron [Ros58], Winnow [Lit87] and many other variations. Besides, a paradigm of prediction with expert advice [CBL06] also inspires some interesting works, such as AddExp [KM05] and DWM [KM03,KM07]. Most of those approaches require to store the entire or partial training data and scan data items multiple times. Recently, one-pass algorithms gradually draw more attentions demanding that each data item should be processed only once. Concretely speaking, after a data item has been processed and relevant statistics have been stored, the raw data item should be discarded and never be accessed any more. Obviously, one-pass constraints impose a higher degree of difficulty on algorithm design. Some efforts have been devoted [WBSD16]. Nonstationary Learning. Owing to the effectiveness and simplicity, sliding window is usually adopted to handle data stream with distribution change. It only uses a fixed or variable number of recent data which are the most informative for current prediction [Lit87,Kli04]. Usually, the model built is updated following two processes: one is a learning process, i.e., updating the model based on the new coming data; the other one is a forgetting process, i.e., discarding data items that are moving out of the window [GZB + 14]. However, how to choose an appropriate window size is of great importance which now mainly depends on heuristics to a certian extent. Some efforts have been paid to select window size adaptively [Kli04,GMCR04,KZ09]. The common strategy to adjust window size is based on the performance or estimate of generalization error. SVM-ada [Kli04] presents a theoretically supported approach , however, the computational efficiency issue makes it not practical in real-world applications. Our proposed approach DFOP, short for Distribution-Free One-Pass, is a one-pass style algorithm, i.e., it could guarantee that only one time will data items been gone through. Besides, DFOP is distribution-free, i.e., different from those traditional approaches dealing with distribution change, we did not explicitly model the dynamics, and no prior information about distribution change is assumed. Preliminaries In this part, streaming regression model in a static scenario is briefly introduced. In a streaming scenario, we denote a labeled dataset as {x(t), y(t)}, where x(t) is the feature of the t-th instance and y(t) is a real-valued output. Furthermore, we assume a linear model as follows, y(t) = x(t) T w(t − 1) + (t) (1) where { (t)} is the noise sequence, {w(t)} is what we desire to estimate. When in a static scenario, the sequence {w(t)} is a constant vector denoted as w 0 . Then, the least square could be adopted to minimize the residual sum of squares, which has a close-form solution. However, it fails when adding an online/one-pass constraint which demands the raw item is discarded after it has been processed. Recursive least square (RLS) and stochastic gradient descent (SGD) are two typical approaches to solve this problem in an online paradigm. When in a non-stationary environment, especially when the distribution underlying changes, traditional approaches are not suitable since we could never expect the typical i.i.d assumption continue to work any longer. In the next sections, we propose to handle this scenario based on exponential forgetting mechanism without explicitly modelling the evolution of data stream, and theoretical support and empirical demonstration are presented. In the following, · denotes the 2 -norm in R n space. Meanwhile, for a bounded real-valued sequence {x(t)}, x * denotes the upper bound of sequence, namely, x * = sup t=1,2,··· x(t). Distribution Free One-Pass Learning Since the sequence {w(t)} is changing over time in a dynamic environment, it is no longer reasonable to estimate current (i.e., at time t) concept via methods introduced previously. Instead, we introduce a sequence of discounted factors {λ(t)} to downweight the loss of older data as follows, w(t) = arg min w∈R d t i=1 ( t j=i+1 λ(j)) y(i) − x(i) T w 2 ,(2) where λ(i) ∈ (0, 1) is a discounted factor to smoothly put less weight on older data. The intuition can be more easily obtained if we simplify all λ(i) as a constant λ ∈ (0, 1), then the target function is,ŵ (t) = arg min w∈R d t i=1 λ t−i y(i) − x(i) T w 2 ,(3) And the quantity µ 1 − λ is named as forgetting factor [Hay08]. The value of forgetting factor is, as a matter of fact, a trade-off between stability of past condition and sensitivity of future evolution. It should be pointed out that the forgetting mechanism based on exponential forgetting factor could be also considered as a continuous analogy to sliding window approach to some extent. The older data items with a small enough weight can be somehow thought as exclusion from the window. Some. More discussions on relation with window size and forgetting factor are provided in Section 5.4. Algorithm For the optimization problem proposed in (3), obviously, by taking derivative of the target function, we can directly obtain the optimal solution in a closed-form, [w(t)] opt = t i=1 λ t−i x(i)x(i) T −1 t i=1 λ t−i x(i)y(i)(4) However, above expression is an off-line estimate, namely, all the data items ahead of t are needed. Instead of repeatedly solving (4), we estimate the underlying concept by adding a correction term Algorithm 1 Distribution Free One-Pass Learning Input: A stream of data with {x(t), {y(t)} t=1···T , forgetting factor µ ∈ (0, 1); Output: Prediction {ŷ(t)} t=1···T (real value for regression and discrete-value for classification). Initialize P 0 > 0; for t = 1 to T do P (t) = 1 1−µ P (t − 1) − µ P (t−1)x(t)x(t) T P (t−1) 1−µ+x(t) T P (t−1)x(t) ; L(t) = P (t)x(t); w(t) =ŵ(t − 1) + µL(t)[y(t) −ŵ(t − 1) T x(t)]; y(t) =ŵ(t) T x(t). // for regression; y(t) = sign[ŵ(t) T x(t)]. // for classification end for to the previous estimate based on the information of new coming data item. With the forgetting factor recursive least square method [Hay08], we could solve the target (3) in a one-pass paradigm. And to the best of our knowledge, this is the very first time to adopt traditional forgetting factor RLS to deal with such tasks with distribution change under the one-pass constraints. And we named this as DFOP(short for Distribution-Free One-Pass) summarized in Algorithm 1. Besides, it should be pointed out that {λ(t)} is by no means necessary chosen as a constant, we provide a generalized DFOP (short as G-DFOP) for a dynamic discount factor sequence {λ(t)}, corresponding to target in (11), which is also provably a one-pass algorithm. Detailed proofs are provided in Section 1 of supplementary material. For the classification scenario, y(t) is no longer a real-valued output but a discrete value, and we assume y(t) ∈ {−1, +1} for convenience. A slight modification on original output step is applied in classification, where the effectiveness is empirically validated in the next section. Assuming that the feature is d-dimension, we only need to keep P (t) ∈ R d×d in memory during the algorithm processing procedure. In other words, the storage is always O(d 2 ), which is independent to the number of training examples. Besides, at the t-th time stamp, the update ofŵ(t) is unrelated to the previous data items, namely every data item can be discarded once it has been scanned. Theoretical Guarantee In this section, we develop an estimate error bound in a non-stationary regression scenario. Consider the additive model of drift in sequence {w(t)}, w(t) = w(t − 1) + s(t), t ≥ 1(5) We assume that the adding term {s(t)} is an E-valued martingale-difference d-dimension sub-Gaussian vector sequence, with corresponding variance proxy sequence {σ t }, whose formal definition will be given following. The E-valued martingale-difference assumption is reasonable, in fact, in many real-world application, the drift of concepts are usually independent. Similar to the analysis in [GLP93], we relax the assumptions to be more realistic in real-world applications and provide a non-deterministic estimate error bound w(t) −ŵ(t) based on vector concentration, showing that the estimate error is tending to convergence with high probability. Now we give the formal definitions of sub-Gaussian random variable and random vector. Definition 1 (sub-Gaussian random variable) A random variable X ∈ R is said to be sub-Gaussian with variance proxy σ 2 if E[X] = 0 and its moment generating function satisfies E[exp(sX)] ≤ exp( s 2 σ 2 2 ), ∀s ∈ R (6) Definition 2 (sub-Gaussian random vector) A random vector x ∈ R d = (x 1 , · · · , x d ) is called sub-Gaussian with variance proxy σ 2 if all its coordinates are sub-Gaussian random variables with variance proxy σ 2 . To exploit concentration property of sub-Gaussian random vector, condition (C α [σ ∞ ]) proposed in∀t ≥ 1 : E exp{ x(t) 2 /γ 2 t } ≤ exp{1}(7) Lemma 1 guarantees the "light tail" assumption of sub-Gaussian random vector. Then we could apply the following vector concentration, which is a corollary of Theorem 2.1 proposed in [JN08]. Theorem 1 (Corollary of Theorem 2.1 in [JN08]) In an Euclidean space (R n , · 2 ), let E-valued martingale-difference sub-Gaussian sequence ξ ∞ with a corresponding bounding sequence σ N = [σ 1 ; · · · ; σ N ]. Let S N = N i=1 ξ i , then for all N ≥ 1 and γ ≥ 0: Pr    S N ≥ √ 2(1 + γ) N i=1 σ 2 i    ≤ exp −γ 2 /3 ,(8) Based on Theorem 23, we could provide Lemma 2 and Lemma 3 to bound a sum of sub-Gaussian random vectors and random variables with exponential decrease, respectively. Lemma 2 Let {x(t)} be an E-valued martingale-difference d-dimension sub-Gaussian random vector sequence, with corresponding bounding sequence {γ t }, and Z(t) ∈ R d×d , Y (t) = (1−µ)Y (t− 1) + µZ(t), t ≥ 1. Then for µ ∈ (0, 1), with a probability at least 1 − δ 2 , we have t k=1 (1 − µ) t−k Y (k)x(k) ≤ √ 2(1 + 3 ln(2t/δ)) · γ * ( Y (0) + Z * )µ − 1 2 where Z * = sup k=1,··· ,t Z(k) and γ * = sup k=1,··· ,t γ k . Lemma 3 Let { (t)} be an independent (or E-valued martingale-difference) sub-Gaussian random variable sequence, with corresponding bounding sequence (i.e., variance proxy sequence) {σ t }, and x(t) ∈ R d , t ≥ 1. Then for µ ∈ (0, 1), with a probability at least 1 − δ 2 , we have t k=1 (1 − µ) t−k x(k) (k) ≤ 2 √ 2 1 + 3 ln 2t δ σ * µ − 1 2 where σ * = sup k=1,··· ,t x(k) · sup k=1,··· ,t σ k . Theorem 2 Assume following conditions be satisfied: (i) drift term {s(t)} is an E-valued martingale-difference sub-Gaussian random vector sequence, with corresponding bounding sequence {γ t }; (ii) output noise { (t)} is an independent (or E-valued martingale-difference) sub-Gaussian random variable sequence, with corresponding bounding sequence (i.e., variance proxy sequence) {σ t }. Then with a probability at least 1 − δ, we have w(t) −ŵ(t) ≤ K { (1 − µ) t R(0) w(0) + √ 2(1 + 3 ln(2t/δ)) ·[2σ * µ 1/2 + γ * ( R(0) + x * 2 )µ −1/2 ] where K = sup k=1,··· ,t P (k) , x * = sup k=1,··· ,t x(k) , σ * = sup k=1,··· ,t x(k) · sup k=1,··· ,t σ k and γ * = sup k=1,··· ,t γ k . Remark. The estimate error bound can be decomposed into three parts, i.e., the first one is (1 − µ) t R(0) w(0) , second one is 2 √ 2(1 + 3 ln 2t/δ)σ * µ 1/2 and third one is √ 2(1 + 3 ln 2t/δ)( R(0) + x * 2 )γ * µ −1/2 . Apparently, the first term is decreasing to zero as t increases to infinity, second term is caused by the output noise which shall not be erased, and the third term is introduced by drift of w(t). Ignoring the poly-logarithmic factors in t and d, then, an asymptotic analysis gives the estimate error bound as, w(t) −ŵ(t) =Õ( √ µ + 1 √ µ ) + o(1), w.h.p. where we use theÕ notation to hide constant factors as well as poly-logarithmic factors in t and d, and o(1) will exponentially decrease to zero as t → ∞. Due to the page limits, we present the proofs of Theorem 1 and Theorem 2 (along with Lemma 1, 2 and 3) in Section 2 and 3 of supplementary material, respectively. Experiments In this section, we examine the empirical performance of the proposed DFOP on both regression and classification scenarios. Then, we analyze the parameter sensitivity in Section 5.4. However, due to the page limits, only results on the classification scenario are provided, and the regression ones are appended in the supplementary materials. Moreover, considering that when dealing with real-world datasets, we could not grasp the evolving distribution, specifically, the start and end time of drift, the underlying distribution. As a consequence, it would be very incomplete to analyze the behaviour of algorithms. Hence, both synthesis and real-world datasets are included in the comparison experiments. Comparisons Methods We compare the proposed approach with six common methods on both synthesis and real-world DWM is incremental style but not one-pass because it needs to use data to update experts pool in addition. datasets Synthetic Datasets First, we present the performance comparisons over synthetic datasets. -SEA [SK01] consists of three attributes x 1 , x 2 , x 3 , and 0.0 ≤ x i ≤ 10.0. The target concept is x 1 + x 2 ≤ b, and there are 50,000 instances with 4 stages where b ∈ {7, 8, 9, 9.5}. -hyperplane [Fan04], is generated uniformly in a 10-dimensional hyperplane with 90,000 instances in total over 9 different stages. Besides, another 11 synthesis datasets for binary classification are also adopted. Detailed information are included in the supplementary materials. The performance is measured by holdout accuracy since underlying joint distribution of synthetic datasets are known. Holdout accuracy is calculated over testing data generated according to the identical distribution as training data at each time stamp. Performance comparisons of seven approaches on SEA and hyperplane datasets are depicted in Figure 1. Since the accuracy curves of SVM-ada, SVM-fix, 1NN-win and SVM-win are so unstable that they would shield all the other curves, we also present a relatively neat figure containing RLS, DWM and DFOP only. As shown in Figure 1, the accuracy of all algorithms falls rapidly when the underlying distribution emerges abrupt drift, and then will rise up with more data coming. DFOP is significantly better than RLS which is a special case of DFOP, this phenomenon validates the effectiveness of forgetting mechanism. Furthermore, the best two algorithms, obviously, are DFOP and DWM, both of them can converge to new stage quickly. DFOP shows a slightly better performance than DWM, both in slope and asymptote. Moreover, DWM requires to dynamically maintain a set of experts and needs previous data to update experts pool and to decide whether to remove poorly performing experts. On the contrary, DFOP demonstrates a desirable performance requiring to scan each data item only once. Real-world Datasets To valid the effectiveness of DFOP in real-world applications, performance comparisons are presented over 8 real-world datasets. Detailed descriptions are provided in Section 5 of supplementary materials. In real-world datasets, we can never expect to foreknow the underlying distribution at each data stamp. Thus, it's not possible to still adopt holdout accuracy as performance measurement. In Left sides presents all the seven approaches, on the right side, only RLS, DWM and DFOP are plotted for clearness. could be found in Section 4.3 in supplementary materials. Parameter Study As stated previously, how to choose an appropriate forgetting factor is an important issue since it reflects a trade-off between stability of past condition and sensitivity to future evolution. To figure out how forgetting factor affects the performance, in classification problem, accumulated accuracy (short as 'AA') is adopted as a performance measurement in the time series and is defined as, AA(t) = t i=1 I[ŷ(i) = y(i)]/t,(9) where I(·) is indicator function which takes 1 if · is true, and 0 otherwise,ŷ(i) and y(i) are predictive and ground-truth label, respectively. Figure 2 shows the impact of different forgetting factor µ over four datasets. We notice that the accumulated accuracy of RLS almost decreases all the time. For a relatively small but not zero µ, the performance is satisfying without a significant gap. However, when µ is too large, say 0.5, the performance is even much worse than RLS. This is consistent with intuition since forgetting factor is so large and older data samples are exponentially downweighted that there are not sufficient effective training samples available to update the model. Now, here comes the question: how to choose an appropriate forgetting factor to adapt the distribution change in the data stream? To answer this question, let's recall the target function in (3), when µ is close to 0, which is often the case in practice and validated in Table 2: Datasets, the number of data items between consecutive distribution change, theoretical recommended value and empirical appropriate value for forgetting factor are listed below, and the last column provides relative proportion between theoretical recommended value and empirical appropriate value. Dataset T 0 1/T 0 µ µT 0 Dataset T 0 1/T 0 µ µT 01CDT(1 − µ) t = e t ln(1−µ) ≈ e −tµ = e −t/T 0 ,(10) where we define T 0 = 1/µ as forgetting period. The contribution for prediction error of data items older than T 0 time will be discounted with a weight less than e −1 ≈ 36.8% comparing to the current data. As a matter of fact, the forgetting period in forgetting mechanism is pretty similar to the window size in sliding window technique. It can be regarded as a soft relaxation of window size. Consequently, the forgetting factor µ shall be chosen according to the forgetting period T 0 , where the data distribution should be relatively smooth and stable during this forgetting period. We validate this idea over synthesis datasets reported in Table 2. Theoretical recommended value 1/T 0 and empirical appropriate value µ for forgetting factor are provides. Also, the relative proportions between them are calculated. We can see that these two value are very close over all datasets with no more than 20 times difference, even no more than 5 times in most datasets. This supports our strategy in choosing forgetting factor. Certainly, the drifting properties of real-world datasets are not as clear as synthetic datasets. Nevertheless, we could still infer the forgetting period T 0 based on the domain knowledge and choose an appropriate value as forgetting factor. For instance, considering the weather forecast dataset, although we cannot foreknow the drifting property of distribution, a relative stable period can still be estimated. Conclusion In this paper, we proposed an approach based on forgetting mechanism called DFOP handling streaming learning problems with distribution change. The main idea is to downweight the older data items by introducing exponential forgetting factor without considering any prior about drifting information. Meanwhile, DFOP meets the one-pass constraints guaranteeing that only once will the data items be scanned without storing the entire dataset. Hence, DFOP. The Besides, how to efficiently reduce the storage and make DFOP paralleled to adapt a even larger scale real-world applications would be an interesting future work. Appendix Generalized DFOP and Proofs Recursive Algorithm for Dynamic Discounted Factors When dynamic discounted factors are introduced to downweight the contribution of older data items, the target function can be written as, w(t) = arg min w∈R d t i=1 ( t j=i+1 λ(j)) y(i) − x(i) T w 2(11) In this part, we provide a provably recursive algorithm to directly solve (11) as shown in Algorithm 2 named generalized DFOP algorithm, short as G-DFOP. Algorithm 2 Generalized DFOP Input: A stream of data with feature {x(t)} t=1···T and {y(t)} t=1···T , discounted factor sequence {λ(t)} t=1···T ; Output: Prediction label {ŷ(t)} t=1···T (real value for regression and discrete-value for classification). Initialize P 0 > 0, for t = 1 to T do P (t) = 1 λ(t) [P (t − 1) − P (t−1)x(t)x(t) T P (t−1) λ(t)+x(t) T P (t−1)x(t) ], L(t) = P (t)x(t), w(t) =ŵ(t − 1) + L(t)[y(t) −ŵ(t − 1) T x(t)]. y(t) =ŵ(t) T x(t). // for regression; y(t) = sign[ŵ(t) T x(t)] . // for classification end for Proof of Generalized DFOP In this part, we will prove the consistency between G-DFOP and target function in (11). Lemma 4 Let A, B, C and D be matrices of compatible dimensions such that the product BCD and the sum A + BCD exist. Then we have [A + BCD] −1 = A −1 − A −1 B[DA −1 B + C −1 ] −1 DA −1 (12) Proof: Multiply the right-hand side of (12) by A + BCD from the right, this gives A −1 − A −1 B[DA −1 B + C −1 ] −1 DA −1 [A + BCD] = I + A −1 BCD − A −1 B[DA −1 B + C −1 ] −1 D − A −1 B[DA −1 B + C −1 ] −1 DA −1 BCD = I + A −1 B[DA −1 B + C −1 ] −1 {0} = I For convenience, let Λ(i, t) = t j=i+1 λ(j), then the close-form solution of optimization (11) can be calculated as followŝ w(t) = t i=1 Λ(i, t)x(i)x(i) T −1 t i=1 Λ(i, t)x(i)y(i)(13) Now we will prove that the solution obtained by G-DFOP is equivalent to close-form solution in (13). Theorem. By the policy in Algorithm 2, we can achieve the same solution as result in (13). Proof: DenoteR(t) = t k=1 Λ(i, t)x(t)x(t) T , obviouslȳ R(t) = λ(t)R(t − 1) + x(t)x(t) T Then the solution in (13) can be rewritten into the following form: w(t) =R −1 (t) λ(t) t−1 i=1 Λ(i, t − 1)x(i)y(i) + x(t)y(t) =R −1 (t) λ(t)R(t − 1)ŵ(t − 1) + x(t)y(t) =R −1 (t) (R(t) − x(t)x(t) T )ŵ(t − 1) + x(t)y(t) =ŵ(t − 1) +R −1 (t)x(t) y(t) − x(t) Tŵ (t − 1)) Now, we introduce P (t) =R −1 (t) and then apply Lemma 4 to (??), this gives P (t) = 1 λ(t) P (t − 1) − 1 λ(t) P (t − 1)x(t) · 1 λ(t) x(t) T P (t − 1)x(t) + 1 −1 1 λ(t) x(t) T P (t − 1) = 1 λ(t) P (t − 1) − P (t − 1)x(t)x(t) T P (t − 1) λ(t) + x(t) T P (t − 1)x(t) Let L(t) = P (t)x(t), we can obtain the policy described in Algorithm 2. Remark. Obviously, DFOP in paper is only a special case when fixing discounted factor sequence {λ(t)} as (1 − µ). Note that, for a simplicity notations in estimate error analysis of DFOP, we slightly modified L(t) and P (t) multiplying by µ. Proof of Theorem 1 Definition of Sub-Gaussian and Sub-Exponential First we give typical definitions of sub-Gaussian random variable and random vector, meanwhile, definition of sub-Exponential random variable is also provided. Definition 3 (sub-Gaussian random variable) A random variable X ∈ R is said to be sub-Gaussian with variance proxy σ 2 if E[X] = 0 and its moment generating function satisfies E[exp(sX)] ≤ exp( s 2 σ 2 2 ), ∀s ∈ R (14) In this case we write X ∼ subG(σ 2 ). Definition 4 (sub-Gaussian random vector) A random vector x ∈ R d = (x 1 , · · · , x d ) is called sub-Gaussian with variance proxy σ 2 if all its coordinates are sub-Gaussian random variables with variance proxy σ 2 . Definition 5 (sub-Exponential random variable) A random variable X ∈ R is said to be sub-Exponential with parameter λ if E[X] = 0 and its moment generating function satisfies E[exp(sX)] ≤ exp( s 2 λ 2 2 ), ∀\|s\| ≤ 1 λ . (15) In this case we write X ∼ subE(λ). Remark. Attention that definitions above all require a zero-mean constraint, which is not necessary in analysis "light tail" property. Hence, for random variable that is not zero-mean but satisfies condition (14) is called generalized sub-Gaussian. And definitions for generalized sub-Gaussian random vector and generalized sub-Exponential random variable are similar. Proof of Theorem 1 Theorem 1 in the paper presents a vector concentration inequality is shown for sub-Gaussian random vector sequence in the following which plays an important role in proving Lemma 2 and Lemma 3 in our paper. To prove Theorem 1 in the paper, first, we present following Lemma 5 to show that the norm of a sub-Gaussian random vector is a generalized sub-Gaussian random variable. Lemma 5 If x ∈ R d is a sub-Gaussian random vector with variance proxy σ 2 , then E exp( x ) ≤ 2 exp d 2 σ 2 2(16) Proof: The conclusion here is a direct corollary from Theorem 3.1 in [BP10], where c = 1 = (1, · · · , 1), G = · , and B(c, x) = d k=1 c k τ (x k ) = dσ. Then we present the following Lemma 6 to show the equivalence between sub-Gaussian random variable and sub-Exponential variable, which plays an important role in proving Theorem 1. Lemma 6 Let X be a sub-Gaussian random variable, i.e., X ∼ subG(σ 2 ). Then the random variable Z = X 2 − E[X 2 ] is sub-Exponential: Z ∼ subE(16σ 2 ). Proof: We prove this lemma by definition. E[e sZ ] = 1 + ∞ k=2 s k E[X 2 − E[X 2 ]] k k! ≤ 1 + ∞ k=2 s k 2 k (E[X 2k − (E[X 2 ]) k ) k! (17) ≤ 1 + ∞ k=2 s k 4 k (E[X 2k ] 2(k!) (18) ≤ 1 + ∞ k=2 s k 4 k 2(2σ 2 k) 2(k!) (19) = 1 + (8sσ 2 ) 2 ∞ k=0 (8sσ 2 ) k = 1 + 128s 2 σ 4 ≤ e 128s 2 σ 4(20) where (17) Remark. Attention that in the proof of Lemma 6, we didn't use the zero mean property of sub-Gaussian random variable X. Hence, Lemma 6 can be applied to a generalized sub-Gaussian random variable without requiring zero-mean condition. Now, let's begin prove the Lemma 2 in the paper stated as follows, For a sub-Gaussian random vector sequence {x(t)} with a variance proxy sequence {σ t }, there exists a corresponding positive bounding sequence {γ t }, such that ∀t ≥ 1 : E exp{ x(t) 2 /γ 2 t } ≤ exp{1}(21) Proof: Consider any vector in the sequence, say x(t). Because it is a sub-Gaussian random vector, directly applying Lemma 5, we have E exp( x(t) ) ≤ 2 exp d 2 σ 2 t 2 which means for a sub-Gaussian random vector, its norm is a generalized sub-Gaussian random variable. Here, "generalized" means it may not meet the zero-mean condition. Because we didn't use the zero mean property of sub-Gaussian random variable in the proof of Lemma 6, it can also be applied to generalized sub-Gaussian random variable. Let Z = x(t) 2 − E[ x(t) 2 ], then Z ∼ subE(16σ 2 t ), specifically, E[exp{s( x(t) 2 − E[ x(t) 2 ])}] ≤ exp{s 2 16σ 2 t /2}, ∀\|s\| ≤ 1 4σ t Thus, for ∀\|s\| ≤ 1/(4σ t ), we have E[exp{s x(t) 2 }] ≤ exp{8s 2 σ 2 t + sE[ x(t) 2 ]},(22) Obviously, we can choose a sufficient small positive constant s as γ t , such that E exp{ x(t) 2 /γ 2 t } ≤ exp{1}. And {γ t } is exactly the bounding sequence as we desired. This completes the proof. Now, we prove the Theorem 1 in the paper stated as follows, Theorem 3 In an Euclidean space (R n , · 2 ), let E-valued martingale-difference sub-Gaussian sequence ξ ∞ with a corresponding bounding sequence σ N = [σ 1 ; · · · ; σ N ]. Let S N = N i=1 ξ i , then for all N ≥ 1 and γ ≥ 0: Pr    S N ≥ √ 2(1 + γ) N i=1 σ 2 i    ≤ exp −γ 2 /3 ,(23) Proof: First of all, for a sub-Gaussian sequence ξ ∞ with corresponding bounding sequence σ ∞ , from Theorem 1 in paper, we have E i−1 exp{ ξ i 2 σ 2 i } = exp{1}. Besides, an Euclidean space (R n , · 2 ) is 1-smooth and 1-regular, which means κ = 1. Then, it follows immediately according to Theorem 2.1 proposed in [JN08]. Proof of Theorem 2 Firstly, the following generalized summation by parts is essential for the proof of main theorem in our paper. Lemma 7 Let f n and g n be two sequences. And denote G n = n k=1 g k , then for m ≥ 2 we have, n k=m f k g k = f n G n − f m G m−1 − n−1 k=m G k (f k+1 − f k ) Proof: The right-hand side can be easily verified by expanding g k in the left-hand as G k − G k−1 . Remark. When m = 1, the same derivation gives the famous Abel transformation, n k=1 f k g k = f n G n − n−1 k=1 G k (f k+1 − f k ) Proof of Lemma 2 in paper Lemma 2 in our paper states as follows, Let {x(t)} be an E-valued martingale-difference d-dimension sub-Gaussian random vector sequence, with corresponding bounding sequence {γ t }, and Z(t) ∈ R d×d , Y (t) = (1 − µ)Y (t − 1) + µZ(t), t ≥ 1. Then for µ ∈ (0, 1), with a probability at least 1 − δ 2 , we have t k=1 (1 − µ) t−k Y (k)x(k) ≤ √ 2(1 + 3 ln(2t/δ)) · γ * ( Y (0) + Z * )µ − 1 And it could be separated into two parts, i.e., (a) = (1 − µ) t Y (0) t i=1 x(i) ; and (b) = Z * t k=1 µ(1 − µ) k−1 t i=t−k+1 x(i) . For k = 1, · · · , t, based on Lemma 3, we have Pr    t i=k x(i) ≤ √ 2(1 + 3 ln 2t δ ) t i=k γ 2 i    ≥ 1 − δ 2t taking by the union bound over t > 0, and let γ * = sup k=1,··· ,t γ k the following holds in a probability at least 1 − δ 2 , (b) ≤ √ 2 1 + 3 ln 2t δ Z * t k=1 µ(1 − µ) k−1 t i=t−k+1 γ 2 i ≤ √ 2 1 + 3 ln 2t δ Z * γ * t k=1 µ(1 − µ) k−1 √ k ≤ √ 2 1 + 3 ln 2t δ Z * γ * µ −1/2 Conditioning on all above concentration inequalities hold, for (a), we have (a) ≤ (1 − µ) t Y (0) √ 2(1 + 3 ln 2t δ ) t i=k γ 2 i ≤ √ 2(1 + 3 ln 2t δ ) Y (0) γ * (1 − µ) t √ t ≤ √ 2(1 + 3 ln 2t δ ) Y (0) γ * µ −1/2 Hence, combining (a) and (b), we complete the proof. Proof of Lemma 3 in paper Lemma 3 in our paper states as follows, Let { (t)} be an independent (or E-valued martingale-difference) sub-Gaussian random variable sequence, with corresponding bounding sequence (i.e., variance proxy sequence) {σ t }, and x(t) ∈ R d , t ≥ 1. Then for µ ∈ (0, 1), with a probability at least 1 − δ 2 , we have t k=1 (1 − µ) t−k x(k) (k) ≤ 2 √ 2 1 + 3 ln 2t δ σ * µ − 1 2 where σ * = sup k=1,··· ,t x(k) · sup k=1,··· ,t σ k . Proof: First, we examine the "light tail" condition, i.e.,condition (C α [σ ∞ ]) in Theorem 2.1 proposed in [JN08], let σ 2 i = (max x(k) 2 )σ 2 i , ∀i ≥ 1, then E i−1 exp{ (k)x(k) 2 σ 2 i } ≤ E i−1 exp{ (k) 2 x(k) 2 σ 2 i } = E x(k) E (k) exp{ (k) 2 x(k) 2 σ 2 i (max x(k) 2 ) } ≤ E x(k) E (k) exp{ (k) 2 σ 2 i } ≤ exp{1} Hence, conclusion in Lemma 3 holds on sequence {x(t) (t)}. Similar to the proof in proof of Lemma 2(original paper), by taking Y (t) ≡ Z(t) ≡ I, we complete the proof. Proof of Theorem 2 Now let's begin to prove the main theorem, i.e., Theorem 2 in our paper states as follows. Theorem 4 Assume following conditions be satisfied: (i) drift term {s(t)} is an E-valued martingale-difference sub-Gaussian random vector sequence, with corresponding bounding sequence {γ t }; (ii) output noise { (t)} is an independent (or E-valued martingale-difference) sub-Gaussian random variable sequence, with corresponding bounding sequence (i.e., variance proxy sequence) {σ t }. Then with a probability at least 1 − δ, we have w(t)−ŵ(t) ≤ K (1 − µ) t R(0) w(0) + √ 2(1 + 3 ln(2t/δ)) · [2σ * µ 1/2 + γ * ( R(0) + x * 2 )µ −1/2 ] where K = sup k=1,··· ,t P (k) , x * = sup k=1,··· ,t x(k) , σ * = sup k=1,··· ,t x(k) · sup k=1,··· ,t σ k and γ * = sup k=1,··· ,t γ k . Proof: Define R(t) P −1 (t), then we have w(t) = P (t)R(t)w(t) ≤ P (t) R(t)w(t) ≤ K R(t)w(t) so we only need to consider R(t)w(t) . Recall model assumption, drifting assumption and update rule, y(t) = x(t) T w(t − 1) + (t) w(t) = w(t − 1) + s(t) w(t) =ŵ(t − 1) + µP (t)x(t) y(t) − x(t) Tŵ (t − 1) we can obtain linear recurrence relations of R(t)w(t), R(t)w(t) = (1 − µ)R(t − 1)w(t − 1) + µx(t) (t) − R(t)s(t) which gives R(t)w(t) = (1 − µ) t R(0)w(0) + t k=1 (1 − µ) t−k [µx(k) (k) − R(k)s(k)] Hence, by Minkowski's inequality, it can be bounded by R(t)w(t) ≤ (1−µ) t R(0)w(0) + t k=1 (1 − µ) t−k µx(k) (k) + t k=1 (1 − µ) t−k R(k)s(k) (24) When output noise { (t)} is an independent (or E-valued martingale-difference) sub-Gaussian random variable sequence, with corresponding bounding sequence (i.e., variance proxy sequence) {σ t }, then directly apply Lemma 3(original paper) on the second term in (24) gives, with at least 1 − δ 2 , µ t k=1 (1 − µ) t−k x(k) (k) ≤ 2 √ 2 1 + 3 ln 2t δ σ * µ 1/2(25) where σ * = sup k=1,··· ,t x(k) · sup k=1,··· ,t σ k . When the drift term {s(t)} is an E-valued martingale-difference sub-Gaussian random vector sequence, with corresponding bounding sequence {γ t }. Besides, it's trivial to verified that R(k) meets the decreasing recursive structure stated, then directly apply Lemma 2 on the last term in (24) gives, with at least 1 − δ 2 , t k=1 (1 − µ) t−k R(k)s(k) ≤ √ 2 1 + 3 ln 2t δ γ * ( R(0) + x * 2 )µ −1/2(26) where x * = sup k=1,··· ,t x(k) and γ * = sup k=1,··· ,t γ k . Combining (25) and (26) by union bound, we get desired estimate error bound in assertion, hence complete the proof. Additional Experiments Regression In this part, we compare the proposed DFOP to state-of-the-art streaming regression approaches The performance of various streaming regression approaches is assessed by MSE (mean square error) between ground-truth and predict values over 10 trails. Both mean and standard deviation are reported in Table 3. It has to be pointed out that, not all comparisons are fair. DFOP is a onepass approach which needs to scan each data item only once. For most ensemble style methods, data items are usually scanned many times for the need of dynamically update ensemble models. Besides, Learn ++ .NSE is batch basis, nevertheless, DFOP is incremental and could update the model by a single data item. From Table 3, we can see that DFOP outperforms RLS, and achieved a satisfying performance with lowest MSE in three datasets. Meanwhile, since the real drifting property of synthesis dataset is clear, we also present the trend chart of DFOP with different forgetting factors in terms of square loss over generated test data and estimate error w(t) = w(t) −ŵ(t) in . And we can see that the error of DFOP is significantly lower than RLS, which is a special case of DFOP when forgetting factor is 0. This proves the effectiveness of forgetting mechanism. Besides, in the right side, we can see that w(t) is tending towards stability as time series go to the infinity which is consistent with our theory proposed in the original paper even the dataset does not exactly meet the assumptions proposed in Theorem 2. Robustness Comparisons Additionally, the robustness of all these different algorithms are compared. Concretely speaking, for a particular algorithm algo, similar to definition in [VDG + 02], the robustness here is defined as the proportion between its accuracy and the smallest accuracy among all compared algorithms, r algo = acc algo min α acc α Apparently, the worst algorithm has r algo = 1, and the others have r algo ≥ 1, the greater the better. Hence, the sum of r algo over all datasets indicates the robustness of for algorithm algo. S V M -w in 1 N N -w in S V M -fi x S V M -a d The greater the value of the sum, the better the performance of the algorithm. We provide a robustness comparison on six compared algorithms and DFOP over 20 datasets in Figure 4. From the figure, we can see that DFOP achieves the best over 20 datasets, and RLS ranks last as expected since it didn't consider the evolving distribution in datasets at all. For sliding window type approaches, i.e., SVM-win and 1NN-win, their performances are not quite satisfying, even if we have chosen a relatively optimal window size by cross-validation with different data splits. For two batch type approaches, i.e., SVM-fix and SVM-ada. Performance of SVM-fix is comparable to DFOP, however, it's curious that SVM-ada demonstrates a frustrating performance, even worse than SVM-fix. This might because the estimate generalization error is not consistent with empirical one. DWM shows a competitive performance, nevertheless, it needs to scan data many times to maintain a dynamic expert pool. As the only one-pass learning algorithm above except for RLS, DFOP shows a satisfying performance and property in handling data stream with distribution change. Detailed Descriptions of Datasets Regression Datasets -hyperplane [KM05] is a benchmark synthesis dataset for regression scenario, where feature consists of 10 variables, each is uniform sampled from [0, 1]. There are in total 2,000 data samples with four stages, target concepts are (x i + x i+1 + x i+2 )/3 > 0.5, with i = 1, 2, 4, 7 in each stage. -Sulfur recovery unit [FGRX07] is, a real-world dataset which is a record of gas diffusion. Feature consists of 5 different chemical and physical indexes, with in total 10,081 data samples. There are two outputs in original dataset represent the concentration of SO 2 and H 2 S, and we split it into SRU-1 and SRU-2, respectively. -Debutanizer column [FGRX07] is, a real-world dataset which is a record of chemical reactions, with in total 2,394 data samples consisting of 7 different features. The output represents C4 content in the debutanizer bottoms. Classification Datasets -Chess [Zli11] is constructed by the data from chess.com portal, which consists of game records of players over a period from December 2007 to March 2010, a total of 533 instances with 7 attributes. The label indicates win or loss in the game. -Usenet [KTV08] is split into Usenet-1 and Usenet-2 which both consist of 1,500 instances with 100 attributes based on 20 newsgroups collection. They simulate a stream of messages from different newsgroups that are sequentially presented to a user, who then labels them according to his/her personal interests. -Luxembourg [Zli11] is constructed using European Social Survey data. There are 1,900 instance with 32 attributes in total, each instance is an individual and attributes are formed from answers to the survey questionnaire. The label indicates high or low internet usage. -Spam Detection [KTB + 09], real-world textual dataset that use email messages from the Spam Assassin Collection, and boolean bag-of-words approach is adopted to represent emails. It consists of 9,324 instances with 500 attributes, and label indicates spam or legitimate. -Weather [EP11] Weather dataset is originally collected from the Offutt Air Force Base in Bellevue, Nebraska. 18,159 instances are presented with an extensive range of 50 years (1949 − 1999) and diverse weather patterns. Eight features were selected based on their availability, eliminating those with a missing feature rate above 15%. The remaining missing values were imputed by the mean of features in the preceding and following instances. Class labels are based on the binary indicator(s) provided for each daily reading of rain with 18,159 daily readings: 5698 (31%) positive (rain) and 12,461 (69%) negative (no rain). -Powersupply [CKH + 15] contains three year power supply records including 29,928 instances with 2 attributes from 1995 to 1998, and our learning task is to predict which hour the current power supply belongs to. We relabeled into binary classification according to p.m. or a.m. -Electricity [HW99] is wildly adopted and is collected from the Australian New South Wales Electricity Market where prices are affected by demand and supply of the market. The dataset contains 45,312 instances with 8 features. The class label identifies the change of the price relative to a moving average of the last 24 hours. The basic information for all datasets mentioned in the original paper is summarized in the following Table 4. Such phenomena above are typical examples of distribution change. Under this scenario, the performance of traditional approaches dramatically drop down and thus are not empirically and theoretically suitable for these tasks. . The comparison methods are (a) RLS, least square approach solved in a recursive manner, (b) Sliding window approach, the classifier is constantly updated by the nearest data samples in the window. Base classifiers are 1NN and SVM, denoted as 1NN-win and SVMwin [dSSGB15], (c) SVM-fix, batch implementation of SVM with a fixed window size [SLS99], (d) SVM-ada, batch implementation of SVM with an adaptive window size [Kli04], (e) DWM, dynamic weighted majority algorithm, an adaptive ensemble based on the traditional weighted majority algorithm Winnow [KM03, KM07].It's noteworthy to emphasize that the above comparisons are not all fair enough, because DFOP requires each data item be processed only once. Moreover, DFOP only needs one instance to update the model. Not all comparison methods can meets these two constraints, specifically, 1NN-win, SVM-win, SVM-fix and SVM-ada are window-based algorithms, hence, they are not one-pass. Besides, SVM-fix and SVM-ada are not incremental but updated in a series of batches. Figure 1 : 1Performance comparison of seven approaches on synthetic datasets in terms of holdout accuracy. 4, we conduct all the experiments for 10 trails and report the overall mean and standard deviation of predictive accuracy over above real-world datasets as well as other 12 synthesis datasets.In a total of 20 datasets, the number of instance vary from 533 to at most 200,000. DFOP achieves the best among all approaches in 15 over 20 datasets. Also, in other 5 datasets, DFOP ranks the second or the third. This validates the effectiveness of DFOP, especially under an unfair comparison condition.Additionally, the robustness[VDG + 02] of all these different algorithms are compared. Briefly speaking, for a particular algorithm algo, the robustness r algo is defined as the proportion between its accuracy and the smallest accuracy among all compared algorithms, i.e., r algo = acc algo / min α acc α . Hence, the sum of r algo over all datasets indicates the robustness of for algorithm algo. The greater the value of the sum, the better the performance. DFOP achieves the best over 20 datasets, and RLS ranks last as expected since it didn't consider the evolving distribution in datasets at all. Due to the page limits, detailed robustness comparison results Figure 2 : 2Accumulated accuracy with different forgetting factors over four datasets with distribution change. Figure 2 , 2then we have storage requirement of DFOP is O(d 2 ), where d is the dimension of data, independent from the number of training examples. Both theoretical supports and empirical demonstrations for DFOP are presented to validate its effectiveness and practicality. and (18) because of Jensen's Inequality. (19) holds becauseof E[ X k ] ≤ (2σ 2 ) k/2 kΓ(k/2)in[Ver10]. The last step (20) holds because the condition in definition of sub-Exponential, i.e., \|s\| ≤ 1/(16σ 2 ). and on both synthesis and real-world datasets. The comparison methods are (a) RLS, least square approach solved in a recursive manner, (b) Online Bagging (OB) [OR01], (c) AddExp.C [KM05], (d) EOS-ELM [LSH09], (e) Learn ++ .NSE [EP11], and (f) OAUE [BS14], where (b)-(f) are all ensemble style approaches which dynamically adapt the models based on the coming data item or data batch, the differences are the strategies on how to update models like adding new models, excluding models or adjusting models' weights. Both synthesis dataset hyperplane and real-world datasets Sulfur recovery unit and Debutanizer column are employed to demonstrate the effectiveness of proposed DFOP. The detailed descriptions of datasets are included in Section 4.1 of supplementary material. Figure 3 : 3A comparison result in terms of square loss over generated test data and w(t) of DFOP with different forgetting factors on hyperplane dataset. Figure 3 3Figure 3. And we can see that the error of DFOP is significantly lower than RLS, which is a Figure 4 : 4Robustness of accuracy on six compared algorithms and DFOP over 20 datasets with distribution change. Table 1 : 1Performance comparison in terms of mean and standard deviation of accuracy (both in percents).Bold values indicates the best performance. Besides, • (•) indicates that DFOP is significantly better (worse) than the compared method (paired t-tests at 95% significance level). And Win/ Tie/ Loss are summarized in the last row.Dataset SVM-win 1NN-win SVM-fix SVM-ada DWM RLS DFOP SEA 73.94 ± 0.12• 77.27 ± 0.04• 86.19 ± 0.06• 83.47 ± 0.09• 87.04 ± 0.03• 84.54 ± 0.47• 87.99 ±0.05 hyperplane 83.74 ± 0.03• 70.66 ± 0.03• 87.98 ± 0.03• 81.94 ± 0.07• 88.36 ± 0.25• 69.67 ± 1.40• 90.14 ±0.05 1CDT 98.71 ± 0.05• 99.96 ± 0.07 99.77 ± 0.06• 99.77 ± 0.08• 99.90 ± 0.09• 98.79 ± 1.65• 99.97 ±0.05 2CDT 94.86 ± 0.06• 94.62 ± 0.10• 95.19 ± 0.13• 95.18 ± 0.15• 90.21 ± 0.67• 62.24 ± 0.23• 96.36 ±0.09 1CHT 98.75 ± 0.18• 99.81 ± 0.22 99.63 ± 0.17• 99.63 ± 0.18• 99.69 ± 0.26• 98.49 ± 1.61• 99.84 ±0.16 2CHT 87.70 ± 0.04• 85.69 ± 0.05• 89.48 ± 0.12• 88.89 ± 0.13• 85.92 ± 0.72• 62.57 ± 0.23• 89.91 ±0.07 1CSurr 97.99 ± 0.04• 98.12 ± 0.11• 94.24 ± 1.08 93.56 ± 1.08 96.31 ± 0.50• 67.82 ± 0.22• 93.24 ±1.44 UG-2C-2D 94.47 ± 0.13• 93.55 ± 0.16• 95.41 ± 0.10• 94.92 ± 0.12• 95.59 ± 0.11 67.02 ± 1.46• 95.59 ±0.10 UG-2C-3D 93.60 ± 0.73• 92.83 ± 0.93• 95.05 ± 0.64• 94.48 ± 0.71 95.14 ± 0.62 61.95 ± 2.60• 95.37 ±0.61 UG-2C-5D 74.82 ± 0.45• 88.04 ± 0.42• 91.74 ± 0.26• 90.37 ± 0.35• 92.82 ± 0.23• 81.20 ± 2.42• 92.51 ±0.25 MG-2C-2D 90.20 ± 0.07• 87.84 ± 0.09• 84.98 ± 0.06• 84.22 ± 0.06• 90.15 ± 0.06• 57.18 ± 3.66• 85.06 ±0.06 G-2C-2D 95.54 ± 0.01• 99.61 ± 0.00• 95.41 ± 0.01• 95.26 ± 0.02• 95.82 ± 0.02 95.84 ± 0.01 95.83 ±0.02 Chess 69.67 ± 1.51• 79.58 ± 0.54 77.73 ± 1.56• 69.18 ± 3.65• 73.77 ± 0.66• 78.70 ± 0.83• 79.15 ±0.62 Usenet-1 68.92 ± 1.12 65.36 ± 1.55• 64.18 ± 2.24• 67.68 ± 1.86• 64.43 ± 4.53• 60.65 ± 0.53• 69.20 ±0.68 Usenet-2 74.44 ± 0.71• 71.03 ± 0.60• 73.99 ± 0.69• 72.64 ± 0.84• 73.37 ± 0.93• 73.16 ± 0.67• 75.60 ±0.57 Luxembourg 88.57 ± 0.28• 77.51 ± 0.44• 98.25 ± 0.19• 97.43 ± 0.42• 92.61 ± 0.40• 99.06 ± 0.14• 99.09 ±0.14 Spam 83.91 ± 2.20• 93.43 ± 0.82• 92.44 ± 0.80• 91.01 ± 0.94• 91.49 ± 1.09• 94.46 ± 0.16• 94.77 ±0.26 Weather 68.54 ± 0.55• 72.64 ± 0.25• 67.79 ± 0.65• 77.26 ± 0.33• 70.86 ± 0.42• 78.35 ± 0.18• 79.23 ±0.12 Powersupply 73.33 ± 0.25• 72.42 ± 0.21• 71.17 ± 0.15• 69.39 ± 0.17• 72.18 ± 0.29• 69.67 ± 0.64• 80.46 ±0.04 Electricity 74.20 ± 0.08• 85.33 ± 0.09• 62.01 ± 0.59• 58.69 ± 0.58• 78.60 ± 0.41• 74.20 ± 0.63• 76.94 ±0.26 DFOP W/ T/ L 18/ 1/ 1 14/ 4/ 2 19/ 1/ 0 18/ 2/ 0 14/ 3/ 3 19/ 1/ 0 - Table Table 3 : 3Mean and standard deviation of MSE over 10 trails, the lower the better, and the all the values have been multiplied by 100. Besides, • (•) indicates that DFOP is significantly better (worse) than the compared method (paired t-tests at 95% significance level). Note that the comparisons are unfair to DFOP, because DFOP is able to only scan data once, whereas most compared methods not.Methods hyperplane SRU-1 SRU-2 Debutanizer RLS 2.291(.045)• 0.306(.006)• 0.392(.005)• 2.467(.003)• OB 1.914(.004)• 0.051(.000) 0.142(.000)• 0.577(.000)• AddExp.C 0.730(.002)• 0.049(.001)• 0.139(.001)• 1.283(.000)• EOS-ELM 1.903(.002)• 0.050(.000) 0.142(.001)• 1.057(.000)• Learn++.NSE 1.662(.009)• 0.098(.028)• 0.232(.024)• 1.040(.000)• OAUE 1.959(.002)• 0.047(.000)• 0.131(.001)• 0.731(.008)• DFOP 0.619(.048) 0.063(.004) 0.064(.004) 0.360(.057) where Z * = sup k=1,··· ,t Z(k) and γ * = sup k=1,··· ,t γ k .Proof: Denote S(t, k) t i=k x(i), then with the recursive property of Y (t), left side can be expanded by Lemma 7, t k=1(1 − µ) t−k Y (k)x(k) = (1 − µ) t Y (0)S(t, 1) + t k=1 µ(1 − µ) k−1 Z(t − k + 1)S(t, t − k + 1) ≤ (1 − µ) t Y (0)S(t, 1) + t k=1 µ(1 − µ) k−1 Z(t − k + 1)S(t, t − k + 1) ≤ (1 − µ) t Y (0) t i=1 x(i) + Z * t k=1 µ(1 − µ) k−1 t i=t−k+1 x(i) Inequalities for the distributions of functionals of subgaussian vectors. 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[ "Semiclassical form factor for chaotic systems with spin 1/2 Semiclassical form factor for chaotic systems with spin 1/2 2", "Semiclassical form factor for chaotic systems with spin 1/2 Semiclassical form factor for chaotic systems with spin 1/2 2" ]	[ "Jens Bolte \nAbteilung Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany\n", "Stefan Keppeler \nSchool of Mathematics\nUniversity of Bristol\nUniversity Walk\nBS8 1TWBristolUnited Kingdom\n\nBRIMS\nAbteilung Theoretische Physik\nHewlett-Packard Laboratories\nFilton RoadBS34 8QZStoke Gifford, BristolUnited Kingdom\n\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany\n" ]	[ "Abteilung Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany", "School of Mathematics\nUniversity of Bristol\nUniversity Walk\nBS8 1TWBristolUnited Kingdom", "BRIMS\nAbteilung Theoretische Physik\nHewlett-Packard Laboratories\nFilton RoadBS34 8QZStoke Gifford, BristolUnited Kingdom", "Universität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany" ]	[]	We study the properties of the two-point spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the so-called diagonal approximation can be recovered. The incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics. Provided this skew product is mixing, we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory.	10.1088/0305-4470/32/50/307	[ "https://arxiv.org/pdf/chao-dyn/9909026v1.pdf" ]	16,174,129	chao-dyn/9909026	18a9833569109e3e181c5bb6025776c7338727e5	Semiclassical form factor for chaotic systems with spin 1/2 Semiclassical form factor for chaotic systems with spin 1/2 2 Sep 1999 September 1999 October 1999 Jens Bolte Abteilung Theoretische Physik Universität Ulm Albert-Einstein-Allee 11D-89069UlmGermany Stefan Keppeler School of Mathematics University of Bristol University Walk BS8 1TWBristolUnited Kingdom BRIMS Abteilung Theoretische Physik Hewlett-Packard Laboratories Filton RoadBS34 8QZStoke Gifford, BristolUnited Kingdom Universität Ulm Albert-Einstein-Allee 11D-89069UlmGermany Semiclassical form factor for chaotic systems with spin 1/2 Semiclassical form factor for chaotic systems with spin 1/2 2 Sep 1999 September 1999 October 1999arXiv:chao-dyn/9909026v1 16 + Address after 1PACS numbers: 0365Sq, 0545Mt We study the properties of the two-point spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the so-called diagonal approximation can be recovered. The incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics. Provided this skew product is mixing, we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory. Introduction One of the major paradigms of quantum chaos is the conjecture of Bohigas, Giannoni and Schmit (BGS) [1] which states that the local statistics of energy spectra of (generic) individual quantum systems, whose classical analogues exhibit (strongly) chaotic behaviour, can be well described by that of ensembles of large random matrices. The symmetry properties of the relevant matrix ensembles have to be chosen according to the symmetries of the quantum system under consideration. In case the system is invariant under time reversal and has integer total angular momentum its local eigenvalue statistics are conjectured to be that of the Gaussian Orthogonal Ensemble (GOE). If time reversal invariance is broken one expects local statistics according to the Gaussian Unitary Ensemble (GUE). However, if the total angular momentum of the system is half-integer and the system is invariant under time reversal all eigenvalues show Kramers' degeneracy [2,3] and their statistics have to be compared with the Gaussian Symplectic Ensemble (GSE). In the GOE-and in the GUE-case there is plenty of numerical evidence available in favour of the BGS-conjecture, see, e.g., [1,4,5], whereas only few examples have been studied in the GSE-case as, e.g., in [6,7]. For quantum systems whose classical limit is integrable, i.e., which shows regular behaviour, one expects the local eigenvalue statistics to follow the laws of a Poisson process [8]. For the analytical treatment semiclassical methods, in particular semiclassical trace formulae, have become the most important tools since Berry and Tabor [8] investigated the behaviour of the spectral form factor for classically integrable systems by means of an appropriate trace formula. By making use of the Gutzwiller trace formula [9,5,10,11,12], Berry provided a semiclassical theory for the spectral form factor [13] of classically chaotic systems without spin. Based on the socalled diagonal approximation he could explain the semiclassical asymptotics of the form factor for small values of its argument, thus recovering the GOE-and GUE-behaviour, respectively. In this paper our aim is to show that Berry's semiclassical treatment of the twopoint form factor can be carried over to quantum systems with spin 1/2, whose classical translational dynamics are chaotic. We base our analysis on the semiclassical trace formula for the Dirac equation that we developed recently [14,15]. In this trace formula the presence of spin is reflected in a modification of the amplitudes with which the periodic orbits of the translational dynamics contribute. This modification arises from a spin dynamics that involves a 'classical' spin precessing along the periodic orbits. The central part of this paper therefore consists of calculating the effect of this spin contribution to the semiclassical form factor. For our analysis we use a two-point form factor whose definition differs slightly from the one that is more commonly used in spectral statistics as, e.g., in [13]. We rather prefer the point of view adopted in [16,17]. Both definitions, however, are equivalent in the limit where infinitely many eigenvalues are taken into account. We also stress that both the form factor and the associated correlation function are distributions and hence have to be evaluated on suitable test functions. This approach makes a spectral average obsolete and enables one to state the BGS-conjecture, specialized to the form factor, in a precise manner. Moreover, the lacking self-averaging property discussed in [18] poses no difficulty in this context. The paper is organized as follows. In section 2 we introduce our definition of the spectral form factor for a finite part of the spectrum and discuss in which sense one can expect to recover the form factors given by random matrix theory. Section 3 is devoted to the definition of a regularized form factor which we evaluate semiclassically using trace formulae for the Dirac as well as for the Pauli equation [14,15]. We also invoke the diagonal approximation and briefly discuss its range of validity. In section 4 a suitable version of the equidistribution principle of long periodic orbits is used in order to obtain the semiclassical asymptotics of the diagonal form factor. In this context we employ the notion of a skew product of the translational and the spin dynamics, the ergodic properties of which determine the semiclassical asymptotics. Our principal results are then summarized in section 5. Namely, depending on the presence or absence of quantum mechanical time reversal invariance, and provided the dynamics of the skew product is mixing, we can recover a GSE-or GUE-behaviour of the diagonal form factor, respectively. The relation between classical and quantum mechanical time reversal as well as the equidistribution of long periodic orbits are discussed in two appendices. The form factor for quantum systems with half-integer spin The two-point correlations of a discrete quantum spectrum are conveniently measured by either the two-point correlation function R 2 or by the two-point form factor K 2 , which is related to R 2 through a Fourier transform. Before defining these quantities one usually unfolds the spectrum, i.e., the eigenvalues E k are rescaled to x k such that the unfolded eigenvalues have a mean separation of one. This means that the spectral density d(x) of the unfolded spectrum allows for a separation d(x) := k δ(x − x k ) = 1 + d f l (x) , (2.1) such that 1 2∆x x+∆x x−∆x d f l (y) dy = 1 2∆x #{k; x − ∆x ≤ x k ≤ x + ∆x} − 1 (2.2) vanishes as x → ∞, ∆x → ∞, ∆x/x → 0. For a finite part of the spectrum, containing N unfolded eigenvalues enumerated as x 1 , . . . , x N , one defines the two-level correlation function by R 2 (s; N) := 1 N k,l≤N δ(s − (x k − x l )) − 1 . (2.3) Accordingly, the two-level form factor is defined as K 2 (τ ; N) := R R 2 (s; N) e −2πiτ s ds = 1 N k,l≤N e −2πiτ (x k −x l ) − δ(τ ) . (2.4) Since both quantities are distributions, which is most clearly seen in the case of the correlation function (2.3), one should evaluate these on test functions φ ∈ C ∞ 0 (R), R K 2 (τ ; N) φ(τ ) dτ = 1 N k,l≤N R φ(τ ) e −2πiτ (x k −x l ) dτ − φ(0) = 1 N k,l≤Nφ (2π(x l − x k )) − φ(0) = R R 2 (s; N)φ(2πs) ds . (2.5) We remark that since the form factor is obviously even in τ , it suffices to consider only even test functions φ. The convention for the Fourier transform that was used, and that will be used in all of what follows, iŝ f (k) = R f (x) e ixk dx and f (x) = 1 2π Rf (k) e −ixk dk . (2.6) After smearing with a test function all expressions occurring in (2.5) are obviously finite. In this form a semiclassical analysis of either the form factor or the correlation function can be carried out. If φ is chosen non-negative, the left-hand side of (2.5) can also be viewed as the mean value of K 2 (τ ; N) when τ is drawn randomly with probability density φ. Since we always understand the form factor in the above sense, the absence of a self-averaging property discussed in [18] is not essential for our further considerations. For a given quantum HamiltonianĤ the unfolding of its discrete spectrum shall proceed in the following manner. We consider a spectral interval I = I(E, ) := [E − ω, E + ω] , ω > 0 ,(2.7) that has no overlap with a possible essential spectrum ofĤ. Then the condition E k ∈ I is equivalent to − ω ≤ E k − E ≤ ω . (2.8) The number of eigenvalues contained in I, N I := #{k; E k ∈ I} ,(2.9) can be estimated semiclassically as N I ∼ 2 ωd(E) = ω π T H (E) , → 0 ,(2.10) whered(E) denotes an appropriate mean spectral density and T H (E) := 2π d (E) is the Heisenberg-time. A convenient definition ofd(E) can be derived from the semiclassical trace formula forĤ in that it shall denote the contribution coming from the singularity of Tr exp[− i Ĥ t] at t = 0 to all polynomial orders in see, e.g., [10,11,12,15,19]. The spectra that we are going to consider below are such that in the semiclassical limit → 0 the Heisenberg-time tends to infinity, T H → ∞. For example, given E in the gap of the essential spectrum of a Dirac-HamiltonianĤ D , i.e., in typical cases −mc 2 < E < mc 2 , the mean spectral density reads [14,15] d(E) = 2 vol Ω + E + vol Ω − E (2π ) 3 [1 + O( )] ,(2.11) where Ω ± E denote the hypersurfaces of energy E in phase space corresponding to the classical Hamiltonians H ± (p, x) = eϕ(x) ± (cp − eA(x)) 2 + m 2 c 4 (2.12) for relativistic particles of positive and negative kinetic energy, respectively, in the static external electromagnetic fields generated by the potentials ϕ and A. Therefore, in the semiclassical limit the number of eigenvalues in the interval I increases, although its length \|I\| = 2 ω shrinks to zero. A completely analogous argument applies to Pauli-Hamiltonians [15]. We now define the unfolded spectrum through x k := E kd (E) and x := Ed(E) . (2.13) The condition E k ∈ I(E, ) is hence equivalent to x k ∈ [x − ∆x, x + ∆x], where ∆x := ω 2π T H . With this choice indeed x → ∞, ∆x → ∞, such that ∆x/x = ω/E → 0 in the semiclassical limit. In this context the quantity (2.2) reads 1 2∆x #{k ; x − ∆x ≤ x k ≤ x + ∆x} − 1 = π ωT H N I − 1 ,(2.14) such that (2.10) ensures its vanishing in the semiclassical limit. From now on we will choose the numbering of the eigenvalues E k and x k , respectively, in such a way that the eigenvalues in I are given by E 1 ≤ E 2 ≤ . . . ≤ E N I . (2.15) Changing the value of therefore alters the numbering of the eigenvalues. As a consequence, the condition E k ∈ I is equivalent to k ≤ N I . At this place we recall that the semiclassical limit → 0, or T H → ∞, implies N I → ∞. For the form factor (2.4) we now obtain K 2 (τ ; N I ) = 1 N I E k ,E l ∈I e −2πiτd(E)(E k −E l ) − δ(τ ) = 1 √ N I k χ [−ω,ω] E k − E e − i τ T H E k 2 − δ(τ ) ,(2.16) where the k-sum extends over all eigenvalues ofĤ and χ [−ω,ω] denotes the characteristic function of the interval [−ω, ω] that occurs due to the condition (2.8). In the preceding discussion we tacitly assumed that the discrete spectrum of the quantum Hamiltonian carries no systematic degeneracies. In order to achieve such a situation one has to remove all symmetries. As opposed to geometric or internal symmetries that have to be realized by unitary representations of the respective symmetry groups, the time reversal operation must be implemented by an anti-unitary operatorT , see [3] and Appendix A. For single particles of spin s the square ofT depends on s being integer or half-integer in thatT 2 = (−1) 2s . In case the quantum system is time reversal invariant, i.e., [Ĥ,T ] = 0, and has half-integer spin this leads to Kramers' degeneracy [2]: SinceT 2 = −1 implies that every vector ψ = 0 in the Hilbert space is orthogonal toT ψ, all eigenvalues ofĤ are (at least) two-fold degenerate, see [20,21] for details. Following the usual practice, we will remove Kramers' degeneracy by replacing each degenerate pair E 2k = E 2k+1 of eigenvalues by one of its representatives. Thus the mean spectral density is lowered by a factor of two. In analogy to (2.13) the unfolding E k →x k of the so modified spectrum can therefore be achieved through the choicex k := x k /2. The modified form factor then reads K 2 (τ ; N) = 2 N k,l≤N k,l odd e −2πiτ (x k −x l ) − δ(τ ) = 1 2N k,l≤N e −2πi τ 2 (x k −x l ) − 1 2 δ τ 2 = 1 2 K 2 τ 2 ; N . (2.17) In this setting the conjecture of Bohigas, Giannoni and Schmit [1] states that for individual (generic) classically chaotic quantum systems of particles with half-integer total spin and no unitary symmetries one should obtain lim N →∞ R K 2 (τ ; N) φ(τ ) dτ ! = R K GSE 2 (τ ) φ(τ ) dτ (2.18) for all test functions φ ∈ C ∞ 0 (R). Here K GSE 2 denotes the two-point form factor of the Gaussian symplectic ensemble of random matrix theory, K GSE 2 (τ ) = 1 2 \|τ \| − 1 4 \|τ \| log \|1 − \|τ \|\| for \|τ \| ≤ 2 , 1 for \|τ \| ≥ 2 , (2.19) see, e.g., [21]. When time reversal invariance is lacking, the respective conjecture reads lim N →∞ R K 2 (τ ; N) φ(τ ) dτ ! = R K GU E 2 (τ ) φ(τ ) dτ , (2.20) where now the form factor of the Gaussian unitary ensemble [21] should occur, K GU E 2 (τ ) = \|τ \| for \|τ \| ≤ 1 , 1 for \|τ \| ≥ 1 . (2.21) In our subsequent semiclassical investigations we will in both cases, i.e., with and without time reversal invariance, consider the form factor K 2 (τ ; N I ) as it is given in (2.16). When dealing with the case of time reversal invariance we appeal to the relation (2.17). The semiclassical form factor Since the work of Berry and Tabor [8] on the distribution of eigenvalues for classically integrable systems, semiclassical trace formulae have found numerous and fruitful applications in the analysis of spectral statistics. A prominent example is Berry's analysis of the spectral rigidity [13], which relies in an essential way on a semiclassical evaluation of the two-point form factor based on the Gutzwiller trace formula. In this work it already became apparent that present semiclassical methods at most allow to study the form factor in the restricted range \|τ \| < 1, see also [19] for a review. Only recently, improved techniques have been developed [22] that might allow to extend the semiclassical analysis of spectral statistics. In this paper, however, we follow the more traditional path in that in the end we consider the so-called diagonal approximation for the form factor. The two-point form factor as given in (2.16) requires to establish a trace formula for the sum k χ [−ω,ω] E k − E e − i τ T H E k . (3.1) However, the general structure of (convergent) semiclassical trace formulae, see, e.g., [10,11,12,15,19], necessitates the use of a smooth test function ρ ∈ C ∞ (R) with Fourier transformρ ∈ C ∞ 0 (R). One therefore has to replace the sharp cut-off, provided by the characteristic function in (3.1), by a smoothened substitute. For this reason we now introduce the regularized form factor K χ,η 2 (τ ; T H ) := π ωT H k χ(E k ) η E k − E e − i τ T H E k 2 − δ(τ ) , (3.2) where η ∈ C ∞ (R) is a test function with Fourier transformη ∈ C ∞ 0 (R), but that is otherwise arbitrary at the moment. Later we will introduce a further normalization condition. In the following we will consider both relativistic and non-relativistic particles with spin 1/2. In the relativistic case, when one is dealing with a Dirac-HamiltonianĤ D , the function χ ∈ C ∞ 0 (R), which is not to be confused with the characteristic function χ [−ω,ω] , is necessary to truncate the essential spectrum ofĤ D . In typical situations χ should therefore be supported in the interval (−mc 2 , mc 2 ), where the eigenvalues E k of H D are located. When these do not accumulate at some point, one could also leave out the truncation χ from (3.2). We are now in a position to use the test function ρ(ε) := η(ε) e −2πid(E)τ ( ε+E) (3.3) in the semiclassical trace formula for the Dirac equation that was developed in [14,15], k χ(E k ) ρ E k − E = χ(E) T H (E) 2πρ (0) [1 + O( )] + χ(E) γ k =0 T γ 2πρ (kT γ ) A γ,k . (3.4) The outer sum on the right-hand side extends over all primitive periodic orbits γ of energy E, with periods T γ , of the two classical flows generated by the Hamiltonians (2.12). The inner sum then is over all k-fold repetitions of primitive orbits, formally including negative ones. The weight attached to each pair (γ, k) reads A γ,k := tr d k γ \| det(M k γ − ½)\| 1 2 e i kSγ(E)−i π 2 kµγ [1 + O( )] . (3.5) Here d γ ∈ SU(2) denotes the semiclassical time evolution operator for the spin degrees of freedom along the primitive periodic orbit γ of the translational dynamics. Furthermore, S γ (E) denotes the action of γ and µ γ is its Maslov index. The (monodromy) matrix M γ is the linearized Poincaré map transversal to γ. In the form given in (3.4) the trace formula is valid for all cases where the classical flows have only isolated and non-degenerate periodic orbits. An analogous trace formula, with appropriate simplifications, is also available for Pauli-Hamiltonians, see [15]. Upon choosing the test function (3.3) in the trace formula (3.4), its left-hand side reads k χ(E k ) ρ E k − E = k χ(E k ) η E k − E e − i τ T H E k , (3.6) and is hence the appropriate starting point for a semiclassical analysis of the form factor, compare (3.2). As a first ingredient on the right-hand side of (3.4) one requires the Fourier transform of the test function (3.3), which is given bŷ ρ(t) = e − i Eτ T Hη (t − τ T H ) . (3.7) For convenience we now choose η to be even and real-valued, which implies thatη also shares these properties. Furthermore, the truncation χ of the essential spectrum shall be such that χ(E) = 1. Thus, the trace formula yields the following semiclassical representation of the regularized form factor, K χ,η 2 (τ ; T H ) = − δ(τ ) + T H 4πω [η(τ T H )] 2 [1 + O( )] + γ k =0 T γ 4πωη (τ T H )η(kT γ − τ T H ) A γ,k [1 + O( )] (3.8) + 1 T H γ,γ ′ k,k ′ =0 T γ T γ ′ 4πωη (kT γ − τ T H )η(τ T H − k ′ T γ ′ ) A γ,k A γ ′ ,−k ′ . In a next step we are going to test the semiclassical form factor with some φ ∈ C ∞ 0 (R), compare (2.5). To this end one needs the integral F (T, T ′ ) := R φ(τ )η(T − τ T H )η(τ T H − T ′ ) dτ ,(3.9) whose leading term in the semiclassical limit T H → ∞ can be calculated by introducing the Fourier representations for φ andη. A straight-forward calculation then yields F (T, T ′ ) = 1 T H R R Rφ (t) η(ε) η(ε ′ ) e i(εT −ε ′ T ′ ) δ t T H + ε − ε ′ dε ′ dε dt . (3.10) Changing variables from ε, ε ′ to u := ε ′ − ε and v := (ε ′ + ε)/2, and employing the expansions η v ± t 2T H = η(v) + O t T H ,(3.11) finally shows that F (T, T ′ ) = 1 T H R Rφ (t) e −it( T +T ′ 2T H ) η(v) 2 e iv(T −T ′ ) dv dt + O 1 T 2 H = 1 T H φ T + T ′ 2T H η * η(T − T ′ ) + O 1 T 2 H ,(3.12) where the convolution η * η(t) := Rη (t − t ′ )η(t ′ ) dt ′ = 2π R η(v) 2 e itv dv (3.13) enters. We therefore conclude that R K χ,η 2 (τ ; T H ) φ(τ ) dτ = φ(0) −1 + 1 4πωη * η(0) + O( ) + 1 T H γ k =0 T γ 4πωη * η(kT γ ) A γ,k φ kT γ 2T H [1 + O( )] + 1 T 2 H γ,γ ′ k,k ′ =0 T γ T γ ′ 4πωη * η(kT γ − k ′ T γ ′ ) × φ kT γ + k ′ T γ ′ 2T H A γ,k A γ ′ ,−k ′ . (3.14) At this point we introduce the normalization of η announced previously. Guided by the simple observation R [χ [−ω,ω] (ε)] 2 dε = 2ω ,(3.15) we require the same normalization for the smooth substitute η of the sharp cut-off χ [−ω,ω] , 1 2πη * η(0) = R η(ε) 2 dε ! = 2ω . (3.16) As a consequence, the leading semiclassical order of the first line on the right-hand side of (3.14) vanishes. Furthermore, since the Fourier transformη of the test function η is required to be compactly supported, the second line is a finite sum, multiplied by 1/T H . A similar argument applies to the third and fourth line, apart from the diagonal contribution with kT γ = k ′ T γ ′ , whereη * η(0) occurs and thus no such cut-off is present. Due to the above reasoning it is tempting to assume that in the semiclassical limit T H → ∞ the right-hand side of (3.14) is completely fixed by the contribution of the diagonal form factor K diag 2 (τ ; T H ) := 1 T 2 H γ k =0 g γ,k T 2 γ \|A γ,k \| 2 δ τ − kT γ T H .(3.17) Here we assumed that k ′ T γ ′ = kT γ implies A γ ′ ,k ′ = A γ,k , see Appendix A for the spin contribution, and then g γ,k denotes the number of pairs (γ, k) such that kT γ has a given value. It is, however, well known that the above assumption is not justified. The reason for this lies in the subtleties of the limits involved. In order to arrive at the left-hand side of (2.18), or of (2.20), one first has to remove the smoothening of the characteristic function χ [−ω,ω] in that a sequence of functions η ∈ C ∞ (R) approaching χ [−ω,ω] has to be considered. Then, in the limit,η * η is no longer compactly supported. Indeed, according to (3.13) one obtainsη * η(t) = 4π sin(ωt)/t. Still, the periodic-orbit sums in (3.14) are truncated by the test function φ. However, in the semiclassical limit this cut-off is being removed. Moreover, for long periodic orbits the differences kT γ − k ′ T γ ′ can become arbitrarily small so thatη * η(kT γ − k ′ T γ ′ ) only provides a modest truncation of near-diagonal contributions. The only example where it could be rigorously shown [16] that the diagonal form factor itself produces the correct limit, if the test functions φ are restricted to those that are supported in the interval [−1, 1], is that of the correlations of the non-trivial zeros of principal L-functions, including the case of the Riemann zeta function, see also [23]. Rudnick and Sarnak [16] even proved an analogous result for general n-point correlations. As announced previously, we now invoke the diagonal approximation, i.e., we leave aside the contribution of K χ,η 2 (τ ; T H ) − K diag 2 (τ ; T H ) to (3.14). This procedure, which goes back to Berry [13], is generally supposed to reveal the correct behaviour of the form factor for small \|τ \|. The reason for this being that if the test function φ is supported in a small interval, the contribution of long periodic orbits to (3.14) is truncated. Furthermore, due to the normalization (3.16) the diagonal form factor is independent of the smoothening η. This convenient fact exempts one from the need to discuss the removal of this smoothening. In order to test now the range of small \|τ \| one should restrict the class of test functions φ to those supported in intervals [−τ ′ , τ ′ ], where τ ′ > 0 is 'small enough'. Recalling that K diag 2 and φ are even in τ an integration by parts yields R K diag 2 (τ ; T H ) φ(τ ) dτ = 2 ∞ 0 K diag 2 (τ ; T H ) φ(τ ) dτ = − τ ′ 0 φ ′ (τ ) 2 T 2 H γ k≥1 kTγ ≤τ T H g γ,k T 2 γ \|A γ,k \| 2 dτ . (3.18) What is required now is the asymptotic behaviour of the periodic-orbit sum in (3.18) as T H → ∞. Since the contributions of repetitions of primitive periodic orbits are asymptotically suppressed due to their stronger instabilities, compare also (4.3) below, in the following we only take the k = 1-term of the sum over the repetitions into account. As a consequence we therefore have to study the asymptotics of the periodic-orbit sum γ, Tγ ≤τ T H g γ,1 T 2 γ (tr d γ ) 2 \| det(M γ − ½)\| (3.19) in the double limit T H → ∞, τ → 0, such that τ T H → ∞. In order to simplify this task we now make two assumptions, which should be verified in all cases that could be considered as 'generic' in any reasonable sense: (i) The periods T γ of primitive periodic orbits shall be such that any finite subset of them is linearly independent over Q. This implies that the multiplicities g γ,k are independent of k, i.e., g γ,k = g γ . (ii) The subset of primitive periodic orbits γ with g γ =ḡ is of density zero in the set of all primitive periodic orbits, lim T →∞ #{γ; g γ =ḡ, T γ ≤ T } #{γ; T γ ≤ T } ! = 0 ,(3.20) whereḡ = 2 in case the classical dynamics are time reversal invariant, andḡ = 1 when time reversal symmetry is absent. In this context time reversal invariance does not only mean that an orbit γ is geometrically identical to its time reversed partner, but also that both orbits yield the same contribution of the spin degrees of freedom to the trace formula, which then implies that A γ,k is invariant under time reversal. In Appendix A we show that this condition is a consequence of quantum mechanical time reversal invariance. Under these assumptions the factors g γ,k can be replaced byḡ and can then be pulled out of the sum (3.19). Classical periodic-orbit sums and the contribution of spin The aim of this section is to obtain the leading semiclassical behaviour of the periodicorbit sum (3.19). Apart from the appearance of the Heisenberg-time only quantities related to the classical flow enter this expression. It therefore seems appropriate to invoke results about the distribution of periodic orbits in phase space. In order to retain a certain convenient generality, we will not specify the classical translational dynamics further, except for the following assumptions: (i) The classical flow Φ t H : Ω E → Ω E on the compact hypersurface Ω E of energy E in the 2d-dimensional phase space is generated by some Hamiltonian function H(p, x) and hence preserves the (normalized) Liouville measure dµ E (p, x) := 1 vol Ω E δ(H(p, x) − E) d d p d d x (4.1) on Ω E . (ii) Φ t H is ergodic with respect to Liouville measure. (iii) Φ t H is hyperbolic on all of Ω E . In the case of the semiclassical form factor for a Dirac-Hamiltonian these requirements shall apply to both classical flows, i.e., to those generated by the two classical Hamiltonians H ± given in (2.12). Moreover, we now assume that for a given energy E there will only be either a contribution coming from the dynamics generated by H + or from the dynamics generated by H − , but never from both at the same time. This is not a strong restriction since it only excludes situations in which Klein's paradox [24] can appear. The hyperbolicity of the classical flows implies that in particular all periodic orbits are either hyperbolic or loxodromic. This means that all monodromy matrices M γ have eigenvalues with moduli strictly different from one. Since the eigenvalues occur in pairs of mutually inverse numbers, we denote them as e ±(u γ,j +iv γ,j ) , u γ,j > 0, v γ,j ∈ [0, 2π), j = 1, . . . , d − 1. Thus \| det(M γ − ½)\| = d−1 j=1 e u γ,j +iv γ,j − 1 e −u γ,j −iv γ,j − 1 = exp d−1 j=1 u γ,j d−1 j=1 1 − e −u γ,j −iv γ,j 2 . (4.2) The stability exponents u γ,j are related to the Lyapunov exponents λ γ,j of γ through u γ,j = λ γ,j T γ so that u γ,j → ∞ as T γ → ∞. Hence, in this limit one obtains 1 \| det(M γ − ½)\| ∼ p γ := exp − d−1 j=1 u γ,j . (4.3) Since the semiclassical limit of the periodic-orbit sum (3.19) is dominated by the contribution of long periodic orbits, (4.3) allows to analyze (3.19) in terms of periodicorbits sums that are familiar from equidistribution theorems of periodic orbits, see, e.g., [25]. For the kind of Hamiltonian flows characterized above mean values of observables on Ω E with respect to Liouville measure can be calculated with the help of appropriate periodic-orbit sums. We postpone a detailed discussion of this matter to Appendix B, from which we here only quote that for any continuous observable a one obtains the representationā E := Ω E a(p, x) dµ E (p, x) = lim T →∞ 1 T γ, Tγ ≤T T γā γ p γ , (4.4) whereā γ denotes an average of a along the periodic orbit γ, a γ := 1 T γ Tγ 0 a Φ t H (p, x) dt , with (p, x) ∈ γ . (4.5) We remark that a heuristic derivation of an analogous identity to (4.4) was given by Hannay and Ozorio de Almeida [26]. One can apply the relation (4.4) to determine the leading semiclassical behaviour of (3.19) once one has chosen a suitable observable a whose averageā γ yields the quantity (tr d γ ) 2 appearing in (3.19). This, however, can only be achieved in an indirect manner. Our choice of the observable requires to recall the semiclassical time evolution of the spin degrees of freedom along the trajectories of the classical flow Φ t H . Let d(p, x, t) ∈ SU(2) denote the solution of the spin transport equation [14,15] d(p, x, t) + i M(Φ t H (p, x)) d(p, x, t) = 0 , d(p, x, 0) = ½ 2 , (4.6) where the time derivative is understood along the trajectory Φ t H (p, x). M is a certain hermitian and traceless 2×2-matrix valued function on Ω E , whose precise form depends on the quantum Hamiltonian under consideration, see [15] for details. Geometrically, d(p, x, t) can also be interpreted as a parallel transporter in some vector bundle so that d(p, x, T γ ), with (p, x) ∈ γ, is the holonomy associated with the periodic orbit γ. Since its trace is invariant under a shift of the initial point (p, x) ∈ γ along the orbit, one can introduce the notation tr d γ := tr d(p, x, T γ ). We are thus in a position to define the observable a(p, x, t) := [tr d(p, x, t)] 2 ,(4.7) which is a function on Ω E that in addition depends on a parameter t. Due to the above remark concerning the interpretation of d(p, x, T γ ) as a holonomy, the average (4.5) of this observable along a periodic orbit γ, when t = T γ is chosen, yields a γ (T γ ) = 1 T γ Tγ 0 a Φ t ′ H (p, x), T γ dt ′ = (tr d γ ) 2 ,(4.8) for any (p, x) ∈ γ. Without the choice t = T γ , however,ā γ (t) is not related to (tr d γ ) 2 . We can hence now employ (4.4) to deduce the asymptotic relation γ, Tγ ≤T T 2 γā γ (t) p γ ∼ 1 2 T 2āE (t) , T → ∞ ,(4.9) which is valid for any t. Notice that here we have introduced an extra power of T γ in the same manner as in (B.7)-(B.8). We now differentiate with respect to T , γ T 2 γā γ (t) p γ δ(T − T γ ) ∼ Tā E (t) , T → ∞ ,(4.10) and then choose t = T . Together with (4.8) this allows to conclude that γ T 2 γ (tr d γ ) 2 p γ δ(T − T γ ) ∼ Tā E (T ) , T → ∞ . (4.11) Thus, at this point we have obtained the asymptotic relation K diag 2 (τ ; T H ) ∼ḡ τā E (τ T H ) (4.12) for the diagonal form factor (3.17) in the limit T H → ∞, τ → 0 such that τ T H → ∞. The remaining task therefore consists of determining the asymptotics ofā E (T ) as T → ∞. In order to achieve this one has to go back to the representation a E (T ) = Ω E [tr d(p, x, T )] 2 dµ E (p, x) (4.13) ofā E (T ) as an average over phase space. Since the T -dependence involves d(p, x, T ) one might anticipate that the limit T → ∞ of (4.13) depends on certain ergodic properties of the spin dynamics. The latter being considered along trajectories of the translational dynamics, one hence has to combine both dynamics in a suitable way. In ergodic theory the relevant construction is known as a skew product, see, e.g., [27]. Appropriate ergodic properties of the skew product dynamics will then allow for a determination of the asymptotic behaviour of (4.13). Let us therefore now construct the skew product of translational and spin dynamics. To this end one defines a flow Y t on the product phase space M := Ω E × SU(2) in the following way, Y t ((p, x), g) := Φ t H (p, x), d(p, x, t)g (4.14) for (p, x) ∈ Ω E and g ∈ SU(2). The initial condition Y 0 ((p, x), g) = ((p, x), g) is obviously fulfilled, and the composition law Y t+t ′ = Y t • Y t ′ immediately follows from the relation d(p, x, t + t ′ ) = d(Φ t ′ H (p, x), t) d(p, x, t ′ ) (4.15) that can be concluded from (4.6). On M one then defines the direct product µ := µ E × µ H of Liouville measure µ E and of the normalized Haar measure µ H of SU (2). We recall that the latter is the unique normalized left-and right-invariant positive Radon measure on the group manifold, see, e.g., [28]. Due to both the invariance of Liouville measure under the Hamiltonian flow Φ t H and the left-invariance of Haar measure, the product measure µ is invariant under Y t . A dynamical system of this kind is known as an SU(2)-extension of Φ t H or, more generally, a skew product. The spin dynamics defined by (4.6) is then called a cocycle for Φ t H with values in SU (2). For further information see, e.g., [27]. In addition to the assumptions made for Φ t H in section 3, in the following we will assume that Y t is (strongly) mixing. This implies that for any F ∈ L 2 (M × M) lim t→∞ M F Y t ((p, x), g), ((p, x), g) dµ((p, x), g) = M M F (((p, x), g), ((ξ, y), h)) dµ((p, x), g) dµ((ξ, y), h) .F (g, h) dµ H (g) dµ H (h) . (4.17) A suitable choice of the function F then allows to determine the asymptotic behaviour ofā E (T ) as T → ∞ from (4.17). In order to achieve this we recall the representation (4.13) ofā E (T ), which obviously can also be written as Substituting g ′ = gh −1 in the inner integral and using the right-invariance of µ H , the integrand does not depend on h any more. Thus we obtain lim T →∞ā a E (T ) = SU(2) Ω E tr(d(p, x, T )gg −1 ) 2 dµ E (p, x) dµ H (g) .E (T ) = SU(2) [tr g ′ ] 2 dµ H (g ′ ) ,(4.20) i.e., in the limit T → ∞ the expectation value of [tr d(p, x, T )] 2 , when averaged over phase space, can be computed by an average over the group SU(2) with respect to Haar measure. The same holds obviously true for any moment of tr d(p, x, T ) so that the asymptotic distribution of tr d(p, x, T ), when the initial points (p, x) ∈ Ω E are uniformly distributed with respect to µ E , can be computed via lim T →∞ µ E {(p, x) ∈ Ω E ; tr d(p, x, T ) ∈ [a, b]} = b a SU(2) δ (tr g − w) dµ H (g) dw . (4.21) In order to evaluate the integral over SU(2) explicitly we remark that any g ∈ SU(2) can be represented as g(u) = u 0 ½ 2 + iσu with u = (u 0 , u) ∈ R 4 and 3 j=0 u 2 j = 1 . (4.22) In this parameterization the Haar measure is given by (see, e.g., [28]) dµ H (g(u)) = 1 π 2 δ 3 j=0 u 2 j − 1 d 4 u . (4.23) A simple calculation now shows that the distribution (4.21) obeys a semicircle law, i.e., its density reads p(w) = SU(2) δ (tr g − w) dµ H (g) =      1 π 1 − w 2 2 −2 ≤ w ≤ +2 , 0 else . (4.24) What is required in (4.20) is the second moment of the distribution (4.21). With the help of (4.24) this can now easily be computed to yield one, i.e., a E (T ) ∼ 1 as T → ∞ . (4.25) The integrated version of (4.11) therefore reads γ, Tγ ≤T T 2 γ (tr d γ ) 2 p γ ∼ 1 2 T 2 , T → ∞ . (4.26) Furthermore, (4.12) and (4.25) imply the asymptotic behaviour K diag 2 (τ ; T H ) ∼ḡτ (4.27) of the diagonal form factor in the regime T H → ∞, τ → 0, τ T H → ∞. Summary and Conclusions In section 3 we demonstrated how the two-point form factor for a quantum system with spin 1/2 can be analyzed semiclassically by making use of an appropriate trace formula. The diagonal approximation led us to the periodic-orbit sum (3.19) whose asymptotics determines the behaviour of the diagonal form factor for small τ in the semiclassical limit. Our conclusion (4.27) now has to be compared with the relevant random matrix results (2.19) and (2.21). To this end we have to distinguish time reversal invariant quantum Hamiltonians from non-invariant ones. But let us first summarize our findings. In Appendix A we show that quantum mechanical time reversal invariance implies that not only the classical translational dynamics are time reversal invariant but, moreover, the spin dynamics behave in such a way that also the amplitudes A γ,k appearing in the semiclassical trace formula are invariant under time reversal. This leads to the occurrence of the multiplicities g γ,k in the expression (3.19). Then, when the generic conditions stated at the end of section 3 are met, we can pull out the factors g γ,k from the sum and replace them by eitherḡ = 2, if (quantum mechanical) time reversal invariance is present, or else byḡ = 1. If, furthermore, the translational dynamics are ergodic and hyperbolic, the equidistribution of periodic orbits allowed us to determine the spin contribution to the amplitudes A γ,k . We further requested the skew product of the translational and the spin dynamics to be mixing. This then enabled us to identify the distribution of the traces of the spin-transport matrices and to calculate its second moment, which enters through the amplitudes A γ,k . Hence, if quantum mechanical time reversal invariance is absent, the diagonal approximation (4.27) states that K diag 2 (τ ; T H ) ∼ τ , which is identical with the diagonal approximation in the case of Schrödinger operators [13] and coincides with the small-τ asymptotics of the GUE-form factor (2.21). If, however, the quantum Hamiltonian is invariant under time reversal so that we have to chooseḡ = 2, the diagonal approximation becomes K diag 2 (τ ; T H ) ∼ 2τ . According to (2.17) Kramers' degeneracy then forces us to compare the random matrix form factor with the modified semiclassical result conventional time reversal operation that leaves the position coordinates unchanged and reverses momentum and spin coordinates as well as the time t. The discussion of generalized time reversal operators that combine conventional time reversal with another unitary symmetry operation is analogous, see [20] for examples. To be specific we consider as a quantum Hamiltonian for a particle of mass m, charge e and spin 1/2 either a Dirac-Hamiltonian K diag 2 (τ ; T H ) = 1 2 K diag 2 τ 2 ; T H ∼ 1 2 τ ,(5.H D = cα i ∇ − e c A(x) + βmc 2 + eϕ(x) , (A.1) or a (generalized) Pauli-Hamiltonian H P =Ĥ S ½ 2 + σC i ∇, x . (A.2) In the relativistic case (A.1) the Dirac algebra is realized by the 4 × 4 matrices α = 0 σ σ 0 , β = ½ 2 0 0 −½ 2 , (A.3) where σ denotes the vector of Pauli matrices and ½ 2 is a 2 × 2 unit matrix. The nonrelativistic Hamiltonian (A.2) is composed of a Schrödinger operatorĤ S and a coupling term of spin to the translational degrees of freedom. The latter has to be understood as the quantization of some R 3 -valued function C(p, x) on phase space. For example, this can be a magnetic field, i.e., C B (p, x) = − e 2mc B(x), or a spin-orbit coupling term C so (p, x) = whereK is the operator of complex conjugation in position representation, see [20]. In the relativistic case, whereT has to act on four component spinors, (A.4) shall mean a block diagonal 4 × 4 matrix with two copies of (A.4) in the diagonal blocks. A quantum system with HamiltonianĤ to be time reversal invariant requiresTĤT −1 =Ĥ. Thus, in the case of a Dirac-Hamiltonian (A.1) TĤ DT −1 = c(−α) − i ∇ − e c A(x) + βmc 2 + eϕ(x) (A.5) shows that conventional time reversal invariance is equivalent to the absence of magnetic forces. For the Pauli-Hamiltonian (A.2) to commute withT we first need the Schrödinger operatorĤ S to be time reversal invariant, i.e.,KĤ SK =Ĥ S . In addition, the condition T σC i ∇, x T −1 = −σC − i ∇, x ! = σC i ∇, x (A.6) has to be met, i.e., the coupling term C(p, x) must be an odd function of momentum p. This requirement is fulfilled by the spin-orbit coupling term C so , but is violated by the coupling C B to an external magnetic field. In both the relativistic and the nonrelativistic situation, however, even the presence of a magnetic field might allow for the existence of an anti-unitary operator representing a generalized time reversal symmetry that commutes with the Hamiltonian, see [3,20]. In a next step one has to identify Liouville measure as the equilibrium measure of some observable f . To this end we consider the tangential map DΦ t H restricted to the unstable subbundle E u of the tangent bundle T Ω E and define f (p, x) := − d dt log det DΦ t H (p, x) E u ,t=0 . (B.3) A direct calculation then yields f γ = − 1 T γ d−1 j=1 u γ,j so that e Tγf γ = p γ , (B.4) compare (4.3). Furthermore, the equilibrium measure associated with the observable (B.3) is called Sinai-Ruelle-Bowen measure µ SRB , for which it is known that for every a ∈ C(Ω E ) lim T →∞ 1 T T 0 a Φ t H (p, x) dt = Ω E a(p ′ , x ′ ) dµ SRB (p ′ , x ′ ) (B.5) holds for a set of initial conditions (p, x) ∈ Ω E with positive Liouville measure, see [29] for more information. Since Φ t H is supposed to be ergodic with respect to µ E , one concludes that µ SRB = µ E . What is still lacking is an asymptotic estimate of the denominator on the right-hand side of (B.2). In order to obtain this we first have to introduce a regularization in that we multiply the observable (B.3) by some factor β < 1. Then we appeal to the relation γ T −ε≤Tγ ≤T +ε T γ p β γ ∼ e P (βf )T P (βf ) e P (βf )2ε − 1 , T → ∞ , (B.6) see [25]. Since P (f ) = 0, in the limit β → 1 one obtains for k ∈ N γ T −ε≤Tγ ≤T +ε T k γ p γ ∼ 2ε T k−1 , T → ∞ . (B.7) Replacing now ε by T , followed by the rescaling 2T → T , this implies that usually the mixing property is defined for pairs of functions F 1 , F 2 ∈ L 2 (M). However, if one views L 2 (M ×M) as L 2 (M) ⊗L 2 (M) and introduces a tensorproduct basis, (4.16) follows immediately because every element of this basis fulfills the usual mixing property. If now the function F does not depend on the translational degrees of freedom, i.e., F : SU(2) × SU(2) → R, the mixing property (4.16) yields lim t→∞ M F (d(p, x, t)g, g) dµ((p, x), g) = SU(2) SU(2) F defining the function F (g, h) := [tr(gh −1 )] 2 , g, h ∈ SU(2), (d(p, x, T )g, g) dµ((p, x), g) = SU(2) SU(2) [tr(gh −1 )] 2 dµ H (g) dµ H (h) . x × p). For systems with spin 1/2 the operator of time reversal is given bŷ T := e i π 2 σyK = i σ yK , (A.4) γ ∼ 2 1−k T k , T → ∞ .(B.8) Introducing then the relations (B.4) and (B.8) in (B.2) finally yields the periodic-orbit representation (4.4) of Liouville measure. 1 ) 1which now coincides with the small-τ asymptotics of the GSE-form factor(2.19). In both cases this is exactly the behaviour that is predicted by the conjecture of Bohigas, Giannoni and Schmit as stated in(2.20) and(2.18). AcknowledgmentsWe would like to thank Prof F Haake for an important remark on Kramers' degeneracy. S K would like to thank Prof J P Keating, Dr J Marklof and Dr J M Robbins for helpful discussions. S K also acknowledges financial support from Deutscher Akademischer Austauschdienst under grant number D/99/02553.Appendix A. Classical and quantum mechanical time reversalIn this appendix we are going to discuss the relation between classical and quantum mechanical time reversal for systems with spin 1/2. We restrict our discussion to theAppendix B. Equidistribution of long periodic orbitsIn this appendix we want to show how the periodic-orbit representation (4.4) of Liouville measure can be obtained from equidistribution properties of periodic orbits. The basic reference for the following is[25]. Our assumptions on the flow Φ t H are as stated in section 4.In order to proceed further we first have to introduce some notation. Let B denote the set of all Φ t H -invariant Borel probability measures on Ω E . Any µ ∈ B can be associated a metric entropy h µ . Then for any Hölder-continuous observable f ∈ C α (Ω E ), with some α > 0, the topological pressure is defined asThe supremum is attained for a unique measure µ f , which is called equilibrium measure for the observable f . The periodic orbits of the flow Φ t H are then equidistributed with respect to µ f in the following sense,for every a ∈ C(Ω E ). The averages over periodic orbits are defined as in(4.5). In order to discuss the latter, one has to study the behaviour of the matrix M entering (4.6) under x → x, p → −p. For a Pauli-Hamiltonian M is given by σC(p, x), where C(p, x) is defined as above, and for a Dirac-Hamiltonian with no magnetic field one finds M = g(\|p\|) x × p, see [15] for details. each case quantum mechanical time reversal invariance therefore implies M(−p, x) = −M(p, x). Substituting now t → −t and p → −. exhibits a certain kind of symmetry under time reversal. p in (4.6) one obtains −ḋ(−p, x, −t) + i M(Φ −t H (−p, x)) d(−p, x, −t) = 0 . (A.7translational dynamics. But furthermore, also the spin dynamics, governed by the spin transport equation (4.6), exhibits a certain kind of symmetry under time reversal. In order to discuss the latter, one has to study the behaviour of the matrix M entering (4.6) under x → x, p → −p. For a Pauli- Hamiltonian M is given by σC(p, x), where C(p, x) is defined as above, and for a Dirac-Hamiltonian with no magnetic field one finds M = g(\|p\|) x × p, see [15] for details. In each case quantum mechanical time reversal invariance therefore implies M(−p, x) = −M(p, x). Substituting now t → −t and p → −p in (4.6) one obtains −ḋ(−p, x, −t) + i M(Φ −t H (−p, x)) d(−p, x, −t) = 0 . (A.7) Since the translational dynamics are time reversal invariant, M being odd in the momentum variable leads tȯ d(−p, x, −t) + i M. Φ t H (p, x)) d(−p, x, −t) = 0 , (A.8Since the translational dynamics are time reversal invariant, M being odd in the momentum variable leads tȯ d(−p, x, −t) + i M(Φ t H (p, x)) d(−p, x, −t) = 0 , (A.8) Moreover, the fact that d(p, x, T γ ) ∈ SU(2) is a holonomy implies d(−p, x, −T γ ) =. Therefore D(−p, X, −t Γ ) = D ; P, X, d(−p, x, T γ )] −1 so that finally tr d(−p, x, −T γ ) = tr d(p, x, T γand therefore d(−p, x, −T γ ) = d(p, x, T γ ). 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